Quasilinear hyperbolic systems and dissipative mechanisms

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

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Quasilinear Hyperbolic Systems and Dissipative Mechanisms

L. Hsiao Academia Sinica, Institute of Mathematics, Beijing, China

World Scientific Singapore • New Jersey • London • Hong Kong

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Library of Congress Cataloging-in-Publication Data Hsiao, Ling, 1940Quasilinear hyperbolic systems and dissipative mechanisms / L. Hsiao. p. cm. Includes bibliographical references and index. ISBN 9810232055 1. Differential equations, Hyperbolic. 2. Quasilineahzation. I. Title. QC20.7.D5H75 1997 531\01'515353"dc21 97-28760 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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Preface The study of quasilinear hyperbolic systems that result from the balance laws of continuum physics was initiated more than a century ago, and considerable progress in this field has been made although many fundamental problems in the analytical theory remain unsolved. The aim of this book is to introduce recent developments on the subject of quasilinear hyperbolic systems with dissipation, particularly systems of hy­ perbolic conservation laws with either lower order dissipation such as frictional damping and relaxation, or higher order dissipation such as viscosity and heat diffusion. Other kinds of dissipative mechanisms, such as those induced by "memory" or boundary conditions, will not be considered. The purpose is, on the one hand, to understand the influence of dissipation on the qualitative behavior of solutions, such as the nonlinear diffusive phenomena, phase trasitions etc. and on the other hand to employ dissipation as a device for approximating solutions of hyperbolic systems by passing to the zero viscosity or zero relaxation limit. The first two chapters are devoted to a research program which connects the area of equations with nonlinear diffusion with the area of quasilinear hyperbolic systems. Nonlinear diffusion problems arising in various applied sciences have been widely investigated in recent years and connections with well understood, more "abstract" subjects in the theory of nonlinear parabolic equations have been uncovered. The class of systems with damping mechanisms has attracted particular attention because they model a number of problems such as the flow of an elastic fluid through a porous medium or in a pipeline, the oscillation of an elastic string immersed in a fluid, etc. In Chapter 1, we show that the large-time behavior of classical solutions of systems of hyperbolic conservation laws with frictional damping is related to the large time behavior of solutions of systems with nonlinear diffusion. Chapter 2 is devoted to weak solutions for such systems, either in the class of piece-wise continuous and piece-wise smooth functions or in the class of BV functions. Many physical situations require consideration of the effects of relaxation. For instance, in the kinetic theory relaxation is related to the mean free path and in viscoelasticity it is induced by memory. In gas dynamics, the relaxation phenomenon occurs when the gas is in thermo-non-equilibrium. It also occurs v

vi in chromatography, river flow, traffic flow, etc. Therefore, a natural approach is to approximate hyperbolic systems of conservation laws by systems that in­ volve relaxation. We investigate the topics of stability of elementary waves and singular limits of zero relaxation time in Chapter 3. Chapter 4 investigates the influence of higher order dissipation on the quali­ tative behavior of solutions, such as viscosity and heat diffusion. In general, dissipative mechanisms may afffect the asymptotic behavior as well as the smooth­ ness of solutions. The stabilizing effect on the solutions will be investigated in Section 4 . 1 . It is shown, for instance, that the combined dissipation of viscos­ ity and heat difffusion will preserve the smoothness of the initial data without restrictions on smallness. Another aim of this Chapter is to understand the influence of dissipation on the large-time behaivor of solutions. Various phe­ nomena, including phase transition phenomena, are exhibited on the large-time behavior of solutions. This is disussed in Section 4.2. The last section, Sec­ tion 4.3, is devoted to the study of behavior of solutions depending on different boundary conditions. To obtain the a priori estimates guaranteeing either the global existence or the expected large-time behavior of solutions, careful analysis based on energy approach has been carried out. The last Chapter, Chapter 5, considers a viscous system of conservation laws. The first topic is to study the relationship between the solutions of the viscous system considered and the distributional solutions of the corresponding hyperbolic system of conservation laws without viscosity, especially the problem of vanishing viscosity which is discussed in Section 5.1. Another closely related topic is to study the similarities and differences of the large-time behavior of solutions for these two systems, particularly, to compare the large-time behavior of solutions for the viscous system to the solutions of the Riemann problem for the corresponding system of quasilinear hyperbolic conservation laws. This problem of nonlinear stability of waves will be discussed in Section 5.2. Section 5.3 investigates the above two topics with regards to multi-space dimensional systems. The bibliography contained in this book provides only a sample of the rele­ vant literature and is far from comprehensive. I am grateful to C. H. Gu and X. X. Ding for their continuous support in my research work. I am indebted to P. D. Lax, J. Glimm and J. Smoller for their enthusiastic support and encouragement in these years. Special thanks are due to C. M. Dafermos, Tai-Ping Liu, D. Serre, J. Greenberg, P. Marcati and Ta-tsien Li for their fruitful and friendly cooperation. I am pleased to thank T. Luo, Z. P. Xin, T . Yang and S. Q. Tang for their contribution in my writing this book. Finally, I would like to thank the National Science Foundation of China for its continuous support.

Contents

Preface 1

2

3

4

v

Frictional damping: Globally defined classical solutions and their nonlinear diffusive phenomena 1.1 The system of compressible isentropic flow with damping . . . . 1.2 The global existence of smooth solutions for the system of com­ pressible adiabatic flow with damping 1.3 The nonlinear diffusive phenomena of smooth solutions for the system of compressible adiabatic flow with damping

15

Frictional damping: Globally defined weak solutions and the nonlinear diffusive phenomena 2.1 Riemann problem and perturbed Riemann problem 2.2 Nonlinear diffusive phenomena for entropy weak solution . . . . 2.3 Globally defined admissible BV solutions

35 36 51 64

Relaxation 3.1 The stability of shock profiles for a rate-type viscoelastic system with relaxation 3.2 The stability of planar rarefaction waves and shock fronts for the relaxation approximations of conservation laws in several dimen­ sions 3.3 The zero relaxation limit

1 3 9

67 67

89 102

The influence of dissipation mechanism on the qualitative be­ havior of solutions 111 4.1 The influence of dissipation mechanism on the smoothness of solutions 114 4.2 The influence of dissipation mechanism on the large-time behav­ ior of solutions 128 4.3 Various large-time behavior corresponding to different boundary conditions 146 vn

Vlll

5

Vanishing viscosity and nonlinear stability of waves 5.1 Vanishing viscosity - the zero dissipation limit 5.2 Nonlinear stability of waves 5.3 The multidimensional case

