Some Mathematical Problems in Geophysical Fluid Dynamics

Handbook of Mathematical Fluid Dynamics, Vol 3 S. Friedlander and D. Serre Editors, Elsevier. To appear SOME MATHEMATIC...

Author: Temam R. | Ziane M.

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Handbook of Mathematical Fluid Dynamics, Vol 3 S. Friedlander and D. Serre Editors, Elsevier. To appear

SOME MATHEMATICAL PROBLEMS IN GEOPHYSICAL FLUID DYNAMICS Roger Temam*# and Mohammed Ziane% ∗

The Institute for Scientific Computing and Applied Mathematics Indiana University, Bloomington, IN 47405 #

Laboratoire d’Analyse Numérique, Université Paris-Sud Bˆ atiment 425, 91405 Orsay, France %

University of Southern California DRB 155; 1042 W. 36 Place; Los Angeles, CA 90049 Abstract. This article addresses some mathematical aspects of the equations of geophysical fluid dynamics namely, existence, uniqueness, and regularity of solutions of the Primitive Equations (PEs) of the ocean, the atmosphere and the coupled atmosphere-ocean. The emphasis is on the case of the ocean which encompasses most of the mathematical difficulties. A guide and summary of results for the physics oriented reader is provided at the end of the Introduction (Section 1).

Contents 1. Introduction 2. The Primitive Equations. Weak Formulation. Existence of Weak Solutions. 2.1. The Primitive Equations of the Ocean 2.2. Weak formulation of the PEs of the ocean. The stationary PEs Date: January 14, 2004. 1

2 9 9 18

2

SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04

2.3. Existence of weak solutions for the PEs of the ocean 27 2.4. The Primitive equations of the atmosphere 32 2.5. The coupled atmosphere and ocean 40 3. Strong Solutions of the Primitive Equations in Dimension 2 and 3. 45 3.1. Strong solutions in space dimension 3 45 3.2. Strong solutions of the two dimensional primitive equations: Physical Boundary Conditions 56 3.3. The space periodic case in dimension 2: Higher regularities 71 4. Regularity for the elliptic linear problems in GFD. 87 4.1. Regularity of solutions of elliptic boundary value problems in cylinder type domains 89 4.2. Regularity of Solutions of a Dirichlet-Robin Mixed Boundary Value Problem 97 4.3. Regularity of Solutions of a Neumann-Robin Boundary Value Problem 104 4.4. Regularity of the Velocity 116 Proof of Theorem 4.4 125 4.5. Regularity of the coupled system 130 References 134

1. Introduction The aim of this article is to address some mathematical aspects of the equations of geophysical fluid dynamics, namely existence, uniqueness and regularity of solutions. The equations of geophysical fluid dynamics are the equations governing the motion of the atmosphere and the ocean, and are derived from the conservation equations from physics, namely conservation of mass, momentum, energy, and some other components such as salt for the ocean, humidity (or chemical pollutants) for the atmosphere. The basic equations of conservation of mass and momentum, that is the three-dimensional compressible Navier–Stokes equations contain however too much information and we can not hope to numerically solve


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these equations with enough accuracy in a foreseeable future. Owing to the difference of sizes of the vertical and horizontal dimensions, both in the atmosphere and in the ocean (10 to 20 kms versus several thousands of kilometers), the most natural simplification leads to the so-called Primitive Equations (PEs) which we study in this article. We continue this Introduction by briefly describing the physical and mathematical backgrounds of the PEs. Physical Background The primitive equations are based on the so-called hydrostatic approximation, in which the conservation of momentum in the vertical direction is replaced by the simpler, hydrostatic equation (see e.g. equation (2.25)). As far as we know, the primitive equations were essentially introduced by L. F. Richardson (1922); when it appeared that they were still too complicated, they were left out and, instead, attention was focused on even simpler models, the geostrophic and quasi-geostrophic models, considered in the late 1940’s by J. von Neumann and his collaborators, in particular J. G. Charney. With the increase of computing power, interest eventually returned to the PEs, which are now the core of many Global Circulation Models (GCM) or Ocean Global Circulation Models (OGCM), available at the National Center for Atmospheric Research (NCAR) and elsewhere. GCMs and OGCMs are very complex models which contain many components, but still, the PEs are the central component for the dynamics of the air or the water. For some phenomena there is need to give up the hydrostatic hypothesis and then non-hydrostatic models are considered, such as in Laprise [18] or Smolarkiewicz, Margolin and Wyszogrodzki [34]; these models stand at an intermediate level of physical complexity between the full Navier–Stokes equations and the PEs-hydrostatic equations. Research on nonhydrostatic models is ongoing and, at this time, there is no agreement, in the physical community, for a specific model. In this hierarchy of models for geophysical fluid dynamics, let us add also the Shallow Water equation corresponding essentially to a vertically integrated form of the Navier–Stokes equations; from the physical point of view they stand as an intermediate model between the primitive and the quasi-geostrophic equations.

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In summary, in term of physical relevance and the level of complexity of the physical phenomena they can account for, the hierarchy of models in geophysical fluid dynamics is as follows: 3D Navier–Stokes Equations ⇓ Nonhydrostatic Models ⇓ Primitive Equations (hydrostatic equations) ⇓ Shallow Water Equations ⇓ Quasi-Geostrophic Models ⇓ 2D Barotropic Equations We remark here also that much study is needed for the boundary conditions from both the physical and mathematical points of views. As we said, our aim in this article is the study of mathematical properties of the primitive equations. Mathematical Background The level of mathematical complexity of the equations above is not the same as the level of physical complexity: at both ends, the quasi-geostrophic models and barotropic equations are mathematically well understood (at least in the presence of viscosity; see S. Wang [40, 41]), and we know the level of complexity of the Navier–Stokes equations to which this handbook is devoted. On the other hand, non-hydrostatic models are mathematically out of reach, and there are much less mathematical results available for the shallow water equations than for the Navier–Stokes equations, even in space dimension two (see however Orenga [31]). As we indicate hereafter, the primitive equations although physically simpler are, in fact, slightly more complicated than the incompressible Navier–Stokes equations.


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Indeed this is due to the fact that the nonlinear term in the Navier–Stokes equations, also called inertial term, is of the form velocity × first order derivatives of velocity, whereas, the nonlinear term for the primitive equations, is of the form first order derivatives of horizontal velocity × first order derivatives of horizontal velocity. The mathematical study of the primitive equations was initiated by J. L. Lions, R. Temam, and S. Wang in the early 1990s. They produced a mathematical formulation of the PEs which resembles that of the Navier–Stokes due to J. Leray, and obtained the existence for all time of weak solutions; see Section 2, and the original articles [21], [22], [24], in the list of references. Further works, conducted during the 1990s and more especially during the past few years, have improved and supplemented the early results of [21], [22], [24] by a set of results which, essentially, brings the mathematical theory of the PEs to that of the 3D incompressible Navier–Stokes equations, despite the added complexity mentioned above; this added complexity is overcome by a non-isotropic treatment of the equations (of certain nonlinear terms), in which the horizontal and vertical directions are treated differently. In summary the following results are now available which will be presented in details in this article: (i) Existence of weak solutions for all time (dimension two and three). (ii) In space dimension three, existence of a strong solution for a limited time (local in time existence). (iii) In space dimension two, existence and uniqueness for all time of a strong solution. (iv) Uniqueness of a weak solution in space dimension two. In the above, the terminology is that normally used for Navier–Stokes equations: the weak solutions are those with finite (fluid) kinematic energy (L∞ (L2 ) and L2 (H 1 )), and the strong solutions are those with finite (fluid) enstrophy (L∞ (H 1 ) and L2 (H 2 )). Essential in the most recent developments (ii)–(iv) above is the H 2 regularity result for a Stokes type problem appearing in the PEs, the

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analog of the H 2 regularity in the Cattabriga–Solonnikov results on the usual Stokes problem; the whole Section 4 is devoted to this problem. Content of this article Because of space limitation it was not possible to consider all relevant cases here. Relevant cases include: The Ocean, The Atmosphere, and The Coupled Ocean and Atmosphere, on the one hand, and, on the other hand, the study of global phenomena on the sphere (involving the writing of the equations in spherical coordinates), and the study of mid-latitude regional models in which the equations are projected on a space tangent to the sphere (the earth), corresponding to the so-called β-plane approximation: here 0x is the west-east axis, 0y the south-north axis, and 0z the ascending vertical. For this more mathematically oriented article we have chosen to concentrate on the cases mathematically most significant. Hence for each case, after a brief description of the equations on the sphere (in spherical coordinates), we concentrate our efforts on the corresponding β-plane case (in Cartesian coordinates). Indeed, in general, going from the β-plane case in Cartesian coordinates to the spherical case necessitates only the proper handling of terms involving lower order derivatives; full details concerning the spherical case can be found also in the original articles [21], [22], [24],... In the Cartesian case of emphasis, generally we first concentrate our attention on the ocean. Indeed, as we will see in Section 2, the domain occupied by the ocean contains corners (in 2D) or wedges (in 3D); some regularity issues occur in this case which must be handled using the theory of regularity of elliptic problems in nonsmooth domains (Grisvard [11], Kozlov, Maz’ya and Rossmann [17], Maz’ya and Rossmann [25]). For the atmosphere or the coupled atmosphere–ocean, the difficulties are similar or easier to handle — hence most of the mathematical efforts will be devoted to the ocean in Cartesian coordinates. In Section 2 we describe the governing equations and derive the result of existence of weak solutions with a different method than in the original articles [21], [22], [24], thus allowing more generality (for the ocean, the atmosphere, and the coupled atmosphere–ocean).


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In Section 3 we study the existence of strong solutions in space dimension three and two (solutions local in time in dimension three, and for all time in dimension two). We establish in dimension three the existence and uniqueness of strong solutions on a limited interval of time(Section 3.2); in dimension two we prove the existence and uniqueness, for all time, of such strong solutions. Finally in Section 3.3, we consider the two-dimensional space-periodic case and prove the existence of solutions for all time, in all H m , m ≥ 2. Section 4 is technically very important, and many results of Sections 2 and 3 rely on it: this section contains the proof of the H 2 regularity of elliptic problems which arise in the primitive equations. This proof relies, as we said, on the theory of regularity of solutions of elliptic problems in nonsmooth domains. More explanations and references will be given in the Introduction of or within each section. As mentioned earlier, the mathematical formulation of the equations of the atmosphere, of the ocean and of the coupled atmosphere ocean were derived in the articles by Lions, Temam and Wang, [21], [22], [24]. For each of these problems, these articles also contain the proof of existence of weak solutions for all time (in dimension three with a proof which easily extends to dimension two). An alternative proof of result, slightly more general is given in Section 2. Concerning the strong solutions, the proof given here of the local existence in dimension three is based on the article by Hu, Temam and Ziane [16]. An alternate proof of this result is due to [12]. In dimension two, the proof of existence and uniqueness of strong solutions, for all time, for the considered system of equations and boundary conditions is new, and based on an unpublished manuscript of M. Ziane [45]. This result is also established, for a simpler system (without temperature and salinity) by Bresch, Kazhikov and Lemoine, [6]. In the space periodic case, existence and uniqueness of solutions in all the spaces H m is proved in Petcu, Temam and Wirosoetisno [29]. Summary of results for the physics oriented reader. The physics oriented reader will recognize in (2.1)- (2.5) the basic conservation laws: conservations of momentum, mass, energy and salt for the ocean, equation of state. In (2.6) and (2.7) appears the simplification due to the Boussinesq approximation, and in (2.11)-(2.16) the simplifications resulting from the hydrostatic assumptions. Hence (2.11)-(2.16) are the Primitive Equations of the ocean.

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The Primitive Equations of the atmosphere appear in (2.116)-(2.121), and those of the coupled atmosphere and ocean are described in section 2.5. Concerning, to begin, the ocean, the first task is to write these equations, supplemented by the initial and boundary conditions, as an initial value problem in a phase space H of the form (1.1)

dU + AU + B(U, U ) + E(U ) = `, dt

(1.2)

U (0) = U0 ,

where U is the set of prognostic variables of the problem, that is the horizontal velocity v = (u, v), the temperature T and the salinity S, U = (v, T, S); see (2.66). The phase space H consists, for its fluid mechanics part, of (horizontal) vector fields with finite kinetic energy. We then study the stationary solutions of (1.1) in Section 2.2.2 and, in Theorem 2.2, we prove the existence for all times of weak solutions of (1.1) - (1.2), which are solutions in L∞ (0, t1 ; L2 ) and L2 (0, t1 ; H 1 ) (bounded kinetic energy and square integrable enstrophy for the fluid mechanics part). A parallel study is conducted for the atmosphere and the coupled atmosphere-ocean in Sections 2.4 and 2.5. In Section 3 we consider in dimension 3 and 2 the strong solutions, which are solutions bounded for all times in the Sobolev space H 1 (“finite enstrophy” space). The main results are Theorems 3.1 and 3.2. Section 4 mathematically very important although technical. It is shown there, that the solutions to certain elliptic problems enjoy certain regularity properties (H 2 regularity, that is the function and their first and second derivatives are square integrable); the problems corresponding to the (horizontal) velocity, the temperature and the salinity are successively considered. The study in Section 4 contains many specific aspects which are explained in details in the long introduction to that section. The study presented in this article is only a small part of the mathematical problems on geophysical flows, but we believe it is an important part. We did not try to produce here an exhaustive bibliography. Further mathematical references on geophysical flows will be given in the text; see also the bibliography of the articles and books that we quote. There is also of course a very large literature in the physical context; we only mentioned some of the books which were very useful to us such as [14, 28, 39, 42, 46].


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Beside the efforts of the authors, we mention in several places that this study is based on joint works with J.L Lions, S. Wang, C. Hu and others. Their help is gratefully acknowledged and we pay tribute to the memory of Jacques-Louis Lions. The authors wish to thank Denis Serre and Shouhong Wang for their careful reading of the whole manuscript and for their numerous comments which significantly improved the manuscript. They extend also their gratitude to Daniele Le Meur and Teresa Bunge who typed significant parts of the manuscript.

2. The Primitive Equations. Weak Formulation. Existence of Weak Solutions. As explained in the Introduction to this article, our aim in this section is first to present the derivation of the Primitive Equations from the basic physical conservation laws. We then describe the natural boundary conditions. Then, on the mathematical side, we introduce the function spaces and derive the mathematical formulation of the PEs. Finally we derive the existence for all time of weak solutions. We successively consider the ocean, the atmosphere and the coupled atmosphereocean.

2.1. The Primitive Equations of the Ocean. Our aim in this section is to describe the Primitive Equations of the ocean (see Section 2.1.1), we then describe the corresponding boundary conditions and the associated initial and boundary value problems (Section 2.1.2).

2.1.1. The Primitive Equations. Generally speaking, it is considered that the ocean is made up of a slightly compressible fluid with Coriolis force. The full set of equations of the large-scale ocean are the following: the conservation of momentum equation, the continuity equation (conservation of mass), the thermodynamics equation (that is the conservation of energy equation), the equation

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of state and the equation of diffusion for the salinity S: (2.1) (2.2) (2.3) (2.4) (2.5)

ρ

dV 3 + 2ρ Ω × V 3 + ∇3 p + ρ g = D, dt dρ + ρ div3 V 3 = 0, dt dT = QT , dt dS = QS , dt ρ = f (T, S, p).

Here V 3 is the three-dimensional velocity vector, V 3 = (u, v, w), ρ, p, T are the density, pressure and temperature and S is the concentration of salinity; g = (0, 0, g) is the gravity vector, D the molecular dissipation, QT and QS are the heat and salinity diffusions. The analytic expressions of D, QT and QS will be given below. The Boussinesq Approximation From both the theoretical and the computational points of view, the above systems of equations of the ocean seem to be too complicated to study. So it is necessary to simplify them according to some physical and mathematical considerations. The Mach number for the flow in the ocean is not large and therefore, as a starting point, we can make the so-called Boussinesq approximation in which the density is assumed constant, ρ = ρ0 , except in the buoyancy term and in the equation of state. This amounts to replacing (2.1), (2.2) by (2.6) (2.7)

ρ0

dV 3 + 2ρ0 Ω × V 3 + ∇3 p + ρg = D, dt div3 V 3 = 0.

Consider the spherical coordinate system (θ, φ, r), where θ (−π/2 < θ < π/2) stands for the latitude on the earth, φ (0 ≤ φ ≤ 2π) on the longitude of the earth, r for the radial distance, and z = r − a for the vertical coordinate with respect to the sea level, and let eθ , eφ , er be the unit vectors in the θ-, φ- and z-directions, respectively. Then we write the velocity of the ocean in the form (2.8)

V 3 = vθ eθ + vφ eφ + vr er = v + w,

where v = vθ eθ +vφ eφ is the horizontal velocity field and w is the vertical velocity.


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Another common simplification is to replace, to first order, r by the radius a of the earth. This is based on the fact that the depth of the ocean is small compared with the radius of the earth. In particular

(2.9)

d ∂ vθ ∂ vφ ∂ ∂ = + + + vr , dt ∂t r ∂θ r cos θ ∂φ ∂r

becomes (2.10)

d ∂ vθ ∂ vφ ∂ ∂ = + + + vz , dt ∂t a ∂θ a cos θ ∂φ ∂z

and, taking the viscosity into consideration, we obtain the equations of the largescale ocean with Boussinesq approximation, which are simply called Boussinesq equations of the ocean (BEs), i.e. equations (2.11)–(2.16) hereafter (for the equation of state (2.16), see Remark 2.1):

(2.11) (2.12) (2.13) (2.14) (2.15) (2.16)

∂v ∂v 1 ∂ 2v + ∇v v + w + ∇p + 2Ω sin θ × v − µv ∆v − νv 2 = 0, ∂t ∂z ρ0 ∂z 2 ∂w ∂w 1 ∂p ρ ∂ w + ∇v w + w + + g − µv ∆w − νv 2 = 0, ∂t ∂z ρ0 ∂z ρ0 ∂z ∂w div v + = 0, ∂z ∂T ∂T ∂ 2T + ∇v T + w − µT ∆T − νT 2 = 0, ∂t ∂z ∂z ∂S ∂S ∂ 2S + ∇v S + w − µS ∆S − νS 2 = 0, ∂t ∂z ∂z ρ = ρ0 (1 − βT (T − Tr ) + βS (S − Sr )),

where v is the horizontal velocity of the water, w is the vertical velocity, and, Tr , Sr are averaged (or reference) values of T and S. The diffusion coefficients µv , µT , µS and νv , νT , νS are different in the horizontal and vertical directions, accounting for some eddy diffusions in the sense of Smagorinsky [33]. The differential operators are defined as follows. The (horizontal) gradient operator grad = ∇ is defined by (2.17)

grad p = ∇p =

1 ∂p 1 ∂p eθ + eφ . a ∂θ a cos θ ∂φ

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The (horizontal) divergence operator div = ∇· is defined by µ ¶ 1 ∂(vθ cos θ) ∂vφ (2.18) div(vθ eθ + vφ eφ ) = ∇ · v = + . a cos θ ∂θ ∂φ ˜ and ∇v T˜ of a vector function v ˜ and a scalar function T˜ The derivatives ∇v v (covariant derivatives with respect to v) are ½ ¾ vθ vφ ∂˜ vθ vφ v˜φ vθ ∂˜ ˜= (2.19) ∇v v + − cot θ eθ a ∂θ a cos θ ∂φ a ½ ¾ vθ ∂˜ vφ vφ ∂˜ vφ v˜θ vφ + + − tan θ eφ , a ∂θ a cos θ ∂φ a vφ ∂ T˜ vθ ∂ T˜ (2.20) ∇v T˜ = + . a ∂θ a cos θ ∂φ Moreover, we have used the same notation ∆ to denote the Laplace–Beltrami operators for both scalar functions and vector fields on Sa2 . More precisely, we have ½ µ ¶ ¾ ∂ ∂T 1 ∂ 2T 1 (2.21) cos θ + , ∆T = 2 a cos θ ∂θ ∂θ cos θ ∂φ2 ½ ¾ 2 sin θ ∂vφ vθ ∆v = ∆(vθ eθ + vφ eφ ) = ∆vθ − 2 (2.22) − 2 eθ a cos2 θ ∂φ a cos2 θ ¾ ½ vφ 2 sin θ ∂vθ − 2 eφ + ∆vφ − 2 a cos2 θ ∂φ a cos2 θ where in (2.22), ∆vθ , ∆vφ are defined by (2.21), and in (2.21), T is any given (smooth) function on Sa2 the two-dimensional sphere of radius a. Remark 2.1. Generally speaking, the equation of state for the ocean is given by (2.5). Only empirical forms of the function ρ = f (T, S, p) are known (see Washington and Parkinson [42], p. 131–132). This equation of state is generally derived on a phenomenological basis. It is natural to expect that ρ decreases if T increases and that ρ increases if S increases. The simplest law is (2.16) corresponding to a linearization around average (or reference) values ρ0 , Tr , Sr of the density, the temperature and the salinity, βT and βS are positive constant expansion coefficients. Much of what follows extends to more general nonlinear equations of state. ¤


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Remark 2.2. The replacement of r by 2.10 in the differential operators implies a change of metric in R3 , where the usual metric is replaced by that of S2a × R, S2a the two-dimensional sphere of radius a centered at O. Remark 2.3. In a classical manner, the Coriolis force 2ρΩ × V 3 produces the term 2 Ω sin θ k × v and a horizontal gradient term which is combined with the pressure, so that p in 2.11 is the so-called augmented pressure. The hydrostatic approximation It is known that for large scale ocean, the horizontal scale is much bigger than the vertical one (5–10 kms versus a few thousands kms). Therefore, the scale analysis (see Pedlosky [28]) shows that ∂p/∂z and ρg are the dominant terms in equations (2.12), leading to the hydrostatic approximation ∂p = −ρg, ∂z which then replaces (2.12). The approximate relation is highly accurate for the large-scale ocean and it is considered as a fundamental equation in oceanography. From the mathematical point of view, its justification relies on tools similar to those used in Section 4.1. It will not be discussed in this article; see however Remark 4.1 in Section 4.1. Using the hydrostatic approximation, we obtain the following equations called the primitive equations of the large-scale ocean (PEs):

(2.23)

(2.24) (2.25) (2.26) (2.27) (2.28) (2.29)

∂v ∂v 1 ∂2v + ∇v v + w + p + 2 Ω sin θ k × v − µv ∆v − νv 2 = Fv , ∂t ∂z ρ0 ∂z ∂p = −ρg, ∂z ∂w div v + = 0, ∂z ∂T ∂ 2T ∂T + ∇v T + w − µT ∆T − νT 2 = FT , ∂t ∂z ∂z ∂S ∂ 2S ∂S + ∇v S + w − µS ∆S − νS 2 = FS , ∂t ∂z ∂z ρ = ρ0 (1 − βT (T − Tr ) + βS (S − Sr )).

Note that Fv , FT , and FS corresponding to volumic sources (of horizontal momentum, heat and salt), vanish in reality; they are introduced here for mathematical generality. We also set Ω = Ωk, where k is the unit vector in the direction of the poles (from south to north).

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Remark 2.4. At this stage the unknown functions can be divided into two sets. The first one, called the prognostic variables, v, T, S (4 scalar functions); we aim to write the PEs as an initial (boundary value problem) for these unknowns, and we set U = (v, T, S). The second set of variables comprises p, ρ, w; they are called the diagnostic variables. In Section 2.1.2, we will see how, using the boundary condition, one can, at each instant of time, express the diagnostic variables in terms of the prognostic variables (a fact which is already transparent for ρ in 2.29). Remark 2.5. Integrating (2.28) over M, using Stokes formula, and taking into account (2.26) and the boundary conditions, we arrive at Z Z d (2.30) SdM = Fs dM; dt M M hence

Z

Z

Z tZ

S dM|t = M

Fs dM dt0 .

S dM|0 + M

0

M

In practical applications, FS = 0 as we said, and the total amount of salt R S dM is conserved. In all cases we write M Z Z 1 1 0 0 S dM, FS = FS − FS dM, (2.31) S =S− |M| M |M| M where |M| is the volume of M, and we see that S 0 satisfies the same equation (2.15), with FS replaced by FS0 . From now on, dropping the primes, we consider (2.15) as the equation for S 0 and we thus have Z Z (2.32) S dM = 0, FS dM = 0. M

M

2.1.2. The initial and boundary value problems. We assume that the ocean fills a domain M of R3 which we describe as follows (see Figure 2.1): The top of the ocean is a domain Γi included in the surface of the earth Sa (sphere centered at 0 of radius a). The bottom Γb of the ocean is defined by (z = x3 = r − a) z = −h(θ, ϕ), ¯ i ; it is assumed also that h is bounded where h is a function of class C 2 at least on Γ from below, (2.33)

0 < h ≤ h(θ, ϕ) ≤ h, (θ, ϕ) ∈ Γi .


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The lateral surface Γ` consists of the part of cylinder (2.34)

(θ, ϕ) ∈ ∂Γi , −h(θ, ϕ) ≤ z ≤ 0.

Figure 2.1: The ocean M. Remark 2.6. Let us make two remarks concerning the geometry of the ocean; the first one is that, for mathematical reasons, the depth is not allowed to be 0 (h ≥ h > 0), and thus “beaches” are excluded. The second one is that the top of the ocean is flat (spherical), not allowing waves; this corresponds to the so-called rigid lid assumption in oceanography. The assumption h > 0 can be relaxed for some of the following results, but this will not be discussed here. The rigid lid assumption can be also relaxed by the introduction of an additional equation for the free surface but this also will not be considered. ¤ Boundary Conditions There are several sets of natural boundary conditions that one can associate to the primitive equations; for instance the following: On the top of the ocean Γi (z = 0) : ∂v + αv (v − v a ) = τv , w = 0, νv ∂z ∂T (2.35) νT + αT (T − T a ) = 0, ∂z ∂S = 0. ∂z

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At the bottom of the ocean Γb (z = −h(θ, ϕ)) : v = 0, w = 0, (2.36) ∂T ∂S = 0, = 0. ∂nT ∂nS On the lateral boundary Γ` (−h(θ, ϕ) < z < 0, (θ, ϕ) ∈ ∂Γi ) : ∂S ∂T = 0, = 0. ∂nT ∂nS Here n = (nH , nz ) is the unit outward normal on ∂M decomposed into its horizontal and vertical components; the conormal derivatives ∂/∂nT and ∂/∂nS are those associated with the linear (temperature and salinity) operators, (2.37)

v = 0, w = 0,

(2.38)

∂ ∂ = µT nH · ∇ + νT nz ∂nT ∂z ∂ ∂ = µS nH · ∇ + νS nz ∂nS ∂z

Remark 2.7. i) The boundary conditions (which are the same) on Γb and Γ` express the no-slip boundary conditions for the water and the absence of fluxes of heat or salt. For Γi , w = 0 is the geometrical (kinematical) boundary condition required by the rigid lid assumption; the Neumann boundary condition on S expresses the absence of salt flux. ii) In general, the boundary conditions on v and T on Γi are not fully settled from the physical point of view. These above correspond to some resolution of the viscous boundary layers on the top of the ocean. Here αv and αT are given ≥ 0, v a and Ta correspond to the values in the atmosphere and τv corresponds to the shear of the wind. iii) The first boundary condition (2.35) could be replaced by v = v a expressing a no-slip condition between air and sea. However such a boundary condition necessitating an exact resolution of the boundary layer would not be practically (computationally) realistic, and as indicated in ii) we use instead some classical resolution of the boundary layer ([32]). iv) As we said the boundary condition of Γi are standard unless more involved interactions are taken into consideration. However for Γb and Γ` different combinations of the Dirichlet and Neumann boundary conditions can be (have been) considered; [22]. Beta-plane approximation


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For midlatitude regional studies it is usual to consider the beta plane approximation of the equations in which M is a domain in the space R3 with Cartesian coordinates denoted x, y, z or x1 , x2 , x3 . In the beta plane approximation, Ω = 2f k, f = f0 + βy, k the unit vector along the south to north poles, ∆ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 , ∇ is the usual nabla vector, (∂/∂x, ∂/∂y), and ∇v = u∂/∂x + v∂/∂y (v = (u, v)). With these notations, the equations (2.24) (2.29) and the boundary conditions (2.35) - (2.38) keep the same form; here the depth h = h(x, y) satisfies, like (2.33) ¯ 0 < h ≤ h(x, y) ≤ h,

(2.39)

and the boundary of M consists of Γi , Γl , Γ` , defined as before. As indicated in the Introduction, we will emphasize in this article the regional model which is slightly simpler, in particular because of the use of cartesian coordinates. Usually the general model in spherical coordinates simply requires the treatment of lower order terms. From now on we consider the regional (cartesian coordinate) case. The diagnostic variables The first step in the mathematical formulation of the PEs consists in showing how to express the diagnostic variables in terms of the prognostic variables, thanks to the equations and boundary conditions. Since w = 0 on Γi and Γb , integration of (2.26) in z gives Z 0 (2.40) w = w(v) = div v dz 0 z

and

Z

0

(2.41)

div vdz = 0. −h

Note that

Z

Z

0

div

0

vdz = −h

div v dz + ∇h · v|z=−h , −h

and since v vanishes on Γb , condition (2.41) is the same as Z 0 (2.42) div vdz = 0 −h

18


Similary, integration of (2.26) in z gives (2.43)

Z

0

p = ps + P, P = P (T, S) = g

ρdz 0 .

z

Here ρ is expressed in terms of T and S through (2.29) and ps = ps (x, y, t) = p(x, y, 0, t) is the pressure at the surface of the ocean. Hence (2.40) and (2.43) provide an expression of the diagnostic variables in terms of the prognostic variables (and the surface pressure); and (2.42) is an additonal equation which, we will see, is mathematically related to the surface pressure. Remark 2.8. The introduction of the nonlocal constraint (2.41) and of the surface pressure ps was first carried out in [21], [22]. This new formulation has played a crucial role in much of the mathematical analysis of the PEs in various cases. 2.2. Weak formulation of the PEs of the ocean. The stationary PEs. We denote by U the triplet (v, T, S) (four scalar functions). In summary the equations that we consider for the subsequent mathematical theory (the PEs) are (2.24), (2.27), (2.28), with w = w(v) given by (2.40), and p given by (2.43) (ρ given by (2.29)); furthermore v satisfies (2.41); hence (2.44) (2.45) (2.46) (2.47) (2.48)

∂v ∂v 1 ∂2v + ∇v v + w + ∇p + 2f k × v − µv ∆v − νv 2 = Fv , ∂t ∂z ρ0 ∂z 2 ∂T ∂T ∂ T + ∇v T + w − µT ∆T − νT 2 = FT , ∂t ∂z ∂z ∂S ∂S ∂ 2S + ∇v S + w − µS ∆S − νS 2 = FS , ∂t ∂z ∂z Z 0 w = w(v) = div v dz 0 , z Z 0 div vdz = 0, −h

(2.49)

Z

0

p = ps + P, P = P (T, S) = g

ρ dz 0 ,

z

(2.50) (2.51)

ρ = ρ0 (1 − βT (T − Tr ) + βS (S − Sr )), Z SdM = 0. M

The boundary conditions are (2.35) - (2.38).


19

2.2.1. Weak formulation and functional setting. For the weak formulation of this problem, we introduce the following function spaces V and H : V = V1 × V2 × V3 , H = H1 × H2 × H3 , Z 0 1 2 V1 = {v ∈ H (M) , div vdz = 0, v = 0 on Γb ∪ Γ` }, −h Z 1 1 1 ˙ V2 = H (M), V3 = H (M) = {S ∈ H (M), SdM = 0}, M Z 0 Z 0 2 2 H1 = {v ∈ L (M) , div vdz = 0, nH · vdz = 0 −h

−h

on ∂Γi ( i.e. on Γ` )}, Z 2 2 2 ˙ SdM = 0}. H2 = L (M), H3 = L (M) = {S ∈ L (M), M

These spaces are endowed with the following scalar products and norms: ˜ 3, ˜ ))1 + KT ((T, T˜))2 + KS ((S, S)) ((U, U˜ )) = ((v, v Z ˜ ∂v ∂ v ˜ ))1 = ((v, v (µv ∇v · ∇˜ v + νv )dM, ∂z ∂z M Z Z ∂T ∂ T˜ ˜ ˜ ((T, T ))2 = (µT ∇T · ∇T + νT )dM + αT T T˜dΓi , ∂z ∂z M Γi Z ∂S ∂ S˜ ˜ 3= )dM, ((S, S)) (µS ∇S · ∇S˜ + νS ∂z ∂z ZM ˜ ˜ + KT T T˜ + KS S S)dM, (U, U˜ )H = (v · v M

||U || = ((U, U ))1/2 ,

1/2

|U |H = (U, U )H .

Here KT and KS are suitable positive constants chosen below. The norm on H is of course equivalent to the L2 - norm and because of the Poincaré inequality, 1/2 v vanishing on Γb ∪ Γ` , and (2.51), || · ||i = ((·, ·))i is a Hilbert norm on Vi , and || · || is a Hilbert norm on V ; more precisely we have, with c0 > 0 a suitable constant depending on M : (2.52)

|U |H ≤ c0 kU k, ∀ U ∈ V.

20


Let V1 be the space of C ∞ (2 Dimensional) vector functions v which vanish in a neighborhood of Γb ∪ Γ` and such that Z

0

div

vdz = 0. −h

Then V1 ⊂ V1 and it has been shown in Lions, Temam and Wang [21], that (2.53)

V1 is dense in V1 .

