Stability of Multi-Dimensional Shock Fronts: A New Problem for Linear Hyperbolic Equations (Memoirs of the American Mathematical Society)

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303731441Q

Memoirs of the American Mathematical Society Number 275

Andrew Majda The stability of multi-dimensional shock fronts

Published by the

AMERICAN MATHEMATICAL SOCIETY

§2.

THE LIlm~RIZATION

OF A CURVED

FOR VARIABLE §3.

A GEl~RAL

3.A

14

OF THE UNIFORGI STABILITY

'iE::: PHYSICAL

EQoUATIONS OF COMPRESSIBLE

Conservation

Laws in a Single

Uniformly

Stable

The Uniform

A.

SPACE COEFFICEINTS: Appendix

B.

KREISS'

for Isentropic

25

and Lax's ·

··············

of Shock Fronts

33

43

HITH SOBOLEV

THE PROOF OF LE~~~ 4.2

SY~{ETRIZER

2 .....

for the Euler

in Three Dimensions

OPERATORS

25

Gas Dynamics

-- the Proof of Proposition

of Gas Dynamics

PSEUDO-DIFFERENTIAL

AND

FLUID FLOH

Space Variable

Shock Fronts

Stability

Equations

Appendix

CONDITIONS

Inequalities

in Two Space Dimensions 3.D

THE MAIN THEOREMS

COEFFICIENTS

DISCUSSION

Shock

3.C

SHOCK FRONT:

AND SOBOLEV

75

SPACE PA~lliTERS:

LE~lI·1A 4.3..... ...... ... ... .. .. .. ... . .. ... ... ........ ... .. ..... ...... 85

AMOS (MOS) subject classification.

Key words and phrases.

Primary 76L05; 35L65 Secondary 35B4o; 35A40

Hyperbolic conservation laws, multi-dimensional fronts, stability, mixed problems.

Majda, Andrew, 1949The stability of multi-dimensional shock fronts. (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 275) Bibliography: p. 1. Shock waves. 2. Differential equations, Hyperbolic--Numerical solutions. I. Title. II. Series. QA3.A57 no. 275 [QA927 J ISBN 0-8218-2275-6

shock

au

-+

at

where F(u)

N

R

3x=

(xl'

...

a L aX

N

j=l

, :ll~ ) , u

j

=

t

Fj (u)

(Ul'

are smooth nonlinear mappings into

(~) ==

(Aj (u)) == A(u)

RM

, the corresponding

'\1) , and the

... j

M x M

1,

...

(Fj(u))

-

, N , with

Jacobian matrices.

Here,

t ~ 0,

noncharacteristic

and two smooth functions domains

G+

and

G-

for (1.1), with space time normal, u+(x, t)

and

u-(x, t) ,

defined on respective

on either side of this hypersurface

N

+

LA. ( u+ )

au + at j=l

auat

+

J

+

au 0 aXj =

N

L

j=l

pw w l 2

+_a_ aX

2

pW2

2

(nt' nx) ,

+ P

so that

where

(wl' w2)

is the velocity,

well-defined function of

p

with

p

is the density, and

p'(p)

>0

p

=

p(p)

is a

and determined by an equation of

(1.7)

where and

v

E

+

and

v

+

+ EV+(X) for x N

u - + EV-(X) for

x

N

>0

where

2

The uniformly

(3.9)

C

l

stable

by choosing

and observing

t + { (rM_j+l,

C

that

and

shock

s = 1 a basis

O)}~ U {t(O J=l ' r;)} ~=l

C

fronts

w

and for

are ~ priori

2

= 0

E+(l,

where

define

coefficients

in the 0)

+ r , r k k

constants. which

definition

is given

of weak

sta-

by

are defined

in (3.3)

and

+

dV dt

L j=l

+

+

N-l +

A.(u+) J

~+

dX. J

(~-

(})

~

F

d~

~ >a ,

t

>a

VI' CPt resulting inthe form (3.11)) where the Bj

are

(m

1) x 1

(m - 1) x (m + 1)

matrices,

matrix.

T

is a

a

j

are scalar functions,

1 x (m + 1)

matrix, and

S

is

We consider the characteristic variety of the

3.3.