165 166 178 193

Bibliography

207

Index

222

Chapter 1 Frictional damping: Globally defined classical solutions and their nonlinear diffusive phenomena

In this chapter, we consider quasilinear hyperbolic systems with damping, such as the system of compressible isentropic or adiabatic flow through porous media. We are interested in the influence of damping mechanisms on the qual­ itative behavior of the processes in consideration. The system which can be used to model the adiabatic gas flow through porous media takes the form vt -ux = 0 ut+p(v,s)x = -au,

{ a a>0

^

^

I e(v, s) + ^ - 1 + (pu)x = -au2 where v, u and s denote the specific volume, velocity and entropy respectively, e stands for the specific internal energy for which es ^ 0 and ev + p = 0 due to the second law of thermodynamics, and p denotes pressure with pv < 0 for v >0. The system (1.1) is equivalent to the following system (1.2) for smooth solutions ( vt — ux = 0 < ut +p(v,s)x = -au, a>0 (1.2)

a U, =0

which is strictly hyperbolic with eigenvalues Ai = —y/—pv, A2 = 0 and A3 = In the case where a — 0, the system (1.1) or (1.2) becomes a system of hyperbolic conservation laws. In general for such kind of system there is no globally defined classical solution for its Cauchy problem no matter how smooth and small the initial data are. Moreover, the large-time behavior of the solution, when it exists, can be described by the corresponding Riemann problem. For the case where a > 0, there is a damping term in the system. We investigate the role of this damping mechanism in this chapter, particularly its influence on the asymptotic behavior of the processes in consideration. 1

2

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

In a special situation where s = s (s is a constant) initially, the system (1.1) or (1.2) becomes

f vt - ux = 0 \ ut+p(v,t)x -

-au,

which is the system for isentropic gas flow with damping. We prove in Section 1.1 that the solution of the Cauchy problem

(

vt - ux = 0 ut +p(v)x = -au, a > 0, p'(v) < 0 for v > 0 v(x,0) = vo{x),u(x,0) = u0(x), with lim (vo(s)> u0(x))

= (vT, u^)

a?—>^oo

(1.3) can be described by the solution of the problem

a u = -±p(v)x v(x,0) — v*(x + Xo) ,x

time-asymptotically, where v* f

j is the similarity solution of (1.4) i, with

boundary condition v* (rj) —>• v* as 77 —> T 0 0 ? mined uniquely by

/

(1-4)

an

d #0 E -ft is a constant, deter­

°° r 1 w + — w~ |vo(aO-v*(z + *o)}ds= — • -OO ^

(1.5)

OL

J

Moreover, it is proved that the Z/2-norm and the Loo-norm of the difference between these two solutions tend to zero with a rate tf-1/2 as time tends to infinity. This shows that the damping mechanism induces certain nonlinear diffusive phenomena in the solutions of the original hyperbolic problem. For the general situation when s is not a constant initially, we have to investigate the system (1.1) or (1.2). The result on global existence of smooth solutions of (1.1) or (1.2) is given in Section 1.2 which shows that the damping dissipation is strong enough to preserve the smoothness of the initial data when they are small. A more important problem is to consider the initial data with lim u0(x) = ti*,

lim v0(x) = vT > 0,

and

lim s0(x) = s

(1.6)

where v~ is not equal to v+, and to compare the solution of (1.2), (1.6) to the solution of

(

v

*-

-hp(v>s)™

u = -h(v,s)x 8t=0

(1.7)

Frictional damping: Globally defined classical solutions... with initial data v(x,0) defined by

= v*(x + ar*,0) and s(x,0)

= SQ(X),

where v*(x,t)

p(v4x,t),s(x))=p(v*(^=j,sy and v* f

.x

3 is

(1.8)

J is the similarity solution of

a I v* (77) —>• v^ as 7/ —y =foo,

while x* £ R is chosen uniquely by

/

°° r ^ w+ - 1 / ~ |vo(a?)-t;Ill(x + x * , 0 ) | d x = — .

,g

(v+ (x, t) = v* (

(1.10)

) if SQ(X) = s in which case ar* = xo and (1.10) becomes

W* + l y

(1.5).) This is discussed in Section 1.3. We show that the difference of these two solutions tends to zero as t —> oo in the Loo-norm, provided the initial data are suitably small. Namely the solution of the original hyperbolic problem with (1.2) converges to nonlinear diffusion waves essentially as time tends to infinity. 1.1

T h e s y s t e m of c o m p r e s s i b l e i s e n t r o p i c flow w i t h d a m p i n g

Consider the system in (1.3). It may be viewed as the isentropic Euler equa­ tions, in Lagrangian coordinates, with friction term (—au) in the momentum equation which models the compressible isentropic flow through porous media. The commonly called porous media equation (1.4)i is obtained by approxi­ mating the second equation in (1.3) with Darcy's law. It is proved by Hsiao and Liu [69] that Darcy's law may be obtained from the more complete equations in (1.3) time-asymptotically. That is, the solutions of (1.3) tend to those of (1.4) as the time t tends to infinity. More precisely, \\v(x,t)

- v(x +

Xo,t)-

m(x,t)\\L2(t)

+ I K * , * ) - v{x + x0,t) - m{z,t)\\Loo{t) \\u(x,t)

-u(x

+ x0,t)

= 0{l)(t

+ 1)"1/2,

as t -> oo (1.11)

- 6(a?,t)|| L a ( t ) + | | « 0 M ) - u(x + *o,t) - £OM)IUoo(0

= 0(l)(t + 1)"1/2,

as*->oo (1.12)

4

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

where (v, u)(x,t) is the solution of (1.3), v(x,f) is any solution of (1.4) with (1.5), u{x,t) is determined by (1.4) 2 , m(x,t) is defined by m{z,t)

u

= J

~u

)mo(x)e-at

(1.13)

for a smooth function mo(x) with compact support and oo

/ u(xjt)

m0{z)dz

= l,

(1.14)

-oo

is defined by u(x,t)

= u'e"at-\-

/

mt{r})t)d'q,

(1.15)

«/—oo

and xo is determined uniquely by the formula (1.5). We first discuss self-similar solutions for (1.4)i. They are time-asymptotic states for general solutions. The decay estimates for such a solution is essential in showing the asymptotic behavior of solutions of (1.3), expressed in (1.11) and (1.12). It can be established by using energy estimates. It is clear that the nonlinear diffusion equation (1.4)i is invariant under (x, i) —>• (ex, c2t), c > 0. Therefore, it possesses solutions of the form v*(r/), 77 = Ix , —oo < T) < oo, for which (1.4)i becomes

a P(V

(77))

We are interested in solutions which have limits at x = =F°o, v*(^oo) = v=F.