¯ and by V3 ⊂ V3 the set We also denote by V2 ⊂ V2 the set of C ∞ functions on M ∞ ¯ with zero average; V = V1 × V2 × V3 is dense in V. of C functions on M To derive the weak formulation of this problem we consider a sufficiently regular ˜ in V. We multiply equation (2.44) by v ˜ , the second test function U˜ = (˜ v , T˜, S) ˜ integrate over M and add the resulting one by KT T˜, the third one by KS S, equations; KT , KS > 0 are two constants to be chosen later on. The term involving grad pS vanishes; indeed by the Stokes formula: Z

Z

Z

∇pS · vdM = M

pS nH · vd(∂M) − ∂M

pS ∇vdM, M

where n = (nH , nz ) is the unit outward normal on ∂M, and nH its horizontal component. The integral on ∂M vanishes because nH · v vanishes on ∂M; the remaining integral on M vanishes too since by Fubini’s theorem, (2.48), and v = 0 on Γb : Z

Z

Z pS ∇vdM =

pS

M

Γi

Z

0

Z

∇˜ v dz dΓi = −h

0

˜ dz) dΓi = 0. v

pS (∇ · Γi

−h

Using Stokes’ formula and the boundary conditions (2.35) - (2.38) we arrive after some easy calculations at: µ (2.54)

d U, U˜ dt

¶ + a(U, U˜ ) + b(U, U, U˜ ) + e(U, U˜ ) = `(U˜ ). H


21

The notations are as follows: Z ˜ ˜ + KT T T˜ + KS S S)dM, (U, U˜ )H = (v · v M

a(U, U˜ ) = a1 (U, U˜ ) + KT a2 (U, U˜ ) + KS a3 (U, U˜ ), Z ˜ ∂v ∂ v ˜ )dM a1 (U, U ) = (µv ∇v · ∇˜ v + νv ∂z ∂z M Z Z ¯ − P (T, S)∇˜ v dM + αv v˜ v dΓi , M

Γi

P¯ (T, S) = ρ0 (−βT T + βS S) (see (2.49), (2.60)), Z Z ∂T ∂ T˜ ˜ ˜ a2 (U, U ) = (µT ∇T · ∇T + νT ) dM + αT T T˜dΓi , ∂z ∂z M Γi Z ∂S ∂ S˜ ˜ a3 (U, U ) = (µS ∇S · ∇S˜ + νS ) dM, ∂z ∂z M b = b1 + KT b2 + KS b3 , Z

˜ ] ∂v )v dM, ∂z M Z ∂ T˜ ] ˜ b2 (U, U , U ) = (v · ∇T˜ + w(v) )T ] dM, ∂z M Z ∂ S˜ b3 (U, U˜ , U ] ) = (v · ∇S˜ + w(v) )S ] dM, ∂z M Z ˜ dM. e(U, U˜ ) = 2 (Ω × v) · v

b1 (U, U˜ , U ] ) =

(v · ∇˜ v + w(v)

M

Z `(U˜ ) =

˜ dM ˜ + KT FT T˜ + KS FS S) (Fv v Z Z + (βT Tr − βS Sr ) ∇˜ v dM + [(gv )˜ v + gT T˜] dΓi ,

(2.55)

M

M

Γi

where (see 2.35): gv = τv + αv v a , gT = αT T a . For ` we observe that, if Tr and Sr are constant, then Z Z (2.56) (1 + βT Tr − βS Sr )∇v dM = (1 + βT Tr − βS Sr ) nH v d(∂M) = 0. M

∂M

22


It is clear that each ai , and thus a, is a trilinear continuous form on V ; furthermore if KT and KS are sufficiently large, a is coercive (a2 , a3 are automatically coercive on V2 , V3 ): a(U, U ) ≥ c1 ||U ||2 ,

(2.57)

∀U ∈ V

(c1 > 0).

Similarly e is bilinear continuous on V1 and even H1 , and (2.58)

e(U, U ) = 0,

∀U ∈ H.

Before studying the properties of the form b, we introduce the space V(2) : V(2) is the closure of V in (H 2 (M))4 .

(2.59)

Then we have the following Lemma 2.1. The form b is trilinear continuous on V ×V ×V(2) and V ×V(2) ×V,1 ( (2.60)

(2.61)

|b(U, U˜ , U ] )| ≤

c2 ||U || ||U˜ || ||U ] ||V(2) , ∀ U, U˜ ∈ V, U ] ∈ V(2) , c2 ||U || ||U˜ ||V(2) ||U ] ||, ∀ U, U ] ∈ V, U˜ ∈ V(2) .

1/2 |b(U, U˜ , U ] )| ≤ c2 kU k|U˜ |H kU˜ k1/2 kU ] kV(2) ,

∀ U, U˜ ∈ V, U ] ∈ V(2) .

Furthermore (2.62)

b(U, U˜ , U˜ ) = 0, for U ∈ V, U˜ ∈ V(2) ,

and (2.63)

b(U, U˜ , U ] ) = −b(U, U ] , U˜ ),

for U, U˜ , U # ∈ V, and U˜ or U ] in V(2) . Proof To show first that b is defined on V × V × V(2) let us consider the typical and most problematic term: Z ∂ T˜ (2.64) w(v) T ] dM. ∂z M We have Z ∂ T˜ ∂ T˜ |w(v) T ] |dM ≤ |w(v)|L2 (M) | |L2 (M) |T ] |L∞ (M) . ∂z ∂z M 1For

(2.60) and (2.61), the specific form of V and V(2) is not important : b is as well trilinear continuous on H 1 (M)4 × H 2 (M)4 × H 1 (M)4 and H 1 (M)4 × H 1 (M)4 × H 2 (M), and the estimates are similar, the H 1 and H 2 norms replacing the V and V(2) norms.


23

The first two terms in the right-hand-side of this inequality are bounded by const. ||v||1 (using (2.40)) and ||T˜||2 . In dimension three H 2 (M) ⊂ L∞ (M) so that the third term is bounded by const.||T ] ||V(2) , and hence the right-hand-side of the last inequality is bounded by c||U || ||U˜ || ||U ] ||V(2) . With similar (and easier) inequalities for the other integral, we conclude that b is defined and trilinear continuous on V × V × V(2) . For the continuity on V × V(2) × V, the typical term above is bounded by |w(v)|L2 (M) |

∂ T˜ |L4 (M) |T ] |L4 (M) , ∂z

which is bounded by c||v||1 ||T˜||H 2 ||T ] ||H 1 ≤ c||U || ||U˜ ||V(2) ||U ] ||, since H 1 (M) ⊂ L6 (M); hence the second bound (2.60). We easily prove (2.62) and (2.63) by integration by parts for U, U˜ , U ] ∈ V; the relations are then extended by continuity to the other cases, using (2.60). To establish the improvement (2.61) of the first inequality (2.60), we observe that b(U, U˜ , U ] ) = −b(U, U ] , U˜ ) and consider again the most typical term Z ∂U ] ˜ w(v) U dM, that we bound by ∂z M ¯ ]¯ ¯ ∂U ¯ ¯ |U˜ |L3 . |w(v)|L2 ¯¯ ∂z ¯ 6 L

1

6

Remembering that H ⊂ L and H this term by

1/2

⊂ L3 in space dimension 3, we bound

ckvkkU ] kV(2) |U˜ |1/2 kU˜ k1/2 , and (2.61) follows. The operator form of the equation. We can write equation (2.54) in the form of an evolution equation in the Hilbert 0 space V(2) . For that purpose we observe that we can associate to the forms a, b, e above, the following operators : • A linear continuous from V into V 0 , defined by < AU, U˜ >= a(U, U˜ ), ∀ U, U˜ ∈ V,

24

SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04 0 • B bilinear continuous from V × V into V(2) defined by

< B(U, U˜ ), U ] >= b(U, U˜ , U ] ), ∀ U, U˜ ∈ V, ∀ U ] ∈ V(2) , • E linear continuous from H into itself, defined by < E(U ), U˜ >= e(U, U˜ ), ∀U, U˜ ∈ H. Since V(2) ⊂ V ⊂ H, with continuous injections, each space being dense in the next one, we also have the Gelfand-Lions inclusions 0 . V(2) ⊂ V ⊂ H ⊂ V 0 ⊂ V(2)

(2.65)

With this we see that (2.54) is equivalent to the following operator evolution equation dU (2.66) + AU + B(U, U ) + E(U ) = `, dt 0 understood in V(2) and with ` defined in (2.55). To this equation we will naturally add an initial condition : (2.67)

U (0) = U0 . ¤

2.2.2. The stationary PEs. We now establish the existence of solutions of the stationary PEs. Beside its intrinsic interest, this result will be needed in the next section for the study of the time dependent case. The equations to be considered are the same as (2.54), with the only difference that the derivatives ∂v/∂t, ∂T /∂t and ∂S/∂t are removed, and that the source terms Fv , FT , FS are given independent of time t. The weak formulation proceeds as before: Given F = (Fv , FT , FS ) in H ( or L2 (M)4 ), and g = (gv , gT ) in L2 (Γi )2 , find (2.68)

U = (v, T, S) ∈ V, such that a(U, U˜ ) + b(U, U, U˜ ) + e(U, U˜ ) = `(U˜ ), for every U˜ ∈ V(2) ;

a, b, e, and ` are the same as above. We have the following result


25

Theorem 2.1. We are given F = (Fv , FT , FS ) in L2 (M)4 (or in H), and g = (gv , gT ) in L2 (Γi )3 ; then problem (2.68) possesses at least one solution U ∈ V such that 1 (2.69) ||U || ≤ ||`||V 0 . c1 Proof. The proof of existence is done by Galerkin method, a priori estimates and passage to the limit. The proof is essentially standard, but we give the details because of some specificities in this case. We consider a family of elements {Φj }j of V(2) which is free and total in V (V(2) is dense in V ); and for each m ∈ N, we look for an approximate solution of P (2.68), Um = m j=1 ξjm Φj , such that (2.70)

a(Um , Φk ) + b(Um , Um , Φk ) = `(Φk ), k = 1, . . . , m.

The existence of Um is shown below. An a priori estimate on Um is obtained by multiplying each equation (2.68) by ξkm and summing for k = 1, · · · , m. This amounts to replacing Φk by Um in (2.70); since b(Um , Um , Um ) = 0 by Lemma 2.1, we obtain a(Um , Um ) = `(Um ), and, with (2.57), c1 ||Um ||2 ≤ ||`||V 0 ||Um ||, 1 ||`||V 0 . c1 From (2.71) we see that there exists U ∈ V and a subsequence Um0 , such that Um0 , converges weakly to U as m0 → ∞. Since we cannot replace U˜ by U in (2.68) it is useful to notice that 1 0 ||U || ≤ lim inf ||Um || ≤ ||`||V 0 , 0 m →∞ c1 so that (2.69) is satisfied. Then we pass to the limit in (2.70) written with m0 , and k fixed ≤ m0 . We observe below that (2.71)

(2.72)

||Um || ≤

b(Um0 , Um0 , Φk ) −→ b(U, U, Φk ),

so that, at the limit, U satisfies (2.68) for U˜ = Φk , k fixed arbitrary; hence (2.68) is valid for any U˜ linear combination of Φk and, by continuity (Lemma 2.1), for U˜ ∈ V(2) .

26


The proof is complete after we prove the results used above. ¤ Convergence of the b term To prove (2.72) we first observe, with (2.63), that b(Um0 , Um0 , Φk ) = −b(Um0 , Φk , Um0 ). We also observe that each component of Um0 converges weakly in H 1 (M) to the corresponding component of Φk . Therefore, by compactness, the convergence takes place in H 3/4 (M), strongly; by Sobolev imbedding, H 3/4 (M) ⊂ L4 (M) in dimension 3, and the convergence holds in L4 (M) strongly. Writing Φk = Φ = (v Φ , TΦ , SΦ ), the typical most problematic term is Z ∂v Φ (2.73) w(v m0 ) v m0 dM. ∂z M Since div v m0 converges weakly to div v in L2 (M), w(v m0 ) converges weakly in L2 (M) to w(v); v m0 converges strongly to v in L4 (M) as observed before, and since ∂v Φ /∂z belongs to L4 (M), the term above converges to the corresponding term where v m0 is replaced by v (U = (v, T, S)). Hence (2.69). Existence of Um Equations (2.70) amount to a system of m nonlinear equations for the components of the vector ξ = (ξ1 , ..., ξm ), where we have written ξjm = ξj for simplicity. Existence follows from the following consequence of the Brouwer fixed point theorem. (See J. L. Lions [19]). Lemma 2.2. Let F be a continuous mapping of Rm into itself such that (2.74)

[F(ξ), ξ] > 0 for [ξ] = k, for some k > 0,

where [·, ·] and [·] are the scalar product and norm in Rm . Then there exists ξ ∈ Rm with [ξ] < k, such that F(ξ) = 0. Proof If F never vanishes, then G = −kF(ξ)/[F(ξ)] is continuous on Rm , and we can apply the Brouwer fixed point theorem to G which maps the ball C centered at 0 of radius k into itself. Then G has a fixed point ξ0 in C and we have [G(ξ0 )] = [ξ0 ] = k, [F(ξ0 ), ξ0 ] = [ξ0 ]2 . [F(ξ0 )] This contradicts the hypotheses on F; the lemma is proven. [G(ξ0 ), ξ0 ] = −k

¤


27

We apply this lemma to (2.70) as follows: F = (F1 , ..., Fm ), with (2.75)

Fk (ξ) = a(Um , Φk ) + b(Um , Vm , Φk ) + e(Um , Vm ) − l(Φk ).

Rm is equipped with the usual Euclidean scalar product, so that [F(ξ), ξ] =

m X

Fk (ξ)ξk

k=1

(2.76)

= a(Um , Um ) + b(Um , Um , Vm ) − l(U ) ≥ ( with (2.57), (2.58), (2.62) and Schwarz’ inequality) ≥ c1 ||Um ||2 − ||`||V 0 ||Um ||.

Since the last expression converges to +∞ as ||Um || ∼ [ξ] converges to +∞, there exists k > 0 such that (2.74) holds. The existence of Um follows. Remark 2.9. A perusal of the proof of Theorem 2.1 shows that we proved the following more general result: Lemma 2.3. Let V, W be two Hilbert spaces with W ⊂ V , the injection being continuous. Assume that a is bilinear continuous coercive on V , and that b is trilinear continuous on V × W × V, V × V × W , and continuous on Vw × Vw × W , where Vw is V equipped with the weak topology. Furthermore b(U, U˜ , U ] ) = −b(U, U ] , U˜ ), if U, U˜ , U ] ∈ V, and U˜ or U ] ∈ W. Then, for l given in V 0 , there exists at least one solution U of (2.77)

a(U, U˜ ) + b(U, U, U˜ ) = l(U˜ ), ∀U˜ ∈ W,

which satisfies (2.78)

¯ ). a ¯(U, U ) ≤ `(U

Lemma 2.3 will be useful in the next section. 2.3. Existence of weak solutions for the PEs of the ocean. In this section we establish the existence, for all time, of weak solutions for the equations of the ocean. The main result is Theorem 2.2 given at the end of the section.

28


We consider the primitive equations in their formulation (2.54), that is, with the notations of Section 2.2 : Given t1 > 0, U0 in H, F = (Fv , FT , FS ) in L2 (0, t1 ; H), and g = (gv , gT ) in L2 (0, t1 ; L2 (Γi ))3 ), ∞ 2 to (2.79) µ find U¶∈ L (0, t1 ; H) ∩ L (0, t1 ; V ), such that d U, U˜ + a(U, U˜ ) + b(U, U, U˜ ) + e(U, U˜ ) = `(U˜ ), ∀U˜ ∈ V(2) . dt (2.80)

U (0) = U0 .

Alternatively, and as explained in the previous section (see (2.66),(2.67)), we can write (2.79),(2.80) in the form of an operator evolution equation dU (2.81) + AU + B(U, U ) + E(U ) = `, dt (2.82)

U (0) = U0 .

To establish the existence of weak solutions of this problem we proceed by finite differences in time. 2 Finite differences in time Given t1 > 0 which is arbitrary, we consider N an arbitrary integer and introduce the time step k = ∆t = t1 /N. By time discretization of (2.79)–(2.80), we are naturally led to define a sequence of elements of V , U n , 0 ≤ n ≤ N, defined by U 0 = U0 ,

(2.83)

and then, recursively for n = 1, · · · , N by 1 (U n − U n−1 , U˜ )H + a(U n , U˜ ) + b(U n , U n , U˜ ) ∆t (2.84) + e(U n , U˜ ) = `n (U˜ ), ∀U˜ ∈ V(2) . Here `n ∈ V(2) is given by (2.85)

1 ` (U˜ ) = ∆t n

Z

n∆t

`(t; U˜ )dt,

(n−1)∆t

where `(t; U˜ ) is defined exactly as in (2.54), the dependence of ` on t reflecting now the dependence on t of F, gv and gT . 2At

this level of generality it has not been possible to prove the existence of weak solutions to the Primitive Equations by any other classical method for parabolic equations. In particular the proofs in the articles [21], [22], [24] based on the Galerkin method, assume the H 2 regularity of the solutions of the GFD-Stokes problem, and this result is not available at this level of generality. We recall that the whole Section 4 is devoted to this regularity question.


29

The existence, for all n, of U n ∈ V solution of (2.84) follows from Lemma 2.3, equation (2.84) being the same as (2.77); the notations are obvious and the verification of the hypotheses of the lemma is easy ; furthermore by (2.78), and after multiplication by 2∆t : |U n |2H − |U n−1 |2H + |U n − U n−1 |2H + 2∆t a(U n , U n )

(2.86)

≤ 2∆t `n (U n ). For (2.86), we also used (2.58) and the elementary relation 2(U˜ − U ] , U˜ )H = |U˜ |2H − |U ] |2H + |U˜ − U ] |2H , ∀U˜ , U ] ∈ H.

(2.87)

A priori estimates We now proceed and derive a priori estimates for the U n and then for some associated approximate functions. Using (2.85), (2.54) and Schwarz’ inequality, we bound ∆t `n (U n ) by ∆t1/2 (ξ n )1/2 kU n k with Z n∆t n ξ = ξ(t)dt, (n−1)∆t £ R (2.88) ξ(t) = cZ01 |F (t)|2H + M |1 + βT Tr (t) − βS Sr (t)|2 dt ¤ + (|gv (t)|2 + gT (t)2 )dt , Γi

c01

where is an absolute constant related to c0 (see (2.52)). Hence using also (2.57), we infer from (2.86) that |U n |2H − |U n−1 |2H + |U n − U n−1 |2H + 2∆t c1 kU n k2 ≤

2∆t1/2 (ξ n )1/2 kU n k

≤

n ∆t c1 kU n k2 + c−1 1 ξ .

Hence |U n |2H − |U n−1 |2H + |U n − U n−1 |2H + ∆t c1 kU n k2 n ≤ c−1 1 ξ , for n = 1, · · · , N.

(2.89)

Summing all these relations for n = 1, · · · , N , we find (2.90)

|U n |2H

+

N X £ n=1

¤ |U n − U n−1 |2H + ∆t c1 kU n k2 ≤ κ1 ,

30


with

Z 1 T κ1 = |U0 | + ξ(t)dt. c1 0 Summing the relations (2.89) for n = 1, · · · , m, with m fixed, 1 ≤ m ≤ N , we obtain as well 2

(2.91)

|U m |2H ≤ κ1 , ∀m = 0, · · · , N.

Approximate functions The subsequent steps follow closely the proof in Temam [36] (Chapter 3; Sec. 4), for the Navier-Stokes equations, and we will skip many details3. We first introduce the approximate functions defined as follows on (0, t1 ) (k = ∆t) : Uk : (0, t1 ) 7→ V, Uk (t) = U n , t ∈ ((n − 1)k, nk), `k : (0, t1 ) 7→ V 0 , `k (t) = `n , t ∈ ((n − 1)k, nk), U˜k : (0, t1 ) 7→ V, U˜k is continuous, linear on each interval ((n − 1)k, nk) and U˜k (nk) = U n , n = 0, · · · N. An easy computation (see [36]) shows that : (2.92)

N ³ k ´1/2 ³ X ˜ |Uk − Uk |L2 (0,t1 ;H) ≤ |U n − U n−1 |2H )1/2 , 3 n=1

and we infer from (2.90) and (2.91) that (2.93)

Uk and U˜k are bounded independently of ∆t in L∞ (0, t1 ; H) and L2 (0, t1 ; V ).

We infer from (2.92) and (2.93) that there exists U ∈ L∞ (0, t1 ; H)∩L2 (0, t1 ; V ), and a subsequence k 0 → 0, such that, as k 0 → 0 (2.94)

(2.95)

Uk0 and U˜k0 * U in L∞ (0, t1 ; H) weak star and in L2 (0, t1 ; V )weakly, Uk0 − U˜k0 → 0 in L2 (0, t1 ; H) strongly .

Further a priori estimates and compactness 3The

proof given here would apply to the Navier-Stokes equations in space dimension d ≥ 4 ; it extends the proof given in [36] which is only valid for the Navier-Stokes equations in dimension d = 2 or 3.


31

With the notations above and those used for (2.66) and (2.67 (or (2.84)–(2.81)– (2.82)), we see that the scheme (2.84) can be rewritten as (2.96)

dU˜k + AUk + B(Uk , Uk ) + E(Uk ) = `k , 0 < t < t1 , dt

(2.97)

U˜k (0) = U0 .

0 ); From (2.93) and (2.61) we see that B(Uk , Uk ) is bounded in L4/3 (0, t1 ; V(2) 2 0 since the other terms in (2.96) are bounded in L (0, t1 ; V ) independently of k, we conclude that dUk 0 (2.98) ). is bounded in L4/3 (0, t1 ; V(2) dt

We then infer from (2.93) and the Aubin compactness theorem (see e.g. [36]), that, as k 0 → 0 U˜k0 −→ U in L2 (0, t1 ; H) strongly ,

(2.99)

and the same is true for Uk because of (2.95). Passage to the limit The passage to the limit k 0 → 0 (k = ∆t) follows now closely that of Theorem 4.1 Chapter 2 in [36], we skip the details. We consider U˜ ∈ V (which is dense in Vr , see (2.59)) and a scalar function ψ in C 1 ([0, t1 ]), such that ψ(t1 ) = 0. We take the scalar product in H of (2.96) with U˜ ψ, integrate from 0 to t1 , and integrate by parts the first term; we arrive at Z t1 Z t1 0 ˜ ˜ − (Uk , U )H ψ dt + [a(U˜k , U˜ ) + b(Uk , Uk , U˜ ) 0 0 (2.100) Z t1 ˜ ˜ + e(Uk , U )]ψdt = (U0 , U )H ψ(0) + `k (U˜ )ψdt. 0 0

We can pass to the limit in (2.100) for the sequence k ; for the b term we proceed somehow as for (2.72). For the nonlinear term we write Z t1 Z t1 b(Uk , Uk , U˜ )ψdt = − b(Uk , U˜ , Uk )ψdt, 0

0

and, considering the typical most problematic term, we show that, as k 0 → 0, Z t1 Z Z t1 Z ∂ T˜ ∂ T˜ (2.101) w(v k ) Tk0 ψdMdt −→ w(v) T ψdMdt; ∂z ∂z 0 M 0 M

32


this follows from the fact that div v k0 converges to divv weakly in L2 (M×(0, t1 )), that Tk0 converges to T strongly in L2 (M × (0, t1 )), and ψ∂ T˜/∂z belongs to L∞ (M × (0, t1 )). From this we conclude that U satisfies Z

t1

− 0

(2.102)

Z

t1

(U, U˜ )H ψ 0 dt +

[a(U, U˜ )

0

+ b(U, U, U˜ ) + e(U, U˜ )]ψdt Z t1 = (U0 , U˜ )ψ(0) + `(U˜ )ψdt, 0

for all U˜ in V and all ψ of the indicated type. Also, by continuity (Lemma 2.1), (2.102) is valid as well for all U in V(2) since V is dense in V(2) by (2.59). It is then standard to infer from (2.102) that U is solution of (2.79)–(2.80); this leads us to the main result of this section. Theorem 2.2. The domain M is as before. We are given t1 > 0, U0 in H, and F = (Fv , FT , FS ) in L2 (0, t1 ; H) or L2 (0, t1 ; L2 (M)4 ); g = gv , gT is given in L2 (0, t1 ; L2 (Γi )3 ). Then there exists (2.103)

U ∈ L∞ (0, t1 ; H) ∩ L2 (0, t1 ; V ),

which is solution of (2.79)–(2.80) (or (2.81)–(2.82)) ; furthermore U is weakly continuous from [0, t1 ] into H. 2.4. The Primitive equations of the atmosphere. In this section we briefly describe the Primitive Equations (PEs) of the atmosphere, introduce their mathematical (weak) formulation and state without proof the existence of weak solutions; the proof is essentially the same as for the ocean. We start from the conservation equations similar to (2.1)–(2.5); in fact (2.1) and (2.2) are the same; the equation of energy conservation is slightly different from (2.3) because of the compressibility of the air ; the state equation is that of perfect gaz instead of (2.5); finally instead of the concentration of salt in the


33

water, we consider the amount q of water in air. Hence we have (2.104) (2.105) (2.106) (2.107) (2.108)

ρ

dV 3 + 2ρ Ω × V 3 + ∇3 p + ρ g = D, dt dρ + ρ div3 V 3 = 0, dt dT RT dp cp − = QT , dt p dt dq = Qq , dt p = RρT.

Here, D, QT and Qq contain the dissipation terms. As we said the difference between (2.106) and (2.3) is due to the compressibility of the air; in (2.106) cp > 0 is the specific heat of the air at constant pressure, and R is the specific gas constant for the air; (2.108) is the equation of state for the air. The hydrostatic approximation We decompose V 3 into its horizontal and vertical components as in (2.8), V 3 = (v, w), and we use the approximation (2.10) of d/dt. Also, as for the ocean, we use the hydrostatic approximation, replacing the equation of conservation of vertical momentum (third equation (2.104)) by the hydrostatic equation (2.23). We find (2.109) (2.110) (2.111) (2.112) (2.113) (2.114)

∂v ∂v 1 ∂ 2v + ∇v v + w + ∇p + 2Ω sin θ k × v − µv ∆v − νv 2 = 0, ∂t ∂z ρ ∂z ∂p = −ρg, ∂z ∂ρ ∂w ∂ρ + ρ(∇v + ) + v∇ρ + w = 0, ∂t ∂z ∂z ∂T ∂T ∂ 2T RT dp + ∇v T + w − µT ∆T − νT 2 − = QT , ∂t ∂z ∂z p dt ∂q ∂ 2q ∂q + ∇v q + w − µq ∆q − νq 2 = 0, ∂t ∂z ∂z p = RρT.

The right-hand side of (2.112), which is different from QT in (2.106) now represents the solar heating. Change of vertical coordinate

34


Since ρ does not vanish, the hydrostatic equation (2.110) implies that p is a strictly decreasing function of z, and we are thus allowed to use p as the vertical coordinate ; hence in spherical geometry the independent variables are now ϕ, θ, p and t. By an abuse of notation we still denote by v, T, q, ρ these functions expressed in the ϕ, θ, p, t variables. We also denote by ω the vertical component of the wind, and one can show (see e.g. [14]) that: (2.115)

ω=

dp ∂p ∂p = + ∇v p + w ; dt ∂t ∂z

in (2.115), p is a dependent variable expressed as a function of ϕ, θ, z and t. In this context, the PEs of the atmosphere become (2.116) (2.117) (2.118) (2.119) (2.120) (2.121)

∂v ∂v + ∇v v + ω + 2Ω sin θ k × v + ∇Φ − Lv v = Fv , ∂t ∂p ∂Φ RT + = 0, ∂p p ∂ω div v + = 0, ∂p ∂T ∂T RT¯ + ∇v T + ω − ω − LT T = FT , ∂t ∂p cp p ∂q ∂q + ∇v q + ω − Lq q = Fq , ∂t ∂p p = RρT.

We have denoted by Φ = gz the geopotential (z is now a function of ϕ, θ, p, t); Lv , LT , Lq are the Laplace operators, with suitable eddy viscosity coefficients, expressed in the ϕ, θ, p variables. Hence e.g. · ¸ ∂ ³ gp ´2 ∂v (2.122) Lv v = µv ∆v + νv , ∂p RT¯ ∂p with similar expressions for LT and Lq . Note that FT corresponds to the heating of the sun, whereas Fv and Fq which vanish in reality, are added here for mathematical generality. In (2.119) T has been replaced by T¯ in the term RT ω/cp p. See in [24] a better approximation of RT ω/cp p involving an additional term. With additional precautions, and using the maximum principle for the temperature as in [8], we could keep the exact term RT ω/cp p. The change of variable gives for ∂ 2 v/∂z 2 a term different from the coefficient of νv . The expression above is a simplified form of this coefficient, the simplification


35

is legitimate because νv is a very small coefficient; in particular T has been replaced by T¯ (known) which is an average value of the temperature. Pseudo geometrical domain For physical and mathematical reasons, we do not allow the pressure to go to zero, and assume that p ≥ p0 , with p0 > 0 ”small”. Physically, in the very high atmosphere (p very small), the air is ionized and the equations above are not valid anymore ; mathematically, with p > p0 , we avoid the appearance of singular terms as e.g. in the expressions of Lv , LT and Lq . The pressure is then restricted to an interval p0 < p < p1 , where p1 is a value of the pressure smaller in average than the pressure on earth, so that the isobar p = p1 is slightly above the earth and the isobar p = p0 is an isobar high in the sky. We study the motion of the air between these two isobars ; as we said, for p < p0 we would need a different set of equations and for the ”thin” portion of air between the earth and the isobar p = p1 , another specific simplified model would be necessary. For the whole atmosphere, the domain is M = {(ϕ, θ, p), p0 < p < p1 }, and its boundary consists first of an upper part Γu , p = p0 and a lower part p = p1 which is divided into two parts: Γi the part of p = p1 at the interface with the ocean, and Γe the part of p = p1 above the earth. Boundary conditions Typically the boundary conditions are as follows : On the top of the atmosphere Γu (p = p0 ) : (2.123)

∂v ∂T ∂q = 0, ω = 0, = 0, = 0. ∂p ∂p ∂p

Above the earth on Γe : v = 0, ω = 0, (2.124)

νT

∂T + αT (T − Te ) = 0, ∂p ∂q = gq ∂p

36


Above the ocean on Γi : ³ gp ´2 ∂v νv + αv (v − v S ) = τv , ω = 0, RT¯ ∂p ³ gp ´2 ∂T (2.125) νT + αT (T − T S ) = 0, RT¯ ∂p ∂q = gq . ∂p Remark 2.10. The equations on Γi (2.128) are similar to those on Γi for the ocean (2.35), with different values of the coefficients νv , νT , · · · ; comparison between these two sets of boundary conditions is made in Section 2.5 devoted to the coupled atmosphere and ocean system. In (2.127) and (2.128) Te is the (given) temperature on the earth, and v S , TS are the (given) velocity and temperature of the sea. The boundary conditions (2.126) on Γu are physically reasonable boundary conditions ; they can be replaced by other boundary conditions (e.g. v = 0), which can be treated mathematically in a similar manner. Regional problems and beta plane approximation It is reasonable to study regional problems, in particular at midlatitudes and, in this case we use the beta plane approximation. In this case, as for the ocean, we use the Cartesian coordinates denoted x, y, z or x1 , x2 , x3 ; and Ω = (f0 +βy)k. The equations are exactly the same as (2.115)–(2.126), but now ∆ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 , ∇ is the usual nabla vector (∂/∂x, ∂/∂y), and ∇v = u∂/∂x +v∂/∂y, v = (u, v). The domain M is now some portion of the whole atmosphere : M = {(ϕ, θ, p), (θ, ϕ) ∈ Γi ∪ Γe , p0 < p < p1 }, where Γi ∪ Γe are only part of the isobar p = p1 . The boundary of M consists of Γu , Γi , Γe defined as before and of a lateral boundary Γ` = {(ϕ, θ, p), p0 < p < p1 , (ϕ, θ) ∈ ∂Γu }. The boundary conditions are the same as before on Γu , Γi , Γe , and, on Γ` the conditions would be as follows : Boundary conditions on Γ` (2.126)

v = 0, ω = 0,

∂q ∂T = 0, = 0. ∂nT ∂nq


37

Here ∂/∂nT and ∂/∂nq are defined as in (2.37). Comparing to (2.37), v = 0, ω = 0, is not a physically satisfactory boundary condition ; we would rather assume that (v, ω) has a non zero prescribed value ; however in the mathematical treatment of this boundary condition we would then recover (v, ω) = 0 after removing a background flow ; the necessary modifications are minor. Below we only discuss the regional case. Prognostic and diagnostic variables The unknown functions are regrouped in two sets : the prognostic variables U = (v, T, q) for which and initial value problem will be defined, and the diagnostic variables ω, ρ, Φ(= gz) which can be defined, at each instant of time as functions (functionals) of the prognostic variables, using the equations and boundary conditions. In fact ω is determined in terms of v very much as in the case of the ocean: Z p (2.127) ω = ω(v) = − divv dp0 , p0

with

Z

p1

(2.128)

div v dp = 0. p0

Then ρ is determined by the equation of state (2.121), and Φ is a function of p and T determined by integration of (2.118) : Z p1 RT (p0 ) 0 (2.129) Φ = ΦS + dp ; p0 p in (2.129), ΦS = Φ|p=p1 is the geopotential at p = p1 , that is g times the height of the isobar p = p1 . Weak formulation of the PEs For the weak formulation of the PEs, we introduce function spaces similar to those considered for the ocean, namely : V = V1 × V2 × V3 , H = H1 × H2 × H3 , Z p1 1 2 V1 = {v ∈ H (M) , div vdp = 0, v = 0 on Γe ∪ Γ` }, p0

V2 = V3 = H 1 (M), Z p1 Z p1 2 2 H1 = {v ∈ L (M) , div vdp = 0, nH · vdp = 0 on ∂Γu (i.e. on Γ` )}, p0

p0 2

H2 = H3 = L (M).

38


These spaces are endowed with scalar products similar to those for the ocean ˜ ))1 + KT ((T, T˜))2 + Kq ((q, q˜))3 , ((U, U˜ )) = ((v, v ¶ Z µ ³ gp ´2 ∂v ∂ v ˜ ˜ ))1 = ((˜ v, v ∇v · ∇˜ v+ dM, RT¯ ∂p ∂p M ! Z Ã Z ³ gp ´2 ∂T ∂ T˜ ((T, T˜))2 = ∇T · ∇T˜ + dM + αT T T˜dΓi , RT¯ ∂p ∂p M Γi Z Z µ ³ gp ´2 ∂q ∂ q˜¶ 0 ((q, q˜))3 = ∇q · ∇˜ q+ dM + Kq q q˜dM, RT¯ ∂p ∂p M M Z ˜ ˜ + KT T T˜ + Kq q q˜)dM, (U, U )H = (v · v M 1/2

kU k = ((U, U ))1/2 , |U |H = (U, U )H . Here KT , Kq are suitable positive constants chosen below, Kq0 > 0 is a constant of suitable (physical) dimension. The norm on H is of course equivalent to the L2 -norm and, thanks to the Poincaré inequality, k·k is a Hilbert norm on V ; more precisely, there exists a suitable constant c0 > 0 (different from that in (2.52)) such that (2.130)

|U |H ≤ c0 kU k, ∀U ∈ V.

