n
I11111111111
m 1111I
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 <0
are smooth functions with compact support in
~ ~ 0
is a small quantity, we would expect that there are functions
u;(x, t) , u~(X, t) xN = WE
(x',
t)
and a corresponding surface, SE(t) , described by
with
x' = (Xl' '"
au+ _E_+
at
small E. that
j
u(x, 0)
u
'~-l)
so that
N
+ A (u ) j j=l E
L
To compute the linearized problem which we study below, we assume
(u;, u~, WE)
+ du
depend smoothly on
I
dE E=O
dUI
dE E=O
E with
~I dE:E=O
satisfies
an evolution
dary values,
equation on the boundary which is coupled to the boun-
(v+, v-),
a half-space
of the solutions
through changing variables
tion in (1.10).
by
of hyperbolic
~
to
-x
N
equations
satisfied
in the second equa-
With the notation
A (u+) n
A. (u+) J
0
0
A.(u-)
A.
-
0
0
~
J
J
v b
j
=
t(v+, v-) ,
=
Fj(U+)
- __ Lv =
Mv
=
-(~(u+)
Fj(u-),
dV
at
0
N +
-(AN(U-) -
0)
)
L
dV
A • -d j=l J xj
bO
=
=
F
- o)v+ + (AN(u-) - o)v- , u+ - u
of (l.ll)?
Standard mixed boundary
that for the uniform
Lopatinski
laI 1+la2tla
we use (v>
2
x,T'
3
value problem theory
problems,
I=s
(see
we estimate the weighted
(Ix..=o
2 Ivlx,T
[6])
j~
to denote the unweighted
dictates 12
with
Pj(~)
boundary
,
waves,
a polynomial arising
in
~.
We consider the one-dimensional
from the surface variations,
Res>O 2 2=1 s +
I I I wi
for all boundary
values
(v+,
A) E E+(s, w)
•
space
l
=
1
p
denote the specific
(1.17)
are always satisfied.
equation
of state,
When M2> the right
1,
Ph)
=
no estimate
In particular, AT-Y ,
Y > 1,
all
of the form in
hand side is replaced
by
(g)~,
for an ideal
n
~
compressible
(1.18) is
valid
for any fixed
when large
s.
We contrast the uniform multi-dimensional for compressive
stability
shocks with the standard wuation
iy(-(u (0, 0, e
-u -
+
that always occurs
of state,
)y+(f2(U ))t) +)
P(T) = AT-Y
,
bas
+
I b.iw. j J J
,
plete type of stability cuss here, developed
for the equations
Professor
Joseph
to variable Oliger
surface
of gas dynamics;
coefficients.
structure
in the final part of §3.
coefficients
The author for making
amount
criteria
only guarantee
would
and do not
like to thank
their unpublished
of the three-dimensional
to him -- this saved a considerable
which we dis-
the stability
fashion,
for constant
and Dr. Arne Sundstrom
for the block
variations,
however,
in an analytic
in the sense of Hadamard
generalize
calculations available
including
in [3], when interpreted
well-posedness readily
analysis
of effort
Euler
equations
for the author
the compact aE
RN-l
MO
parametrizing
MO
the outside
of
the outward
unit normal
The above mapping
IT I <1\ with
a tubular x
with
of
of the shock surface,
S(t)
d(x, M ) O
,
1
<:2
We let
RO}
vector S"l
+
denote
MO
We also let
n(o.) denote
MO ,
and consider
the mapping
neighborhood
the equation
through
with the variable
{xl Ixl
the inside
rarametrizes
< 02 , in
of
provided
M
O
there
(2.1).
that
a
is a unique
We parametrize
the
at later times by a function,
so that
(2.2)
8(a, 0)
S(t)
in
where the time interval max O";;;t ";;;T
RN
of
to the hyper surface ,
so that given any
,
S(o., t)
MO c
and
S"l
and
o.(x) well defined
position
MO ,
smooth hypersurface,
18(0., t)!
< 01
(2.2)
0
{a + 8(0., t)n(o.)}
is suitably
restricted
As in the introduction,
to
O";;;t";;;T 0
we consider
where
a perturbed
o
family
of shock front
solutions,
differentiable
with respect
to
E,
and
where
SS(t)
troduce
=
{a + $s(a, t)n(a)},
a change of variables
and (1.2), (1.3) are satisfied
which maps both the perturbed
shock fronts to the fixed surface, We choose
p
d(x, M ) O
°2 <2
distance
of a point to a set in
MO x [0, T],
to be a fixed smooth cut-off and vanishing
for
for
function with
T
and unperturbed sufficiently
.