(1.17)

It has been shown by Van Duyn and Peletier [39] that (1.16) and (1.17) have a unique solution which is strictly monotone. We are also interested in the dissipative nature of the solution, particularly the convergence rate of V v * at x = =foo and as t —> 00. It can be easily shown, from (1.16), that

V :M (,?) _

-

» " W K M ) exp exD f r

PVW)

+ - v~)2(t + I ) " *

(1.21)

oo

/

| V ; t (x, oo with rate (t + 1 ) " 1 / 2 in the Z^-norm and (t + U-'H1-*)

These give the result in (1.11) and (1.12), since v(x + x0,t)

\ v t -+-1 / in the L°°-norm for 0 < e < 1 if the initial data v(x, 0) for (1.4)i tend to v* as x -» =Foo fast enough. We refer to Van Duyn and Peletier [38], Luo and Xiao [134] for details. 1.2

T h e g l o b a l e x i s t e n c e of s m o o t h s o l u t i o n s for t h e s y s t e m of c o m ­ p r e s s i b l e adiabatic flow w i t h d a m p i n g

Let us consider the system (1.1) which can be written in a different form (1.2), equivalent to (1.1) for smooth solutions. For the case a — 0, it has been shown that there is no globally defined classical solution for a Cauchy problem in general. In fact, considering the following initial data u(x,0)

= euo(x),

v — rJ-f evo(x),

s = eso(x)

where v is a positive constant and (UQ(X), VO(X), SO(X)) £ C1 with compact support, it was proved by Liu in [117] and Li, Zhou and Kong in [103] that there exists a small So > 0 such that for any s G (0,£o] the C 1 solution of this Cauchy problem blows-up at a finite time, moreover the life span T(e) = 0(e~1). However, for the system (1.1) or (1.2) with damping term, the situation is different. Taking any initial data u(x,Q) = u0(x),v(x,0)

= t7+ vo(x),s(x,0)

= s + s0(x)

(1.41)

l

where v > 0, v and s are constants, (uo(#), ^o(^)) £ C with compact support and so(x) G C2 with compact support, it is proved by Hsiao and Serre [80] that there exists a globally defined classical solution for the Cauchy problem if the C ^ n o r m of (uo(x), v0(x)) and C 2 - n o r m of s0(x) are small. Denote c = (— pv(v,s))*. Let r\ = u —

h(v,s)

r2 = u +

h(v,s),

(1.42) where h(v,s)

satisfies the equation chv-pv=0,

(1.43)

10

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

then the system (1.1) or (1.2) can be written as

{

r\t - crlx = - f ( r 2 H-ri) + {chs -

ps)sx

r2t + cr2x = - f (r 2 + n) + [chs -ps)sx

(1.44)

*t = 0. For the system of gas dynamics where p(v, s) = (7 — l)v~1es(l is easy to see that h(v.s)

= 2«/ -v1^ V 7-1

< 7 < 3), it

-e2

and chs - ps = v ~ 7 e 5 . Thus, the system (1.44) takes the form ( D"ri

= - f (r 2 + r i ) + / ( r 2

-rus,8x)

I D+r2 = - « ( r 2 + n ) + / ( r 2 - r i , * , * * ) l*(t,x) = s(0,z), where D = dt - cdXl D+ = dt + cdx, f{r2 — r i , «, «a.) = i4(7) - f a - r i ) ? - * - e ^ - «a-, c = c(r2 -n,s)

= B(j)

• ey=* • (r 2 - r x ) ? - * ,

a For simplicity, we will discuss the system (1.1) or (1.2) for gas dynamics. Introduce g(r2 - n , s, sx) = gx(r2 - n , 5) + g2(r2 - ri,s)sx in which gx and g2 are chosen so that

02-

4(7 - 1 ) C

'

, where g'{ denotes a (-^l' 7 l )(» = 1,2). We may choose

g,

+ j ^ " . ^ '^

0. Moreover, m2F — 41GF > 0 if ls.nl and I sxxI are small. Denoting cll2r2x + g = y, (1-45) implies that D+[y] = -l(y-yi)(y-y2),

(1.47)

where

2/1 = tt(7+7V1^mF ~ \/m^~4/GF^' and

y2 = ^,~+{\gi{™F

yJm2F-UGF}.

+

Thus, it follows that 0 < 2/i (r 2 - ri,sysx,sxx)

< y2{r2 - r i , s , sX} sxx),

if r2 —ri > 0 and both \sx\ and \sxx\ are small. Similarly, denoting cxl2r\+g — f, (1.46) implies that Zr[*] = - Z ( z - * i ) ( * - s 2 ) , (1.48) where Zl =

a{^+l)gi^mB

" \/mB-4/G£},

a 2

m| -4/GB = ^

+ c1B(r2

- rXl s)sx + c2B(r2 - ri, s)sl

+C23B(^2 -

ri,s)sxx,

c 1 B(r 2 -r 1 , S ) = - ( 3 _ y ( 7 _ 1 ) C - ^ j l + i l . Q 1 B ,

a /

x

CzB(r2-r1,s) =

16-L 2

2

2a(7+l)

_

-2°(2+V.Q3B.

It is easy to see that mB — AIGB > 0 if \sx\ and I s ^ l are small. Thus, it follows that 0 < zi(r2 - ri,s,sx,sxx) < z2(r2 - r l 3 s , ^ , s r a : ), if r2 — ri > 0 and both \sx\ and \sxx\ are small. Moreover, it can be shown that (y(0,x)>y1(r2-r1,s,sx,sxx)(0,x), and \ i ( 0 , x ) >zi(r2-ri,«,*a.,«ara.)(0,a?),

. *'

'

Frictional damping: Globally defined classical solutions...