We denote by V1 the space of C ∞ (R2 valued) vector functions v which vanish in a neighborhood of Γe ∪ Γ` and such that Z p1 div vdp = 0. p0

Let V2 = V3 be the space of C ∞ functions on M, and V = V1 × V2 × V3 ; then, as in (2.53): (2.131)

V1 is dense in V1 , V is dense in V.

We also introduce V(2) the closure of V in H 2 (M)4 . The weak formulation of the PEs of the atmosphere takes the form: ¶ µ dU ˜ ,U + a(U, U˜ ) + b(U, U, U˜ ) + e(U, U˜ ) = `(U˜ ), ∀U˜ ∈ V(2) . (2.132) dt H


39

Here b = b1 + KT b2 + Kq b3 , and e are essentially as for the ocean, replacing, for b, ∂/∂z by ∂/∂p and w(v) by ω(v). Then a = a1 + KT a2 + Kq a3 , with ¶ Z µ ³ gp ´2 ∂v ∂ v ˜ a1 (U, U˜ ) = µv ∇v · ∇˜ v + νv dM RT¯ ∂p ∂p M ¶ Z µZ p1 Z RT 0 − dp ∇˜ v dM + αv v˜ v dΓi , p0 M p Γi ! Z Ã ³ gp ´2 ∂T ∂ T˜ dM a2 (U, U˜ ) = µT ∇T · ∇T˜ + νT RT¯ ∂p ∂p M Z Z RT¯(p) ˜ − ω(v)T dM + αT T T˜ dΓi , cp p M Γi Z µ ³ gp ´2 ∂q ∂ q˜¶ ˜ µS ∇q · ∇˜ q + νS dM. a3 (U, U ) = RT¯ ∂p ∂p M Finally

Z `(U˜ ) =

(Fv v˜ + KT FT T˜ + Kq Fq q˜)dM ZM Z ˜ + gv v˜ + gT T dΓi + gT T˜dΓe , Γi

Γe

gv = τv + αv v S , gT = αT TS on Γi , gT = αT Te on Γe . We find that there exist c1 , c2 > 0 such that Z a(U, U ) + c2 q 2 dM ≥ c1 kU k2 , ∀U ∈ V, M

and that e(U, U ) = 0, ∀U ∈ V. Properties of b are similar to those in Lemma 2.1, with V(2) defined as the closure of V in H 2 (M)4 . The boundary and initial value problem As for the ocean, the weak formulation reads : We are given t1 > 0, U0 in H, F = (Fv , FT , Fq ) in L2 (0, t1 ; H) (or L2 (0, t1 ; L2 (M)4 ), gv in L2 (0, t1 ; L2 (Γi )2 ), gT in L2 (0, t1 ; Γi ∪Γe )). We look for U : (2.133) such that

U ∈ L∞ (0, t1 ; H) ∩ L2 (0, t1 ; V ),

40


µ (2.134)

dU ˜ ,U dt

¶ + a(U, U˜ ) + b(U, U, U˜ ) + e(U, U˜ ) H

= `(U˜ ), ∀U˜ ∈ V(2) , (2.135)

U (0) = U0 .

Remark 2.11. We can introduce the operators A, B, E and write equation 2.134 in an operator form, as (2.66). The analog of Theorem 2.2 can be proved in exactly the same way : Theorem 2.3. The domain M is a before. We are given t1 > 0, U0 in H, F = (Fv , FT , Fq ) in L2 (0, t1 ; H) (or L2 (0, t1 ; L2 (M)4 ), gv in L2 (0, t1 ; L2 (Γi )2 ), gT in L2 (0, t1 ; Γi ∪ Γe )). Then there exists U which satisfies (2.134) and (2.135). Furthermore U is weakly continuous from [0, t1 ] in H. 2.5. The coupled atmosphere and ocean. After considering the ocean and the atmosphere separately, we consider in this section the coupled atmosphere and ocean (CAO in short). The model presented here was first introduced in [24]; it is amenable to the mathematical and numerical analysis and is physically sound. The model was derived by carefully examining the boundary layer near the interface Γi between the ocean and the atmosphere. Although some processes are still not fully understood from the physical point of view, the derivation of the boundary condition is based on the work of A. F. Gill [10] and R.L. Haney [13]. We will present the equations and boundary conditions, the variational formulation and arrive to a point where the mathematical treatment is the same as for the ocean and the atmosphere. The pseudo-geometrical domain Let us first introduce the pseudo-geometrical domain. Let h0 be a typical length (height) for the atmosphere; for harmonization with the ocean we introduce the vertical variable η = z, for z < 0 (in the ocean) and µ (2.136)

η = h0

p1 − p p1 − p0

¶ , 0 < η < h0 ,


41

for the atmosphere. The pseudo-geometrical domain is M = Ma ∪ Ms ∪ Γi , where Ms is the ocean defined as in Section 2.1.2, Ma is the atmosphere, 0 < η < h0 , and Γi is, as before the interface between the ocean and the atmosphere. All quantities will now be defined as for the atmosphere alone, adding when needed, a superscript a or as for the ocean alone, adding a superscript s .4 Hence, with obvious notations, the boundary of M consists of

(2.137)

Γs` ∪ Γb ∪ Γe ∪ Γu .

The governing equations In Ma , the variable is U a = (v a , T a , q) and in Ms the variable is U s = (v s , T s , S); we set also U = {U a , U s } , or alternatively v = {v a , v s } , T = {T a , T s } . The equations for U s are exactly as in (2.24) - (2.29), introducing only a superscript s for w, p, ρ0 , Fv , FT , FS , Tr , Sr , as well as the eddy viscosity coefficients µv , νv , etc. The equations for U a are exactly as in (2.116) - (2.126), introducing again a superscript a for ω, p, ρ, Fv , FT , Fq , as well as the various coefficients.5 Of course the variable p is replaced by η following (2.136), and the differential operators are changed accordingly. Boundary Conditions Except for Γi , the boundary conditions are the same as for the ocean and the atmosphere taken separately. Hence we recover the conditions (2.126) on Γu and (2.127) on Γe , adding the superscripts a. Let us now describe the boundary conditions on Γi . These conditions were introduced in [24]; we refer the reader to this monography for justification and a detailed discussion. 4s

for sea, rather than o for ocean which could be confused with zero. additional term linear in ω a with a coefficient depending on pa appears in the equation a for T in [24]; see equation (1.11) p.14 and the footnote (2 ) p.15 of [24]. This term does not affect the discussion hereafter. 5An

42


We first have the geometrical (kinematical) condition: ws = ω a = 0 on Γi ,

(2.138)

which expresses that η = 0 (z = 0) is indeed the upper limit of the ocean (under the rigid lid hypothesis) and η = 0 is the lower limit of the atmosphere (the isobar p = p1 ). Then for the velocity we express the fact that the tangential shear-stresses exerted by the atmosphere on the ocean have opposite values and vice-versa, and this value is expressed as a function of the differences of velocities v a − v s using a classical empirical model of resolution of boundary layers (see e.g. [10, 13, 24]; the boundary layer model is used to model the boundary layer of the atmosphere that is most significant). These conditions read: ∂v s ∂v a a = −¯ ρa νva = ρ¯a CD (α)(v a − v s )|v a − v s |α , 6 ∂z ∂z a Here α ≥ 0 and CD (α) ≥ 0 are coefficients from boundary layer theory, and a ρ¯ > 0 is an averaged value of the atmosphere density. Similar conditions hold for the temperatures, the salinity S in the ocean, and the humidity q in the atmosphere. For the sake of simplicity, to keep the boundary condition linear, we take α = 0; we also need to replace z by η(p) in the atmosphere, see [24] for the details. In the end we arrive at the following conditions on Γi (2.139)

ρs0 νvs

(2.140)

wa = ω s = 0, ∂v s gpa ∂v a ρs0 νvs = −¯ ρa νva ( ¯ )2 a = αv (v a − v s ), ∂z RT ∂p s a a s s S ∂T a a a gp 2 ∂T = −cp ρ¯ νT ( ¯ ) = αT (T a − T s ), cp ρ0 νT ∂z RT ∂z ∂q ∂S = = 0. ∂pa ∂z

Weak formulation of the PEs For the sake of simplicity we restrict ourselves to a regional problem using the beta-plane approximation. 6The

same equation appears in [24] with ρ¯a replaced by ρa . Replacing ρa by ρ¯a is a necessary simplification for the developments below.


43

The function spaces that we introduce are similar to those used for the ocean and the atmosphere; hence V = V1 × V2 × V3 ,

H = H1 × H2 × H3 ,

where Vi = Via × Vis , Hi = Hia × His , the spaces Via , Hia , Vis , His being exactly like those of the atmosphere and the ocean respectively. Alternatively we can write, with obvious notations, V = V a × V s , H = H a × H s . These spaces are endowed with the following scalar products: ((U, U˜ )) = ((U, U˜ ))a + ((U, U˜ ))s , ˜ a ))a,1 + KT ((T a , Tã ))a,2 ((U, U˜ ))a = ((v a , v +Kq ((q, q˜))a,3 , ˜ s ))s,1 + KT ((T s , Tã ))s,2 ((U, U˜ ))s = ((v s , v ˜ s,3 , +Ks ((S, S)) (U, U˜ )H = (U, U˜ )a + (U, U˜ )s , Z ˜ a ) + KT T a Tã + Kq q q˜)dM, (U, U˜ )a = (v a · v a ZM ˜ ˜ s + KT T s T˜s + KS S S)dM. (U, U˜ )s = (v s · v MS

Note that KT is chosen the same in the atmosphere and the ocean. By the Poincaré inequality, there exists a constant c0 > 0 (different than those for the ocean and the atmosphere), such that (2.141)

|U |H ≤ c0 ||U ||,

∀U ∈ V.

From this we conclude that || · || is a Hilbert norm on V. We also introduce, in a very similar way the spaces V and V(2) . With this, the weak formulation of the PEs of the coupled atmosphere and ocean takes the form: dU ( , U˜ )H + a(U, U˜ )+b(U, U, U˜ ) + e(U, U˜ ) dt (2.142) = `(U˜ ), ∀U˜ ∈ V(2) , (2.143)

U (0) = U0 .

44


Here a = a1 + a2 + a3 , b = b1 + b2 + b3 , e = ea + es , where

Z |v a − v s |2 dΓi , Γi Z a a a s s s a2 = KT cp ρ¯ a2 + KT cp ρ0 a2 + αT |T a − T s |2 dΓi , a1 =

ρ¯a aa1

a3 =

Kq aa3

+

ρs0 as1

+

+ αv

Γi

Ks as3 ,

b1 = ρ¯a ba1 + ρs0 bs1 , b2 = KT cap ρ¯a ba2 + KT csp ρs0 bs2 , b3 = Kq ba3 + Ks bs3 . Here, of course, the forms aai , bai , ea are those of the atmosphere, and asi , bsi , es are those of the ocean. The form ` is defined as for the ocean and the atmosphere, the terms concerning Γi being omitted. Hence ` = à + `s , Z a ˜ ˜ a + KT cap ρ¯a FTa Tã + ` (U ) = (¯ ρa Fva v Ma Z a +Kq Fq q˜)dM + gTa Tã dΓe , Γe Z ˜ s + KT csp ρs0 T˜s `s (U˜ ) = (ρs0 Fvs v Ms Z s ˜ +Ks Fs S)dM + (βT Tr − βS Sr )∇˜ v s dMs . MS

With these definitions, the properties of a, b, e, ` are exactly the same as for the ocean and atmosphere (separately) and we prove, exactly as before, the existence, for all time, of weak solutions: Theorem 2.4. The domain M is a before. We are given t1 > 0, U0 in H and F in L2 (0, t1 ; H) (or L2 (0, t1 ; L2 (M)8 ), and gTa in L2 (0, t1 ; Γe ). Then there exists U which satisfies (2.142) and (2.143), and U ∈ L∞ (0, t1 ; H) ∩ L2 (0, t1 ; V ).


45

Furthermore U is weakly continuous from [0, t1 ] into H.

3. Strong Solutions of the Primitive Equations in Dimension 2 and 3. In this section we first show, in Section 3.1 the existence, local in time, of strong solutions to the PEs in space dimension 3, that is solutions whose norm in H 1 remains bounded. Then, in Section 3.2 we consider the PEs in space dimension 2 in view of adapting to this case the results of Sections 2 and 3.1. The 2D-PEs are presented in Section 3.2.1 as well as their weak formulation (Section 3.2.2). Strong solutions are considered in Section 3.2.3 and we show that the strong solutions already considered in dimension 3 are now defined for all time t > 0. Essential in all this section is the anisotropic treatment of the equations, the vertical direction 0z playing a different role than the horizontal ones (0x in 2D, 0x and 0y in 3D).

3.1. Strong solutions in space dimension 3. In this section we establish the local, in time, existence of strong solutions of the primitive equations of the ocean. The result that we obtain is similar to that for the 3D-Navier-Stokes equations. The analysis given in this section also applies to the primitive equations of the atmosphere and the coupled atmosphere-ocean equations using the notations and equations given in Section 2. We first state the main result of this section (Theorem 3.1). We then prepare its proof in several steps : in Step 1, we consider the linearized primitive equations and establish the global existence of strong solutions. In Step 2 we use the solution of the linearized equations in order to reduce the primitive equations to a nonlinear evolution equation with zero initial data and homogeneous boundary conditions. We also provide the necessary a priori estimates for this new problem with zero initial data and homogeneous boundary conditions. Finally, in the last step, we actually prove Theorem 3.1; in particular we show how one can establish the existence of solutions for this problem using the Galerkin approximation with

46


basis consisting of the eigenvectors of A (which are in H 2 , thanks to the regularity results of Section 4). We use the previous estimates and then pass to the limit. The main result of this section is as follows: ¯ i → R+ is of Theorem 3.1. We assume that Γi is of class C 3 and that h : Γ class C 3 ; we also assume (see (2.92)) that (3.1)

∇h · nΓi = 0 on ∂Γi ,

where nΓi is the unit outward normal on ∂Γi (in the plane 0xy). Furthermore, we are given U0 in V , F = (Fv , FT , FS ) in L2 (0, t1 ; H) with ∂F/∂t in L2 (0, t1 ; L2 (M)4 ), and g = (gv , gT ) in L2 (0, t1 ; H01 (Γi )3 ) with ∂g/∂t in L2 (0, t1 ; H01 (Γi )3 ). 7 Then there exists t∗ > 0, t∗ = t∗ (kU0 k), and there exists a unique solution U = U (t) of the primitive equations (2.79) such that (3.2)

U ∈ C([0, t∗ ]; V ) ∩ L2 (0, t∗ , H 2 (M)4 ).

Step 1 The first step in the proof of Theorem 3.1 is the study of the linear primitive equations of the ocean. Hence we consider the equations (compare to (2.44)–(2.51)) :

(3.9)

∂v ∗ ∂ 2v∗ ∗ ∗ ∗ + ∇p + 2f k × v − µv ∆v − νv 2 = Fv , ∂t ∂z ∂p∗ = −ρg, ∂z ∂T ∗ ∂ 2T ∗ − µT ∆T ∗ − νT = FT , ∂t ∂z 2 ∂S ∗ ∂ 2S ∗ + µS ∆S ∗ − νS = FS , ∂t ∂z 2 Z 0 div v ∗ dz 0 = 0, Z −h S ∗ dM = 0, M Z 0 ∗ ∗ p = ps + g ρ∗ dz 0 ,

(3.10)

ρ∗ = ρ0 (1 − βT (T ∗ − Tr ) + βS (S ∗ − Sr )),

(3.3) (3.4) (3.5) (3.6) (3.7) (3.8)

z

7The

hypotheses on ∂F/∂t and ∂g/∂t can be weakened.


47

with the same initial and boundary conditions as for the full nonlinear problem, that is (see (2.35)–(2.38) :

(3.11)

∂v ∗ + αv (v ∗ − v a ) = 0, ∂z ∂S ∗ ∂T ∗ νT + αT (T ∗ − Ta ) = 0, = 0 on Γi , ∂z ∂z

(3.12)

v ∗ = 0, ∂T ∗ ∂S ∗ = 0, = 0 on Γb ∪ Γ` , ∂nT ∂nS

(3.13)

v ∗ = v 0 , T ∗ = T0 , and S ∗ = S0 at t = 0.

νv

Comparing with the nonlinear problems (2.44)–(2.51), and using the same notations as in (2.66), (2.67), we see that U ∗ = (v ∗ , T ∗ , S ∗ ) is solution of the following equation written in functional form:

(3.14)

dU ∗ + AU ∗ + E(U ∗ ) = `, dt

(3.15)

U ∗ (0) = U0 .

The right-hand side ` of (3.14) is exactly the same as in (2.66) (see (2.44)–(2.51) and the expression of ` in (2.55)).

48


¯ of the linear stationary problem, We also consider the solution U¯ = (¯ v , T¯, S) namely −µv ∆¯ v − νv

¯ ∂2v ¯, + ∇¯ p = Fv − 2f k × v ∂z 2

∂ p¯ = −¯ ρg, ∂z ∂ T¯ −µT ∆T¯ − νT 2 = FT , ∂z (3.16)

∂ 2 S¯ −µT S∆S¯ − νS 2 = FS , ∂z Z 0 0 ¯ dz = 0, div v −h Z ¯ SdM = 0, M Z 0 p¯ = p¯s + g ρ¯ dz 0 , z ρ¯ = ρ0 (1 − βT (T¯ − Tr ) + βS (S¯ − Sr )),

with the boundary conditions ¯ ∂v + αv (¯ v − v a ) = τv , ∂z ∂ T¯ ∂ S¯ νT + αT (T¯ − Ta ) = 0, = 0 on Γi , ∂z ∂z ∂ S¯ ∂ T¯ ¯ = 0, = 0, = 0 on Γb ∪ Γ` . v ∂nT ∂nS νv

(3.17)

Note that the left and right-hand sides in (3.16) and (3.17) depend on the time t (as well as in (3.3)–(3.12)). The existence and uniqueness for (almost) every time t for (3.16)–(3.17), follows from the Lax-Milgram theorem as e.g. explained, for ¯ , in Section 4.4.1 and Proposition 4.1. Furthermore, the regularity the velocity v results of Section 4 (in particular Theorems 4.1 to 4.5) show that the solution belong to H 2 (M), and that (3.18)

v k2H2 (M) ≤ c0 κ1 , k¯ v k2H2 (M) + kT¯k2H2 (M) + k¯ κ1 = κ1 (F, τv , v a , Ta ) = |F |2H + kτv k2H1 (Γi ) + kv a k2H1 (Γi ) + kTa k2H1 (Γi )


49

Note that, again, each side of (3.18) depends on t, and (3.18) is valid for (almost) every t. The constant C0 depends only on M according to the results of Section 4; in particular C0 is independent of t. Hypothesis (3.1) is precisely what is needed for the utilization of Theorem 4.5 used for (3.18). It is noteworthy that ¯ ¯ /∂t, ∂ T¯/∂t, ∂ S/∂t, we have the same estimates as (3.18) for the derivatives ∂ v κ1 0 being replaced by κ1 which is defined similarly in terms of the time derivatives ∂F/∂t, etc. ˜ = v∗ − v ¯ , p˜ = p∗ − p¯, ρ˜ = ρ∗ − ρ¯, T˜ = T ∗ − T¯ and S˜ = S ∗ − Now let v ¯ The equations satisfied by (˜ ˜ are the same as (3.3)–(3.13) but with S. v , T˜, S) (Fv , FT , FS ) = −dU¯ /dt, v a = τv = Ta = Tr = Ts = 0, and with initial data (3.19)

˜ t=0 = S0 − S(0). ¯ ˜ |t=0 = v 0 − v ¯ (0), T˜|t=0 = T0 − T¯(0) and S| v

Comparing with the nonlinear problem (2.44)–(2.51), and using the same no˜ is solution of the following tation as in (2.66), (2.67), we see that U˜ = (˜ v , T˜, S) equation written in functional form : dU˜ dU¯ (3.20) + AU˜ + E(U˜ ) = − , dt dt U˜ (0) = U˜0 = U0 − U¯ (0).

(3.21)

Note that the contribution from ` vanishes (see the expression in the equation preceeding (2.56)). The existence for all time of a strong solution U˜ to (3.20)–(3.21) is classical, and we recall the estimate obtained by multiplying (3.20) by AU˜ and integrating in time : Z t1 2 ˜ sup kU (t)k + |AU˜ (s)|2 ds ≤ ckU˜ (0)k2 + cκ0 . 0≤t≤t1

0

H

1

Here we have used the analog of (3.18) for dU¯ /dt, see the comments after (3.18). From this we obtain Z t1 ∗ 2 sup kU (t)k + |U ∗ (s)|2H 2 (M)4 ds ≤ c(kU (0)k2 + kU¯ (0)k2 ) 0≤t≤t1 0 (3.22) Z t1 Z t1 2 κ01 (s)ds, +c |U¯ (s)| 2 4 ds + c 0

H (M)

0

and, with (3.18), we bound the right-hand side of (3.22) by an expression κ2 of the form : Z t1 2 (3.23) κ2 = ckU0 k + c0 (κ1 (0) + (κ1 + κ01 )(s)ds). 0

50


Step 2 We will now use U ∗ = (v ∗ , T ∗ , S ∗ ) and write the primitive equations of the ocean using the decomposition U = U ∗ + U 0 ; we note that U 0 (0) = 0 and that U 0 = (v 0 , T 0 , S 0 ) satisfies homogeneous boundary conditions of the same type as U . More precisely, starting from the functional form (2.66)–(2.67) of the equation for U and using (3.14)–(3.15), we see that U 0 satisfies

(3.24)

dU 0 + AU 0 + B(U 0 , U ∗ ) + B(U ∗ , U 0 ) dt + B(U 0 , U 0 ) + E(U 0 ) = −AU ∗ − B(U ∗ , U ∗ ),

(3.25)

U 0 (0) = 0.

The existence of solution in Theorem 3.1 is obtained by proving the existence of solution for this system (on some interval of time (0, t∗ )). As usual this proof of existence is based on a priori estimates for the solutions U 0 of (3.24)–(3.25). Some a priori estimates can be obtained by proceeding exactly as for Theorem 2.2, but additional estimates are needed here. Essential for these new estimates is another estimate on the bilinear operator B (or the trilinear form b), which is obtained by an anisotropic treatment of certain integrals. We have the following result (compare to Lemma 2.1) : Lemma 3.1. In space dimension three, the form b is trilinear continuous on H 2 (M)4 × H 2 (M)4 × L2 (M)4 and we have (3.26)

|b(U, U [ , U ] )| 1/2

1/2

1/2

1/2

1/2

1/2

≤ c3 (kU kH 1 kU [ kH 1 kU [ kH 2 + kU kH 1 kU kH 2 kU [ kH 1 kU [ kH 2 )|U ] |L2 ,

for every (U, U [ , U ] ) in this space. The proof of this Lemma is given below. Using Lemma 3.2, we obtain a priori estimates on the solution U 0 of (3.24)–(3.25). We denote by A1/2 the square root of A so that (A1/2 U, A1/2 U˜ )H = a(U, U˜ ), ∀ U, U˜ ∈ V.


51

Taking the scalar product of (3.24) with AU 0 in H, we obtain

(3.27)

1 d 1/2 0 2 |A U |H + |AU 0 |2H 2 dt = −b(U 0 , U ∗ , AU 0 ) − b(U ∗ , U 0 , AU 0 ) − b(U 0 , U 0 , AU 0 ) − b(U ∗ , U ∗ , AU 0 ) − (AU ∗ , AU 0 )H − (E(U 0 ), AU 0 )H .

We bound each term in the right-hand side of (3.27) as follows, using Lemma 3.2 for the b-terms : 1 |(AU ∗ , AU 0 )H | ≤ |AU 0 |2H + c|U ∗ |2H 2 , 8 12 |(E(U 0 ), AU 0 )H | ≤

1 |AU 0 |2H + c|U 0 |2H , 12

|b(U 0 , U ∗ , AU 0 ) ≤ ckU 0 kkU ∗ kH 2 |AU 0 |H 1/2

1/2

3/2

+ ckU 0 k1/2 kU ∗ kH 1 kU ∗ kH 2 |AU 0 |H 1 ≤ |AU 0 |2H + ckU 0 k2 kU ∗ k2H 2 (1 + kU ∗ k2H 1 ), 12 1/2 1/2 3/2 ∗ 0 0 |b(U , U , AU )| ≤ ckU ∗ kH 1 kU ∗ kH 2 kU 0 k1/2 |AU 0 |H 1 2 ≤ |AU 0 |2H + ckU 0 k2 kU ∗ k2H 1 kU ∗ kH 2, 12 |b(U 0 , U 0 , AU 0 )| ≤ c4 kU 0 k|AU 0 |2H , |b(U ∗ , U ∗ , AU 0 )| ≤ ckU ∗ kH 1 kU ∗ kH 2 |AU 0 |H 1 ≤ |AU 0 |2H + ckU ∗ k2H 1 kU ∗ k2H 2 . 12 Here we used the fact that the norm |AU 0 |H is equivalent to the norm |U 0 |H 2 , thanks to the results of Section 4, and (this is easy), the fact that the norm |A1/2 U 0 |H is equivalent to the norm kU 0 k = kU 0 kH 1 . Taking all these bounds into account, we infer from (3.27) that (3.28)

8U ∗

d 1/2 0 2 |A U |H + (1 − c4 |A1/2 U 0 |H )|AU 0 |2H dt ≤ λ(t)|A1/2 U 0 |2H + µ(t),

is not in D(A) and AU 0 ∈ V 0 , because U ∗ does not satisfy the homogeneous boundary conditions; however this bound is valid, see the details in [16].

52


with λ(t) = ckU ∗ k2H 2 + µ(t), µ(t) = ckU ∗ k2H 1 kU ∗ k2H 2 . We infer from (3.22)–(3.23) (and the precise expression of κ2 in (3.23)), that λ and µ are integrable on (0, t1 ) and we set Z t1 κ3 = λ(t)dt. 0 0

By Gronwall’s Lemma and since U (0) = 0, we have, on some interval of time (0, t∗ ), and as long as 1 − c4 |A1/2 U 0 |H ≥ 0 : µZ t ¶ µZ t ¶ 1/2 0 2 |A U (t)|H ≤ µ(s)ds exp λ(s)ds , 0 0 (3.29) µZ t ¶ 1/2 0 2 |A U (t)|H ≤ exp λ(s)ds . 0

In fact (3.28) is valid as long as 0 < t < t∗ where t∗ is the smaller of t1 and t∗ , where t∗ is either +∞ or the time at which µ ¶ Z t∗ 1 λ(s)ds = log (3.30) . 4c24 κ3 0 We then have (3.31)

|A1/ U 0 (t)|2H ≤

1 , for 0 < t < t∗ , 4c24

and returning to (3.29) we find also a bound Z t∗ (3.32) |AU 0 (t)|2H dt ≤ const. . 0

Step 3 - Proof of the existence in Theorem 3.1 As we said, the existence for Theorem 3.1 is shown by proving the existence of a solution U 0 of (3.24)(3.25) in C([0, t∗ ]; V ) ∩ L2 (0, t∗ ; D(A)). For that purpose we implement a Galerkin method using the eigenvectors ej of A : Aej = λj ej , j ≥ 1, 0 < λ1 ≤ λ2 ≤ · · · The results of Section 4.1 show that the ej belong to H 2 (M)4 (D(A) ⊂ H 2 (M)4 ). We look, for each fixed m > 0, for an approximate solution 0 Um

=

m X j=1

ξjm (t)ej ,


53

satisfying (compare to (3.24)(3.25)) : µ 0 ¶ dUm 0 0 , ek + a(Um , ek ) + b(Um , U ∗ , ek ) dt H (3.33) 0 0 0 0 + b(U ∗ , Um , ek ) + b(Um , Um , ek ) + e(Um , ek ) = −a(U ∗ , ek ) − b(U ∗ , U ∗ , ek ), k = 1, · · · , m, and 0 (0) = 0. Um

Multiplying (3.33) by ξkm (t)λk , and adding these equations for k = 1, · · · , m, 0 we obtain the analogue of (3.27) for Um . The same calculations as above show 0 that Um satisfies the same estimates independent of m as (3.31)–(3.32), with the same time t∗ (also independent of m). It is then straightforward to pass to the limit m → ∞ and we obtain the existence. Step 4 - Proof of uniqueness in Theorem 3.1 The proof of uniqueness is easy. Consider two solutions U1 , U2 of the primitive equations ; let U = U1 − U2 , and consider as above the associated functions Ui0 = Ui − U ∗ , U 0 = U10 − U20 . Then U 0 satisfies dU 0 + AU 0 + B(U 0 , U2 ) + B(U2 , U 0 ) + B(U 0 , U 0 ) + E(U 0 ) = 0, dt U 0 (0) = 0. Treating this equation exactly as (3.24) we obtain an equation similar to (3.28) but with µ = 0 and a different λ : d 1/2 0 2 |A U |H + (1 − c4 |A0 U |H )|AU 0 |2H dt 1/2 0 2 ˜ ≤ λ(t)|A U |H . The uniqueness follows using Gronwall’s Lemma. To conclude this section and the proof of Theorem 3.1, we now prove Lemma 3.2. Proof of Lemma 3.2 We need only to show how the different integrals in b are bounded by the expressions appearing in the right-hand side of (3.26) and, in fact, we restrict ourselves to two typical terms, the other terms being treated in the same way.

54


The first one is bounded as follows : ¯Z ¯ ¯ ¯ b ] b ] ¯ ¯ (3.34) ¯ [(v · ∇)v ]v dM¯ ≤ |v|L6 |∇v |L3 |v |L2 . M

By Sobolev imbeddings and interpolation, we bound the right-hand side by 1/2

ckvkH 1 kv b kH 2 |v ] |L2 , which corresponds to the first term in the right-hand side of (3.26). The second typical term is bounded as follows : ¯Z ¯ ¯Z Z 0 ¯ ∂v b ] ∂v b ] ¯ ¯ ¯ ¯ w(v) v dM¯ = ¯ w(v) v dzdΓi ¯ ¯ ∂z ∂z M Γi −h ¯ b¯ Z ¯ ∂v ¯ ¯ ¯ |v ] |L2 dΓi ≤ |w(v)|L∞ z ¯ z ¯ 2 ∂z Γi Lz ¯ b¯ Z ¯ ∂v ¯ (3.35) 1/2 ¯ ¯ |v ] |L2 dΓi | div v|L2z ¯¯ ≤h z ¯ 2 ∂z Γi Lz ≤ (with Holder’s inequality) ¯ b¯ ¯ ∂v ¯ 1/2 ¯ ¯ ≤ h | div v|L4Γ L2z ¯¯ |v ] |L2Γ L2z . i i ∂z ¯ 4 2 LΓ Lz i

For a (scalar or vector) function ξ defined on M, we have written (1 ≤ α, β ≤ ∞) : ¶1/β µZ 0 β |ξ(x, y, z)| dz , |ξ|Lβz = |ξ|Lβz (x, y) = −h(x,y)

|ξ|Lα Lβz Γi

¯ ¯ ¯ ¯ = ¯|ξ|Lβz ¯

Lα (Γi )

µZ

¶1/α

= Γi

|ξ|αLβ (x, y)dΓi z

.

Notice also that |ξ|L2Γ L2z = |ξ|L2 (M) and that, from the expression (2.40) of w (v), i and (2.39), ¯Z 0 ¯ ¯ ¯ 0¯ ¯ 1/2 | div v|L2 . ¯ |w(v)|L∞ =¯ div vdz ¯ ≤ h z z ∞ z

Lz

Now we remember that (in space dimension two) there exists a constant c = c(Γi ) such that, for every function ζ in H 1 (Γi ), (3.36)

1/2

1/2

|ζ|L4 (Γi ) ≤ c|ζ|L2 (Γi ) |ζ|H 1 (Γi ) ,

from which we infer, for a function ξ as above (setting ζ = |ξ|L2z ) : ¯ ¯2 1/2 (3.37) |ξ|L4Γ L2z ≤ c|ξ|L2 L2 ¯|ξ|L2z ¯H 1 (Γi ) . i

Γi

z


As before |ξ|L2Γ (3.38)

i

L2z

55

= |ξ|L2 (M) , whereas ¯ ¯ ¯|ξ|L2 ¯2 1 z

=

H (Γi )

|ξ|2L2 L2z Γ i

¯ ¯2 ¯ ¯ + ¯∇|ξ|L2z ¯

L2 (Γi )

= |ξ|2L2 (M) + |∇θ|2L2 (Γi ) , with

µZ θ = θ(x, y) = |ξ|L2z (x, y) =

¶1/2

0

2

|ξ(x, y, z)| dz

.

−h

We intend to show that for ξ in H 1 (M) (scalar or vector ξ) : (3.39)

|ξ|L4Γ

i

L2z

1/2

1/2

≤ c|ξ|L2 (M) |ξ|H 1 (M) ,

for some suitable constant c = c(M). For that purpose we note that µZ 0 ¶1/2 ½Z 0 ¾ 2 2 ∇θ = |ξ| dz ξ∇ξdz + |ξ(−h)| ∇h , −h

−h

where ξ(−h) and, below, ξ(z) are simplified notations for ξ(x, y, −h(x, y)) and ξ(x, y, z). With the Schwarz inequality we find that pointwise a.e. (for a.e. (x, y) ∈ Γi ). (3.40)

2 |∇θ| ≤ |∇ξ|L2z + c|ξ|−1 L2z |ξ(−h)| .