PROPOSITION with

2

Is/

2

+ Iwl

min Res ~O 2 2 IsI +lwI =1

=

1

A shock

and

2

IQ(s, w)v+1

front is uniformly

Res ~ 0

~

v

y2/ +12

stable at points

in the complement

of

CV

(s, w)

if and only if

~

MO

)1/2

is a constant

coefficient

(L: g ..w.w.)1/2 ij

lJ

advantage

iw. J

.

s+lawk+E !\:

(

g .. W.W.

lJ

l J

)1/2 vII

where

l J

of the boundary

given initially

in the region,

xl

> o.

Thus, the basic unperturbed

has the form,

d 1/2 wl - (~p)

I p_ >

1/2

+ 0

>

wl - (~) dp

I p+

state

fied, we use the remark

in (3.10) so that it is sufficient

to consider 1/2 (QE) dp

per-

I

+ P

Xl > 0 +

+

Also,

(w , w , p) 2 1

by

(the

(w~,w{, p') . From

(1.4) and (3.10), the linear equation

dW~

dW~

1, 2

t > 0

for

P

,

+ (A - (JA )u bO0

(1) The boundary value problem in (3.31), (3.32) is uniform Lopatinski

al+0~

a ..£. '> d'-

0

I (c al

-

2

+ 2 + (wI - 0) ) +

-2

c(w

l

points where

s ~ 0,

s ~ w(w~

- 0) ,

W2Jl < l pJ

P

p

- 0)

if and only if

I+ P

+ (w+ 1

0)2

+ W( w 1 - 0)

+ l

w(w this directly

but we omit the tedious

is violated.

All compressive

py condition

straightforward

shocks with

but, by continuity,

Then,

0);

calculations

T* < T+

0,

the analogue of

ANDREW MAJDA

46

(3.42)

Ml(s,

0

Iwl)

~

dv dx l

v

-s + wl -

---1

0

where

Ml(s,

structure

Iwl)

is the

assumption

3 x 3

matrix

in a neighborhood

0

in (3.22).

This verifies

of any point with

Iwl

1 0;

With the form in (3.42), it is a simple matter to determine E+(s, w) For

s

1

by using our previous 0

and

s

1

calculations

Iwl(w~ - 0) ,

(3.24) - (3.31) above to translate

provided

we can proceed the uniform

that

s 1

the block the

a basis for Iwl(w~ - 0)

in the same fashion

stability

as in

of shock fronts to

2 + 2 (~ - l)c + (wl - 0) (-------) 9,

Since

f(O)

2[~J

,

and

+ 7 ,

>

EO

0

so that if

lal T + (b) T TO

and

Ixl TO

It

'

min Res ~O

\ e+ (x',

t, s,

w)

I ~

y

is 12+lw12 = 1 Ixl+lt\ ";;;2TO

vllien(4.11) is satisfied,

the matrix projection

+ e ) e+ 2 \e+1

(v,

min Res>O

IsI2+lwI2=1

Ip(x',

t,

s,

w)Mv)

P

is well-defined

by

for all

v+ E E+(s, w)

•

By setting

v+

o ,

we obtain the fact in (4.12)

(Mv+, e~)

le;12

Below, we temporarily the simplified (ll

2

+ 1~12)1/2

xN

regard

notation,

x

=

(x

for I

We suppress the

,

t) xN

and

A (~) = II

L

mal:

161';;;;13 (~,n) h;;>-l

DS_ (t;,n)

Here

is short-hand

for

D

130 131 Dt;l

Sn Dt;n

n

the square of the Sobolev norm of order

s

(b( •, t;, II ) )2 s

and

in the

x

HS Sm,n comp

II

II ,m,B'

of the symbols involved; this dependence

s

dix A but plays no essential role here.

(1)

L2

continuity

S SO,n s ~ [~] + 1 and a(x, t;, n) E H comp 2 Hs+l S-l,ll then comp , 2 2 (a(x, D ' n)v)O ,;;;; C(v)O x n2( b(i, D ' n )v)~ + (b(i,

If

x

variables

denotes

with

-

(t;,n)

to denote symbols

is untangled

in Appen-

(2)

If

s 1

;;;. s [!!.] + 6 2 or

with

m2

0

(a)

The Adjoint

1

satisfying

m

l

I b EH

, m

+ m2

l

s

m2,T)

comp

=

and

8

0

or

1

Formula

(b(x,

*

D , T))) x

m -1 R 2

where

s ml,T) a E H 1 8 comp

,

b

is an operator

*

m -1 T)) + R 2

(x, D

x'