13

if the C i - n o r m of (uo(x), vo(x)) is small and the C2-norm of SQ(X) is small, (see [80] for the details). In view of (1.47)-(1.49), it turns out that L e m m a 1.2.1. Suppose that the solution of (1.2) and (1.41), defined for 0 < t < T, satisfies ( r 2 , r ! , s ) 6 D — { ( r 2 , r i , s ) : aih(v,s)S < r2 — n < a 2 /i(i7,s), \s — s\ < 1, where a\ and a2 are constants such that 0 < a\ < 2 < a2 < oo}. Then, the following estimates hold if the Ci-norm of (uo(x), vo(x)) is sufficiently small and the C 2 -norm of SQ(X) is small enough. For any (t,x) G [ 0 , T ] x R,

mini

{c^r^

+ g)^xsiO^x)),

I V

inf

y2 \

x 3 (r;*,x) 0 0 and ui — i«2 — u —> 0 in the L2-norm and the Loo-norm as t —>• oo. To establish the result mentioned above, it is essential to solve the Cauchy problem in v^ and to obtain the required L 2 -estimates particularly on t^- Turn­ ing to gas dynamics and taking a = 1 for convenience, the Cauchy problem for V2 becomes

fvt = -p{v,s)xx \v(x,0) = (7 — l)v~1es(l

where p(v,s)

= e#'W-7lv*(x

khj [l b6) '

+ x*)

< 7 < 3 ) , s = s(x) and v* (

.x

j is the

similarity solution of (vt = -p(vs)xx [ V(T)) —> v* as 77 —>■ Too,

khjbnh v '

and v* E Ris determined by oo

/

{v0(x) - « # • < * > - V (at + x*)}dx = - ( « + - « ~ ) . -OO

(1.55)

Frictional damping: Globally defined classical solutions...

17

For convenience, we only give a proof for the case u~ = u+ = 0 in which m(x,t) = 0,«(x,^) = 0 and (1.55) becomes oo

/

{v0(x)-eyM*)-7lv*{x

+ x*)}dx = 0.

-oo

For the initial data (1.6), we assume that v0(x) — v+ = 0(x~ f c l ),

as x —> +oo

v0(#) — v = 0((—x)-^ 2 ),

(1.56) as x - » —oo

for certain positive constants Ar,- > l ( i = 1,2), while (so(x) — s) possesses compact support, say support(s 0 (x) - 5) C [-N, N]

(1.57)

for a positive constant AT. Furthermore, so(x), v + and v" are given so that there are constants Ar0 > AT, ao > 0 and x £ R such that if V3 is a positive solution of the following problem (vt =

-p(v,s(x))xx

\n(*>0) = c ^ W - V ( x + *)

(L58)

then it holds either v3(x,t)-* ■ t , ' ( * ^ ± a ° )

(1.61)

where x and a 0 are the constants in the condition (1.59), and v*(r}) is the similarity solution of (1.54), for which it has been shown (see Atkinson, Van Duyn and Peletier in [5] [6] [39]) that V*(TJ) - v~ = O ( e r f c — = J L = = =

V ( and

V*(T}) - v+ = O

as

77 - » - 0 0 , n R9v

\

erfc — .

V

]

2y/-Pv{v-,8)J "

_

)

' as

'

77 -> + 0 0 .

2y/-pv(v+,s)J

Let t; 3 (x,t) > 0 be a smooth solution of (1.58) a n d 0 ( z , * ) = v 3 ( z , * ) - i ) ( : c , t ) . It holds that

a a \e(a;,0)=e^^-5l.[t;*(a;

+

£ ) - v * ( x + x + ao)].

l

'

Frictional damping: Globally defined classical solutions...

19

We seek a smooth solution 0(x,t) € C2(t > 0, x £ R) with ||0(-,t)||2 = \H;t)\c*

+ \0t(;t)\c* + \0tt{;t)\c < oo

(1.64)

The uniqueness of such a solution 0(xyt) satisfying v(x,t) + 9(x,t) > 0 (x £ R,t E [0,T] for any T > 0) can be easily obtained by the maximum principle. Therefore, v3(x,t) = v(x,t) + 0{x,t) would be the unique positive solution of (1.58) if one could show that there exists a solution 9(x,t) for (1.63) such that v(x, t) + 0(x, t) > 0. We show the existence of such a solution 0(x, i) by solving the Cauchy problem (1.63) in the Banach space X3, using the Z,2-energy method. Assume {s0{x)-s)eH5(R). (1.65) We can obtain that

| | ^ , 0 ) | | ^ < ^ { K ( 0 ) ] 2 + ||5 X ||^}, by a similar argument as used in [69] and [70] to get the L 2 -estimates on the corresponding derivatives of the similarity solution. Where K > 0 is a constant depending only on the quantities 6, di(i = 1, 2, 3), s, and 7 defined as follows &=supe^(x)-J),

a di =

inf \-pv(v,8)]

> 0,

vG[v~,v+J

d2=

sup

[-pv(v>s)]>0,

d3=

sup 2 ^

^t

•

We can prove, by a routine manner that the classical local existence theorem gives the solution for the Cauchy problem (1.63) in the space X3 locally in time. For the global existence in t > 0 we only need the a priori estimates in the norm (1.64) for which the a priori estimates in the norm of X3 is sufficient by Sobolev's lemma, i.e.,

III'WIIIS = ll'WIIlr. + \\0t(t)\\h + IIMln < +00 for t > 0. For establishing the L 2 -estimates on 0(t), we need a sequence of Lemmas. Defining QT = R x [0,T], for any T > 0, we introduce the transformation

77= J L _ r = log(l + *) vt 4-1

20

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

under which QT becomes ST> = R x [0,7"] on the plane (77,7-) where T' = log(l -f T). Let us use the notation v(r), r) to denote the solution v3(x,t) of (1.58). It follows that

a where s = s(x) =

s(rje^).

L e m m a 1.3.1 Under the assumptions (1.6), ( 1 5 7 ) and (1.65) on so(x), there are constants 770 > N, rji < —N,pi > 0 and m, > l ( i = 1,2) such that for any positive smooth solution of (1.66) h(r), T) < £(77, r ) < 7I(T7, T)

where the continuous functions h and h are defined as follows I eyl

w

;

A

v

5

* ( l ^ ) = S max{e^ ^ I

77 < 7/0

, ef

>-

5l

t;-,r;+-A(r?-7;o)-

= max{ir,t;+-/^(r/-770) ~

mi

},

mi

}

77 > *7o

and

a M » l > r ) = S m i n { e ^ 5 ^ ' e f ) - ^ + , v - + / ? 2 ( - 7 7 + 77i)- m 2} I

=min{v+,v-+^2(77-77i)~m2},

*? < *7i

Proof. It is easy to check that the functions c*M*«*>-*].tr

and

c*^*)-

3

are lower and upper solutions for (1.66) respectively. principle implies that c ±Mi«*)-31

!.^ Thus, the Maximum

. v- < j ^ ) < c ^[*(ie*)-il . v+

for (77, r ) G S T ' . Thus, the lemma 1.3.1 can be easily obtained by the property of similar solution and the maximum principle used by Cosner in [21] (the details can be found in [73]).