We have classically :

Z

2

z

∂ξ dz −h ∂z ¯ ∂ξ ¯ ¯ ¯ ≤ |ξ(z)|2 + 2|ξ|L2z ¯ ¯ , ∂z L2z 2

|ξ(−h)| = |ξ(z)| − 2

ξ

and by integration in z from −h(x, y) to 0 : ¯ ¯ ¯ L2 ¯¯ ∂ξ ¯¯ . h|ξ(−h)|2 ≤ |ξ|2L2z + 2h|ξ| z ∂z L2z Hence (3.40) yields pointwise a.e. ¯ ∂ξ ¯ ¯ ¯ |∇θ| ≤ |∇ξ|L2z + c|ξ|L2z + c¯ ¯ . ∂z L2z By integration on Γi , we find |∇θ|L2 (Γi ) ≤ c|ξ|H 1 (M) , and then (3.37) and (3.38) yield (3.39).

56


Having established (3.39), we return to (3.35) : applying (3.39) with ξ = div v ˜ b /∂z, we can bound the left-hand side of (3.35) by the second term in and ξ = ∂ v the right-hand side of (3.26), thus concluding the proof of Lemma 3.2. 3.2. Strong solutions of the two dimensional primitive equations: Physical Boundary Conditions. In this section, we are concerned with the global existence and the uniqueness of strong solutions of the two-dimensional primitive equations of the ocean. We will first derive the equations formally from the 3Dprimitive equations under the assumption of invariance with respect to the y− variable, i.e., we will assume that the initial data, the forcing terms, as well as the depth function h are independent of the variable y. The uniqueness of weak solutions implies that the solution will be independent of y. In Section 3.2.1 and 3.2.2 we introduce the 2D-PEs and present their weak formulation. In Section 3.2.2 and 3.2.3 we show that the strong solutions provided by an analog of Theorem 3.1 are in fact defined for all t > 0; this result is based on improved and more involved a priori estimates which are described hereafter. 3.2.1. The 2D-Primitive Equations. We assume that the domain occupied by the ocean is represented by {(x, y, z) ∈ R3 , x ∈ (0, L), y ∈ R, −h(x) < z < 0}, and we denote by M its cross section: (3.41)

M = {(x, z), x ∈ (0, L), −h(x) < z < 0}.

Here L is a positive number and h : [0, L] → R satisfies h ∈ C 3 ([0, L]), (3.42)

h(x) ≥ h > 0 for x ∈ (0, L), h0 (0) = h0 (L) = 0.

By dropping all the terms containing a derivative with respect to y in the 3Dprimitive equations (2.44)-(2.50), we obtain the following system: (3.43)

(3.44)

∂u ∂u ∂u ∂ 2u ∂ 2u ∂ps +u +w − µv 2 − ν v 2 − f v + =g ∂t ∂x ∂z ∂x ∂z ∂x

Z

0 z

∂v ∂v ∂v ∂ 2v ∂2v +u +w − µv 2 − νv 2 + f u = Fv , ∂t ∂x ∂z ∂x ∂z

∂ρ 0 dz + Fu , ∂x


(3.45)

∂T ∂T ∂T ∂ 2T ∂2T +u +w − µT 2 − νT 2 = FT , ∂t ∂x ∂z ∂x ∂z

(3.46)

∂S ∂S ∂S ∂ 2S ∂ 2S +u +w − µS 2 − ν S 2 = F S , ∂t ∂x ∂z ∂x ∂z

(3.47)

∂u ∂w + = 0, ∂x ∂z

(3.48)

ρ = 1 − βT (T − Tr ) + βS (S − Sr ),

57

Z (3.49)

SdM = 0. M

Here u and v are the two components of the horizontal velocity v. Note that, despite y− invariance, v does not vanish in the problem of physical relevance (unlike the 2D Navier-Stokes equations). The quantity ps above is the same as p in (2.49), R0 whereas the expression P in (2.49) has been replaced by g z ρ dz 0 , ρ function of T and S through (3.48). Finally, as in the 3D case, F = (Fu , Fv , FT , FS ) vanishes in the physical problem and it is added here for mathematical generality. Boundary Conditions These equations are supplemented with the same set of boundary conditions and initial data as in Section 2. On the top boundary of M, denoted Γi , Γi = {(x, y); x ∈ (0, L); z = 0}, we have (see also after (2.54)): ∂u ∂v ∂T ∂S + αv u = gu , νv + αv v = gv , νT + αT T = gT , = 0. ∂z ∂z ∂z ∂z On the remaining part of the boundary, we assume the Dirichlet boundary condition for the velocity and the Neumann condition for the temperature and the salinity. That is (3.50)

(3.51)

νv

(u, v, w) = (0, 0, 0) on Γ` ∪ Γb ,

∂S ∂T = = 0 on Γ` ∪ Γb , ∂nT ∂nS

58


where (3.52)

Γ` = {(x, y); x = 0 or L, −h(x) < z < 0}, Γb = {(x, z); x ∈ (0, L), z = −h(x)}.

We also have the initial data given by (3.53)

u|t=0 = u0 , v|t=0 = v0 ,

T |t=0 = T0 , and S|t=0 = S0 .

3.2.2. Weak formulation. The main result. We now proceed, as in Section 2 towards the weak and functional formulations of this problem (3.43) - (3.16) with some simplifications due to the invariance with respect to y, and some other aspects which are specific to dimension two. We introduce as in Section 2.2.1, the following spaces V = V1 × V2 × V3 , H = H1 × H2 × H3 , Z 0 1 2 ∂ V1 = {v = (u, v) ∈ H (M) , u(x, z)dz = 0, v = 0 on Γ` ∪ Γb }, ∂x −h(x) Z 1 1 1 V = H (M), V3 = H˙ (M) = {S ∈ H (M), SdM = 0}, (3.54) 2 M Z 0 2 2 H1 = {v = (u, v) ∈ L (M) , u(x, z)dz = 0, u = 0 on Γ` }, −h(x) Z 2 2 2 H2 = L (M), H3 = L˙ (M) = {S ∈ L (M), SdM = 0}. M

The scalar products are defined exactly as in Section 2.2.1, ∇ being replaced by R0 ∂/∂x. The condition −h(x) u(x, z)dz = 0 comes from the fact that the derivative in x of this quantity vanishes (the 2D analog of (2.48)) and that this quantity vanishes at x = 0 and L (see Section 2.2.1 for the 3D analog). We also introduce the spaces V1 , V2 , V3 and V = V1 × V2 × V3 with a similar definition. Similarly we consider the forms a, b, c, `, defined exactly as in dimension 3, just deleting all quantities involving a y− derivative; the associated operators A, B, E are defined in the same way. With these notations, the weak formulation is exactly as in dimension 3 (see (2.79), (2.80) or, in operational form, (2.81), (2.82)). There is no new difficulty in proving the analogue of Theorem 2.4 giving the existence, for all time, of weak solutions.


59

Similarly we can prove, exactly as in Section (3.1), an analogue of Theorem 3.1. Our aim in this section is to show that t∗ = t1 in space dimension two, for the t∗ appearing in the statement of Theorem 3.1. More precisely we will prove the following (compare to Theorems 3.1 and 2.2): Theorem 3.2. We assume that M is as in (3.41) and that (3.42) is satisfied. We are given t1 > 0, U0 ∈ V, F = (Fv , FT , FS ), and g = (gv , gT ) such that F and dF/dt are in L2 (0, t1 ; H) (or L2 (0, t1 ; L2 (M)4 )and g and dg/dt are in L2 (0, t1 ; H01 (Γi )3 ). Then there exists a unique solution U of the Primitive Equations (2.79), (2.80) such that (3.55)

U ∈ C([0, t1 ]; V ) ∩ L2 (0, t1 ; H 2 (M)4 ).

Proof. The proof of uniqueness is easy and done as in Theorem 3.1 for dimension three. To prove the existence of solutions, we start from the strong solution given by the 2D analogue of Theorem 3.1 and prove by contradiction that t∗ = t1 . Indeed let us denote by [0, t0 ] the maximal interval of existence of a strong solution, that is9 (3.56)

U ∈ L∞ (0, t0 ; V ),

for every t0 < t0 and (3.56) does not occur for t0 = t0 , which means in particular that (3.57)

lim sup ||U (t)|| = +∞. t→t0 −0

We will show that (3.57) can not occur: we will derive a finite bound for ||U (t0 )|| on [0, t0 ], thus contradicting (3.57). The bounds for ||U (t)|| will be derived sequentially: we will show successively that uz , ux , are in L∞ (0, t0 ; L2 (M)) and L2 (0, t0 ; H 1 (M)) where ϕx = ∂ϕ/∂x and ϕz = ∂ϕ/∂z; then we will prove at once that v, T and S are in L∞ (0, t0 ; H 1 ) and L2 (0, t0 ; H 2 ). In fact we will give the proofs for uz , ux , T the other quantities being estimated in exactly the same way. For the sake of simplicity, we assume hereafter that g = (gv , gT ) = 0. When g 6= 0, we need to ”homogenize” the boundary conditions by considering U 0 = U − U ∗ , with U ∗ 9It

is easy to see that, if U ∈ L∞ (0, t0 ; V ), then U ∈ L2 (0, t0 ; H 2 (M)4 ) as well.

60


defined exactly as in Section (3.1), then perform the following calculations for U 010. Before we proceed, let us recall that we have already available, the a priori estimates for U in L∞ (0, t1 ; L2 ) and L2 (0, t1 ; H 1 ) used to prove the analog of Theorem 2.2, (that is (2.93) in the discrete case). ¤ 3.2.3. Vertical Averaging. To derive the new a priori estimates, we need some operators related to vertical averaging that we now define. For any function ϕ defined and integrable on M we set Z 0 ˜ P ϕ(x) = ϕ(x, z)dz, −h(x) (3.58) 1 P ϕ = P¯ ϕ, Qϕ = ϕ − P ϕ. h We now establish some useful properties of these operators, some simple, some more involved. We first note that P Q = 0, that is Z

0

(3.59)

Qϕ(x, z)dz = 0,

∀ϕ ∈ L1 (M),

−h(x)

and

Z

(3.60)

(P ϕ)(x)(Qψ)(x, z)dxdz = 0,

∀ϕ, ψ ∈ L2 (M),

M

Also, for all ϕ sufficiently regular: ∂ϕ ∂ ¯ P¯ = P ϕ − h0 (x)ϕ(x, −h(x)), ∂x ∂x 2 ∂ ϕ ∂2 ¯ ∂ϕ (3.61) P¯ 2 = P ϕ − 2h0 (x) (x, h(x)) 2 ∂x ∂x ∂x ∂ϕ +h0 (x)2 (x, −h(x)) − h00 (x)ϕ(x, −h(x)). ∂z Now, if ϕ vanishes on Γb , ϕ(x, −h(x)) = 0, 0 < x < L, then (3.62) 10Note

∂ϕ ∂ϕ (x, −h(x)) = h0 (x) (x, −h(x)), ∂x ∂z

that, in (3.1), U ∗ was chosen so that the initial and boundary conditions for U 0 vanish. Here we do not need to homogenize the initial condition, but we can use the same U ∗.


61

and hence

∂ϕ ∂ ¯ P¯ = P ϕ, ∂x ∂x ∂ 2ϕ ∂2 ¯ ∂ϕ P¯ 2 = P ϕ − h0 (x) (x, −h(x)), 2 ∂x ∂x ∂x

(3.63)

P

∂ 2ϕ h0 ∂ϕ h02 ∂ϕ (x) = − (x, −h(x)) = − (x, −h(x)). ∂x2 h ∂x h ∂z

Finally the following Lemma will be needed. Lemma 3.2. For any v ∈ H 2 (M) such that

ν

∂v + αv = 0 on Γi , v = 0 on Γ` ∪ Γb , ∂z

we have Z (3.64)

M

Z ∂ 2v ∂ 2v ∂ 2v 2 dzdx = | | dzdx ∂x2 ∂z 2 M ∂x∂z Z L Z ∂v 1 L 00 ∂v 2 +α | (x, 0)| dx − h (x)| (x, −h(x))|2 dx. ∂x 2 0 ∂z 0

Proof. We give the proof for v smooth, say v ∈ C 3 (M); the result extends then to v ∈ H 2 (M) using a density argument (that we skip). We write

(3.65)

∂ ∂v ∂ 2 v ∂2v ∂ 2v = ( )− ∂x2 ∂z 2 ∂x ∂x ∂z 2 ∂ ∂v ∂ 2 v = ( )− ∂x ∂x ∂z 2

∂v ∂ 3 v ∂x ∂x∂z 2 ∂ ∂v ∂ 2 v ∂ 2v 2 ( )+| |. ∂z ∂x ∂x∂z ∂x∂z

We first integrate in z and, taking into account (3.61) we obtain

62


Z

0 −h

(3.66)

∂ 2v ∂2v ∂v ∂2v 0 dz = I − h (x) (x, −h(x)) (x, −h(x)) ∂x2 ∂z 2 ∂x ∂z 2 ∂ 2v ∂v ∂2v ∂v − (x, 0) (x, 0) + (x, −h(x)) (x, −h(x)) ∂x ∂x∂z ∂x ∂x∂z Z 0 ∂ 2v 2 + | | dz, −h ∂x∂z Z 0 ∂ ∂v ∂ 2 v I= dz. ∂x −h ∂x ∂z 2

We now integrate in x. The integral of I = I(x) vanishes because v(0, z) = v(L, z) = 0, ∀z, so that (

∂ 2v ∂2v )(0, z) = ( )(L, z) = 0, ∀z. ∂z 2 ∂z 2

The third term in the right hand side of (3.62) is equal to α The sum of the second and fourth terms is equal to Z

L

(3.67) 0

RL 0

|(∂v/∂x)(x, 0)|2 dx.

∂v ∂2v ∂ 2v (x, −h(x))( − h0 2 )(x, −h(x))dx. ∂x ∂x∂z ∂z

Setting ϕ(x) = (∂v/∂z)(x, −h (x)), we see that ϕ0 (x) =

∂ 2v ∂ 2v (x, −h(x)) − h0 (x) 2 (x, −h(x)), ∂x∂z ∂z

and since v(x, −h(x)) = 0, we have ∂v ∂v (x, −h(x)) − h0 (x) (x, −h(x)) = 0, ∂x ∂z and the integral in (3.67) is equal to Z

L 0

1 h (x)ϕ(x)ϕ (x)dx = − 2 0

Z

L

0

h00 (x)ϕ2 (x)dx;

0

for the last relation we have used ϕ(0) = ϕ(L) = 0. The lemma is proved.

¤


63

3.2.4. Estimates for uz . To show that uz ∈ L∞ (0, t0 ; L2 (M)) ∩ L2 (0, t0 ; H 1 (M)), we multiply (3.43) by Quzz , integrate over M and integrate by parts, and remember that Qu = u. Thus, for each term successively, omitting the variable t, we find: Z

Z ut Q(−uzz )dM = − M

ut uzz dM M Z L

=− 0

Z ut uz ]0−h dx

+

utz uz dM M

Z αv L 1d |uz |2L2 = ut (x, 0)u(x, 0)dx + νv 0 2 dt αv 1d = (|uz |2L2 + |u(x, 0)|2L2 (Γi ) ). 2 dt νv

Z

Z

Z

uux Q(−uzz )dM = M

uux uzz dM +

uux P uzz dM Z Z 0 0 uux uz ]−h dx + uuxz uz dM =− L M Z Z 2 + ux uz dM + uux P uzz dM M M Z Z αv L 2 1 = u (x, 0)ux (x, 0)dx + unx u2z d(∂M) ν 2 ∂M Z v 0 Z 1 ux u2z dM + uuz P uzz dM + 2 M M Z Z 1 αv 3 L 2 u (x, 0)]0 + = ux uz dM + uux P uzz dM. 3νv 2 M M Z Z 1 2 uux P uzz dM. = ux uz dM + 2 M M M

M

In the relations above, n = (nx , nz ) is the unit outward normal on ∂M, and we used the fact that unx = 0 on ∂M, and that u(0, 0) = u(L, 0) = 0 because u = 0 on Γ` .

64


Z

Z

Z

wuz Q(−uzz )dM = − M

wuz uzz dM + M

wuz P uzz dM M

Z Z 1 L 20 1 =− wz u2z dM wuz ]−h dx + 2 0 2 M Z + wuz P uzz dM M

= (since w = 0 on Γi and Γb and wz = −uz ) Z Z 1 2 wuz P uzz dM. =− ux uz dM + 2 M M Since P uzz is independent of z and w vanishes on Γi and Γb , we have Z

Z

L

wuz P wzz dM = M

Z0 =

Z wu]0−h P uzz dx

−

uwz P uzz dM M

uuz P uzz dM. M

Finally the last two terms add up in the following way Z (uux + wuz )Q(−uzz )dM Z =2 uux P uzz dM ZM 1 =2 uux [uz (x, 0) − uz (x, h)]dM, M h Z psx Q(−uzz )dM = 0 since ps is independent of z, M

M

Z uzz Q(−uzz )dM = νv |Quzz |2L2 = νv |uzz |2L2 − νv |P uzz |2L2 , M Z Z Z − µv uxx Q(−uzz )dM = µv uxx uzz dM − µv (P uxx ) uzz dM.

−νv

M

M

M

Using 3.63 and Lemma 3.2, we see that this expression is equal to


Z µv |uxz |2L2 +αv µv Z −µv

0 L

0

αv − µv νv

L

µ

Z

65

|ux (x, 0)|2 dx

¶ h0 (x)2 1 00 + h (x) |uz (x, −h(x))|2 dx h(x) 2

L

uz (x, −h)u(x, 0)dx. 0

The other terms are left unchanged, then gathering all these terms we find, 1d αv |u(x, 0)|2L2 (Γi ) )+ (|uz |2L2 (M) + 2 dt νv + µv |uxz |2L2 (M) + αv µv |ux (x, 0)|2L2 (Γi )

(3.68)

+ νv |uzz |2L2 (M) = νv |P uzz |2L2 + Z L 02 h 1 + µv ( + h00 )|uz (x, −h(x))|2 dx h 2 0 Z L Z αv uz (x, −h)u(x, 0)dx + f vQuzz dM + µv νv 0 M Z Z 0 −g ( ρx dz 0 )uzz dM. M

z

We estimate the right-hand-side of (3.68) as follows, c denoting a constant depending only on M and on the coefficients αv , µv , νv : 1 (uz (x, 0) − uz (x, −h)) h 1 αv 1 = u(x, 0) − uz (x, −h(x)) h νv h

P uzz =

(3.69)

|P uzz |L2 (M) ≤ c|u(x, 0)|L2 (Γi ) + c|uz (x, −h(x))|L2 (Γi ) .

The last two norms are bounded by the trace theorems:

(3.70)

1/2

|u|L2 (Γi ) ≤ c|u|L2 (M) ||u||1/2 ,

66


1/2

(3.71)

1/2

|uz (x, −h(x))|L2 (Γi ) ≤ c|uz |L2 (Γb ) ≤ c|uz |L2 (M) |∇uz |L2 (M) ³ ´1/4 ≤ c||u||1/2 |uzz |2L2 (M) + |uzx |2L2 (M) , |P uzz |2L2 (M) ≤ c|u|L2 (M) ||u|| + c||u||||uz ||.

We write also Z | f vQuzz dM| ≤ c|v|L2 (M) |uzz |L2 (M) ≤ c|v|L2 (M) ||uz ||. M

Finally using again (3.70) and (3.71) for the other terms in the right-hand-side of (3.68), we obtain 1 d αv (|uz |2L2 (M) + |u(x, 0)|2L2 (Γi ) )+ 2 dr νv ν||uz ||2 + αv µv |ux (x, 0)|2L2 (Γi ) ≤ ≤ c|u|L2 (M) ||u|| + c||u||||uz || + c|v|L2 ||uz || + c(|Tx |L2 + |Sx |L2 )|uzz |L2 + |Fu |L2 |uzz |L2 ν ≤ ||uz ||2 + c||u||2 + c|v|2L2 + c||U ||2 + c|Fu |2L2 , 2 where ν = min(µv , νv ). Hence

(3.72)

αv d (|uz |2L2 (M) + |ux (x, 0)|2L2 (Γi ) )+ dt νv + ν||uz ||2 ≤ c||U ||2 + c|Fu |2L2 (M) .

Taking into account the earlier estimates of U in L∞ (0, t1 ; H) and L2 (0, t1 ; V ), we obtain an a priori bound of uz in L∞ (0, t0 ; L2 (M)), a first step in proving that t0 can not be less than t1 . Remark 3.1. We recall that the estimates above were made under the simplifying assumption that g = (gv , gT ) = 0. When this is not the case, we explained that we ought to consider U 0 = U − U ∗ , U ∗ defined as in Section 3.1. Then the calculations above are made for the equation for u0 . This equation will involve some additional terms such as u∗ ∂u0 /∂x, u0 ∂u∗ /∂x, etc.; these additional terms are estimated in a similar way, leading to the same conclusions. The same remark applies for the estimates below concerning ux , vz , etc.


67

3.2.5. Estimates for ux . To show that ux is bounded in L∞ (0, t0 ; L2 (M)∩ L2 (0, t0 ; H 1 (M)), we multiply (3.43) by −uxx , integrate over M and integrate by parts. Omitting again the variable t, we find for each term: Z Z Z − ut uxx dM = − ut ux nx d(∂M) + utx ux dM M ∂M M Z 1d u2x dM, = 2 dt M Z Z Z 1 1 2 − uux uxx dM = − uux nx d(∂M) + u3x dM 2 2 M M Z ∂M 1 = u3 dM, 2 M x Z Z Z − wuz uxx dM = − wuz ux nx d(∂M) + wx uz ux dM M ∂M M Z + wuzx ux dM M Z Z 1 L 20 wux ]−h dx = wx uz ux dM + 2 0 Z M 1 − wz u2x dM 2 M Z Z 1 u3x dM. = wx uz ux dM + 2 M M Z νv uzz uxx dM = (thanks to Lemma 3.2) M Z L 2 = νv |uzx |L2 + αv νv |ux (x, 0)|2 dx 0 Z L 1 − h00 (x)|uz (x, −h(x))|2 dx. 2 0 The other terms are left unchanged and, with ν = min(µv , νv ), we arrive at: Z L 1 d 2 2 |ux (x, 0)|2 dx |ux |L2 +ν||ux || + αv νv 2 dr 0 Z Z 3 wx uz ux dM =− ux dM − M M Z Z 1 L 00 − f vuxx dM + h (x)|uz (x, −h(x))|2 dx 2 M 0 Z Z 0 Z 0 −g ( ρx dz )uxx dM + Fu uxx dM. M

z

M

68


We write Z | M

u3x dM| = |ux |3L3

≤ (by Sobolev imbedding and interpolation) ≤ c|ux |3H 1/3 ≤ c|ux |2L2 |ux |H 1 ≤ c||u||2 ||ux || ν ≤ ||ux ||2 + c||u||4 , Z 10 |

wx uz ux dM| ≤ |wx |L2 |uz |L4 |ux |L4 M

≤ (by Sobolev imbedding and interpolation) 1/2

1/2

≤ c|uxx |L2 |uz |L2 ||uz ||1/2 |ux |L2 ||ux ||1/2 1/2

1/2

≤ c|uz |L2 ||uz ||1/2 |ux |L2 ||ux ||3/2 ν ≤ ||ux ||2 + c|uz |2L2 ||uz ||2 |ux |2L2 , Z 10 |

f vuxx dM| ≤ |f v|L2 |uxx |L2 ≤ M

νv ||ux ||2 + c|v|2L2 . 10

The next terms are bounded as before and the last term is easy. Hence Z L d 2 2 |ux |L2 (M) +ν||ux || + αv νv |ux (x, 0)|2 dx dt 0 (3.73) ≤ c||u||4 + c|v|2L2 + c|uz |2L2 (M) ||uz ||2 |wz |2L2 (M) +c||u|| ||uz || + c ||U ||2 + c |Fu |2L2 (M) . Remembering that ||u||2 = |ux |2L2 (M) + |uz |2L2 (M) , we see that the right hand side of (3.73) is of the form ξ(t) + η(t)|ux |2L2 (M) , where ξ, η are in L1 (0, t1 ); for η = c||u||2 +c|uz |2L2 (M) ||uz ||2 , this follows from the previous estimates on U and on uz ; similarly the contribution of c||u||||uz || to ξ is in L1 (0, t1 ) due to the previous results on uz and U . Therefore the Gronwall lemma applied to (3.73) provides an a priori bound of ux in L∞ (0, t0 ; L2 (M)) and in L2 (0, t0 ; H 1 (M)).

3.2.6. Estimates for v, T and S. We now prove at once that T is bounded in L∞ (0, t0 ; H 1 (M)) and L2 (0, t0 ; H 2 (M)); the proof is similar for v and S, and this will thus conclude the proof of Theorem 3.2.


69

For this, we multiply each side of (3.45) by A2 T = −µT ∂ 2 T /∂x2 − νT ∂ 2 T /∂z 2 . We recall that gT = 0 here, see the end of Section 3.2.2. We have

Z −

Tt (µT Txx + νT Tzz )dM = Z =− Tt (µT Tx nx + νT Tz nz )d(∂M) ∂M Z 1 + (µT Ttx Tx + νT Ttz Tz )dM 2 M = (see the notations in (2.38) and after (2.54)) Z ∂T 1d =− Tt d(∂M) + a2 (T, T ) ∂nT 2 dt ∂M = (with (2.35), (2.55) and gT = 0) Z 1d a2 (T, T ) + αT T Tt dΓi = 2 dt Γi 1d = a2 (T, T ) + αT |T (x, 0)|2L2 (Γi ) 2 dt M

Hence, we find

(3.74)

1d 1/2 (|A2 T |2L2 +αT |T (x, 0)|2L2 (Γi ) ) 2 dt Z +|A2 T |2L2

=−

Z

(uTx + wTz )A2 T dM + M

FT A2 T dM. M

70


Each term in the right-hand side of (3.74) is bounded as follows: Z | uTx A2 T dM| ≤ |u|L4 |Tx |L4 |A2 T |L2 M 1/2

1/2

≤ c|u|L2 ||u||1/2 |Tx |L2 ||Tx ||1/2 |A2 T |L2 1/2

1/2

1/2

3/2

≤ c|u|L2 ||u||1/2 |A2 T |L2 |A2 T |L2 1 1/2 ≤ |A2 T |2 + c|u|2L2 ||u||2 |A2 T |2L2 , 6

Z

wTz A2 T dM| ≤ |w|L4 |Tz |L4 |A2 T |L2

| M

≤ c|ux |L4 |Tz |L4 |A2 T |L2 1/2

1/2

≤ c||u||1/2 ||ux ||1/2 |A2 T |L2 |A2 T |3/2 1 1/2 ≤ |A2 T |2 + c||u||2 ||ux ||2 |A2 T |2L2 , 6

Z |

FT A2 T dM| ≤ |FT |L2 |A2 T |L2 M

1 ≤ |A2 T |2L2 + c|FT |2L2 . 6 1/2

Here we have used the fact (easy to prove), that |A2 T |L2 is a norm equivalent to ||T || in V2 , and the much more involved result, proved in Section 4.3, that |A2 T |L2 (M) is, on D(A2 ), a norm equivalent to |T |H 2 (M) ; for the application of Theorem 4.3, we required (3.42) which is the one-dimensional analog of (4.54). Note that, as explained in Remark 4.1, we believe that this purely technical hypothesis can be removed. With this we infer from (3.74) that d 1/2 (|A2 T |2L2 (M) +αT |T (x, 0))|2L2 (Γi ) ) dt (3.75) 1/2 +|A2 T |2L2 (M) ≤ ξ(t) + η(t)|A2 T |2L2 (M) , with ξ = c|FT |2L2 (M) and η = c(|u|2L2 (M) +||ux ||2 )||u||2 . By assumption ξ ∈ L1 (0, t0 ) and the earlier estimates on U, ux and uz show that η ∈ L1 (0, t0 ). Then, Gron1/2 wall’s lemma implies that |A2 T |L2 (M) is in L2 (0, t0 ), which means that T is in L∞ (0, t0 ; H 1 (M)) and L2 (0, t0 ; H 2 (M)). This concludes the proof of Theorem 3.2.


71

3.3. The space periodic case in dimension 2: Higher regularities. 11 Our aim in this section is to present some existence, uniqueness and regularity results for the Primitive Equations of the ocean in space dimension two with periodic boundary conditions. We prove the existence of weak solutions for the PEs, the existence and uniqueness of strong solutions and the existence of more regular solutions, up to C ∞ regularity. For the sake of simplicity and to follow [29], we do not consider the salinity; introducing the salinity would not produce any additional technical difficulty. In this case ρ is a linear function of T 12, and, in what follows, ρ is the prognostic variable instead of T . Because of the hydrostatic equation it is not possible to produce a solution that is space periodic in all variables; for that reason ρ, p and T below represent the deviation from a stratified solution ρ¯ for which N 2 = −(g/ρ0 )(d¯ ρ/dz) is ¯ a constant, and, as usual d¯ p/dz = −g ρ¯ and ρ¯ = ρ0 (1 − α(T − T0 )), ρ0 , T0 being reference values of ρ and T (of the same order as ρ¯ and T¯). Furthermore the periodic (disturbance) solutions that we consider present certain symmetries that are described below (see (3.77) below). We refer the reader to [29] for more details on the physical background. Unlike the preceding sections, (but this is not important), we consider here the PEs written in nondimensional form, that is (see [29]): ∂u ∂u ∂u 1 1 ∂p +u +w − v+ = νv ∆u + Fu , ∂t ∂x ∂z R0 R0 ∂x ∂v ∂v ∂v 1 (3.76b) +u +w + u = νv ∆v + Fv , ∂t ∂x ∂z R0 ∂p (3.76c) = −ρ, ∂z ∂u ∂w (3.76d) + = 0, ∂x ∂z ∂ρ ∂ρ ∂ρ N 2 (3.76e) +u +w − w = νρ ∆ρ + Fρ . ∂t ∂x ∂z R0 Here (u, v, w) are the three components of the velocity vector and, as usual, we denote by p and ρ the pressure and density deviations, respectively, from (3.76a)

11This

section is based on [29], with the authorization of the Editors. fact ρ is an affine function of T but the deviation from the density considered below is a linear function of the deviation from the temperature. 12In

72


the background state mentioned above. The (dimensionless) parameters are the Rossby number R0 , the Burgers’ number N , and the inverse (eddy) Reynolds numbers νv and νρ . We notice easily that if u, v, ρ, w, p are solutions of (3.76) for F = (Fu , Fv , Fρ ), then u˜, v˜, ρ˜, w, ˜ p˜ are solutions of (3.76) for F˜u , F˜v , F˜ρ where:

(3.77)

u˜(x, z, t) =u(x, −z, t),

v˜(x, z, t) =v(x, −z, t),

w(x, ˜ z, t) = − w(x, −z, t),

p˜(x, z, t) =p(x, −z, t),

ρ˜(x, z, t) = − ρ(x, −z, t), F˜u (x, z, t) =Fu (x, −z, t), F˜ρ (x, z, t) = − Fρ (x, −z, t).

F˜v (x, z, t) =Fv (x, −z, t),

Therefore if we assume that Fu , Fv , Fρ are periodic and Fu , Fv are even in z and Fρ is odd in z, then we can anticipate the existence of a solution of (3.76) such that: (3.78)

(3.79)

u, v, w, p, ρ are periodic in x and z with periods L1 and L3 and,

u, v and p are even in z; w and ρ are odd in z,

provided the initial conditions satisfy the same symmetry properties. Our aim is to solve the problem (3.76) with the periodicity and symmetry properties above and with initial data u = u0 , v = v0 , ρ = ρ0 at t = 0. One motivation for considering periodic boundary conditions is that they are needed in numerical studies of rotating stratified turbulence (see e.g. [3]) and also for the study of the renormalized equations considered in [30]. The two spatial directions are 0x and 0z, corresponding to the west–east and vertical directions in the so-called f -plane approximation for geophysical flows; ∆ = ∂ 2 /∂x2 + ∂ 2 /∂z 2 . The rest of this section is organized as follows: We start in Section 3.3.1 by recalling the variational formulation of problem (3.76) under suitable assumptions and we say a few words about (the now standard) proof of existence of weak solutions for the PEs. We continue in Section 3.3.2 by proving the existence and


73

uniqueness of strong solutions. Finally in Section 3.3.3 we prove the existence of more regular solutions, up to C ∞ regularity. 3.3.1. Existence of the Weak Solutions for the PEs. We work in the limited domain (3.80)

M = (0, L1 ) × (−L3 /2, L3 /2),

and, as mentioned, we assume space periodicity with period M, that is, all functions are taken to satisfy f (x+L1 , z, t) = f (x, z, t) = f (x, z+L3 , t) when extended to R2 . Moreover, we assume that the symmetries (3.77) hold. Our aim is to solve the problem (3.76) with initial data (3.81)

u = u0 , v = v 0 , ρ = ρ 0

at t = 0.

Hence the natural function spaces for this problem are as follows: (3.82)

1 V = {(u, v, ρ) ∈ (H˙ per (M))3 ,

u, v even in z, ρ odd in z, (3.83)

R L3 /2 −L3 /2

u(x, z 0 ) dz 0 = 0},

H = closure of V in (L˙ 2 (M))3 .

1 Here the dot above H˙ per or L˙ 2 denotes the functions with average in M equal to zero. These spaces are endowed with Hilbert scalar products; in H the scalar product is

(3.84)

(U, U˜ )H = (u, u˜)L2 + (v, v˜)L2 + κ(ρ, ρ˜)L2 ,

1 and in H˙ per and V the scalar product is (using the same notation when there is no ambiguity):

(3.85)

((U, U˜ )) = ((u, u˜)) + ((v, v˜)) + κ((ρ, ρ˜));

where we have written dM for dx dz, and Z ³ ∂φ ∂ φ˜ ∂φ ∂ φ˜ ´ ˜ = + (3.86) ((φ, φ)) dM. ∂x ∂x ∂z ∂z M The positive constant κ is defined below. We have (3.87)

|U |H ≤ c0 kU k, ∀ U ∈ V,

where c0 > 0 is a positive constant related to κ and the Poincaré constant in 1 (M). More generally, the ci , c0i , c00i will denote various positive constants. H˙ per Inequality (3.87) implies that kU k = ((U, U ))1/2 is indeed a norm on V .