T)

with the following

continuity

T)

m2-1

( RT)

v ) 0 ,,-;;; C( v )0

m -1 m -1 T)(R 2 v )0 + (RT)2 v )1 ,,-;;; C(v )0

a(x, D-, x if

D_, T)) = (aob)(x, D-, T)) + x x m1+m2-1 1, (RT) v)O ,,-;;; c(v)o

T))obex, m1 + m

2

=

m +m2-1 m +m -1 T)(R 1 v)O+(R12) T)

If

s a E H 1 80,T) comp

for

?o v) 2a

Ixl C2

81,T) with and

a(x,~,

0

>0

s;;;.2[!!.] 2 then

-m1

T));;;. 01, (a(x, Dx'

,,-;;;C(v)o

0

>0

T))v, v)o;;;.

also valid

for uniformly

well-posed

pseudo-differential

and that the symbol of this symmetrizer

L

a

Ct.:

+

-1
N

+ [A2(1-(~+m2))

-

+ s +

n 2 .

m +m 2 R 1 (x, ~, 11) satisfies the estimate 2

2

x 11 (m1+m2-N\1

+ 1~12)N+s1-S2])

s

2

;;;'3+s+~

2

For the product s2 b E H

comp

formula

in (2) of Lemma 4.2, we first remark that when

m2,n

S

m -1 R 2 v

n

does not have a symbol with compact

[%J

+ 2.

support, this

From (A-IT) we

II

e

i(x-y)·~a(x,

n)b(y, ~, n)v(y) dy d~

~,

+ (a

0

m -1 R 2 )v

n

TI

m -1 ,,;;; C( R 2

n

m -1

,,;;; C( R 2

n

)2 v 0

m -1

,,;;; C(R2

n

v)2 0

)2 v 0

sl,m, N '

If

(A-2O)

C

S 80,1l ll) E H loc

with

A(x, t"

ll)

~OI

then

Al!2(x,

t"

II

and

where

A(x, t"

,

is a Hermitian 8

>

°

and

s ~

of

[%J

+ 1 ,

0 ) E HSloc 8 ,1l

A E HS. 8l,1l comp

s ~

[%J

+ 3 ,

TIAlI

depends on only a finite number of seminorms,

independent

matrix

s ,1,6

II

sit ion A.2 to the operator,

A

=

a - 8 111, l

and then choosing

II

sufficient-

ly large.

Choose

J

¢2(x) dx

=

¢

to be a positive

smooth rapidly

1

and define the multiple

symbol,

decreasing AG(x,

t"

even function with Y, ll)

,

by

Then

AG

belongs

e 0.,6

depend on

Furthermore,

where RG(x,

BG(X,

E;"

E;"

s~:i/2'

to the symbol class

n)

=

A

the quantities

shows that

(2TI)n(iD~D )AG(x, D , x + y, n)j sy x y=O

n) E s~:nl/2

e

only through

Nagase

From standard

since only a finite number of seminorms constant

and satisfies

2 L -continuity

for

AG

and

estimates,

are needed for estimating

and the remarks in (A-22) and just below (A-22) apply.

Rn

G

has a symbol which satisfies the estimate

eTI All

So

+2 1 N

"

the

We claim

clA (x + tz, a:2 x

seminorms in

HS Sm,n comp

follow by simple approximation argu~ents.

~, n) dz dt

d

N-l

~

+

o

L j =1

d

d

t) -"- + A __ "x..

A. (x, J

oX

j

-11

0

j~

N-l

~

dt

+

L

j=l

(A.(x, t) + a.(x, t)) _d_ J

3xj

J

+ (AN +~)

WE assume that the coefficients for some

aj

{(x, t,

11, ~,

s

with

E HSlac

w)/Ixl

+

It I

Aj(x, t)

s ~ [Il+lJ + 1 2

d

dXN

belong to the

and that these coeffi-

N-l -(~ + aN)-l((i~ +

n)I

+

L

j=l

The matrices

M

a

l (zO)

so that

hw.) J

have the property that given any point

there is an invertible transformation E

(A. + a. J J

V(z, a) defined for

Zo

E S

Iz - zol

+

la\
a

is

(We set that

IaI

Bj ~ Fj

So

+

0

- II KII

the composition

for 0 0

sO'

,

j

< min

=

1 .

-£

E

Then, with

~ EO'

So =

0 0

SO' ,