Frictional damping: Globally defined classical solutions... L e m m a 1.3.2.

21

For any positive smooth solution of (1.58) withu^-,*) E

a f°° I , fx + x + a0\ /

\v

.

J-oo\
+ a 0 (v+ - v~) L xe[-N,N] J +(4fc0 + 2iV)v+ .

sup l e ^ ' t o - 1 ) - ll xe[-N,N] ' '

where Ar0 is the constant in (1.59). Proof. Let us still use the notation V*(T],T) to denote v*(x*ftt°Q), v* (77, T) = v* (TJ -h (x -f ao)e~ 2) which satisfies

namely,

a I v*(17,0) = t;*(x-hx + a ). 0

In view of lemma 1.3.1 and (1.62), we can show that the function $ ( r ) oo

/

[V*{TJ+ (X + a 0 )e~^) - £(77, T)]G?77 -00

is well defined for any r £ [0, X"]. Subtracting (1.66)i from the above equation on v* and integrating it over (—00, 00), with the help of integration by parts and the facts that v* -> 0 as 77 -> =poo (see [5], [6], [38], [39]), vv -> 0 as 77 -» =foo, and s ^ e ^ ) = s for 1771 > TV, we obtain ^

= |»?K-«)l~oo-5*W

for

r€[0,n

which, with lemma 1.3.1 and (1.62), implies *(r) = * ( 0 ) e - *

for

r€[0,T'],

where CO

_

a / [v*(x + x + o 0 ) - e * ( ' ( r ) - , V 0 B + *)] + i»;iei + i«?iei + i»?t|eo < cK(o)i, T r M2/ , w «. -&f a n d &•

24

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

Lemma 1.3.4. Under the assumptions (1.65), (1.6) and (1.57), if the solution 0(t) eX3 (t E [0,T]) to the Cauchy problem (1.63) satisfies l|0(*)l|22x(x,t)dx)

.

Frictional damping: Globally defined classical solutions...

27

Using a similar method as in the proof of Lemma 1.3.4, we have +oo

pt

/»+oo

a /

1>l{x,t)dx+-£

-oo

/

^

JO

2

/

tf(x,T)dzdT

J-oo

) _ l| + |,,+ - , r | i , xe[-N,N] J

where K$ depends only on the quantities in Lemma 1.3.4. The proof of Lemma 1.3.7 can be finished then, following the same lines as in Lemma 1.3.4. By a similar approach as used in Lemma 1.3.5 and Lemma 1.3.6, we prove Lemma 1.3.8. Under the assumptions of Lemma 1.3.5, if \v+ — v~\ and \\s(x) — sHIJ-S are suitably small, then there exists a unique globally defined smooth solution ip(t) G X3 for the Cauchy problem with ip. The function V2(x,t), defined by 1*2(2>i) = v{xjt) + i/j(x,t), is the unique positive smooth solution of the Cauchy problem (1.53) for which the estimates in (1.60) hold. By using the result on V2, one is able to solve the Cauchy problem (1.51), (1.52) for y in the space X3. One can prove that the classical local existence theorem gives this solution in the space X3 locally in time. For the global existence in t > 0 one only needs the a priori estimates in the C 2 -norm of y for which the a priori estimates in the norm of X3 is sufficient. Choose r > 0 such that 0 < f < a\{a\ = infirG(_00)00) ey(s(<x'~s'v~) and introduce 3

£(*) = $ > ; ( * )

(1.69)

for the solution y with yx E [—r, f] at each (x, t) 6 R x [0, T], where

a /•OO

£2( 0 such that inf

[-Pv{v,8]\ >Po > 0 .

•€[inf so(z),supso(z)] x£R xeR We assume (t0 < 1 for convenience. Then ^-yl

the same kind of discontinuous initial value problem (2.5) (2.4) has been studied by Li [106], and by Li and Zhao in [105] where uT(x) and Vzp(x) are considered as perturbed Riemann data. It has been proved under certain restriction on the initial data (2.4) that the problem (2.5) and (2.4) admits a unique global solution in a class of piecewise continuous and piecewise smooth functions. Also, the solution has a global structure similar to that of the corresponding Riemann problem with Riemann data (w~, v~) and (u+, v+). The Riemann problem for (2.5) has been studied in great detail, (see Chang and Hsiao [12], Liu [112], Dafermos and Diperna [27], Smoller [169] and etc.)

Frictional damping: Globally defined weak solutions...

37

For the system (2.3) with damping term, however, the Riemann problem is much more complicated since there is no self-similar solution anymore. In this section, we investigate the perturbed Riemann problem (2.3) and (2.4) and construct the globally defined piece-wise continuous and piece-wise smooth solutions, showing the qualitative behavior in the process. For simplic­ ity, we discuss a typical form of the state function p(v) instead of a general form, namely, p(v) — av'1 with a > 0 and 1 < 7 < 3 which is the state function for polytropic gas. We deal with the case in which the two states (u~,v~) and ( u + , V+) are connected by a backward shock curve and a forward shock curve subsequently in the phase plane. Here, we only discuss the case when 7 is 1 or nearly 1 (see Hsiao and Tang [78]). The investigation for the case 1 < 7 < 3 can be found in [79] given by Hsiao and Tang. For other kind of relation between (u~,v~) and ( u + , v + ) , the result has been obtained by the same authers (see [173]). Under certain restriction on the perturbation in (w_(#), v-(x)) and (u+(#), v + ( x ) ) , we prove that the problem (2.3) and (2.4) admits a unique global dis­ continuous solution on t > 0 in a class of piecewise continuous and piecewise smooth functions. The solution only contains two entropy shocks - a backward shock and a forward shock. The shocks do not disappear for any finite time and do disappear with the strength decay exponentially when time t tends to infinity. First, we prove the result for unperturbed Riemann data with 7 = 1 . Then, we establish the result for perturbed Riemann data with 7 = 1 . Finally we discuss the case when 7 is nearly 1. Consider the unperturbed initial data, namely,

K(*w*))={j:;;:;j; m

m

which is called Riemann data in the theory for systems of hyperbolic conserva­ tion laws. Now, consider the system (2.3) where p(v) - § . Without loss of generality, we take a = 1 and a — 1 in (2.3) for convenience. Thus, the characteristic speed takes the form that A = - \ and \i — \ respectively, corresponding to the backward and forward family. Introduce Riemann invariants r\ — u — (j> r 2 = u + <j)

where is defined by <j) = — \nv. Then, the system (2.3) can be written as r u + Ari a: = - ^ ( r 1 - h r 2 ) r2t + f*r2x

= - J j ( r i + r2)

a

38

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

which is equivalent t o (2.3) wherever the solution is smooth. We may use the following notation

a < -

d

d

- J - it

with which (2.7) becomes r

i = - \ i r i +r2)

r2 = ~\(ri

+ r2).