74


We first show how we can express the diagnostic variables w and p in terms of the prognostic variables u, v and ρ, the situation being slightly different here due to the boundary conditions. For each U = (u, v, ρ) ∈ V we can determine uniquely w = w(U ) from (3.76d), Z z (3.88) w(U ) = w(x, z, t) = − ux (x, z 0 , t) dz 0 , 0

since w(x, 0) = 0, w being odd in z. Furthermore, writing that w(x, −L3 /2, t) = w(x, L3 /2, t), we also have Z

L3 /2

(3.89)

ux (x, z 0 , t) dz 0 = 0.

−L3 /2

As for the pressure, we obtain from (3.76c), Z z (3.90) p(x, z, t) = ps (x, t) − ρ(x, z 0 , t) dz 0 , 0

where ps = p(x, 0, t) is the surface pressure. Thus, we can uniquely determine the pressure p in terms of ρ up to ps . It is appropriate to use Fourier series and we write, e.g., for u, X 0 0 uk1 ,k3 (t)ei(k1 x+k3 z) , (3.91) u(x, z, t) =

Z

(k1 ,k3 )∈

where for notational conciseness we set k10 = 2πk1 /L1 and k30 = 2πk3 /L3 . Since u is real and even in z, we have u−k1 ,−k3 = u¯k1 ,k3 = u¯k1 ,−k3 , where u¯ denotes the complex conjugate of u. Regarding the pressure, we obtain from (3.76c): Z z X 0 0 0 p(x, z, t) = p(x, 0, t) − ρk1 , k3 ei(k1 x+k3 z ) dz 0 =

X k1

0 (k , k ) 1 3

0

ps k1 eik1 x −

X

(k1 , k3 ), k3 6=0

ρk1 , k3 ik10 x ik30 z e (e − 1) ik30

[using the fact that ρk1 , 0 = 0, ρ being odd in z] X¡ X ρk , k ¢ 0 X ρk1 , k3 i(k10 x+k30 z) 1 3 ik1 x = ps k1 + e − e 0 ik3 ik30 k1 k3 6=0 (k1 , k3 ), k3 6=0 X X ρk1 , k3 i(k10 x+k30 z) 0 e , = p? k1 eik1 x − ik30 k 1

(k1 , k3 ), k3 6=0


where we denoted by ps the surface pressure and p? = the average of p in the vertical direction, is defined by X ρk , k 1 3 p?,k1 = ps k1 + . ik30 k 6=0

P

75 0

ik x Z p? k1 e 1 , which is

k1 ∈

3

Note that p is fully determined by ρ, up to one of the terms ps or p? which are connected by the relation above. We now obtain the variational formulation of problem (3.76). For that purpose we consider a test function U˜ = (˜ u, v˜, ρ˜) ∈ V and we multiply (3.76a), (3.76b) and (3.76e), respectively by u˜, v˜ and κ ρ˜, where the constant κ (which was already introduced in (3.84) and (3.85)) will be chosen later. We add the resulting equations and integrate over M. We find: (3.92)

1 d (U, U˜ )H + b(U, U, U˜ ) + a(U, U˜ ) + e(U, U˜ ) = (F, U˜ )H , dt R0

∀ U˜ ∈ V.

Here we set a(U, U˜ ) = νv ((u, u˜)) + νv ((v, v˜)) + κνρ ((ρ, ρ˜)), Z Z ˜ e(U, U ) = (u˜ v − v˜ u) dM + (ρw˜ − κN 2 wρ˜) dM, M M Z ³ Z ³ ] ∂u] ´ ∂v ] ∂v ] ´ ∂u ] ˜ + w(U ) u˜ dM + u + w(U ) v˜ dM b(U, U , U ) = u ∂x ∂z ∂x ∂z M M Z ³ ∂ρ] ∂ρ] ´ + u + w(U ) ρ˜ dM. ∂x ∂z M We now choose κ = 1/N 2 and this way we find e(U, U ) = 0. Also it can be easily seen that: a : V × V → R is bilinear, continuous, coercive, a(U, U ) ≥ c1 kU k2 , (3.93)

e : V × V → R is bilinear, continuous, e(U, U ) = 0, b is trilinear, continuous from V × V2 × V into R, and from V × V × V2 into R,

2 2 (M))3 . Furthermore, (M))3 in (Hper where V2 is the closure of V ∩ (Hper

(3.94)

b(U, U˜ , U ] ) = −b(U, U ] , U˜ ), b(U, U˜ , U˜ ) = 0,

when U, U˜ , U ] ∈ V with U˜ or U ] in V2 . We also have the following:

76


Lemma 3.3. There exists a constant c2 > 0 such that, for all U ∈ V , U˜ ∈ V2 and U ] ∈ V : 1/2 1/2 |b(U, U ] , U˜ )| ≤ c2 |U |L2 kU k1/2 kU ] k|U˜ |L2 kU˜ k1/2 (3.95) 1/2 1/2 + c2 kU kkU ] k1/2 |U ] |V2 |U˜ |L2 kU˜ k1/2 . Proof. We only estimate two typical terms; the other terms are estimated exactly in the same way. Using the Hölder, Sobolev and interpolation inequalities, we write: ¯Z ¯ ¯

¯ ¯ ∂u] ¯ ∂u] ¯ ¯ ¯ u|L 4 u u˜ dM¯ ≤ |u|L4 ¯ ¯ |˜ ∂x ∂x L2 M ¯ ∂u] ¯ ¯ ¯ 1/2 1/2 ≤ c01 |u|L2 kuk1/2 ¯ u| 2 k˜ uk1/2 , ¯ |˜ ∂x L2 L ¯Z ¯ ¯ ∂u] ¯ ∂u] ¯ ¯ ¯ ¯ 2 w(U ) u ˜ dM ≤ |w(U )| u|L 4 ¯ ¯ ¯ 4 |˜ L ¯ ∂z ∂z L M ¯ ∂u] ¯1/2 ° ∂u] °1/2 ¯ ° ° ¯ 0 u|1/2 k˜ uk1/2 ; ≤ c2 kuk¯ ¯ 2° ° |˜ ∂z L ∂z (3.95) follows from these estimates and the analogous estimates for the other terms. ¤ We now recall the result regarding the existence of weak solutions for the PEs of the ocean; the proof is exactly the same as that of Theorem 2.2 in space dimension 3 (see also Theorem 3.1 for the 2D case with different boundary conditions). Theorem 3.3. Given U0 ∈ H and F ∈ L∞ (R+ ; H), there exists at least one solution U of (3.92), U ∈ L∞ (R+ ; H) ∩ L2 (0, t? ; V ), ∀ t? > 0, with U (0) = U0 . As for Theorem 2.2, the proof of this theorem is based on the a priori estimates given below, which gives, as in [22], that U ∈ L∞ (0, t? ; H), ∀ t? > 0; however, as shown below, we have in fact13, U ∈ L∞ (R+ ; H). Taking U˜ = U in equation (3.92), after some simple computations and using (3.93), we obtain: d d (3.96) |U |2H + c0 c1 |U |2H ≤ |U |2H + c1 kU k2 ≤ c01 |F |2∞ , dt dt 13The

same holds in the previous cases in dimension 3 and 2, although the result was not stated in this form. At all orders, we present here results uniform in time, t ∈ R+ .


77

where |F |∞ is the norm of F in L∞ (R+ ; H). Using the Gronwall inequality, we infer from (3.96) that: (3.97)

|U (t)|2H ≤ |U (0)|2H e−c1 c0 t +

c01 (1 − e−c1 c0 t ) |F |2∞ , c1 c0

∀ t > 0.

Hence

c01 |F |2∞ =: r02 , c1 c0 t→∞ 0 0 and any ball B(0, r0 ) in H with r0 > r0 is an absorbing ball; that is, for all U0 , there exists t0 = t0 (|U0 |H ) depending increasingly on |U0 |H (and depending also on r00 , |F |∞ and other data), such that |U (t)|H ≤ r00 , ∀ t ≥ t0 (|U0 |H ). Furthermore, integrating equation (3.96) from t to t + r, with r > 0 arbitrarily chosen, we find: Z t+r (3.98) kU (t0 )k2 dt0 ≤ K1 , for all t ≥ t0 (|U0 |H ), lim sup |U (t)|2H ≤

t

where K1 denotes a constant depending on the data but not on U0 . As mentioned before, (3.97) implies also that U ∈ L∞ (R+ ; H),

|U (t)|H ≤ max(|U0 |H , r0 ).

Remark 3.2. We notice that, in the inviscid case (νv = νρ = 0 with F = 0), taking U˜ = U in (3.92), we find, at least formally, ´ d ³ 2 1 (3.99) |u|L2 + |v|2L2 + 2 |ρ|2L2 = 0. dt N The physical meaning of (3.99) is that the sum of the kinetic energy (given by 1 (|u|2L2 +|v|2L2 )) and the available potential energy (given by 2N1 2 |ρ|2L2 ) is conserved 2 in time. This is the physical justification of the introduction of the constant κ = N −2 in (3.84). 3.3.2. Existence and Uniqueness of Strong Solutions for the PEs. The solutions given by Theorem 3.3 are called weak solutions as usual. We are now interested in strong solutions (and even more regular solutions in Section 3.3.3). We use here the same terminology as before: weak solutions are those in L∞ (L2 ) ∩ L2 (H 1 ), strong solutions are those in L∞ (H 1 )∩L2 (H 2 ). We notice that as for Theorem 3.1, we cannot obtain directly the global existence of strong solutions for the PEs from a single a priori estimate. Instead, we will proceed as for Theorem 3.1 and derive the necessary a priori estimates by steps: we successively derive estimates in L∞ (L2 ) and L2 (H 1 ) for uz , ux , vz , vx , ρz and ρx (here the subscripts t, x, z

78


denote differentiation). Notice that the order in which we obtain these estimates cannot be changed in the calculations below.14 Firstly, using (3.90) we rewrite (3.76a) as: Z z ∂u ∂u ∂u 1 ∂ps 1 (3.100) +u +w − + ρx (x, z 0 , t) dz 0 = νv ∆u + Fu . ∂t ∂x ∂z R0 ∂x R0 0 We differentiate (3.100) with respect to z and we find, with wz = −ux : 1 1 vz − ρx − νv uxxz − νv uzzz = Fu, z , utz + uuxz + wuzz − R0 R0 where Fu, z = ∂z Fu = ∂Fu /∂z. After multiplying this equation by uz and integrating over M, we find: Z Z 1d 2 2 |uz |L2 + νv kuz k + uuz uxz dM + wuz uzz dM 2 dt M M Z Z Z 1 1 uz Fu, z dM. vz uz dM − ρx uz dM = − R0 M R0 M M Integrating by parts and taking into account the periodicity and the conservation of mass equation (3.76d) we obtain: (3.101) Z Z Z 1d 1 1 2 2 |uz |L2 + νv kuz k − vz uz dM − ρx uz dM = uz Fu, z dM. 2 dt R0 M R0 M M In all that follows K, K 0 , K 00 , ..., denote constants depending on the data but not on U0 ; we use the same symbol for different constants. We easily obtain the following estimates: Z Z ¯ ¯ 1 ¯¯ 1 ¯¯ ν ¯ ¯ vz uz dM¯ = vuzz dM¯ ≤ K|v|2L2 + v kuz k2 , ¯ ¯ R0 M R0 M 6 Z Z ¯ ¯ ¯ ¯ 1 ¯ 1 ¯ ¯ ¯ ν ρx uz dM¯ = ρ uxz dM¯ ≤ v kuz k2 +K|ρ|2L2 , ¯ ¯ R0 M R0 M 6 ¯Z ¯ ¯Z ¯ ν ¯ ¯ ¯ ¯ Fu,z uz dM¯ = ¯ Fu uzz dM¯ ≤ v kuz k2 +c01 |Fu |2L2 ; ¯ 6 M M applied to (3.101), these give: d |uz |2L2 + νv kuz k2 ≤ K(|v|2L2 + |ρ|2L2 ) + c01 |Fu |2L2 . (3.102) dt We apply the Poincaré inequality (3.87) and we find: d |uz |2L2 + c0 νv |uz |2L2 ≤ K(|v|2L2 + |ρ|2L2 ) + c01 |Fu |2L2 . (3.103) dt 14However,

as for Theorem 3.2 we could, at once, obtain the estimates for vx and vz , and then for ρx and ρz .


79

Using Gronwall’s Lemma, we infer from (3.103) that: (3.104) |uz (t)|2L2

Z ≤

|uz (0)|2L2 e−c0 νv t

≤

|uz (0)|2L2 e−c0 νv t

+ Ke

−c0 νv t

0

+ K (1 − e

t

0

(|v(t0 )|2L2 + |ρ(t0 )|2L2 )ec0 νv t dt0 + c02 |Fu |2∞

0 −c0 νv t

)(|v|2∞ + |ρ|2∞ ) + c02 |Fu |2∞

≤ |uz (0)|2L2 e−c0 νv t + K 0 (|v|2∞ + |ρ|2∞ ) + c02 |Fu |2∞ , where |v|∞ = |v|L∞ (R+ ; L2 (M)) , and similarly for ρ and Fu . We obtain an explicit bound for the norm of uz in L∞ (R+ ; H): |uz (t)|2L2 ≤ |uz (0)|2L2 + K 0 (|v|2∞ + |ρ|2∞ ) + c02 |Fu |2∞ .

(3.105)

For what follows, we recall here the uniform Gronwall lemma (see e.g., [35]): If ξ, η and y are three positive locally integrable functions on (t1 , ∞) such that 0 y is locally integrable on (t1 , ∞) and which satisfy (3.106)

y 0 ≤ ξy + η, Z t+r Z ξ(s) ds ≤ a1 , t

Z

t+r

t+r

η(s) ds ≤ a2 ,

t

y(s) ds ≤ a3 ,

∀ t ≥ t1 ,

t

where r, a1 , a2 , a3 are positive constants, then ³a ´ 3 (3.107) y(t + r) ≤ + a2 ea1 , t ≥ t1 . r The bound (3.105) depends on the initial data U0 . In order to obtain a bound independent of U0 we apply the uniform Gronwall lemma to the equation: d |uz |2L2 ≤ K(|v|2L2 + |ρ|2L2 ) + c01 |Fu |2L2 . (3.108) dt to obtain |uz (t)| ≤ K 0 (r, r00 ),

(3.109)

∀ t ≥ t01 ,

where t01 = t0 (|U0 |L2 ) + r and r > 0 is fixed. Integrating equation (3.102) from t to t + r with r > 0 as before, we also find: Z t+r ∀ t ≥ t01 . (3.110) kuz (s)k2 ds ≤ K 00 (r, r00 ), t

We now derive the same kind of estimates for ux : We differentiate (3.100) with respect to x and we obtain Z 0 1 1 2 utx + ux + uuxx + wuxz + wx uz − vx + ps,xx + ρxx (z 0 ) dz 0 (3.111) R0 R0 z − νv uxxx − νv uzzx = Fu, x ;

80


multiplying this equation by ux and integrating over M we find, using (3.76d): (3.112) Z Z Z Z 1d 1 1 2 3 |ux |L2 + ux dM + wx uz ux dM − vx ux dM − ps, x x ux dM 2 dt R0 M R0 M M M ¶ Z µZ 0 Z 0 0 2 + ρx x (z ) dz ux dM + νv kux k = ux Fu, x dM. M

z

M

Based on the Hölder, Sobolev and interpolation inequalities, we derive the following estimates: ¯ ¯Z ¯ ¯ u3x dM¯ ≤ |ux |3L3 (M) ≤ c04 |ux |3H 1/3 (M) ≤ c05 |ux |2L2 kux k ¯ M

¯Z ¯ ¯ M

ν ≤ v kux k2 + c06 |ux |4L2 , 12 ¯ ¯ 1/2 1/2 wx uz ux dM¯ ≤ c07 |wx |L2 |uz |L2 kuz k1/2 |ux |L2 kux k1/2 1/2

1/2

≤ c08 |uxx |L2 |uz |L2 kuz k1/2 |ux |L2 kux k1/2 ν ≤ v kux k2 + c09 |uz |2L2 kuz k2 |ux |2L2 , 12 By the definition of V , and since ps is independent of z, we find: Z Z Z L3 /2 ¯ ¯ 1 ¯¯ 1 ¯¯ L ¯ ¯ ps, x x ux dM¯ = ps, xx ux dz dx¯ = 0. ¯ ¯ R0 M R0 0 −L3 /2 We can also prove the following estimates: Z ¯ ν 1 ¯¯ ¯ vx ux dM¯ ≤ v kux k2 +K 0 |v|2L2 , ¯ R0 M 12 Z Z Z Z ¯ ¯ ´ ´ 1 ¯¯ ³ 0 1 ¯¯ ³ 0 ¯ ¯ 0 0 0 0 ρx x (z ) dz ux dM¯ = ρx (z ) dz uxx dM¯ ¯ ¯ R0 M z R0 M z ν ≤ v kux k2 + K 00 |ρx |2L2 , 12 ¯ ν ¯Z ¯ ¯ ux Fu, x dM¯ ≤ v kux k2 +c010 |Fu |2∞ . ¯ 12 M With these relations (3.112) implies: (3.113)

d |ux |2L2 + νv kux k2 ≤ ξ |ux |2L2 + η, dt

where we denoted ξ = ξ(t) = 2c06 |ux |2L2 + 2c09 |uz |2L2 kuz k2 ,


81

and η = η(t) = 2K 0 |v|2L2 + 2K 00 |ρx |2L2 + 2c010 |Fu |2∞ . We easily conclude from (3.113) that ux ∈ L∞ (0, t? ; L2 ) ∩ L2 (0, t? ; H 1 ),

(3.114)

∀ t? > 0.

However, for later purposes, (3.114) is not sufficient, and we need estimates uniform in time. We will apply the uniform Gronwall lemma to (3.113) with t1 = t01 as in (3.109). Noting that Z t+r Z t+r 0 0 ξ(t ) dt = [2c06 |ux |2L2 + 2c09 |uz (t0 )|2L2 kuz (t0 )k2 ] dt0 t t Z t+r Z t+r (3.115) 0 0 0 2 0 2 ≤ 2c6 |ux (t )|L2 dt + 2c9 |uz |∞ kuz (t0 )k2 dt0 t

t+r t

t

(3.116)

[2K 0 |v|2L2 + 2K 00 |ρx |2L2 + 2c010 |Fu |2∞ ] dt0

≤ K + 2c010 r|Fu |2∞ Z

t+r

(3.117) t

= a2 ,

∀ t ≥ t01 ,

|ux (t0 )|2L2 dt0 ≤ a3 ,

∀ t ≥ t01 ,

(3.107) then yields: (3.118)

t

∀ t ≥ t01 ,

≤ a1 , Z Z t+r 0 0 η(t ) dt =

|ux (t)|2L2

≤

³a

3

r

´

+ a2 ea1 ,

∀ t ≥ t01 + r,

and thus |ux |L2 ∈ L∞ (R+ ).

(3.119)

Note that in (3.115)–(3.117) we can use bounds on |uz |∞ (and other similar terms) independent of U0 , since t ≥ t0 (|U0 |L2 ) + r. Integrating equation (3.113) from 0 to t01 + r where t01 = t01 (|U0 |L2 ), we obtain a bound for ux in L2 (0, t01 + r; H 1 ) which depends on kU0 k. A bound independent of U0 is obtained if we work with t ≥ t01 + r = t001 = t001 (|U0 |L2 ): Integrating equation (3.113) from t to t + r with r as before, we find: Z t+r (3.120) kux (s)k2 ds ≤ K, ∀ t ≥ t001 . t

82


We perform similar computations for vz : We differentiate (3.76b) with respect to z, multiply the resulting equation by vz and integrate over M. Using again the conservation of mass relation, we arrive at: (3.121) Z Z Z 1 1d 2 2 |vz |L2 + uz vx vz dM + wz vz dM + uz vz dM + νv kvz k2 2 dt R 0 M M Z M = vz Fu, z dM. M

We notice the following estimate: ¯ ¯Z ¯ ¯ 1/2 1/2 uz vx vz dM¯ ≤ c011 |uz |L2 kuz k1/2 |vx |L2 |vz |L2 kvz k1/2 ¯ M

ν 2/3 4/3 2/3 ≤ v kvz k2 + c012 |uz |L2 kuz k2/3 |vx |L2 |vz |L2 8 νv 2/3 4/3 ≤ kvz k2 + c012 |uz |L2 kuz k2/3 |vx |L2 (1 + |vz |2L2 ). 8

We also see that ¯Z ¯ ¯Z ¯ ¯ ¯ wz vz vz dM¯ = ¯ ¯ M

M

¯ ¯ 1/2 3/2 ux vz vz dM¯ ≤ c013 |ux |L2 kux k1/2 |vz |L2 kvz k1/2

ν 2/3 ≤ v kvz k2 + c014 |ux |L2 kux k2/3 |vz |2L2 , 8 Z Z ¯ ¯ ν 1 ¯¯ 1 ¯¯ ¯ ¯ uz vz dM¯ = u vzz dM¯ ≤ v kvz k2 +K|u|2L2 , ¯ ¯ R0 M R0 M 8 Z Z ¯ ¯ ¯ ¯ ν ¯ ¯ ¯ ¯ Fv, z vz dM¯ = ¯ Fv vzz dM¯ ≤ v kvz k2 +c015 |Fv |2∞ , ¯ 8 M M which gives: d |vz |2L2 + νv kvz k2 ≤ ξ|vz |2 + η, dt

(3.122) where we denoted

2/3

4/3

η = η(t) = 2c012 |uz |L2 kuz k2/3 |vx |L2 + 2K|u|2 + 2c015 |Fv |2∞ , and 2/3

4/3

2/3

ξ = ξ(t) = 2c012 |uz |L2 kuz k2/3 |vx |L2 + 2c014 |ux |L2 kux k2/3 . From (3.122), using the estimates obtained before and applying the classical Gron2 wall lemma we obtain bounds depending on the initial data for vz in L∞ loc (0, t? ; L ) and L2loc (0, t? ; H 1 ), valid for any finite interval of time (0, t? ).


83

To obtain estimates valid for all time, we apply the uniform Gronwall lemma observing that: (3.123) Z t+r ³ Z t+r ´1/3 ³ Z t+r ´2/3 0 0 0 2/3 0 0 η(t ) dt ≤ 2c12 |uz |∞ kuz (t )k dt |vx (t0 )|2L2 dt0 t

t

+

2K|u|2∞ r

0

0

ξ(t ) dt ≤ t

2/3 2c012 |uz |∞

+

³Z

t+r

0

kuz (t )k dt t

0

´1/3 ³ Z

t+r

|vx (t t

Z

2c014 |ux |2/3 ∞

t+r

0

)|2L2

dt

0

´2/3

kux (t0 )k2/3 dt0

t

∀ t ≥ t001 ,

≤ a2 , (3.125) Z t+r

+

∀ t ≥ t001 ,

≤ a1 , (3.124) Z t+r

t

2c015 r|Fv |2∞

|vz (t0 )|2 dt0 ≤ a3 ,

t

∀ t ≥ t001 .

Then the uniform Gronwall lemma gives: ³a ´ 3 2 (3.126) |vz (t)|L2 ≤ + a2 ea1 , ∀ t ≥ t001 + r, r with a1 , a2 , a3 as in (3.123), (3.124) and (3.125). Integrating equation (3.122) from t to t + r with r > 0 as above and t ≥ t001 + r, we find: Z t+r (3.127) kvz (s)k2 ds ≤ K, ∀ t ≥ t001 + r. t

The same methods apply to vx , ρz and ρx , noticing that at each step we precisely use the estimates from the previous steps, so the order can not be changed in this calculations. With these estimates, the Galerkin method as used for the proof of Theorem 2.2 gives the existence of strong solutions: Theorem 3.4. Given U0 ∈ V and F ∈ L∞ (R+ ; H), there is a unique solution U of equation (3.92) with U (0) = U0 such that (3.128)

U ∈ L∞ (R+ ; V ) ∩ L2 (0, t? ; (H˙ 2 (M))3 ), ∀ t? > 0.

Proof. As we said, the existence of strong solutions follows from the previous estimates. The proof for the uniqueness follows the same idea as of the Theorem 3.1 so we skip it. ¤

84


3.3.3. More Regular Solutions for the PEs. In this section we show how to obtain estimates on the higher order derivatives from which one can derive the existence of solutions of the PEs in (H˙ m (M))3 for all m ∈ N, m ≥ 2 (hence up to C ∞ m regularity). In all that follows we work with U0 in (H˙ per (M))3 . ¡P ¢ α 2 1/2 We set |U |m = . We fix m ≥ 2 and, proceeding by induc[α]=m |D U |L2 tion, we assume that for all 0 ≤ l ≤ m − 1, we have shown that U ∈ L∞ (R+ ; (H˙ l (M))3 ) ∩ L2 (0, t? ; (H˙ l+1 (M))3 ),

(3.129) with

Z

t+r

(3.130) t

|U (t0 )|2l+1 dt0 ≤ al ,

∀ t? > 0,

∀ t ≥ tl (U0 ),

where al is a constant depending on the data (and l) but not on U0 , and r > 0 is fixed (the same as before). We then want to establish the same results for l = m. In equation (3.92) we take U˜ = ∆m U (t) with m ≥ 2 and t arbitrarily fixed, and we obtain: ³ dU ´ 1 , ∆m U e(U, ∆m U ) + a(U, ∆m U ) + b(U, U, ∆m U ) + 2 dt R0 L (3.131) = (F, ∆m U )L2 . Integrating by parts, using periodicity and the coercivity of a and the fact that e(U, U ) = 0, we find: 1d |U (t)|2m + c1 |U |2m+1 ≤ |b(U, U, ∆m U )| + |(F, ∆m U )L2 |. 2 dt We need to estimate the terms on the right hand side of (3.132). We first notice that c1 (3.133) |(F, ∆m U )2L | ≤ c|F |2m−1 + |U |2 , 2(m + 3) m+1 (3.132)

and it remains to estimate |b(U, U, ∆m U )|. By the definition of b we have: (3.134)

Z m

b(U, U, ∆ U ) =

Z m

(uux + w(U )uz )∆ u dM + (uvx + w(U )vz )∆m v dM M M Z + (uρx + w(U )ρz )∆m ρ dM. M

The computations are similar for all the terms, and, for simplicity, we shall only estimate the first integral on the right hand side of (3.134).


85

We notice that b(U, U, ∆m U ) is a sum of integrals of the type Z Z ∂u 2α1 2α3 ∂u u D1 D3 u dM, w(U ) D12α1 D32α3 u dM, ∂z M ∂x M where αi ∈ N with α1 + α3 = m. By Di we denoted the differential operator ∂/∂xi . Integrating by parts and using periodicity, the integrals take the form Z Z ³ ³ ∂u ´ ∂u ´ α α α D u dM, D u dM, (3.135) Dα w(U ) D u ∂x ∂z M M where Dα = D1α1 D3α3 . Using Leibniz’ formula, we see that the integrals are sums of integrals of the form Z Z ∂u α α ∂u α (3.136) uD D u dM, w(U ) Dα D u dM, ∂x ∂z M M and of integrals of the form Z ∂u α (3.137) δ k u δ m−k D u dM, ∂x M

Z δ k w(U ) δ m−k M

∂u α D u dM, ∂z

with k = 1, ..., m, where δ k is some differential operator Dα with [α] = α1 +α3 = k. For each α, after integration by parts we see that the sum of the two integrals in (3.136) is zero because of the mass conservation equation (3.76d). It remains to estimate the integrals of type (3.137). We use here the Sobolev and interpolation inequalities. For the first term in (3.137) we write: (3.138) ¯ ¯ ¯Z ¯ ¯ ¯ ¯ m−k ∂u ¯ α k m−k ∂u k 4 D u dM δ u δ ≤ |δ u| δ ¯ ¯ 4 |Dα u|L2 ¯ L ¯ ∂x ∂x L M ¯ ∂u ¯¯1/2 ¯¯ ∂u ¯¯1/2 1/2 1/2 ¯ ≤ c01 |δ k u|L2 |δ k u|H 1 ¯δ m−k ¯ ¯δ m−k ¯ |Dα u|L2 ∂x L2 ∂x H 1 1/2 1/2 1/2 1/2 0 ≤ c1 |U |k |U |k+1 |U |m−k+1 |U |m−k+2 |U |m , where k = 1, ..., m. The second term from (3.137) is estimated as follows: ¯ ¯ ¯Z ∂u ¯¯ ∂u α ¯ ¯ ¯ δ k w(U ) δ m−k D u dM¯ ≤ |δ k w(U )|L2 ¯δ m−k ¯ |Dα u|L4 ¯ ∂z ∂z L4 M ¯ ¯ ¯ ¯ ∂u ¯1/2 ∂u ¯1/2 ¯ ¯ (3.139) 1/2 1/2 ≤ c02 |δ k w(U )|L2 ¯δ m−k ¯ ¯δ m−k ¯ |Dα u|L2 |Dα u|H 1 ∂z L2 ∂z H 1 1/2 1/2 1/2 ≤ c03 |U |k+1 |U |m−k+1 |U |m−k+2 |U |1/2 m |U |m+1 , where k = 1, ..., m.

86


From (3.138) and (3.139) we obtain that: |b(U, U, ∆m U )| ≤ c3

m X

k=1 m X

(3.140)

1/2

1/2

1/2

1/2

|U |k |U |k+1 |U |m−k+1 |U |m−k+2 |U |m

+ c3

1/2

1/2

1/2

|U |k+1 |U |m−k+1 |U |m−k+2 |U |1/2 m |U |m+1 .

k=1

We now need to bound the terms on the right hand side of (3.140). The terms corresponding to k = 2, ..., m − 1 in the first sum do not contain |U |m+1 and we leave them as they are. For k = 1 and k = m, we apply Young’s inequality and we obtain: c1 1/2 2/3 2/3 1/2 1/2 3/2 |U |m+1 ≤ (3.141) c3 |U |1 |U |2 |U |m |U |2m+1 + c04 |U |1 |U |2 |U |2m . 2(m + 3) For the terms in the second sum in (3.140) we distinguish between k = 1, k = m and k = 2, ..., m − 1. The term corresponding to k = 1 is bounded by: c1 (3.142) c3 |U |2 |U |m |U |m+1 ≤ |U |2 + c05 |U |22 |U |2m . 2(m + 3) m+1 For k = m we find: (3.143)

1/2

1/2

3/2

c3 |U |1 |U |2 |U |1/2 m |U |m+1 ≤

c1 |U |2 + c06 |U |21 |U |22 |U |2m . 2(m + 3) m+1

For the terms corresponding to k = 2, ..., m − 1 we apply Young’s inequality in the following way: (3.144) 1/2 1/2 1/2 c3 |U |k+1 |U |m−k+1 |U |m−k+2 |U |1/2 m |U |m+1 c1 4/3 2/3 2/3 ≤ |U |2 + c07 |U |k+1 |U |m−k+1 |U |m−k+2 |U |2/3 m . 2(m + 3) m+1 Gathering all the estimates above we find: d |U |2m + c1 |U |2m+1 ≤ ξ + η|U |2m , dt where the expressions of ξ and η are easily derived from (3.132), (3.141), (3.142), (3.143) and (3.144). Using the Gronwall lemma and the induction hypotheses (3.129)–(3.130) we obtain a bound for U in L∞ (0, t? ; H m ) and L2 (0, t? ; H m+1 ), for all fixed t? > 0, this bound depending also on |U0 |m . We also see that, because of the induction hypotheses (3.129)–(3.130), we can apply the uniform Gronwall lemma and we obtain U bounded in L∞ (R+ ; H m ) with a bound independent of |U0 |m when t ≥ tm (U0 ); we also obtain an analogue of (3.130). The details


87

regarding the way we apply the uniform Gronwall lemma and derive these bounds are similar to the developments in Section 3.3.2. In summary we have proven the following result: m Theorem 3.5. Given m ∈ N, m ≥ 1, U0 ∈ V ∩(H˙ per (M))3 and F ∈ L∞ (R+ ; H ∩ m−1 (H˙ per (M))3 ), equation (3.92) has a unique solution U such that

(3.145)

m

m+1 (M))3 ), U ∈ L∞ (R+ ; (H˙ per (M))3 ) ∩ L2 (0, t? ; (H˙ per

∀ t? > 0.

m ∞ ∞ Remark 3.3. Since ∩m≥0 H˙ per (M) = C˙per (M), given U0 ∈ (C˙per (M))3 and F ∈ ∞ L∞ (R+ ; (C˙per (M))3 ), equation (3.92) has a unique solution U belonging to L∞ (R+ ; m ∞ (H˙ per (M))3 ) for all m ∈ N; that is, U is in L∞ (R+ ; (C˙per (M))3 ). Regularity (differentiability) in time can be also derived if F is also C ∞ in time. However the ∞ (M))3 . arguments above do not provide the existence of an absorbing set in (C˙per

4. Regularity for the elliptic linear problems in GFD. We have used many times, in particular in Section 3 the result of H 2 regularity of the solutions to certain linear elliptic problems. Following the general results of Agmon, Douglis and Nirenberg, [1, 2], we know that the solutions of second order elliptic problems are in H m+2 , if the right-hand sides of the equations are in H m , m ≥ 0, and the other data are in suitable spaces; see also Lions and Magenes [20] for m < 0. Results of this type are proved in this section. There are several specific aspects and several specific difficulties which justify the lengthy and technical developments of this section which do not allow us to directly refer to the general results of [1] and [2]: For the whole atmosphere (not studied in detail here) and for the space periodic case studied in Section 3.3 The domains are smooth, making the results of this section easy. (i)The (linear, stationary) GFD-Stokes problem (see Section 4.4.1), involves an integral equation (the second equation in (4.96)), which prevents from a purely local treatment, like for the classical Stokes problem of incompressible fluid mechanics. (ii)The boundary conditions of the problem can be a combination of Dirichlet, Neumann and/or Robin boundary conditions. (iii)The domains that we have considered and that we consider in this section are not smooth, they have angles in 2D and edges in 3D. This is automatically the

88


case for the ocean and for regional atmosphere or ocean problems. For this reason, technics pertaining to the theory of elliptic problems in nonsmooth domains (see e.g. [11, 25, 17]), are needed and used here. (iv)Because the domain is not smooth, only the H 2 regularity is proved here, m = 0. The H 3 regularity, m ≥ 1 is not expected in general. (v)Another aspect of the study in this section concerns the shape of the ocean or the atmosphere (shallowness). A ”small” parameter ε is introduced, the depth being called εh instead of h, 0 < ε < 1, and we want to see how the regularity constants (which depend on the domain) depend on ε. The small depth hypothesis was considered in [16] and is not considered in this article. Introducing the parameter ε makes the proofs of this section generally more involved than needed for this article. However these results usefully complement the article [15] used in [16]. Many of the results presented in this section are new although some related results appeared in [43] and [44]. The results are fairly general, except for the orthogonality condition appearing in (4.54) (Γb orthogonal to Γ` ). This condition is not physically desirable; and it is not either mathematically needed (most likely), as no such condition appears for the regularity theory of elliptic problems in angles or edges [11, 25, 17]. We believe that it can be removed, but this problem is open. Let us recall also that all the results in Section 3 are valid whenever the necessary H 2 regularity results can be proved. For the notations, the basic domain under consideration is Mε : Mε = {(x, y, z), (x, y) ∈ Γi , −εh(x, y) < z < 0}. For ε = 1, we recover the domain M(M1 = M) used in Sections 2 and 3. Below, ˆ ε and Qε . A number of for technical reasons, we introduce auxiliary domains M unspecified constants are generically denoted by C0 . Note also that this section is closer to PDE theory than to geoscience and therefore we stay closer to the PDE notations than to the geoscience notations. Hence the notations are not necessarily the same as in the rest of the article; in particular we do not use bold faces for vectors and the current point of R2 or R3 is denoted x = (x1 , x2 ) or x = (x1 , x2 , x3 ) instead of (x, y) or (x, y, z).


89

4.1. Regularity of solutions of elliptic boundary value problems in cylinder type domains. We study in this section the H 2 regularity of solutions of elliptic problems of the second order in a cylinder type domain; the boundary condition is either of Dirichlet or of Neumann type on all the boundary. Since the domain contains wedges, it is not smooth and we rely heavily on the results of [11] about regularity for elliptic problems in nonsmooth domains. However a convexity assumption of the domain is essential in [11], that we want to avoid: this section is mainly devoted to the implementation of a suitable technic, corresponding to a tubular (cylindrical) covering of the domain under consideration. ˆ ε = {(x1 , x2 , x3 ) ∈ R3 ; (x1 , x2 ) ∈ Γi , −εh(x1 , x2 ) < x3 < εh(x1 , x2 )}, Let M ¯ i → R+ where Γi is a bounded open subset of R2 , with ∂Γi a C 2 -curve, and h : Γ ¯ i ) and there exist two positive constants h and h is a positive function, h ∈ C 4 (Γ such that h ≤ h(x1 , x2 ) ≤ h for all (x1 , x2 ) in Γi . Define the elliptic operator A by

(4.1)

3 X

Au = −

k,

∂ ∂xk `=1

µ

∂u a (x) ∂x` k`

¶ +

3 X

bk (x)

k=1

∂u + c(x)u, ∂xk

ˆ ε ), bk , k = 1, 2, 3, are where the coefficients ak,` , k, ` = 1, 2, 3 are of class C 2 (M ˆ ε ) and c is of class C 0 (M ˆ ε ). Furthermore, we assume that A is of class C 1 (M uniformly strongly elliptic, i.e. there exists a positive constant α independent of x and ε such that

(4.2)

3 X

ak` (x)ξk ξ` > α|ξ|2 ,

ˆ ε, ∀x ∈ M

∀ξ ∈ R3 .

k, `=1

We also assume that the function ak` , bk , c, k, ` = 1, 2, 3 are independent of ε. We aim to study the regularity and the dependence on ε of the solutions to the Dirichlet problem

(4.3)

( ˆ ε, Au = f in M ˆ ε, u = 0 in ∂ M

90


and the solutions of the Neumann problem,  ˆ ε, Au = f in M    (4.4)

where

 ∂u  ˆ ε,  = 0 on ∂ M ∂nA ∂ denotes the conormal boundary operator defined by ∂nA X ∂u ∂u = ak` nk , ∂nA ∂x ` k,`

(4.5)

and n = (n1 , n1 , n3 ) denotes the unit vector in the direction of the outward normal ˆ ε . Our goal is to prove the H 2 − regularity of solutions to the Dirichlet to ∂ M problem (4.3) or the Neumann problem (4.4), and to obtain the dependence on ε of the constant Cε appearing in the inequality X ∂ 2u 2 | |2L2 (M (4.6) ˆ ε ) ≤ Cε |Au|L2 (M ˆ ε). ∂xk ∂x` k,` In fact we will show that Cε = C0 is independent of ε and, more precisely, we will prove the following. ˆ ε = {(x1 , x2 , x3 ) ∈ R3 ; (x1 , x2 ) ∈ Γi , −εh(x1 , x2 ) < x3 < Theorem 4.1. Let M εh(x1 , x2 )}, where Γi is a bounded subset of R2 , with ∂Γi a C 2 -curve, and h ∈ C 4 (Γi ), and there exists two positive constants h, h such that h ≤ h(x1 , x2 ) ≤ h ˆ ε ) and u ∈ H01 (M ˆ ε ), with ||u|| 1 ˆ ≤ C0 |f |ε for all (x1 , x2 ) in Γi . Let f ∈ L2 (M H (Mε ) with C0 independent of ε. If u satisfies µ ¶ X 3 3 X ∂ ∂u ∂u k` Au = − a (x) + bk (x) + c(x)u = f, ∂xk ∂x` ∂xk k,`=1 k=1 ˆ ε ), ci , d in C 1 (M ˆ ε ) and A is uniformly strongly elliptic where aij , bi are in C 2 (M ˆ ε ) and there exists a constant C0 independent in the sense of (4.2), then u ∈ H 2 (M of ε such that (4.7)

X i,j=1

|

∂ 2u 2 | ˆ ε ) ≤ C0 |f |2L2 (M ˆ ε). ∂xi ∂xj L2 (M


91

The proof of Theorem 4.1 is divided into four steps. Step 1: Flatening the top and bottom boundaries. We straighten the bottom ˆ ε into the cylinder Qε = Γi × and top boundaries and transform the domain M (−ε, ε), and the operator A is transformed to an operator A˜ of the same form and satisfying the same assumptions as A. In fact let Ψ : (x1 , x2 , x3 ) 7→ (y1 , y2 , y3 ), x3 . y1 = x1 , y2 = x2 , y3 = h(x1 , x2 )

(4.8)

ˆ ε ). Furthermore, we note that Au may Since h ∈ C 4 (Γ¯i ), we have Ψ ∈ C 4 (M be written as

(4.9)

Au = −

3 X

3

ak` (x)

k,`=1

X ∂ 2u ∂u dk (x) + + c(x)u, ∂xk ∂x` k=1 ∂xk

where

k

(4.10)

k

d (x) = b (x) −

3 X ∂a`k (x) `=1

∂x`

ˆ ε ). ∈ C 1 (M

ˆ ε ) and Now let u˜ : Qε → R, u˜(y1 , y2 , y3 ) = u(x1 , x2 , x3 ). Since Ψ ∈ C 4 (M Ψ is independent of ε, the H 2 -norm of u˜ is equivalent to the H 2 -norm of u with constants independent of ε. More precisely, there exists a constant C0 independent of ε such that C0−1 (4.11)

X k,`

|

X ∂ 2u ∂ 2 u˜ 2 | |L2 (Qε ) ≤ |2L2 (M ˆ ε) ∂yk ∂y` ∂x ∂x k ` k,` ≤ C0

X k,`

|

∂ 2 u˜ 2 . | 2 ∂yk ∂y` L (Qε )

Furthermore, we can easily check that X ∂ 2 u˜ ˜u(y) = − A˜ a ˜k` (y) ∂yk ∂y` k,` (4.12) X ∂ u˜ + d˜k (y) + c˜(y)˜ u(y) = f˜(y), ∂y k k

92


where

(4.13)

 X ∂Ψk ∂Ψ`  k`  ars (Ψ−1 (y)), a ˜ (y) =   ∂x ∂x  r s  r,s      X ∂ 2 Ψk X ∂Ψk sr −1 ˜k (y) = − d a (Ψ (y)) + dr (Ψ−1 (y))   ∂x ∂x ∂x  r s r  r,s r        c˜(y) = c(Ψ−1 (y)), and f˜(y) = f (Ψ−1 (y)).

ˆ ε ), that a It is clear, since Ψ ∈ C 4 (M ˜k,` ∈ C 2 (Q), d˜k ∈ C 1 (Q) and c˜ ∈ C 0 (Qε ). ˆ ε then u˜ = 0 on ∂Qε and if ∂u/∂nA = 0 on ∂ M ˆ ε , then Finally if u = 0 on ∂ M ∂ u˜/∂nA˜ = 0 on ∂Qε where, as in (4.5), X ∂ u˜ ∂ u˜ a ˜k` = nk . ∂nA˜ ∂y ` k,`

(4.14)

This is classical, but we include the verification here at the bottom boundary x3 = −εh or equivalently y3 = −ε. By (4.5) we have

(1 + ε2 |∇h|2 )1/2 (4.15)

3 3 X X ∂u ∂u ∂h ∂u ∂h =ε a1j +ε a2j ∂nA ∂xj ∂x1 ∂xj ∂x2 j=1 j=1

+

3 X

a3j

j=1

∂u ; ∂xj

3

but

X ∂ u˜ ∂Ψr ∂u = , and thus ∂xj ∂y ∂x r j r=1 ∂u ∂nA 3 3 µ XX

(1 + ε2 |∇h|2 )1/2 (4.16) =

r=1 j=1

∂h ∂h a (ε ) + a2j (ε ) + a3j ∂x1 ∂x2 1j

¶

∂Ψr ∂ u˜ . ∂xj ∂yr

On the other hand, by definition, we have

(4.17)

3 X ∂ u˜ ∂ u˜ =− a ˜3r (−ε) ∂nA˜ ∂yr r=1

(the normal is in the direction of y3 < 0);


93

but

(4.18)

a ˜3r

3 X ∂Ψ3 ∂Ψr X ∂Ψr mn = a = ∂xm ∂xn ∂xn m,n n=1

Ã

3 X

m=1

amn

∂Ψ3 ∂xm

! ,

and since Ψ3 (x1 , x2 , x3 ) = x3 /h(x1 , x2 ), we have

(4.19)

µ ¶ µ ¶ ¶ 3 µ ∂h ∂h ∂Ψr 1 X 1j 2j 3j a ² +a ε +a . a ˜ (−ε) = h j=1 ∂x1 ∂x2 ∂xj 3r

Hence (4.20)

∂ u˜ = 0 at y3 = −ε. ∂nA˜

A similar computation yields ∂ u˜/∂nA˜ = 0 at y3 = ε. Now we check the Neumann condition at the lateral boundary: First write

(4.21)

3 3 2 X 2 X X X ∂ u˜ ∂Ψr ∂u k` ∂u a ak` = nk = nk , ∂nA ∂x ∂y ` r ∂x` k=1 `=1 k=1 `,r=1

and (4.22)

3 3 2 X 2 X X X ˜ ∂Ψk ∂Ψ` ∂ u˜ ∂ u˜ k` ∂ u a ˜ asr = nk = nk ; ∂nA˜ ∂y` ∂xs ∂xr ∂y` k=1 `=1 k=1 `,r,s=1

but since for k = 1, 2, Ψk (x1 , x2 , x3 ) = xk , we have ∂Ψk /∂xs = δsk (the Kronecker symbol). Hence

(4.23)

3 X ∂ u˜ ∂Ψ` ∂ u˜ = akr nk ; ∂nA˜ k,`,r=1 ∂xr ∂y`

interchanging ` and r, we obtain ∂ u˜/∂nA˜ = ∂u/∂nA = 0 on the lateral boundary. From now on we concentrate on the Dirichlet boundary condition, the Neumann condition case follows in the same manner. Step 2. Interior regularity. Let BR be an open ball, with BR ⊂⊂ Γi ; without loss of generality we assume that BR is centered at 0. By Step 1, we may assume ˆ ε is a right cylinder i.e. M ˆ ε = Γi × (−ε, ε). Now let θ ∈ C ∞ (BR ) (θ that M 0 independent of x3 ) with θ ≡ 1 in BR/2 and 0 ≤ θ ≤ 1. Then

94


˜ u) = θf˜ + Eθ u˜ A(θ˜ where Eθ is a first order differential operator, which implies that |Eθ u˜|L2 (Qε ) ≤ C0 |f |ε with C0 independent on ε. Hence ˜ u) = floc , with |floc |L2 (B ×(−ε,ε)) ≤ C0 |f |ε , A(θ˜ R and, for either boundary condition (Dirichlet, Neumann), θ˜ u = 0 on ∂(BR × (−ε, ε)). We quote the following theorems from [11], we start first with the case of the Dirichlet boundary condition. Theorem A (Grisvard [11]: Theorem 3.2.1.2) Let Ω be a convex, bounded and open subset of Rn . Then for each f ∈ L2 (Ω), there exists a unique u ∈ H 2 (Ω), the solution of Au = f in Ω, u = 0 on ∂Ω. The proof of the theorem, given in [11] pages 148-149, is based on a priori bounds for solutions in H 2 (Ω). These bounds depend in the case of general domains on the curvature of ∂Ω, however in the case of a convex domain the curvature is negative and the constants in the bounds are therefore independent on the domain. Similarly in the case of Neumann boundary condition, we have Theorem B (Grisvard [11]: Theorem 3.2.1.3) Let Ω be a convex, bounded and open subset of Rn . Then for each f ∈ L2 (Ω) and for each λ > 0, there exists a unique u ∈ H 2 (Ω), the solution of Z Z Z 3 X ∂u ∂v k` (4.24) − a (x) dx + λ uv dx = f v dx ∂x ` ∂xk Ω Ω Ω k, `=1 for all v ∈ H 1 (Ω). We note that (4.24) is the weak form of the Neumann problem for the equation µ ¶ 3 X ∂u ∂ k` a (x) + λu = f in Ω − ∂xk ∂x` k, `=1 ∂u = 0 on ∂Ω. Again, here, the convex∂nA ity of the domain implies that the curvature of the boundary of the domain is negative, and therefore the constants in the bounds on the L2 norm of the mixed

together with the boundary condition


95

second derivatives in terms of the L2 norm of f are independent on the domain. For more details, see [11]. Now we use Theorem A above by first rewriting A˜ in a divergence form and moving the extra terms to the right hand side. As above, since θ˜ u ∈ H 1 (BR × (−ε, ε)) the extra terms are in L2 (BR × (−ε, ε)) and X ∂ 2 (θ˜ u) 2 | |L2 (BR ×(−ε,ε)) ≤ C0 |f |2ε . ∂y ∂y i j i,j Finally since θ ≡ 1 in BR /2, we have u˜ ∈ H 2 (BR /2 × (−ε, ε)) and X ∂ 2 u˜ | |2L2 (BR /2×(−ε,ε)) ≤ C0 |f |2ε . ∂yi ∂yj i,j

© ª 2 2 Step 3. Boundary regularity. Let R+ = {x ∈ R2 ; x2 > 0} and let Br+ = x ∈ R+ ; |x| < r be the open half-ball with center at the origin and radius r contained in R2r . By the assumption on Γi , for all x0 ∈ ∂Γi , there exists a neighborhood V of x0 in R2 ˜ such that and a diffeomorphism Ψ ˜ Ψ(V ∩ Γi ) = B + , r

˜ ) = 0. Ψ(x 0

˜ we can construct a (tubular) diffeomorphism Ψ in Using the diffeomorphism Ψ, 3 R such that Ψ(V × (−ε, ε) ∩ Qε ) = Br+ × (−ε, ε), ˜ i (y1 , y2 ), i = 1, 2, and Ψ3 (y1 , y2 , y3 ) = y3 . Following by setting Ψi (y1 , y2 , y3 ) = Ψ the same procedure as in Step 1 and Step 2, let W be an open set of R2 containing x0 such that W ⊂ V and let θ ∈ C0∞ (V ) be such that 0 ≤ θ ≤ 1 and θ ≡ 1 in W. Then ˜ u) = floc with |floc |L2 (V ×(−ε,ε)∩Q ) ≤ C0 |f |ε , A(θ˜ ε and, in the case of the Dirichlet boundary condition θ˜ u = 0 on ∂(V ×(−ε, ε)∩Qε ). Next we use the transformation Ψ which is independent of ε and which transforms u into u∗ , and A˜ into A∗ with the domain V × (−ε, ε) ∩ Qε into Br+ × (−ε, ε), θ˜ A∗ u∗ given as in Step 1. Now u∗ = 0 on ∂(BR+ × (−ε, ε)) and BR+ × (−ε, ε) is convex; hence rewriting A∗ in a divergence form we obtain using [11], u∗ ∈ H 2 (BR+ × (−ε, ε)) and

96


X ∂ 2 u∗ |2L2 (B + ×(−ε,ε)) ≤ C0 |f |2ε . | R ∂z ∂z i j i,j Going back to V × (−ε, ε) ∩ Qε using Ψ−1 , we obtain X ∂ 2 (θ˜ u) 2 | |L2 (V ×(−ε,ε)∩Qε ) ≤ C0 |f |2ε . ∂y , ∂y i j i,j Hence X

|

i,j

∂ 2 u˜ 2 | 2 ≤ C0 |f |2ε . ∂yi ∂yj L (V ×(−ε,ε)∩Qε )

Step 4. Partition of unity and conclusion. Let V0 , V1 , · · · , VN and W0 , · · · , WN ¯ i satisfying V¯0 ⊂ Γi ; Vk , k ≥ 1 is contained in the be two finite open coverings of Γ ˜ (k) such that domain of a local map Ψ ˜ (k) (Vk ∩ Γi ) = B + , Ψ r W0 = V0 , ¯ k ⊂ Vk W

for all k ≥ 1.

Finally let {ϕk }k be a partition of unity subordinated to the covering {Wk }k of P Γi . Then u˜ = N ˜ and by Steps 1,2,3, ϕk u˜ ∈ H 2 (Qε ) and k=1 ϕk u 3 X ∂ 2 (ϕk u˜) 2 | |L2 (Qε ) ≤ C0 |f |2ε . ∂y ∂y i j i,j=1

N X k=0

Therefore u˜ ∈ H 2 (Qε ) and 3 X

|

i,j=1

∂ 2 u˜ 2 ≤ C0 |f |2ε . | 2 ∂yi ∂yj L (Qε )

ˆ ε and conclude that u ∈ H 2 (M ˆ ε ) and Finally, we go back to the domain M 3 X i,j=1

Theorem 4.1 is proved.

|

∂ 2u 2 | ˆ ε ) ≤ C0 |f |2ε . ∂xi ∂xj L2 (M


97

4.2. Regularity of Solutions of a Dirichlet-Robin Mixed Boundary Value Problem. We now want to derive a result similar to that of Section 4.1, for a boundary value problem with mixed Dirichlet-Robin boundary conditions, the elliptic operator being the same as in (4.1). The proof consists in reducing the boundary condition to a full Neumann boundary condition and then use Theorem 4.1. From now on we will consider the actual domain © ª Mε = (x1 , x2 , x3 ) ∈ R3 ; (x1 , x2 ) ∈ Γi , −εh(x1 , x2 ) < x3 < 0 ; | · |ε will denote the norm in L2 (Mε ) (or product of such spaces), and | · |i will denote the norm in L2 (Γi ) (or product of such spaces), ||g||i = |∇g|i . We will prove the following: Theorem 4.2. Assume that Γi is an open bounded set of R2 , with C 3 -boundary ∂Γi and h ∈ C 4 (Γi ). Then, for f ∈ L2 (Mε ) and g ∈ H01 (Γi ), there exists a unique Ψ ∈ H 2 (Mε ) solution of  −∆ Ψ = f in Mε ,    ∂Ψ 3 + αΨ = g on Γi ,  ∂x3   Ψ = 0 on Γb ∪ Γ` .

(4.25)

Furthermore, there exists a constant C = C(h, Γi , α) independent on ε such that

(4.26)

3 X i,j=1

|

∂2Ψ 2 | ≤ C(h, Γi , α)[|f |2ε + ||g||2H 1 (Γi ) ]. ∂xi ∂xj ε

Proof. The proof is divided into several steps. First we construct a function Ψ∗ satisfying the boundary conditions in (4.25), and find the precise dependence on ε of the L2 -norm of the second order derivatives of Ψ∗ (see Lemma 4.1). Then we set (4.27)

ˆ = eαx3 (Ψ − Ψ∗ ) Φ

ˆ satisfies the homogeneous Neumann condition on Γi (∂ Φ/∂x ˆ and verify that Φ 3 = ˆ 0 on Γi ) and the homogenous Dirichlet boundary condition on Γ` ∪ Γb (Φ = 0

98


on Γ` ∪ Γb ). By a reflection argument, we extend f to x3 > 0 to be an even function and consider a homogeneous Dirichlet problem on the extended domain ˆ ε = {(x1 , x2 , x3 ); (x1 , x2 ) ∈ Γi , −εh(x1 , x2 ) < x3 < εh(x1 , x2 )}, the solution of M ˆ , coincides with Φ ˆ on Mε . Finally we invoke Theorem 4.1 to conclude which, W ˆ and thus of Φ, ˆ along with an estimate of the type (4.26) the H 2 -regularity of W ˆ we therefore obtain the H 2 -regularity of Ψ and (4.26) by simply using for Φ; (4.27). Thus the whole proof of Theorem 4.2 hinges on the following lifting lemma. Lemma 4.1. Let h ∈ C 4 (Γi ) and g ∈ H01 (Γi ). There exists Ψ∗ ∈ H 2 (Mε ) such ∂Ψ∗ + αΨ∗ = g on Γi , Ψ∗ = 0 on Γ` ∪ Γb , and there exists a constant that ∂x3 C = C(h, Γi ) independent on ε, such that for 0 < ε ≤ 1 : X

(4.28)

i,j

|

∂ 2 Ψ∗ 2 |ε ≤ C(h, Γi )||g||2H 1 (Γi ) . ∂xk ∂xj

˜ as a solution of the heat Proof of Lemma 4.1 First we construct a function Ψ equation with −x3 corresponding to time     

˜ ∂Ψ ˜ in Γi × (−∞, 0), = −∆Ψ ∂x3 ˜ = 0 on ∂Γi × (−∞, 0), Ψ     Ψ(x ˜ 1 , x2 , 0) = g(x1 , x2 ) on Γi .

(4.29)

Here ∆ = ∆2 = (∂ 2 /∂x21 + ∂ 2 /∂x22 ) and, below, ∇ = ∇2 = (∂/∂x1 , ∂/∂x2 ). The function Ψ∗ is then constructed as Z (4.30)

∗

Ψ (x1 , x2 , x3 ) = e

−αx3

x3

˜ 1 , x2 , z)dz. Ψ(x

−εh(x1 ,x2 )

˜ 1 , x2 , x3 ) in It is clear that Ψ∗ ≡ 0 on Γ` ∪ Γb , and ∂Ψ∗ /∂x3 + αΨ∗ = e−αx3 Ψ(x Γi × (−∞, 0), which implies ∂Ψ∗ /∂x3 + αΨ∗ = g on Γi . We only need to check that Ψ∗ ∈ H 2 (Mε ) and that the inequality (4.28) is valid: This will be done using the classical energy estimates on the solution of the heat equation which are recalled in Lemmas 4.2 and 4.3 below. We note that for k = 1, 2,


αx3 ∂Ψ

e (4.31)

Z

∗

∂xk

x3

=

+ε

−εh(x1 ,x2 )

99

˜ ∂Ψ (x1 , x2 , z)dz ∂xk

∂h ˜ Ψ(x1 , x2 , −εh(x1 , x2 )), ∂xk

and, for k, j = 1, 2,

(4.32)

Z x3 ˜ ∂ 2 Ψ∗ ∂ 2Ψ e = (x1 , x2 , z)dz ∂xk ∂xj −εh(x1 ,x2 ) ∂xk ∂xj ˜ ˜ ∂h ∂ Ψ ∂h ∂ Ψ +ε (x1 , x2 , −εh(x1 , x2 )) + ε (x1 , x2 , −εh(x1 , x2 )) ∂xj ∂xk ∂xk ∂xj ∂ 2h ˜ +ε Ψ(x1 , x2 , −εh(x1 , x2 )) ∂xk ∂xj ˜ ∂h ∂h ∂ Ψ (x1 , x2 , −εh(x1 , x2 )). − ε2 ∂xk ∂xj ∂x3 αx3

˜ Here we need bounds on the L2 -norm (on Γi ) of (∂ Ψ/∂x k )(x1 , x2 , −εh(x1 , x2 )) ˜ and (∂ Ψ/∂x3 )(x1 , x2 , −εh(x1 , x2 )) which are provided by Lemma 4.3. We have:

(4.33)

 R ˜ 1 , x2 , −εh(x1 , x2 )|2 dx1 dx2 ≤ C||g||2i ,  |Ψ(x  Γi  R  ˜ 1 , x2 , −εh(x1 , x2 ))|2 dx1 dx2 ≤ C||g||2i , |∇Ψ(x Γi ˜  R ∂Ψ 1    Γi | (x1 , x2 , −εh(x1 , x2 ))|2 dx1 dx2 ≤ C||g||2i . ∂x3 ε

Now using (4.30) - (4.32) and (4.33), we obtain (4.34)

|

∂ 2 Ψ∗ 2 | ≤ C||g||2i · ∂xk ∂xj ε

k, j = 1, 2.

Similar relations hold for Ψ∗ and ∇Ψ∗ , using (4.30), (4.31). Furthermore, since ˜ we have ∂Ψ∗ /∂x3 = −αΨ∗ + e−αx3 Ψ, (4.35)

(4.36) Finally

|

|

∂Ψ∗ 2 ¯ ˜ 2 ≤ ε2 C(h)||g||2 , |ε ≤ 2α2 |Ψ∗ |2ε + 2e2αh |Ψ| i ε ∂x3

∂ ¯ ˜ 2ε ≤ ε2 C(h)||g||2i . ∇Ψ∗ |2ε ≤ 2α2 |∇Ψ∗ |2ε + 2e2αh |∇Ψ| ∂x3

100

(4.37)


˜ ∂Ψ∗ ∂ 2 Ψ∗ −αx3 ˜ −αx3 ∂ Ψ = −α − αe Ψ + e , ∂x23 ∂x3 ∂x3

implies ∂ 2 Ψ∗ 2 | ≤ C(h)||g||2i . 2 ε ∂x3 The proofs of Lemma 4.1 and Theorem 4.2 are complete. ¤ The proof of Theorem 4.2 relied on estimates given by Lemma 4.2 and 4.3 which we now state and prove (4.38)

|

Lemma 4.2. (Estimates on solutions of the heat equation.) Let Ψ be the solution of the heat equation ∂Ψ = −∆Ψ in Γi × (−∞, 0), ∂x3 (4.39) Ψ(x1 , x2 , 0) = g(x1 , x2 ) on Γi , with either Dirichlet or Neumann boundary condition, and g ∈ H01 (Γi ) and Ψ = 0 on ∂Γi × (−∞, 0), or g ∈ H 1 (Γi ) and Then

(4.40)

and

(4.41)

∂Ψ = 0 on ∂Γi × (−∞, 0). ∂nΓi

Z 0 j ∂j Ψ 1 k ∂ Ψ 2 |z|k |∇ j |2i (z)dz |x3 | | j |i (x3 ) + 2 ∂x3 ∂x3 x3 1 2 ≤ |g|i for k = j = 0, and 2 ≤ C|x3 |k−2j+1 ||g||2i , for k ≥ 2j − 1, j ≥ 1, Z 0 ∂j Ψ 2 1 ∂ j+1 Ψ k |x3 | |∇ j |i (x3 ) + |z|k | j+1 |2i (z)dz 2 ∂x3 ∂x3 x3 1 ≤ ||g||2i for k = 0, j = 0, and 2 ≤ C|x3 |k−2j ||g||2i , for k ≥ 2j, j ≥ 1.

In (4.40) and (4.41), C is a constant depending on k, j and h. As before, ∇ = ∇2 = (∂/∂x1 , ∂/∂x2 ).


101

Proof. Denote by ek,j and fk,j the left-hand sides of (4.40) and (4.41). We differentiate (4.39) j times in x3 , multiply the resulting equation by xk3 ∂ j Ψ/∂xj3 and integrate over Γi . Using Stokes formula and observing that ∂ j Ψ/∂xj3 satisfies the same boundary condition on ∂Γi as Ψ, we obtain after multiplication by (−1)k : Z k 0 k−1 ∂ j Ψ 2 |z| | j |i (z)dz, for k ≥ 1, and, ek,j = 2 x3 ∂x3 (4.42) j 1 1 ∂ Ψ 1 = | j |2i (0) = |∆j Ψ|2i (0) = |∆j g|2i , for k = 0. 2 ∂x3 2 2 Similarly, if we differentiate (4.39) j times in x3 , multiply the resulting equation by xk3 ∂ j+1 Ψ/∂xj+1 and integrate over Γi , we find 3 Z k 0 k−1 ∂ j Ψ 2 fk,j = |z| |∇ j |i (z)dz, for k ≥ 1, and, 2 x3 ∂x3 (4.43) j 1 1 ∂ Ψ = |∇ j |2i (0) = |∇∆j g|2i , for k = 0. 2 ∂x3 2 Using (4.42), (4.43) with k = j = 0, (4.42) with k = j = 1 and (4.43) with k = 2, j = 1, we find some of the relations (4.40), (4.41), namely: 1 |Ψ(x3 )|2i + 2

(4.44)

Z

0 x3

1 |∇Ψ(x3 )|2i + 2

Z

1 |∇Ψ|2i (z)dz ≤ |g|2i , 2 0

∂Ψ 2 1 |i (z)dz ≤ ||g||2i , 2 x3 ∂x3 Z 0 1 ∂Ψ 2 1 ∂Ψ 2 |x3 || |i (x3 ) + |i (z)dz ≤ ||g||2i , |z||∇ 2 ∂x3 ∂x3 4 x3 Z 0 2 1 ∂Ψ 2 1 ∂ Ψ |x3 |2 |∇ |i (x3 ) + z 2 | 2 |2i (z)dz ≤ ||g||2i . 2 ∂x3 ∂x3 4 x3 |

To derive the other relations (4.40) and (4.41), we first integrate (4.42) from x3 to 0, with x3 < 0, k ≥ 1 and j ≥ 0; we obtain Z

Z 0Z 0 ∂j Ψ 2 ∂j Ψ |z| | j |i (z)dz ≤ k |z|k−1 | j |2i (z)dzdt ∂x3 ∂x3 x3 x3 t Z 0 ∂j Ψ ≤k (z − x3 )|z|k−1 | j |2i (z)dz. ∂x3 x3 0

k

102


Thus, for k ≥ 1, j ≥ 0 Z 0 Z 0 j k ∂j Ψ k ∂ Ψ 2 |x3 | |z|k−1 | j |2i (z)dz, (4.45) |z| | j |i (z)dz ≤ k+1 ∂x3 ∂x3 x3 x3 and, by induction on k and using (4.41): Z (4.46)

Z ∂j Ψ 2 |x3 |k 0 ∂ j Ψ 2 |z| | j |i (z)dz ≤ | | (z)dz, k + 1 x3 ∂xj3 i ∂x3 x3 |x3 |k+1 |x3 |k ≤ |g|2i for j = 0, and ≤ ||g||2i for j = 1, (k ≥ 1). 2(k + 1) 2(k + 1) 0

k

Now combining (4.42) and (4.43) we find for k ≥ 2, j ≥ 1 : ek,j ≤

k(k − 1) ek−2,j−1 . 22

Hence by induction k! ek−2r,j−r , − 2r)! k! ≤ 2j−2 es,1, 2 (k − 2j + 2)!

ek,j ≤ ek,j

22r (k

s = k − 2j + 2, for k ≥ 2j − 2, j ≥ 1. For s ≥ 1, i.e. k ≥ 2j − 1 and j ≥ 1, we have, thanks to (4.42) and (4.46):

es,1 (4.47) ek,j

Z 1 0 s−1 ∂Ψ 2 = | (z)dz |z| | 2 x3 ∂x3 i 1 ≤ |x3 |s−1 ||g||2i , 2 k! ≤ 2j−1 |x3 |k−2j+1 ||g||2i , 2 (k − 2j + 2)!

for k ≥ 2j − 1, j ≥ 1. The relations (4.40) are proven; the relations (4.41) follow from (4.44) for j = 0, and from fk,j ≤ (k/2)ek−1,j for j ≥ 1. Lemma 4.2 is proved. ¤ From Lemma 4.2 we easily infer the following Lemma 4.3. Under the hypotheses of Lemma 4.2,


103

Z

∂j Ψ 2 | (x1 , x2 , −εh(x1 , x2 ))dx1 dx2 ∂xj3 Γi ≤ Cε−2j ||g||2H 1 (Γi ) for j ≥ 0, |∇

(4.48)

Z

∂j Ψ 2 j | (x1 , x2 , −εh(x1 , x2 ))dx1 dx2 Γi ∂x3 ≤ C||g||2H 1 (Γi ) for j = 0, and |

(4.49)

≤ Cε−2j+1 ||g||2H 1 (Γi ) for j ≥ 1, Z |∇j Ψ|2 (x1 , x2 , −εh(x1 , x2 ))dx1 dx2

(4.50)

Γi

≤ Cε−j+1 ||g||2H 1 (Γi ) , Z |

(4.51)

Γi

j = 2, 3,

∂ 2 2 ∇ Ψ| (x1 , x2 , −εh(x1 , x2 ))dx1 dx2 ∂x3

≤ Cε−3 ||g||2H 1 (Γi ) , where ||g||2H 1 (Γi ) = |g|2i + ||g||2i , and C is a constant depending on j and h, and independent of ε. Proof. We write µ ¶2 ∂j Ψ ∇ j (x1 , x2 , −εh(x1 , x2 )) = ∂x3 µ ¶2 ∂j Ψ ¯ (x1 , x2 , −εh)+ = ∇ j ∂x3 ¶µ ¶ Z −εh(x1 ,x2 ) µ ∂j Ψ ∂ j+1 Ψ +2 ∇ j ∇ j+1 (x1 , x2 , z)dz. ¯ ∂x3 ∂x3 −εh Integrating in x1 , x2 on Γi we find Z µ Γi

¶2 ∂j Ψ ∇ j (x1 , x2 , −εh(x1 , x2 ))dx1 dx2 ∂x3 ∂j Ψ ¯ ≤ |∇ j |2i (−εh) ∂x3 µZ −εh ¶1/2 µZ −εh ¶1/2 ∂j Ψ 2 ∂ j+1 Ψ 2 +2 |∇ j |i (z)dz |∇ j+1 |i (z)dz . ¯ ¯ ∂x3 ∂x3 −εh −εh

104


By (4.41), |∇

∂j Ψ 2 ¯ ¯ −2j ||g||2 1 H (Γi ) j |i (−εh) ≤ C(εh) ∂x3

(j ≥ 0),

and by integration of (4.41), Z −εh ∂j Ψ ¯ −k (εh) ¯ k−2j+1 ||g||2 1 |∇ j |2i (z)dz ≤ C(εh) H (Γi ) ¯ ∂x3 −εh ≤ Cε−2j+1 ||g||2H 1 (Γi ) (j ≥ 0); (4.48) follows. The proof of (4.49) is similar. For (4.50) and (4.51) we observe that, by the regularity property for the Neumann problem in Γi , 2 X

(4.52)

∂2Ψ 2 |L2 (Γi ) (x3 ) ≤ ∂x k ∂x` k,`=1 ³ ´ ≤ C |Ψ|2L2 (Γi ) (x3 ) + |∆Ψ|2L2 (Γi ) (x3 ) ¶ µ ∂Ψ 2 2 | 2 (x3 ) . ≤ C |Ψ|L2 (Γi ) (x3 ) + | ∂x3 L (Γi ) |

By repeating the argument above, it appears that the bounds for the left-handsides of (4.50) and (4.51) are the same as those of (4.48) and (4.49), for j = 1 and 2 respectively; (4.50) and (4.51) are proved. ¤

4.3. Regularity of Solutions of a Neumann-Robin Boundary Value Problem. We now want to derive a result similar to that of Sections 4.1 and 4.2. for a mixed Neumann-Robin type boundary condition, that is for the problem (4.53) below. Our result is quite general except for the restrictions (4.54) below. We will prove the following: Theorem 4.3. Assume Γi is an open bounded set of R2 , with a C 3 – boundary ¯ i ). For f ∈ L2 (Mε ) and g ∈ H 1 (Γi ), there exists a unique ∂Γi and h ∈ C 4 (Γ Ψ ∈ H 2 (Mε ) solution of


105

− ∆3 Ψ = f, ∂Ψ + αΨ = g on Γi , ∂x3 ∂Ψ = 0 on Γb ∪ Γ` . ∂n

(4.53)

Furthermore if: (4.54)

∇h · nΓi = 0 on ∂Γi ,

then there exists C = C(h, Γi , α) such that: 3 X k,`=1

|

£ ¤ ∂2Ψ 2 |ε ≤ C |f |2ε + ||g||2H 1 . ∂xk ∂x`

Remark 4.1. Condition (4.54) means that Γb and Γ` meet at a wedge angle of π/2. This is a technical condition needed in the method of proof used below, this condition is not required by the theory of regularity of elliptic problems in non smooth three-dimensional problems [11]. It might be possible to remove this condition with a different proof.