Next, turn t o a discontinuous solution of (2.3). A discontinuity in a solution (u,v) of (2.3) at (t,x(t)) should satisfy the Rankine-Hugoniot compatibility condition which implies that either

f dx(t) __ - 1 J j

dt ~ yJvRvL — UL __ 1

UR

I VR-VL

y/vRVL (2.8)

or

( dx(t) _ i I dt ~ y/lJRVL j UR - UL _ 1 i VR~VL y/vRVL respectively, where («#, VR) = (u, v)(t, #(tf)+0) and (UL, VL) = (w, v)(t, x(t)—0). A discontinuity is called a backward shock (or forward shock) if (2.8),- holds and the entropy condition is satisfied as well, namely, VR

< vL

for i — 1,

VR

> VL

for i = 2.

(2.9) or

It is easy to write (2.8) in ( r i , r 2 ) variables as J -

^ = /i(r2jR-rljR,r2L-r1L)

[ V2R - r2L = hi(riR

(2 10)i

- riL)

or J -

^ = / 2 ( r 2 j R - r l j R , r 2 L - r1L)

(2 10

)2

U^-r^M^-r^) where hi(riR-rii)

is determined by # ( r 2 j R - r 2 L , r i ^ - r ^ ) = 0,/i 2 (r 2 j R -r 2 j £ ,)

is determined by H(riR - rix,, r 2jR - r 2 £,) = 0, # ( £ , 77) = ^ 3 ^ + e * ~ e * » ,

r

and fi = -e

(r2R-r1R)+(r2L-r1L)

4

= -/2.

Frictional damping: Globally defined weak solutions...

39

Furthermore, the entropy condition (2.9) can be expressed in ( r i , r 2 ) vari­ ables as r2R - rm > r2L - r1L for i = 1, (2.11) or r2R - rm < r2L - r\L for i = 2. We m a y use either (u,v) or (r\,r2) in future discussion depending on conve­ nience. The initial data (2.6) satisfy the following condition. There exists a state ( ^ I O , r 2 0 ) (or (UQ,VO) in {u,v)

f I I I

variables) such that

r2o - r j = Ai(rio - r f ) , r+ - n o = M r f - r 20), r20 - n o > r J - r f 7*20 - 7*10 > ^ J _ r i "

and

(^ 2 1.3 )^

where ( r f , rj") and (rj", r j ) are corresponding to ( u + , v + ) and (u~, v~) respec­ tively. This means that the two states (u~~>v~) and ( u + , t ; + ) are connected by a backward shock curve and a forward shock curve subsequently in the phase plane. Now, we construct the global discontinuous solution for (2.3) (2.6) (2.12). First of all, by using the initial data on x > 0, we solve the corresponding initial value problem for (2.3). It is clear that this problem admits a unique global C1 solution (ri+(tyx),r2+(t,x)) on the domain R+ = {{t,x)\ t > 0,x > £+(t), V ' = ii(v+(t,£+(t)))}. Similarly, we solve the corresponding initial value problem for (2.3) by means of the initial data on x < 0 and get a unique global C1 solution (n-(*> x), r2-(t, x)) on the domain R- = {(t,x) | t > 0,x
#), r 2 +(2, x)) respectively. And, one is required to solve the following free boundary problem on the angular domain R = {{i,x) \ t > Q,xi(t) < x < x2(t)} for (2.3) in order to construct the globally defined discontinuous solution for (2.3) (2.6) which contains only two shocks.

40

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

( F B P ) : On the free boundary x = z i ( i ) ( * i ( 0 ) = 0) r2(t,xi(t))=r3-(t,xi(t))

+ h1(ri(t,x1(t))-r1-(t,*i(t))),

and

a 4 ^

On the free boundary x = x2(t)(x2(Q) r1(t,x2(t)) ^

1

(

= Mr2(t,xi(t))-r1(t,z1(t)),r2-(t,z1(t))-r1-(t,x1(t))).

= ri+(t,x2(t))

= 0)

- h2(r2+(t,x2(t))

- r2(t,x2(i))),

and

khmjb {2M)

= / 2 ( r 2 + ( t , * 2 ( t ) ) - r 1 + ( * , * 2 ( i ) ) , r 2 (t, * 2 ( t ) ) - n (t, x2(t))),

where ( / i , Ai) and (f2,h2) are defined in (2.10)i and (2.10) 2 respectively. More­ over, the entropy condition has to be satisfied on the free boundaries, namely, on x = xi(t),(r2 on x = x2(t),

-ri)(t,xi(t))

> (r 2 _ - ri_)(*, xi(t)),

(r2 - n ) ( t , ar2(*)) > (^2+ - r1+)(t,

and

^kgjh ^

x2(t)).

a Next, we prove the main theorem. For definiteness, we assume that kgjgh

u * > 0 and 0 < v± < 1,

(2.16)

moreover, v+ >v~. The other cases can be discussed similarly. C o n d i t i o n A: \/v+ /y/vo < gi/ali

wnere v

o is given in (2.12).

T h e o r e m 2 . 1 . 1 . Under the condition A, the above free boundary problem (2.3) (2.13) (2.14) admits a global classical solution ( r i , r 2 ) G C1 in R with Xi(t) G C2(i = 1,2) on which (2.15) hold. Furthermore, this solution possesses the following properties. For any (t, x) G R, r2* < r2(t,x)

min< ^


and At- is a positive constant (i = 1,2,3,4). In the present discussion for (2.3) and (2.6) it is easy to know that v_ = v~ and v+ = v+. To prove this theorem, we need the following lemmas. L e m m a 2.1.2. Suppose that the classical solution of (2.3) (2.13) (2.14) exists in R(T) : {(t,x) | 0 < t < T,xi(t) <x< x2(t)}, then it holds along x = xx(t) that _ 4Vy/vZ(y/vZ

- y/v)

_ /' y/vZ' -

y/v\

~ u/^+V^) \y/vZ+^J a a [ \y/v-+y/vj J y/v~+yjv r2x

_i

I

2

,

,

rix

/y/vZ— \Jv\

/VftJZ

- y/v.3/

_i

i

while along # = x2(t) that r

v'hlx

_

l*

=

4 T ; V / £ J : ( V ^ - y/v) / t 7=T3

+ vl = v^ + [l - ( ^ P ^ ) *

y/v^+x/v

3

ty/v^-y/v^

(

,

,

7=J r2ar

] - ( ^ p f e ^ " W + v*). y/v^+y/v

Proof. The first one can be obtained by using (2.8)i, the relation between (u,v) and (ri,r 2 ), (2.7) and the knowlogy on (ri_ (*,#), r2_(tf, ar)). The one along a? = #2(2) c a n D e obtained in a similar way. Let us denote the following statements by (Hi) and (H2) respectively. (Hi) (ri,r 2 ) e C1 on Xi{t) and Xi(t) € C 2 (i = 1,2)(< > 0).