Proof of Theorem 4.3. The proof is divided into several steps. First we reduce the problem to the case α = 0, then we reduce the problem to the case where g = 0 and (α = 0). Thus we obtain a homogeneous Neumann problem. Then by a reflection around x3 = 0, we can use Theorem 4.1 and conclude the proof. Step 1. Reduction to the case g = 0. Our goal now is to find an explicit function T¯ that satisfies the nonhomogeneous boundary conditions imposed on the temperature. Since we are interested in obtaining a sharp dependence on the thickness ε, we have to construct T¯ explicitly instead of the classical method of lifting by localization and straightening the boundary which yields constants which do not have the right dependence on ε. We will carry the computations only in the case where ∇h · nΓi = 0 on ∂Γi . ¯ i ) with ∇h · nΓ = 0 on ∂Γi and let g ∈ H 1 (Γi ). There Lemma 4.4. Let h ∈ C 4 (Γ i 2 ¯ exists a function T ∈ H (Mε ) such that

106


∂ T¯ + αT¯ = g on Γi , ∂x3 ∂ T¯ = 0 on Γ` ∪ Γb ∂n

(4.55) Furthermore 3 X

∂ 2 T¯ 2 |ε ≤ C||g||2H 1 (Γi ) , | ∂xk ∂xl k,l=1

(4.56)

where C is a constant independent on ε. ¯ be a solution of the heat equation, where −x3 corresponds to time: Proof. Let Ψ            (4.57)

         

¯ ∂Ψ ¯ in Γi × (−∞, 0), = −∆Ψ ∂x3 ¯ ∂Ψ = 0 on ∂Γi × (−∞, 0), ∂nΓi ¯ 1 , x2 , 0) = g(x1 , x2 ), Ψ(x

and define T¯ by Z (4.58)

x3

¯ 1 , x2 , z)dz T¯(x1 , x2 , x3 ) = e Ψ(x −εh(x1 ,x2 ) µ ¶ 1 − x3 − θ1 (x1 , x2 ) + x23 (x3 + εh)θ2 (x1 , x2 ), α −αx3

where (4.59)

¯ 1 , x2 , −εh(x1 , x2 ))(1 + ε2 |∇h|2 ), θ1 (x1 , x2 ) = eαεh(x1 ,x2 ) Ψ(x

and (4.60)

−(εh + α1 ) ∇θ1 · ∇h. θ2 (x1 , x2 ) = 2 εh (1 + ε2 |∇h|2 )

Then (4.61)

∂ T¯ = −αe−αx3 ∂x3

Z

x3

¯ 1 , x2 , x3 ) ¯ 1 , x2 , z)dz + e−αx3 Ψ(x Ψ(x

−εh

−θ1 (x1 , x2 ) + 2x3 (x3 + εh)θ2 (x1 , x2 ) + x23 θ2 (x1 , x2 ),


(4.62)

107

∂ T¯ ¯ 1 , x2 , 0) = g, + αT¯|x3 =0 = Ψ(x ∂x3 Z ∇T¯ =e−αx3

x3

¯ 1 , x2 , z)dz ∇Ψ(x

−εh

(4.63)

¯ 1 , x2 , −εh)∇h − (x3 − + εe−αx3 Ψ(x

1 )∇θ1 α

+ x23 (x3 + εh)∇θ2 + εx23 θ2 ∇h, (4.64) ¯ 1 , x2 , −εh)|∇h|2 ε∇T¯ · ∇h|x3 =−εh = ε2 eαεh Ψ(x 1 +ε(εh + )∇θ1 · ∇h + ε4 h2 θ2 |∇h|2 , α ¯ ∂T ¯ 1 , x2 , −εh) − θ1 (x1 , x2 ) + ε2 h2 θ2 , |x =−εh = eαεh Ψ(x ∂x3 3

∂ T¯ ¯ 1 , x2 , −εh)(1 + ε2 |∇h|2 ) + ε∇T¯ · ∇h|x3 =−εh = eαεh Ψ(x ∂x3

+ ε2 h2 θ2 (1 + ε2 |∇h|2 ) − θ1 + ε(εh +

1 )∇θ1 · ∇h. α

Hence with θ1 , θ2 as in (4.59), (4.60), we have ∂ T¯ + ε∇T¯ · ∇h = 0 ∂x3

on

Γb .

Now we use the assumption (4.54) and prove that

(4.65)

∇θ1 · nΓi = 0, ∇θ2 · nΓi = 0

on

∂Γi ,

which implies that ∇T¯ · nΓi = 0 on ∂Γi . First, by working in local coordinates (s, t) where s is the coordinate in the normal direction of ∂Γi and t the coordinate in the tangential direction, the condition ∇h · nΓi = 0 on Γi implies (since ∂Γi is smooth) (4.66)

∂h =0 ∂s

and

∂h =0 ∂s∂t

on

∂Γi .

108


Therefore

µ ¶ ∂ ∂h 2 ∂ ∂h 2 2 |∇h| = | | +| | ∂s ∂s ∂s ∂t µ ¶ 2 ∂h ∂ h ∂h ∂ 2 h =2 + = 0. ∂s ∂s2 ∂t ∂s∂t

(4.67)

Now ¯ 1 , x2 , −εh)(1 + ε2 |∇h|2 )∇h · nΓ ∇θ1 · nΓi = αεeαεh Ψ(x i αεh 2 2 ¯ + e (1 + ε |∇h| )∇Ψ · nΓ i

(4.68)

¯ ∂Ψ (x1 , x2 , −εh)(1 + ε2 |∇h|2 )∇h · nΓi − εeαεh ∂x3 αεh ¯ + e Ψ(x1 , x2 , −εh)∇(|∇h|2 ) · nΓi

¯ · nΓ = 0 and ∇(|∇h|2 ) · nΓ = 0 on ∂Γi , we have and since ∇h · nΓi = 0, ∇Ψ i i ∇θ1 · nΓi = 0 on ∂Γi . Next we check that ∇θ2 · nΓi = 0 on ∂Γi . Here we only need to show that ∇(∇θ1 · ∇h) · nΓi = 0 on ∂Γi . Again, this can be done by working in local coordinates. We have (4.69)

∇θ1 · ∇h =

∂θ1 ∂h ∂θ1 ∂h + ∂s ∂s ∂t ∂t

and therefore ∂ ∂θ1 ∂ 2 h ∂ 2 θ1 ∂h + (∇θ1 · ∇h) = ∂nΓi ∂s ∂s2 ∂s2 ∂s (4.70) ∂ 2 θ1 ∂h ∂θ1 ∂ 2 h + + , ∂t∂s ∂t ∂t ∂s∂t but since ∂h/∂s = 0 and ∂θ1 /∂s = 0 we have ∂ 2 h/∂s∂t = 0 and ∂ 2 θ1 /∂s∂t = 0. Thus ∇(∇θ1 · ∇h) · nΓi =

(4.71)

∇(∇θ1 · ∇h) · nΓi = 0 on ∂Γi .

Finally since θ2 is the product of functions each of which has normal derivative to ∂Γi vanishing on ∂Γi , we have (4.72)

∇θ2 · nΓi = 0

on

∂Γi .


109

This concludes the verification of T¯ satisfying the boundary conditions. We now use estimates on solutions of the heat equation and the explicit expression of T¯ to establish the inequality (4.56) of the lemma. Here we fully rely on the estimates ¯ provided by Lemmas 4.2 and 4.3. for Ψ With ∂ T¯/∂x3 and ∇T¯ given by (4.61) and (4.63), we write ∂ 2 T¯ = α2 e−αx3 ∂x23

Z

x3

¯ 1 , x2 , z)dz Ψ(x

−εh

¯ ∂Ψ (x1 , x2 , x3 ) ∂x3 + 2(x3 + εh)θ2 (x1 , x2 ) + 4x3 θ2 (x1 , x2 ),

(4.73)

¯ 1 , x2 , x3 ) + e−αx3 − 2αe−αx3 Ψ(x

and, for k = 1, 2, Z x3 ¯ ∂ 2 T¯ ∂Ψ −αx3 ¯ 1 , x2 , −εh) ∂h = − αe (x1 , x2 , z)dz − αεe−αx3 Ψ(x ∂xk ∂x3 ∂xk −εh ∂xk (4.74) ¯ ∂θ2 ∂θ1 ∂h ∂θ2 ∂Ψ − + 2εx3 θ2 + 2x3 (x3 + εh) + x23 ; + e−αx3 ∂xk ∂xk ∂xk ∂xk ∂xk finally for k, ` = 1, 2 ∂ 2 T¯ =e−αx3 ∂xk ∂x`

Z

x3 −εh(x1 ,x2 )

¯ ∂Ψ ∂h (x1 , x2 , −εh(x1 , x2 )) ∂x` ∂xk ¯ ∂Ψ ∂h + εe−αx3 (x1 , x2 , −εh(x1 , x2 )) ∂xk ∂x` ¯ ∂Ψ ∂h ∂h − ε2 e−αx3 (x1 , x2 , −εh) ∂x3 ∂xk ∂x` 2 ¯ 1 , x2 , −h) ∂ h + εe−αx3 Ψ(x ∂xk ∂x` 2 ∂ 2 θ2 1 ∂ θ1 − (x3 − ) + x23 (x3 + εh) α ∂xk ∂x` ∂xk ∂x` 2 ∂h ∂ h ∂θ 2 + εx23 + εx23 θ2 . ∂xk ∂x` ∂xk ∂x` + εe−αx3

(4.75)

¯ ∂2Ψ (x1 , x2 , z)dz ∂xk ∂x`

110


To estimate the L2 – norms of the second derivatives of T¯, we need to bound the L2 – norms of θ1 , θ2 and their derivatives, which we do in Lemma 4.5 below.

Using Lemma 4.5, we estimate as follows, the norm, in L2 (Mε ), of ∂ 2 T¯/∂xk ∂x` , k, ` = 1, 2, as given by (4.75). The first term in the right-hand-side of (4.75) is bounded by a constant times ¯ ¯ the norm of ∂ 2 Ψ/∂x k ∂n` in Qε = Γi × (−εh, 0); using (4.51) and (4.41), this term is bounded as in (4.56). We then use Lemma 4.3 to estimate the four subsequent terms, and the bounds are consistent with (4.56). The remaining terms involve θ1 , θ2 and their derivatives; the integration over Γi of these functions provide the bounds given by Lemma 4.5, and, for each of these terms there is a factor of Cεm , m ≥ 2, which is due to the integration in x3 ; the bound (4.56) follows. We proceed similarly for ∂ 2 T¯/∂x23 given by (4.75) and for ∂ 2 T¯/∂xk ∂x3 , k = 1, 2, given by (4.74). Lemma 4.5 follows. ¤ We now conclude the proof of Lemma 4.3 by proving Lemma 4.5. Lemma 4.5. The function θ1 and θ2 introduced in (4.59), (4.60), are bounded as follows:

(4.76)

(4.77)

|θ1 |L2 (Γi ) + |∇θ1 |L2 (Γi ) + ε1/2 |∇2 θ1 |L2 (Γi ) ≤ C||g||2H 1 (Γi ) ,

|θ2 |L2 (Γi ) + ε1/2 |∇θ2 |L2 (Γi ) + ε|∇2 θ2 |L2 (Γi ) ≤ Cε−1 ||g||2H 1 (Γi ) ,

where C is a constant independent of ε.

Proof of Lemma 4.5. The proof strongly relies on the definition (4.59), (4.60) of θ1 and θ2 , and on ¯ given by Lemmas 4.2 and 4.3. the estimates on Ψ ˜ 1 , x2 ) = Ψ(x ¯ 1 , x2 , −εh(x1 , x2 )), and observe that, pointwise: We write Ψ(x


(4.78)

111

¯ ˜ = ∇Ψ ¯ + εξ ∂ Ψ , ∇Ψ ∂x3 2¯ ¯ ¯ ˜ = ∇2 Ψ ¯ + εξ ∂∇Ψ + εξ ∂ Ψ + ε2 ξ ∂ Ψ , ∇2 Ψ ∂x3 ∂x3 ∂x23 2¯ ¯ ˜ = ∇3 Ψ ¯ + εξ ∂∇ Ψ + εξ ∂∇Ψ ∇3 Ψ ∂x3 ∂x3 2 2¯ ¯ ¯ ∂ ∇Ψ ∂ Ψ ∂Ψ + ε2 ξ + εξ 2 + εξ 2 ∂x3 ∂x3 ∂x3 ¯ ¯ ¯ ∂ 2Ψ ∂ 2Ψ ∂∇Ψ + ε3 ξ 3 + εξ + ε2 ξ 2 . ∂x3 ∂x3 ∂x3

Here the ξ are (different) continuous (scalar, vector or tensor) functions bounded ˜ and its derivatives are evaluated at (x1 , x2 ) ∈ on Γi independently of ε (ε ≤ 1); Ψ ¯ and its derivatives are evaluated at (x1 , x2 , −εh(x1 , x2 )). Γi , Ψ It follows then from Lemma 4.3 that

(4.79)

˜ L2 (Γ ) ≤ C, |∇Ψ| ˜ L2 (Γ ) ≤ C, |Ψ| i i 2˜ 1/2 ˜ L2 (Γ ) ≤ Cε−1 . |∇ Ψ|L2 (Γi ) ≤ Cε , |∇3 Ψ| i

Now, similarly, θ1 and its first, second and third derivatives are of the following form: ˜ θ1 = ξ Ψ,

˜ + ξ∇Ψ, ˜ ∇θ1 = ∇ξ · Ψ ˜ + 2∇ξ ⊗ ∇Ψ ˜ + ξ∇2 Ψ, ˜ ∇2 θ 1 = ∇2 ξ · Ψ ˜ + 3∇2 ξ ⊗ ∇ψ˜ + 3∇ξ ⊗ ∇2 Ψ ˜ + ξ∇3 Ψ, ˜ ∇3 θ 1 = ∇3 ξ · Ψ where ξ and its first, second and third derivatives are uniformly bounded on Γi (for ε ≤ 1); hence (4.76) using (4.78). To obtain (4.77), we observe that, with a different ξ, θ2 is of the form ε−1 ξ · ∇θ1 . Lemma 4.5 is proved. ¤ Step 2. Reduction to the case α = 0 (and g = 0). Let T˜ be the solution of

112


− ∆3 T˜ = f2 in Mε , ∂ T˜ + αT˜ = 0 on Γi , ∂x3 ∂ T˜ = 0 on Γb ∪ Γ` . ∂n

(4.80)

We first note that ∂ T˜ 2 |ε + α|T˜|2i ≤ |f2 |ε |T˜|ε . ∂x3 Also by a density argument and, since Z x3 ∂ T˜ |T˜(x1 , x2 , x3 )| ≤ |T˜(x1 , x2 , 0)| + | |dx03 ∂x 3 0 ÃZ !1/2 (4.82) p 0 ˜ ∂ T | ≤ |T˜(x1 , x2 , 0)| + εh |2 dx3 , ∂x 3 −εh(x1 ,x2 ) |∇T˜|2ε + |

(4.81)

and (4.83)

Z

˜ ¯ T˜|2 + 2εh| ¯ ∂ T |2 . |T˜(x1 , x2 , x3 )|2 dx1 dx2 dx3 ≤ 2εh| i ∂x3 ε Mε

We infer from (4.80) that |∇T˜|2ε + | (4.84)

p p ˜ ∂ T˜ 2 ¯ 2 |ε |T˜|i + 2εh|f ¯ 2 |ε | ∂ T |ε |ε + α|T˜|2i ≤ 2εh|f ∂x3 ∂x3 ¯ εh 1 ∂ T˜ 2 α ¯ 2 |2 . |f2 |2ε + | | + εh|f ≤ |T˜|2i + ε 2 2α 2 ∂x3 ε

Therefore (4.85)

1 ∂ T˜ 2 α ˜ 2 ¯ 2 |2 ( 1 + 1). |∇T˜|2ε + | |ε + |T |i ≤ εh|f ε 2 ∂x3 2 2α

Hence |T˜|2i ≤ Cε|f2 |2ε

(4.86) and

∂ T˜ 2 |ε ≤ Cε|f2 |2ε , ∂x3 where C is a constant independent of ε. Then by (4.82), |

(4.87)

|T˜|2ε ≤ Cε2 |f2 |2ε .


113

Next transform (4.79) into a homogeneous Neumann condition. Let T ∗ = η T˜ where ¸ · αx23 2 + αx3 (x3 + εh(x1 , x2 ))ϕ(x1 , x2 ) . (4.88) η(x1 , x2 , x3 ) = exp αx3 + 2εh(x1 , x2 ) with (4.89)

ϕ=

|∇h|2 . 2h2 (1 + ε2 |∇h|2 )

Noting that ∂T ∗ ∂η ˜ ∂ T˜ ∂T ∗ ∂η ˜ ∂ T˜ = , T +η = T +η , ∂x3 ∂x3 ∂x3 ∂n ∂n ∂n (4.90)

h i ∂η αx3 2 =η α+ + 2αx3 (x3 + εh)ϕ + αx3 ϕ , ∂x3 εh

and at x3 = 0, ∂η/∂x3 = αη and η = 1, which implies that ³ ´ ∂T ∗ /∂x3 = η ∂ T˜/∂x3 + αT˜ = 0. Furthermore at x3 = −εh(x1 , x2 ), we have (4.91)

∂η = αηε2 h2 (x1 , x2 )ϕ(x1 , x2 ). ∂x3

Now we compute ∇η, where ∇ = (∂/∂x1 , ∂/∂x2 ): · ¸ αx23 2 2 (4.92) ∇η = η − ∇h + εαx3 ϕ∇h + αx3 (x3 + εh)∇ϕ . 2εh2 £ ¤ Hence at x3 = −εh(x1 , x2 ), ∇η = η − 21 αεξ∇h + ε3 αh2 ξ∇h , and

(4.93)

· ¸ ∂η ∂η ε2 α 2 2 2 4 2 2 = + ε∇h∇η = η αε h ϕ − |∇h| + ε αh ϕ|∇h| ∂n ∂x3 2 ¸ · 1 2 2 2 2 2 = ε ηα h ϕ(1 + ε |∇h| ) − |∇h| = 0 2

That is ∂η/∂n|Γb = 0, and ∂T ∗ /∂n|Γb = 0. Assume now (4.54), that is (4.94)

∇h · nΓi = 0

on

∂Γi .

One can easily check that ∇ϕ · nΓi = 0 : using a system (s, t) of local coordinates on ∂Γi , with s normal to ∂Γi and t tangential, and using (4.66) and (4.67). Hence

114


(4.95)

∇h · nΓi = 0

and

∇ϕ · nΓi = 0

on

∂Γi ,

and we have immediately ∇η · nΓi = 0

on

∂Γi ,

and therefore ∂T ∗ = 0 on ∂Γi . ∂nΓi The conclusion of these computations and of Step 2 is summarized by the following ¯ i → R+ belong to C 4 (Γ ¯ i ) and Lemma 4.6. Assume that ∂Γi is of class C 2 , h : Γ ∇h · nΓi = 0 on ∂Γi . Let T˜ be a solution of (4.79) and T ∗ = η T˜, with · η = exp where

¸ αx23 2 αx3 + + αx3 (x3 + εh(x1 , x2 ))ϕ(x1 , x2 ) 2εh(x1 , x2 )

|∇h(x1 , x2 )|2 ϕ(x1 , x2 ) = 2 . 2h (x1 , x2 )(1 + ε2 |∇h(x1 , x2 )|2 )

Then −∆T ∗ = f ∗ , ∂T ∗ = 0 on Γi , ∂x3 ∂T ∗ = 0 on Γb ∪ Γ` , ∂n where f ∗ = ηf2 − 2∇3 η · ∇3 T˜ − T˜∆3 η, and |f ∗ |ε ≤ C0 |f2 |ε , with C0 independent on ε. Proof. It remains only to estimate the L2 -norm of f ∗ . First note that there exists a constant C0 independent of ε (depending on α and h) such that 1 ≤ η(x1 , x2 , x3 ) ≤ C0 C0

for

(x1 , x2 , x3 ) ∈ Mε ,


115

and using (4.90) and (4.92), there exists another constant still denoted C0 such that ∂η | |L∞ (Mε ) ≤ C0 and |∇η|L∞ (Mε ) ≤ C0 . ∂x3 Now we compute ∆η (∆ = ∂ 2 /∂x21 + ∂ 2 /∂x22 ) : ∆η = div (∇η) · 2 ¸ αx3 1 2 2 =η ∆( ) + εαx3 div (ϕ∇h) + αx3 div ((x3 + εh)∇ϕ) , 2ε h ¯ i ), we have and therefore since h ∈ C 4 (Γ |∆η|L∞ (Mε ) ≤ C0 , with C0 independent on ε. Finally, we compute ∂ 2 η/∂x23 : ∂ 2η ∂η αx3 = [α + + 2αx3 (x3 + εh)ϕ + αx23 ϕ] 2 ∂x3 ∂x3 εh hα i +η + 2α(x3 + εh)ϕ + 6αx3 ϕ . εh Therefore |

∂2η C | ∞ ≤ , 2 L (Mε ) ∂x3 ε

where C is independent on ε. Now |f ∗ |ε ≤ |η|L∞ |f2 |ε + |∇3 η|L∞ |∇3 T˜|ε + |∆η|L∞ |T˜|ε + | and since by (4.84) and (4.87) |∇T˜|ε ≤ C|f2 |ε and |T˜|ε ≤ Cε|f2 |ε , we have |f ∗ |ε ≤ C|f2 |ε , where C is independent of ε.

∂ 2η |L∞ |T˜|ε , ∂x23

116


4.4. Regularity of the Velocity. In this section, we study the H 2 -regularity of the velocity, solution of the GFDStokes problem:  ∂2v    ) + ∇p = fv in Mε , −(∆v +  2  ∂x  3 Z  0 (4.96) div v dz = 0 in Γi ,   −εh   ∂v   v = 0 on Γl ∪ Γb , + αv v = gv on Γi . ∂x3 The H 2 -regularity of problems similar to (4.96) are given in Ziane [43] where ε = 1 and gv = 0, and in Hu, Temam and Ziane [15] in the case of constant depth function and under a convexity condition of Mε . We study here the H 2 regularity of solutions to (4.96), and give the dependence on ε of the constant appearing in the Cattabriga-Solonnikov type inequality associated to the H 2 -regularity of solutions. By contrast with the articles quoted above, our analysis here will be carried out in the case where Mε is not necessarily convex, and with a varying bottom topography. This regularity result is discussed in Section 4.4.2, and we start in Section 4.4.1 with a discussion of the weak formulation of the GFD-Stokes problem (4.96). 4.4.1. Weak Formulation of the GFD-Stokes Problem. In this section 4.4.1 we drop the index ε which is irrelevant (ε = 1, Mε = M). For the weak formulation of (4.96) we consider the space Z 1

2

0

V = {v ∈ H (M) , div

v dz = 0 on Γi , v = 0 on Γl ∪ Γb }; −h

thanks to the Poincaré inequality, this space is Hilbert for the scalar product ((v, v˜)) =

2 X 3 Z X i=1 j=1

M

vi ∂vi ∂˜ dM. ∂xj ∂xj

To obtain the weak formulation, we multiply the first equation (4.96) by a test function v˜ ∈ V and integrate over Mε ; assuming regularity for v, p and v˜, the term involving p (independent of x3 ) disappears:


Z

Z

117

Z

∇p v˜ dM = p v˜ · n d(∂M) − p div v˜ dM (by Stokes formula) ∂M M Z Z Z 0 =− p div v˜ dM = − p( div v˜ dx3 ) dx1 dx2 M

Z

M

=−

Z

Γi

p (div Γi

−h

0

v˜ dx3 ) dx1 dx2 = 0 (by the properties of v˜). −h

We also have, with Stokes formula, and since v˜ vanishes on Γb ∪ Γl : Z

Z ∂ 2v ∂v − (∆v + 2 ) v˜ dM = − v˜dΓi + ((v, v˜)) ∂x3 M Γi ∂xj Z Z = αv v v˜ dΓi − gv v˜ dΓi + ((v, v˜)). Γi

Ti

Hence the weak formulation of (4.96): To find v ∈ V such that (4.97)

a(v, v˜) = `(˜ v ),

with

(4.98)

∀˜ v ∈ V, Z

a(v, v˜) = ((v, v˜)) + αv v v˜ dΓi , Γi Z `(˜ v ) = (fv , v˜)H + gv v˜ dΓi . Γi

Existence and uniqueness of a solution v ∈ V of (4.97) follow promptly from the Lax–Milgram theorem. More delicate is the question of showing that, conversely, v is, in some sense, solution of (4.96). The second equation (4.96) and v = 0 on Γ` ∪ Γb follow from the fact that v ∈ V ; hence we need to show that there exists a distribution p independent of x3 such that

(4.99)

−(∆v +

∂ 2v ) + ∇p = fv in M, ∂x23

and also that (4.100)

∂v + αv v = gv on Γi . ∂x3

118


For (4.99), consider a test function ϕ ∈ C0∞ (M) (C ∞ with compact support in M), and observe that v˜ = (˜ v1 , 0), v˜1 = ∂ϕ/∂x3 , belongs to V . Writing (4.97) with this v˜, we conclude that ∂ (∆3 v1 + fv1 ) = 0; ∂x3 in the same way we prove that ∂ (∆3 v2 + fv2 ) = 0, ∂x3

(4.101)

showing that each component of ∆v + fv is a distribution on Mε independent of x3 . Distributions independent of x3 Now we can identify a distribution G on Mε independent of x3 , with a distribution on Γi as follows: let θ be any C ∞ scalar function with compact support in (−h, 0), and such that Z

0

(4.102)

θ(z)dz = 1. −h

Then, if ϕ ∈ C0∞ (Γi ) is a C ∞ scalar function with a compact support in Γi , ϕ θ ∈ ˜ on Γi by setting C0∞ (Mε ), and we associate to G a distribution G ˜ ϕiΓ = hG, ϕ θiMε . hG, i

(4.103)

The right-hand side of (4.103) is independent of θ; indeed if θ1 and θ2 are two such functions then hG, ϕ θ1 i = hG, ϕ θ2 i because Z

0

(θ1 − θ2 )(z)dz = 0, so that θ0 (x3 ) = (−h, 0), and

R0 x3

−h

(θ1 − θ2 )(z)dz is a C ∞ function with compact support in

∂G ∂ (ϕ θ0 )iM = h , ϕ θ0 iMε = 0. ∂x3 ∂x3 ˜ as a distribution on Γi . It is then easy to see that (4.103) defines G hG, ϕ(θ1 − θ2 )iM = −hG,


119

˜ is a distribution on Γi , and let ϕ ∈ C ∞ (Mε ). It Now, conversely, assume that G 0 R0 ˜ a distribution is clear that ϕ˜ = −εh ϕdx3 belongs to C0∞ (Γi ) and we associate to G G on Mε by setting, ˜ ϕ˜ >Γ , < G, ϕ >Mε =< G, i

∀ϕ ∈ C0∞ (Mε ).

Introduction of p Thanks to the previous discussion, we can now consider ∆v+fv as a distribution on Γi . As in the theory of Navier-Stokes equations, consider now a vector function v ∗ ∈ V(Γi ), that is v ∗ is (two dimensional) C ∞ with compact support in Γi , and div v ∗ = 0. It is clear that v˜ = v ∗ θ belongs to V, where θ is a function as above (see (4.102)). Writing (4.97) with this v˜, we obtain ((v, v ∗ θ)) = (fv , v ∗ θ)H , (4.104)

< ∆v + fv , v ∗ θ >M = 0, < ∆v + fv , v ∗ >Γi = 0,

∀v ∗ ∈ V(Γi ).

The last equation, which is well known in the theory of Navier-Stokes equations (see e.g. Lions [19], Temam [36]), implies that there exists a distribution p on Γi such that ∆v + fv = ∇p in Γi (or M), and (4.99) is proven. A Trace Theorem The proof of (4.100) necessitates establishing first a trace theorem: we need to show that, for a function v in V , such that (4.99) holds, one can define the trace 1/2 1/2 of ∂v/∂x3 on Γi , as an element of (H00 (Γi ))0 , the dual of H00 (Γi ) (that is the 1/2 interpolate between H01 (Γi ) and L2 (Γi )). Observe first that the trace on Γi of any function Φ in H 1 (Mε ) which vanishes 1/2 on Γ` belongs to H00 (Γi ). Indeed by odd symmetry and truncation one can extend such a Φ as a function Φ∗ in H01 (Γi × R), vanishing for |x3 | sufficiently 1/2 large, and the trace of such a function on any plane x3 = c0 , belongs to H00 (Γi ). 1/2 Conversely if ϕ belongs to H00 (Γi ), there exists Φ∗ in H01 (Γi × R) such that the trace of Φ∗ on Γi is ϕ, the mapping ϕ → Φ∗ being linear continuous (lifting

120


operator). From the remark above we infer that the trace on Γi of a function in 1/2 V belongs to H00 (Γi ). 1/2 We then show that the traces on Γi of the functions of V are all in H00 (Γi )2 . 1/2 ˜ ∈ Indeed let ϕ ∈ H00 (Γi )2 ; using the previous lifting operator, there exists Φ ˜ ∈ H 1 (Γi × ˜ Γ = ϕ; by truncation we can assume that Φ H01 (Γi × R)2 , such that Φ| i ˜ vanishes on ∂Γi and at x3 = −h. Let (−h, 0))2 and Φ Z

0

˜ 3, Φdx

ξ = div −h 2

and observe that ξ ∈ L (Γi ) and Z

Z

Z ˜ dM = div Φ

ξ dΓi =

(4.105)

Qh

Γi

˜ · nh d(∂M) = 0, Φ ∂Qε

where nh is the horizontal component of the unit outward normal n on ∂M. Because of (4.105), we can solve in Qh = Γi × (−h, 0), the (usual) Stokes problem ∆Φ∗ + ∇π = 0 in Qh , 1 div3 Φ∗ = ξ in Qh , h ∗ Φ = 0 on ∂Qh ,

(4.106)

and Φ∗ ∈ H 1 (Qh )3 , π ∈ L2 (Qh ). Now Z

0

Z ∗

0

div3 Φ dx3 = div −h

−h

(φ∗1 , φ∗2 )dx3 = ξ,

˜ − (Φ∗ , Φ∗ ) extended by 0 in M\Qh and it is easy to see that the function Φ = Φ 1 2 belongs to V, and its trace on Γi is precisely ϕ. We can furthermore observe that with the construction above, the mapping ϕ 7→ Φ is linear continuous from 1/2 H00 (Γi ) into V. Finally, (4.100) follows promptly from (4.97), (4.99) and the following proposition. Proposition 4.1. Let v be a function in H 1 (M)2 which vanishes on Γb ∪ Γ` and assume that −∆v + ∇p ∈ L2 (M)2 , for some distribution p independent of x3 . 1/2 Then there exists γ1 v ∈ (H00 (Γi ))2 , such that


(4.107)

γ1 v =

121

∂v ¯ 2, |Γ if v ∈ C 2 (M) ∂x3 i

and γ1 v is defined by Z (4.108)

< γ1 v, ϕ >= ((v, Φ)) −

(−∆v + ∇p) Φ dM, M

1/2

where ϕ is arbitrary in H00 (Γi ) and Φ is any function of V such that Φ|Γi = ϕ. Proof. We first show that the right hand side X(Φ) of (4.108) depends on ϕ and not on Φ. Indeed let Φ1 and Φ2 be two functions of V such that Φ1 |Γi = Φ2 |Γi = ϕ. R0 Then Φ∗ = Φ1 − Φ2 belongs to H01 (M)2 and div −h Φ∗ dx3 = 0. It was shown in Lions, Temam, and Wang [21]that Φ∗ is limit in H01 (M)2 of C ∞ functions Φ∗n R0 with compact support in M such that div −h Φ∗n dx3 = 0. It is easy to see that X(Φ∗n ) = 0 and, by continuity, X(Φ∗ ) = 0, i.e. X(Φ1 ) = X(Φ2 ). After this observation we choose Φ as constructed above, so that the mapping 1/2 ϕ → Φ is linear continuous from H00 (Γi )2 into V. It then appears that the right1/2 hand side of (4.108) is a linear form continuous on H00 (Γi )2 , and thus γ1 v is 1/2 defined and belongs to (H00 (Γi ))2 . Finally (4.107) follows from the fact that (4.108) is easy when v and Φ are smooth and γ1 v is replaced by ∂v/∂x3 |Γi . ¤ Remark 4.2. We have shown the complete equivalence of (4.96) with its variational formulation (4.97).