42

Quasilinear Hyperbolic Systems and Dissipative Mechanisms (H 2 ) Along x = xi(t)(t

.

~T —7=r • ^

> 0), it holds that 0 < v~^r2x

",.

Along x = x2(t)(t

+ v* < {v~)i

and

' G W 4 , A3], for positive constants A 3 > A4 > 0.

> 0), it holds that 0 < v~irix

+ v i < ( v + ) a and

, ~^—7=7 ^ t i T " £ [^2,^4iL f ° r positive constants Ai > A2 > 0. dt (y/v+ ~ y/v) R e m a r k 2.1.3. (H 2 ) implies that (r 2 —ri)(a, £ i ( a ) ) > ( r 2 _ - r i _ ) ( a , # i ( a ) ) > 0 and (r 2 - ri)(/?,* 2 (/?)) > (^2+ - r 1 + ) 0 M 2 ( / ? ) ) > 0 , for a > 0 and /? > 0 respectively. L e m m a 2.1.4. The classical solution defined in R(T) satisfies the estimates cited in (2.17), (2.18), (2.20) and (2.22) if (Hi) and (H 2 ) hold for t G [0,T]. Proof. Due to the characteristic equations (2.7), it follows that

n{t,x)

= e^±r1{fi,x2{/3))

- j

f* 1 -r2(r; ^(r;*, z ) ) e ^ r

(2.23)

a - r i (r; x 2 ( r ; *, x^e^dr

(2.24)

for any (/, x) G R{T), where # I ( T ; t, x) and x 2 ( r ; t, x) are defined as in Theorem 2.1.1, a and /3 have the same sense as in Theorem 2.1.1 as well. Denote M = max#(T){|ri|, | r 2 | } . If M is reached at some (tyx) G R{T) with | r i ( t , x ) | , it reads that ft

i

M<e*\ri{P,x2{P))\+

-Me^dr Jp *

which implies that M < \ri(/3,x2(/3))\. Namely, either M < ri(f3,x2(/3)) if n ( / ? , x2{p)) > 0, or M < - n ( / ? , x 2 ( / ? ) ) if r1(0,x2(0)) < 0. However, the first one gives a contradiction since ri(/?, x2(/3)) < r 2 (/?, x2{j3)) (due to Remark 2.1.2 and (2.16)) while the second implies a contradiction too (see [78]). Thus, M must be reached at some (t,x) G R{T) with | r 2 ( t , x ) | . Similar to the above argument, it follows that M < |r2(a,a?i(a))| = r2(a,a?i(a)). This shows that M < r J . Let P 1 _i . 2

_i

1

2/

.

_i

1

2/

Frictional damping: Globally defined weak solutions...

43

it is easy to find out that w' = u>(A — fw)

? = H\ - h) by which the estimates (2.18) can be obtained with the help of (Hi) and (H 2 ). Now, we finish the estimate (2.17). Let N(T) — max{\/t;} be reached at

(t,x)eR{T). Since (y/v)' = \{^

+ y/v) ~ \V^,

we can show that

a
/F}

due to (2.18) and H2. Thus, N(T) < max{\/v+, y/ir} in view of Remark 2.1.2, namely, v < max{v + ,i;'} = v+. Furthermore, it is clear that the region r2 < r2* can not be reached due to (2.7)2 and the fact of v < v+. Hence, r2* < r2 < r j follows. This, combined with (2.23) and H2l implies that r\(t,x) < ri 0 . The proof of Lemma 2.1.4 is finished. By using Lemma 2.1.2, it is easy to prove that Lemma 2.1.5. Suppose that the classical solution of (2.3) (2.13) (2.14) exists in R(t), then it holds along x = xi(t) that —1

(V^-V^)

d(y/vZ — y/v)

dt

_

=

y/v

or+^

1

2 ( v lriI +

1

"

!,+

y/VV-

(^Tv?'

while along x = x2(t) that

a Lemma 2.1.6. There exists a small constant So such that the solution of (2.3) (2.13) (2.14) exists in R(S0) for which (Hi) and (H2) hold. Proof. The existence follows from the local existence as mentioned before and it is easy to verify (Hi). Moreover, it can be shown that v G [vo(l — So)1vo(l-\-So)] for some small positive constant SQ(0 < So < 1) if ^o is small enough.

44

Quasilinear Hyperbolic Systems and Dissipative Mechanisms Due to Lemma 2.1.2, we can claim that \im(v~ir2x

+ v*)(t,xi(t))

> 0,

\im(v~*rix

+ v^)(tix2{t))

> 0.

and

On the other hand, we can easily obtain that lim(r;"2r2:r + v*)(t,xi(t))

< Vv~

\im(v~*rix

< >/v+.

+ v*)(t,x2(i))

and

In view of the above estimates and Lemma 2.1.5, we are able to find constants A3 > A4. > 0, Ax > A2 > 0, such that -1

(y/v

-y/v)

^ [^ 4 ^ 3 ]

a

}ong

jg^&^&^jU

x

-

Xl(t)t

and

„„„g ,_„rlx + v* > V5+[1 - 2 ( ^ ~ V S 3 ] > 0. VV+ +

y/v

Similarly, for any point (?i,xi(7i)) with t\ G (TQ,T*}, the corresponding back­ ward characteristic xi(T',ti,x\(t\)) will meet x = #2(*) a ^ (?25#2(?2)) with ?2 < To, and it holds at (?i, a?i(?i)) that 0 < v" 27*1* + V 2 < ( v + ) 2.

(2.26)

This, with Lemma 2.1.2 and the fact of v G [vo,v~] on x\(t) for t G (To,T*], implies, under condition A, that v~br2x + vh > V t F - (y/tr+Vv+)

*(-^=—^—5.) 3 , v r + v^o

at any (t,*i(*)) with t G (T 0 ,T*]. DEFINE

IT

IS

EASY

TO

SEE

T H A ITS

EASY

T O S ET H A T

Define F(£) - f [(fzy) 3 - 1], it is easy to see that DUE

TO

AND

F

FOR

IT

READS

y/vo

THAT

UDUADF

DUFYADIUYFUIUOIUYDASFJHJUIEHB

and Ffa) = 6 for & = ^ ± 1 , it reads that

Due to Vir < Vv+ < 2jkX-

Therefore bcgfd V ^ V^o It turns out then that


0 at (t,a?i(—7-f ' r 2-(*) + Vv-(x))>

ro

l

(

/

• r' 1+ (:r) +

SU

P >A-(*)}

r+(x) in

.£( /

TT * r 2+ (*) + \ A + ( * ) ) '

in

j , \A+(*)}

Frictional damping: Globally defined weak solutions... q+ = max{sup

( ,

x>0

l

• r' 1+ (x) +

47

y/v+(x)),

\/V+(x) SU

P(

/—T-r * r 2+(^) + \A+(*))>

a:>0 y/v+(x)

sup \A+(*)}x>0

Condition B. For some positive constants e and e, the following inequalities are satisfied: 1. Vv~e < y/vo < y/v-(l - e) /v+(l -e)

1. 4. For any £ G [1, ^TWv

2

it holds that

(£ 3 + ^ + 3 £ - l )

1 +

4(£+l)

= ^ V v ° and x/^ 7 = ^J-y/vo respectively,

2

( 1 - g ) tf + i ) ( i + g) ' ( 1 + e)

4

(l-e)

(Z+l) 4

(1 + e) '(1-e)

> U

5. For any £ 6 [1, ' f f l + i h , it holds that 1 , (£3+£2 + 3 £ - l ) ( 1 - g ) 2+ 4(£ + l ) 2 ' ( 1 + e) 6. Denote ^

= ( ^ - ) 3 , # 1 ( 6 , 6 , 1 7 1 , ^ ) = 6 - 171(6 + 6 ) + 1711726,

#2(fi,6,i7i> m^i,S2)

= £i€i(l + »7i) + £2*716(1 + 92), it holds that

jhjfhugju ^ i ( ^ , € o " , % ^ o " ) + ^ 2 ( ^ , ^ 0 " , % ,»7o",?,e) < ( l - » ? o " % ) m a x { ^ , ^ }

jfhugjjfnvh hfgyhcgfh lklfkgjuhjythyjhdfhgfjhf R e m a r k 2.1.8. When £ and e tend to zero, the inequalities B^,B^ and Be hold automatically. £1 holds as well if r l T ( x ) = 0 and r 2q: (x) = 0, and the

48

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

Bn and B$ hold under condition A. This means that the condition B is a direct generalization of condition A. Moreover, it is easy to claim that for any fixed number 6, 0 < 6 < -i 23-1

there exist positive constant e, i and SQ such that all of the above inequalities in condition B hold for this pair of (e, e) if max{\Ar% vv+}

,

and osci; T (x)| + | r l T ( x ) | + |r 2 : f (x)| < £o (for x < 0 and x > 0 respectively). By using the initial data on x > 0 and x < 0 we solve the corresponding initial value problem for (2.3). Furthermore, by the similar argument as used in Lemma 2.1.4, it is not difficult to prove T h e o r e m 2.1.9. Suppose ( n ^ x ) , r 2:p (x)) are C 1 functions with bounded Clnorm such that q* > 0, then there exists a globally defined classical solution (ri+(t1x),r2+(t,x)) € Cl and (ri_(/, x), r 2 _(

ri+ = ri + ( 4/ — sup a (a = v 4 ;) would Rm y 7 + 1 R{^yyy 3~7 make a much more strict restriction if 7 is not nearly 1. We omit the detail of the discussion which follows the same lines of arguement used to establish the result for the case when 7 is nearly one. Instead, we introduce a complete investigation for 1 < 7 < 3 in [79] by different methods. 2.2

Nonlinear diffusive phenomena for entropy weak solution

Consider the problem (2.3) (2.6) where v* > 0 and p(v) = av"1 with a > 0 , l < 7 < 3 , which is the state function for poly tropic gas. We compare the solution of (2.3) (2.6) with those of (1.4), namely [vt = -aP{v)xx {P(v)x = — t/_. It is known from [78] and [79] that if the initial shock is suitably weak, then the problem (2.3) (2.6) with (2.28) has a unique entropy weak solution globally

52

Quasilinear Hyperbolic Systems and Dissipative Mechanisms

in time which is piece wise smooth and contains a forward shock x = X2{t)(0 < t < +00), satisfying the entropy condition, namely, it holds that

hdgftbcvhfgyhfg V VR >

VLy

where (uR,vR) = (u, v)(x2{t) + 0,0, ("L, VL) = (u,v)(x 2 (*) - 0 , t ) . Moreover, it is proved in [78] and [79] that (i) the solution (iz, v) is smooth in the domain Qi and Q2 up to the boundary respectively, where fti =:{(*, 0 } , fi2=:{(x,*)

:x>x2(*),*>0};

(ii) in the domain fi2, (tx, v) is defined as (tz,t;) = {ti+(x,*),v + (;M)} = { w + e ~S v +}> and in the domain i i , (u, v) is defined as

(2.30)

(ti,v) = {ir(x,*),v~(x,*)} = {ti_e~',i;_}, where ii =: {(x, 0} and xi(tf) is the backward characteristic

/ ^ = A(._) = - V f ( t > - ) - * \ l « i ( 0 ) = 0;

(2.31)

(iii) along the shock curve x = x 2 (f), it holds (t>+ - v-)e~Alt

< v+ - v < (v+ - v_)e" A 2 t ,

*> 0

(2.32)

where v = t;(x2(f) — 0,*)>^4» is a positive constant (i = 1,2); (iv) t>-< v(x,*) < v+, f o r x G f l , * > 0 . (2.33) On the other hand, it is known from section 1.1 that (2.27) has a unique similarity solution v*(rj),ri = .x , satisfying v*(±oo) = v±, and V*(T;) is strictly monotone for T) £ R. The main result contained in this section, established by Hsiao and Luo in [74], states that there exists a unique x0 G R such that if S = \u+—tz_ | + |v+— v_ | is suitably small, then sup |ti(x, *) - u*(x + x 0 , *)l + sup |v(x, t) - v*(x + x 0 ,01 *€*

= 0(l)(* + l ) - i ,

r€*

as*->+oo

(2.34)

Frictional damping: Globally defined weak solutions... where v*{x + x0,t) = v* (j**\)

,t**(s+ *o,