4.4.2. H 2 Regularity for the GFD-Stokes problem. For convenience, we use hereafter the classical notation L2 , H1 , etc., for spaces of vector functions with components in L2 , H 1 , etc. The main result of this section is ¯ i ), h ≥ h > 0 and Theorem 4.4. Assume that h is a positive function in C 4 (Γ that fv ∈ L2 (Mε ) and gv ∈ H10 (Γi ). Let (v, p) ∈ H1 (Mε ) × L2 (Γi ) be a weak solution of (4.96). Then (4.109)

(v, p) ∈ H2 (Mε ) × H 1 (Mε ).

122


Moreover the following inequality holds: (4.110)

£ ¤ |v|2H2 (Mε ) + ε|p|2H 1 (Γi ) ≤ C |fv |2ε + |gv |2L2 (Γi ) + ε|∇gv |2L2 (Γi )

The approach to the proof of the H 2 – regularity in Theorem 4.4 is the same as in the articles of Ziane [43, 44], and Hu, Temam and Ziane [15]; it is based on the following observation: the weak solution of (4.96) satisfies p ∈ L2 (Γi ); assume ∂v ¯¯ ∂v ¯¯ 2 ∈ L2 (Γb ), further that the solution v of (4.96) satisfies ∈ L (Γ ), i Γ Γ i b ∂x3 ∂x3 ∂v ¯¯ 2 ∈ L (Γb ), k = 1, 2, then an integration of the first equation in (4.96) and ∂xk Γb with respect to x3 over (−εh, 0) yields a 2D- Stokes problem on the smooth domain Γi with a homogeneous boundary condition. By the classical regularity theory of the 2D-Stokes problem in smooth domains, see for instance Ghidaglia [9], Temam [36], Constantin and Foias [7], p belongs to H 1 (Γi ). Then, by moving the pressure term to the right hand side, problem (4.96) reduces to an elliptic problem of the type studied in Section 4.2, and the H 2 regularity of v follows. The estimates on the L2 norms of the second derivatives are then obtained using the trace theorem and the estimates in Section 4.2. ∂v ¯¯ ∂v ¯¯ 2 We start this proof by showing that ∈ L (Γ ), ∈ L2 (Γb ), and i ∂x3 Γi ∂x3 Γb ∂v ¯¯ ∈ L2 (Γi ), k = 1, 2. The following lemma is just a rewriting of Theorem ∂xk Γi 4.2. ¯ i ). For f ∈ L2 (Mε ) and g ∈ H 1 (Γi ), there Lemma 4.7. Assume that h ∈ C 2 (Γ 0 2 exists a unique Ψ ∈ H (Mε ) solution of   −43 Ψ = f in Mε ,    ∂Ψ (4.111) + αΨ = g on Γi ,  ∂x3   Ψ = 0 on Γ ∪ Γ . b l Furthermore, there exists a constant C(h, α) depending only on α and h (and Γi ), such that 3 X ¯ ∂ 2 Ψ ¯2 ¯ ¯ ≤ C(h, α)[|f |2ε + |g|2i + |∇g|2i ]. ε ∂x k ∂xj k,j=1 As we said, this lemma is just a rewriting of Theorem 4.2. We will also need the following intermediate result simply obtained by interpolation between H 1 and H 2 .


123

Lemma 4.8. Under the assumptions of Lemma 4.7, and with g = 0 : if f ∈ H −1/2+δ (Mε ) with − 21 < δ < 12 , δ 6= 0, then Ψ ∈ H 3/2+δ (Mε ). Before we start the proof of the main result of this section (Theorem 4.1), we first prove ¯ i ). For f ∈ L2 (Mε ), g ∈ H 1 (Γi ), and Lemma 4.9. Assume that h ∈ C 3 (Γ 0 1+γ 1 1 ψi ∈ H0 (Γb ), − 2 < γ < 2 , γ 6= 0, there exists a unique Ψi ∈ H 3/2+γ (Mε ) solution of  −43 Ψi = f in Mε ,       ∂Ψi + α Ψ = g − α ψ on Γ , v i v i i (4.112) ∂x3    Ψi = −ψi on Γb    Ψi = 0 on Γl . Proof Using Lemma 4.7, we reduce the problem to the case f = 0 and g = 0, by replacing Ψi with Ψi − Ψ, where Ψ is the function constructed in Lemma 4.7. Thus, without loss of generality, we will assume from now on that f = 0 and g = 0. Our next step is to construct a function v˜p which agrees with Ψi on ∂Mε . This will be done by first constructing an auxiliary function vp on a straight cylinder and then the explicit expression of v˜p will be given. Let Qε be the cylinder Qε = Γi × (−ε, 0), and let vp be the unique solution of

(4.113)

 ∆3 vp = 0, in Qε ,    v = 0 on ∂Γ × (−ε, 0), p i  vp = −ψi on Γi × {−ε},    vp = εhαv ψi on Γi × {0}.

We will show that vp ∈ H 3/2+γ (Qε ) for all − 12 < γ < 12 , γ 6= 0. Then, setting x3 x3 (4.114) v˜p (x1 , x2 , x3 ) = − vp (x1 , x2 , ) for (x1 , x2 , x3 ) ∈ Mε , εh(x1 , x2 ) h(x1 , x2 ) it is obvious that v˜p ∈ H 3/2+γ (Mε ), v˜p (x1 , x2 , −εh(x1 , x2 )) = −ψi (x1 , x2 ), and ∂˜ vp + αv v˜p = −αv ψi on Γi . Therefore setting V˜ = Ψi − v˜p , we have ∂x3   ∆3 V˜ = −∆3 v˜p ∈ H −1/2+γ (Mε ),   ˜ V = 0 on Γl ∪ Γb , (4.115)  ∂ V˜   + αv V˜ = 0 on Γi .  ∂x3

124


Hence, thanks to Lemmas 4.7 and 4.8, we see that V˜ and thus Ψi are in H 3/2+γ (Mε ) for − 21 < γ < 12 , γ 6= 0. To complete the proof of Lemma 4.9, it remains only to show that vp ∈ 3/2+γ ˆ ε be any C 2 H (Qε ) for all − 12 < γ < 12 , γ 6= 0. To this end, let Q ˆ ε . Since ψi (resp. hαv ψi ) is in domain containing Qε such that Γi × {−ε, 0} ⊂ ∂ Q ˆ ε) H01+γ (Γi × {−ε}) (resp H01+γ (Γi × {0})), we can define a function Vi ∈ H 1 (∂ Q by setting Vi = −ψi on Γi × {−ε}, Vi = εhαv ψi on Γi × {0}, and Vi = 0 on ˆ ε \Γi × {−ε, 0}. Now let Vp be the unique solution of ∆3 Vp = 0 in Q ˆ ε and ∂Q ˆ ε . Since ∂ Q ˆ ε is of class C 2 , the classical regularity results for Vp = Vi on ∂ Q ˆ ε ) for elliptic problems (see e.g. Lions and Magenes [20]) yield Vp ∈ H 3/2+γ (Q 1 1 − 2 < γ < 2 , γ 6= 0. Now let V˜i be the trace of Vp on ∂Γi × (−ε, 0). It is easy to see that V˜i ∈ H01+γ (∂Γi × (−ε, 0)). Let V˜p = Vp − vp , we have ∆3 V˜p = 0 in Qε , (4.116)

V˜p = 0 on Γi × {−ε, 0}, V˜p = V˜i on ∂Γi × (−ε, 0).

Using a reflection argument around x3 = 0 (resp. x3 = −ε) by extending V˜i in a “symmetrically” odd function defined on ∂Γi × (−ε, ε) (resp. ∂Γi × (−2ε, 0)) , and using the classical local regularity theory (see e.g. Lions and Magenes [20]), we conclude that V˜p ∈ H 3/2+γ (Qε ) for − 12 < γ < 21 , γ 6= 0. Therefore, since Vp ∈ H 3/2+γ (Qε ), we have vp = Vp − V˜p ∈ H 3/2+γ (Qε ). ¯ i ), with h ≥ h1 > 0 on Γ ¯ i . Let (v, p) be the Lemma 4.10. Assume that h ∈ C 3 (Γ weak solution of (4.96), then v ∈ H2−δ (Mε ) for 0 < δ < 12 and consequently ¯ ¯ ∂v ¯¯ 2 ¯ and ∂v ¯ ∈ L2 (Γb ). (4.117) ∈ L (Γ ), ∇v i Γb ∂x3 Γi ∂x3 Γb Proof By integration by parts, we infer from (2.95) that (4.118)

|∇3 v|2ε + αv |v|2i = (fv , v)ε + (gv , v)i ,

and therefore the existence and uniqueness of the weak solution (v, p) of (2.95) follows from the Lax-Milgram theorem. Hence ∇p belongs to H −1 (Mε ) and thus to H −1 (Γi ) since p is independent of x3 (see Section 4.4.1). Let vi ∈ H01 (Γi ) be the unique solution of the 2D Dirichlet problem on Γi : ( ∆vi = ∇p in Γi , (4.119) vi = 0 on ∂Γi .


Let v˜ = v − vi , then v˜ satisfies  ∆3 v˜ = fv in Mε ,     v˜ = 0 on Γl , (4.120) v˜ = −vi on Γb ,    v   ∂˜ + αv v˜ = gv − αv vi ∂x3

125

on Γi .

Thanks to Corollary ??, with g = gv , ψi = vi and γ = −δ for some 0 < δ < 21 , we have v˜ ∈ H 3/2−δ (Mε ). Hence, Z 0 1 (4.121) gi = − div v˜ dx3 ∈ H 1/2−δ (Γi ), . εh −εh Therefore, since div vi = gi , we rewrite the equation for vi in the form of a 2D Stokes problem:   −∆vi + ∇p = 0 in Γi , 1 (4.122) div vi = gi ∈ H 2 −δ (Γi ),  v = 0 on ∂Γ , i i and thanks to the classical regularity result for the non-homogeneous Stokes prob3 lem on Γi , (see e.g. Ghidaglia [9], Temam [36]), we have vi ∈ H 2 −δ (Γi )∩H01 (Γi ) = 3

−δ

H02 (Γi ). With this new information on the regularity of vi , we return to Problem (4.120) and using Lemma 4.9 with ψi = vi and γ = 12 − δ, 0 < γ < 2,1 , we conclude that v˜ ∈ H 2−δ (Mε ). Therefore gi , given by (4.121), belongs to H 1−δ (Γi ). This in turn implies by the classical regularity of the 2D-Stokes problem, that the solution vi of (4.122) is in H 2−δ (Γi ). Therefore the solution v˜ of (2.101) belongs to H 2.5−δ (Mε ) and v = v˜ + vi belongs to H 2−δ (Mε ). Consequently the trace 1 on Γi of the normal derivative ∂v/∂x3 |Γi belongs to H 2 −δ (Γi ), hence to L2 (Γi ), taking e.g. δ = 1/4. Similarly the traces on Γb of v and its normal derivative ∂v/∂n belong to H 3/2−δ (Γb ) and H 1/2−δ (Γb ), from which we infer that ∇v|Γb and ∂v/∂x3 |Γb are in H 1/2−δ (Γb ) and therefore in L2 (Γb ). The proof of the lemma is now complete. Proof of Theorem 4.4. The proof is divided into two steps. In Step 1, we prove the H 2 – regularity of solutions, i.e., v ∈ H 2 (Mε ) and p ∈ H 1 (Γi ). Then, in Step 2, we establish the Cattabriga-Solonnikov type inequality on the solutions, i.e., establish the bounds (2.97) on the L2 -norms of the second derivatives of v and

126


the H 1 -norm on the pressure, in particular we establish their (non-) dependence on ε. R0 Step 1: The H 2 -regularity of solutions. Let v¯ = −εh v dz; we have Z 0 ∂2 ∂2 (4.123) , x , x ) = v ¯ (x v(x1 , x2 , z) dz + Ik (v), 1 2 3 2 ∂x2k −εh ∂xk ∂h ∂v (x1 , x2 , −εh(x1 , x2 )) ∂xk ∂x3 (4.124) ∂h 2 −ε2 ( ) v(x1 , x2 , −εh(x1 , x2 )), k = 1, 2. ∂xk Therefore, by integrating the first equation in (4.96) with respect to x3 we obtain the 2D Stokes problem: ( −∆¯ v + ∇(εhp) = f¯ in Γi , (4.125) div v¯ = 0 in Γi , v = 0 on ∂Γi , Ik (v) = 2ε

where (4.126)

¯ ¯ ¯ ¯ ∂v ∂v ¯ ¯ fv dz + f¯ = − ∂x3 ¯x3 =0 ∂x3 ¯x3 =−εh −εh Z

0

+ I1 (v) + I2 (v) + εp∇h. Thanks to Lemma 4.10, each term on the right hand side of (4.126) is in L (Γi ), which implies f¯ ∈ L2 (Γi ): this is stated in (4.117) for ∂v/∂x3 |Γi and ∂v/∂x3 |Γb ; similarly each term in I1 and I2 belongs to L2 (Γb ) (and thus L2 (Γi )) because v ∈ H 2−δ (Mε ), 0 < δ < 1/2; finally for p we recall from (4.119) that ∇p ∈ H −1 (Γi ), and thus p ∈ L2 (Γi ). Therefore from the classical regularity theory of the 2D Stokes problem, we conclude that ∇(hp) ∈ L2 (Γi ), and then ∇p ∈ L2 (Γi ). We return to Problem (4.96), and move the gradient of the pressure to the right hand side and obtain, thanks to Lemma 4.7, v ∈ H 2 (Mε ) and 2

(4.127)

3 X ¯ ∂ 2 v ¯2 ¯ ≤ C(h, αv )[|fv |2ε + |gv |2i + |∇gv |2i ] + C(h, α)ε|∇p|2i . ¯ ∂xk ∂xj ε k,j=1

Note that we have the pressure term on the right hand side of (4.127). Removing that term is done in the second step below. Step 2: The Cattabriga-Solonnikov type inequality. Our aim is now to bound |∇p|i properly and to derive (4.110) from (4.127). First we homogenize the


127

boundary condition in (4.96). Let vl = (Ψ1 , Ψ2 ) where Ψ1 and Ψ2 are constructed using Lemma 4.7, i.e.,  −4 Ψ = fv,k in Mε , k = 1, 2,    ∂Ψ3 k k + αv Ψk = gv,k on Γi , k = 1, 2,  ∂x3   Ψk = 0 on Γb ∪ Γl , k = 1, 2, where fv = (fv,1 , fv,2 ) gv = (gv,1 , gv,2 ). Thanks to Lemma 4.7, we have (4.128)

3 X ¯ ∂ 2 vl ¯2 ¯ ¯ ≤ C(h, αv )(|fv |2ε + |gv |2i + |∇gv |2i ). ε ∂x ∂x k j k,j=1

Setting v ∗ = v − vl , it suffices to establish (4.128) with vl replaced by v ∗ . We have:  ∂ 2v∗  ∗  −(∆v + ) + ∇p = 0 in Mε ,    ∂x23  Z  0   div v ∗ dz = g ∗ on Γi , (4.129) −εh   ∗  v =   ∗ 0 on Γl ∪ Γb ,   ∂v   + αv v ∗ = 0 on Γi , ∂x3 where Z 0

g ∗ = −div

vl dx3 . −εh

Note that inequality (4.128) together with the Cauchy-Schwarz inequality imply (4.130)

||g ∗ ||2H 1 (Γi ) ≤ C(h, αv )ε[|f1 |2ε + |gv |2i + |∇gv |2i ].

Define Z ∗

0

V =

v ∗ dx3 ;

−εh ∗

V is the solution of the 2D-Stokes problem  ∗ ∗  −∆V + ∇(εp) = F (4.131) div V ∗ = g ∗  V ∗ = 0 on ∂Γ , i where F∗ =

in Γi ,

∂v ∗ ∂v ∗ |x3 =0 − |x3 =−εh + I1 (v ∗ ) + I2 (v ∗ ), ∂x3 ∂x3

128


with I1 and I2 as in (4.124). Hence £ ∂v ∗ 2 ¤ ∂v ∗ 2 |F ∗ |2L2 (Γi ) ≤ C(h) | |L2 (Γi ) + | |L2 (Γb ) . ∂x3 ∂x3

(4.132)

∂v ∗ ∂h ∂v ∗ =ε on Γb and, by the Poincaré ∂xk ∂xk ∂x3 ∂v ∗ 2 inequality and the boundary condition satisfied by v ∗ on Γi , we have | | 2 ≤ ∂x3 L (Γi ) ∗ ¯ ∂v |2 . Furthermore, we can write 2αv2 εh| ∂x3 ε ¯ ∗ ¯2 ¯ ∗¯ ¯ 2 ∗¯ ¯ ∗ ¯2 ¯ ∂v ¯ ¯ ∂v ¯ ¯ ∂ v ¯ ¯ ∂v ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ≤¯ + 2 ¯¯ 2 ¯ ¯ ¯ ¯ ¯ ∂x3 ¯ 2 ∂x ∂x ∂x 3 L2 (Γi ) 3 ε 3 ε L (Γb ) (4.133) ¯ ¯ 2 ∗ ∗ ¯ 2 ∗¯ ¯ ∂v |2 + θε ¯ ∂ v ¯ + Cε | ∂v |2 . ≤ 2αv2 εh| ε ¯ ∂x2 ¯ ∂x3 θ ∂x3 ε 3 ε Now, since v ∗ = 0 on Γb , we have

where θ is a positive constant independent of ε, that will be chosen below. Therefore ¯ 2 ∗ ¯2 ¯∂ v ¯ ∂v ∗ 2 ∗ 2 (4.134) |F |L2 (Γi ) ≤ C| |ε + C(h)θε ¯¯ 2 ¯¯ . ∂xk ∂x3 ε We estimate the H 1 -norm of v ∗ , using v ∗ = v − vl and the H 1 -estimates of v and vl . Therefore we can easily obtain ¯ 2 ∗ ¯2 ¯∂ v ¯ £ ¤ (4.135) |F ∗ |2L2 (Γi ) ≤ C(h)ε |f1 |2ε + |gv |2i + |∇gv |2i + C(h)θε ¯¯ 2 ¯¯ . ∂x3 ε Now using the Cattabriga-Solonnikov inequality for the 2D Stokes problem (4.131), there exists a constant C independent of ε such that |V ∗ |2H 2 (Γi ) + ε2 |∇(hp)|2L2 (Γi ) ≤ C|F ∗ |2L2 (Γi ) .

(4.136)

From this we obtain (4.137)

ε

2

|∇(hp)|2L2 (Γi )

¯ 2 ∗ ¯2 ¯∂ v ¯ £ 2 ¤ 2 2 ≤ C(h, θ)ε |fv |ε + |gv |i + |∇gv |i + C(h)θε ¯¯ 2 ¯¯ , ∂x3 ε

ε2 |∇p|2L2 (Γi ) ≤ C(h, θ)ε2 |p|2L2 (Γi ) + £ ¤ ∂ 2v∗ C(h, θ)ε |fv |2ε + |gv |2i + |∇gv |2i + C|h|θε| 2 |2ε . ∂x3 From (4.96) and the weak formulation (4.98) of (4.96), we see that


129

|p|L2 (Γi )/R ≤ C|∇p|H −1 (Γi ) ≤ C||v||H 1 (Mε ) ≤ C|fv |ε , so that we actually have the same type of estimate (4.137) for ∇p as for ∇(hp). ∂v ∗ + α∗ v = 0 on Γi , we Finally, since ∆3 v ∗ = ∇p, in Mε , v ∗ = 0 on Γb ∪ Γl and ∂x3 have thanks to Lemma 4.7, 3 X ¯ ∂ 2 v ∗ ¯2 ¯ ¯ ≤ C(h, αv )ε|∇p|2 2 L (Γi ) ε ∂x k ∂xj k,j=1

¯ 2 ∗ ¯2 ¯∂ v ¯ £ 2 ¤ 2 2 ≤ C(h, αv ) |f1 |ε + |gv |i + |∇gv |i + C(h, αv )θ ¯¯ 2 ¯¯ , ∂x3 ε

and therefore for θ small enough, so that C(h, αv )θ ≤ 21 , we conclude that 3 X ¯ ∂ 2 v ∗ ¯2 £ 2 ¤ 2 2 ¯ ¯ ≤ C(h, α)ε|∇p|2 2 ≤ C(h, α) |f | + |g | + |∇g | 1 v v ε i i L (Γ ) i ∂xk ∂xj ε k,j=1

The proof of Theorem4.4 is now complete.

¤

By interpolation, it is easy to derive from Theorem 4.4 the following result : ¯ i ). Let (v, p) ∈ Theorem 4.5. Assume that h is a positive function in C 3 (Γ H1 (Mε ) × L2 (Γi ) be a weak solution of  ∂2v   − (∆v + ) + ∇p = fv in Mε ,   ∂x23    Z 0    v dz = 0 on Γi , div (4.138) −εh    v=0 on Γ` ∪ Γb ,        ∂v + αv v = gv on Γi . ∂x3 Then, if fv ∈ L2 (Mε ) and gv ∈ Hs (Γi ), 0 ≤ s ≤ 1, (4.139)

(v, p) ∈ Hs+1 (Mε ) × H s (Mε ).

Moreover the following inequality holds: (4.140)

¤ £ |v|2H1+s (Mε ) + ε|p|2H s (Γi ) ≤ C0 |fv |2ε + ε1−s ||gv ||2H s (Γi ) ,

130


where C0 is a constant depending on the data but not on ε. 4.5. Regularity of the coupled system. In this subsection we prove the H 2 – regularity of the solution of a coupled system of equations corresponding to the linear part of the primitive equations of the coupled atmosphere–ocean. We will concentrate on the velocity part; the temperature and salinity parts follow in the same manner. The unknown is v = (v a , v s ), with v a , v s corresponding to the horizontal velocities in the air and in the ocean15. These functions satisfy the following equations and boundary conditions :

(4.141)

and

(4.142)

 ∂ 2va  a   − ∆v − in Maε , + ∇pa = fva  2  ∂x3    Z L     v a (x1 , x2 , z) dz = 0 (x1 , x2 ) ∈ R2 , div    0 va = 0 on Γu ∪ Γa` ,     ∂v a   + αv (v a − v s ) = gv on Γi ,   ∂x  3    ∂v a   + αv v a = gv on Γe ,  ∂x3  ∂ 2vs s   − ∆v − + ∇ps = fvs  2  ∂x  3   Z 0    div v s dz = 0 −εh   vs = 0      s    − ∂v + αva (v a − v s ) = gv ∂x3

in Msε , in Γi , on Γs` ∪ Γb , on Γi .

The domain Msε is the domain occupied by the ocean while Maε is the domain occupied by the atmosphere and Mε = Maε ∪ Msε : © ª Msε = (x1 , x2 , x3 ) ∈ R3 ; (x1 , x2 ) ∈ Γi , −εh(x1 , x2 ) < x3 < 0 , © ª Maε = (x1 , x2 , x3 ) ∈ R3 ; (x1 , x2 ) ∈ Γ, 0 < x3 < εL . 15We

recall that we use the superscript s as sea, instead of o as ocean which can be confused with a zero.


131

Here, Γ, which is a bounded domain in the plane x3 = 0, is the lower boundary of the atmosphere; it consists of the interface Γi with the ocean and Γe , the interface with the earth, Γ = Γi ∪ Γe (Γi ∩ Γe = ∅); (see Section 2.5); furthermore, and as in Section 2.5: Γb = {(x0 , x3 ); x0 ∈ Γi , x3 = −εh(x0 )}, Γa` = {(x0 , x3 ), x0 ∈ ∂Γ, 0 < x3 < εL}, Γs` = {(x0 , x3 ); x0 ∈ ∂Γi , −εh(x0 ) < x3 < 0}, Γu = {(x0 , x3 ); x0 ∈ Γi , x3 = εL}, Γe = Γ\Γi , Γ and Γi as above. The coefficient αv is a positive number, and gv is a function defined on Γ. Problem (4.141)–(4.142) is the stationary linearized form of the primitive equations of the coupled system atmosphere-ocean. Besides its intrinsic interest, the study of this problem is needed for the study of the full nonlinear (stationary or time dependent) coupled atmosphere–ocean system. 4.5.1. Weak formulation of the coupled system. As in Section 4.4.1 we start with the weak formulation of (4.141)-(4.142). In this section we drop the index ε which is irrelevant (ε = 1). We are given fv in L2ε (M), fv in L2 (Msε ), and gv in H1/2 (Γ). For the weak formulation of (4.141)-(4.142) we consider the space Z 0 a s 1 a 1 S V = {v = (v , v ) ∈ H (M ) × H (M ), div v s dz = 0, −h a

v = 0 on Γu ∪

Γa` ,

s

v = 0 on Γb ∪

ΓS` }.

Here v a = v|Ma and v S = v|MS ; note that the traces of v a and v s on Γi are not necessarily equal, as explained in Remark 2.7, iii). We set, with obvious notations: ((v, v˜)) = ((v a , vã ))a + ((v s , v˜s ))s , 2 X 3 Z X vi ∂vi ∂˜ dM; = ∂xj ∂xj i=1 j=1 M because of the Poincaré inequality, kvk = ((v, v))1/2 is a Hilbert norm on V .

132


To obtain the weak formulation, we consider a test function v˜ = (˜ v a , v˜s ) ∈ V ; we multiply the first equation (4.141) by v˜s . We integrate over Ma and Ms respectively and add the resulting equations; we proceed exactly as in Section 4.4.1, using the boundary condition in (4.141), (4.142) and we arrive at the following: To find v ∈ V such that a(v, v˜) = `(˜ v ), ∀˜ v ∈ V,

(4.143) with

Z

Z a

a(v, v˜) = ((v, v˜)) + Γi

s

a

αv (v − v )(˜ v − v˜ )dΓi +

Z Ma

fva vã dMa +

αv v a vã dΓe , Γe

Z

Z

(4.144) `(˜ v) =

s

Ms

fva v˜S dMS +

Z a

S

gv vã dΓe .

gv (˜ v − v˜ )dΓi + Γi

Γe

The existence and uniqueness of a solution v ∈ V of (4.143) is elementary, and follows from the Lax–Milgram theorem. The more delicate question of showing that v = v a , v S actually satisfies all the equations (4.141) and (4.142) is handled as follows : we find pa and pS such that the first equation (4.141) and (4.142) are valid exactly as we did in Section 4.4.1, for the ocean and the atmosphere. Using also Proposition 4.1 for the ocean and the atmosphere, we obtain the boundary conditions on Γi and Γe ; the other equations and boundary conditions follow from v ∈ V . Remark 4.3. Setting v˜ = v in (4.143), we find

(4.145)

|∇v a |2L2 (Maε )4 + |∇v S |2L2 (MSε )4 Z Z a S 2 + αv |v − v | dΓi + αv |v a |2 dΓe Γe ZΓi = fva v a dMaε a Z Z ZM a S S S S gv (v − v )dΓi + gv v a dΓe . + fv v dMε + MS

Γi

Γe

4.5.2. H 2 Regularity for the Coupled System. Having established the complete equivalence of (4.141)-(4.142) with (4.143), we now want to show that the solution of this system possesses the H 2 -regularity, namely


(4.146)

133

(v a , pa ) ∈ H2 (Ma ) × H 1 (Ma ) and (v s , ps ) ∈ H2 (Ms ) × H 1 (Ms ),

whenever f a ∈ L2 (Ma ), f s ∈ L2 (Ms ), and gv ∈ H1/2 (Γ). More precisely, we will prove the following : ¯ i ). Let (v a , pa ) ∈ Theorem 4.6. Assume that h is a positive function in C 3 (Γ H1 (Maε ) × L2 (Γi ∪ Γe ), and (v s , ps ) ∈ H1 (Msε ) × L2 (Γi ) be a weak solution of (4.141)-(4.142) (or 4.143). If fva ∈ L2 (Maε ),fvs ∈ L2 (Msε )2 , and gv ∈ H1 (Γ), gv = 0 on ∂Γe , then (4.147)

(v a , pa ) ∈ H2 (Maε ) × H 1 (Γi ∪ Γe ) and (v s , ps ) ∈ H2 (Msε ) × H 1 (Γi ).

Moreover the following inequality holds: (4.148) £ ¤ |v a |2H2 (Maε ) + |v s |2H2 (Msε ) + ε|pa |2H 1 (Γi ) + ε|ps |2H 1 (Γi ) ≤ C0 |fva |2ε + |fvs |2ε + |∇gv |2L2 (Γ) . Proof. Since v a ∈ H1 (Maε ), and v s ∈ H1 (Msε ), v a |Γi and v s |Γi belong to H1/2−δ (Γi ), ∀δ, 0 < δ < 1/2, and there exists a constant C0 independent of ε such that i h (4.149) |v a |2H1/2−δ (Γi ) + |v s |2H1/2−δ (Γi ) ≤ C0 kv a k2H1 (Maε ) + kv s k2H1 (Msε ) . Furthermore (4.145) implies that the right-hand side of (4.149) can be bounded by an expression identical to the right-hand side of (4.147). The boundary conditions on Γi imply then that ∂v a ∂v s + αv v a and − + αv v s , ∂x3 ∂x3 belong to H1/2−δ (Γi ) and their norm in these spaces are bounded similarly. Therefore, by Theorem 4.5 applied separately to Maε and Msε , we conclude that (v a , pa ) ∈ H3/2−δ (Mεa ) × H 1/2−δ (Γ), (v s , ps ) ∈ H3/2−δ (Mεs ) × H 1/2−δ (Γi ), and (4.150)

|v a |2H3/2−δ (Maε ) + |v s |2H3/2−δ (Msε ) + ε|pa |2H 1/2−δ (Γi ) + ε|ps |2H 1/2−δ (Γi ) ≤ κ ˜,

where κ ˜ is the right-hand side of (4.140) with a possibly different constant C0 .

134


Using the Trace theorem again, we see that ∂v a ∂v s 2 + αv v a and − + αv v s belong to H1−δ 0 (Γi ) , ∀δ, ∂x3 ∂x3 and there exists a constant C0 independent of ε such that ´ ³ (4.151) kv a k2H1−δ (Γ) + kv s k2H1−δ (Γi ) ≤ C0 kv a k2H3/2−δ (Maε ) + kv s k2H3/2−δ (Msε ) . Therefore, by Theorem 4.5, we conclude that (v a , pa ) ∈ H2−δ (Maε ) × H 2−δ (Γ), (v s , ps ) ∈ H2−δ (Msε ) × H 1−δ (Γi ), and kv a k2H2−δ (Maε ) + kv s k2H2−δ (Msε ) + ε|pa |2H 1−δ (Γi ) + ε|ps |2H 1−δ (Γi ) ≤ κ ˜, κ ˜ as above. A final application of the Trace theorem and of Theorem 4.5 to Maε and Msε yields : (v a , pa ) ∈ H2−δ (Maε ) × H 2−δ (Γ), (v s , ps ) ∈ H2−δ (Msε ) × H 1−δ (Γi ), and ˜, |v a |2H2 (Maε ) + |v s |2H2 (Msε ) + ε|pa |2H 1 (Γi ) + ε|ps |2H 1 (Γi ) ≤ κ κ ˜ as above. The proof is complete. Acknowledgements. This work was partially supported by the National Science Foundation under the grants NSF-DMS-0074334, NSF-DMS-0204863, NSF-DMS0305110 and by the Research Fund of Indiana University. References [1] S.Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math., 12, 1959, 623–727. [2] S. Agmon,A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math., 17, 1964, 35–92. [3] P. Bartello, Geostrophic adjustment and inverse cascades in rotating stratified turbulence, J. Atmos. Sci., 52, (24), 1995, 4410-4428. [4] T. Bewley, R. Temam and M. Ziane, A general framework for robust control in fluid mechanics, Physica D, 138, 2000, 360-392. [5] T. Bewley, R. Temam and M. Ziane, Existence and uniqueness of optimal control to the Navier-Stokes equations, C.R. Acad. Sci. Paris, 330, Serie J, 2000, 1007-1011.


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