N
d
dxi
L -
S d (p$ n.)J dXj
small.
p == 1 for
) (Here, d(x, d(x, MO) > 02 n ) we change variables via R
j=l
indepen-
is the
The explicit
form of
+ BO'
since with the analogue
+
B
j
,
j
=
0, 1, ... , N
are of no interest
of the norms in (1.12) and the estimate
N
1- + I at j=l
+
A-' J
"I
a
aXj
below
in (1.13),
M
o
x
(n~,
where
W(a)
is the Weingarten
into itself, and Riemannian
E
VtanS
metric on
MO
map, mapping the tangent
is the gradient of
SE
induced by the Euclidean
With this fact and the notation
space of
MO
at
a
with respect to the inner product in
from below (1.1), the Rankine-Hugoniot
RN jump
dS dt I t=O
For any
a
coefficient
(2.7) away from ficients
+
Aj
MO'
t
with
It
I <~
TO'
planar shock from problems
without
are constant
the coefficients
and
for
loss of generality, Ixl ~ RO
the frozen from (2.7)
we assume that the coef-
and have been extended together
in (2.10) in a fixed fashion as smooth functions
for
with
(2.15)
IF+I~,n,T
J: J~
e-2ntIF+12
J: J~
e-2nt IF-12 dx dt
dx dt
+
IF-I~,n,T
(g }2 O,n,T
as in
(1.12)
~+ x [0,
T] ,
to denote ~
these
x [0, T] ,
_
rJ
2nt
o Mo
lgl2
e-
norms relative and with
(¢}2 s+l,n,T
V
ds dt
to the
=
(v+,
+ (v + }2 s,n,T
sets
v-,
¢)
M x [0, T] , O ,
we set
+ (v -}2 s,n,T
(F+, F-, g)
for ()2.7
and ( 2.10 ) , vanishing for
F+
in
t
~+ x [0,
<0
00)
and
T
> TO
with
+ + L v a
8v8t
N
+
L
j=l
=
a
where (a,
b)
let
b
= (bO' vanish
bl, b2,
I s,T,R
for
Ixl denote
m)
.
> RO
We assume or
It I
the ordinary
(a~, a~) J J
that
>
TO
are not arbitrary
all perturbed and with
Sobolev
norm
relative
to the open
set
Ixl
O
It
I
< T
coefficients
the obvious
of order
O We set
but con-
s
notation,
we
of a quantity
THEORE},!2.
Assume that
L+ 8.
+
-
La' La assume that
satisfy
the block structure
(2.13) and (2.14) are satisfied
tions are valid for the perturbed In particular,
the unique
boundary
strong solution,
•
assumption
when
(a,
b)
value problem V
=
of Appendix
B.
=
Then
in [2.18) and (2.19).
¢) ,
(v+, v-,
2
II vii 0 ,n,
T
(B + b)
s,RO ' and
I (A
+ a, B + b)
I s
where
s
is a fixed integer with
s # 2[~J + 7 .
(0, 0) .
of (4.18)
X[O,T](t)
is the characteristic
function of the interval
fine cut-off data for (2.18) and (2.19) by
this solution
coincides
in (2.21) is valid.
with
X[O,T]V
[0, T]
and we de-
<3 + + ...Y-
<3t
~(u
+
) -
0
has exactly
p
positive eigenvalues,
~ A+ - 0 M-p_l +
>
0
with right
+
r , ... , r _ + M M p l
~(u-)
- 0
has exactly
A~ - 0';;; ., . .;;;A~ - 0
<0
~
negative eigenvalues with right eigenvectors,
M
L
0)t)r~ J
j=M-p+l
~
L
j=l
where the
a~(s) J
aJ~(x
0)t)r~ J
are arbitrary functions vanishing for
s
>
O.
The follow-
+
rM_p+1
r
- ~
p
=
+ r p
for any
p
and
+ u - u
is almost parallel satisfied
where
s
when
=
n
i~ +
so that the condition
and
n>
0
and discussed
and there are no explicit
special with
form,
in
(3.6)
(v
Res >0
+
,
e
st+iw·x
in (1.15) is satisfied.
E+(s, w)
above equation
91.14).
Following
provided
that
solutions
,
)
is automatically
Recall that the space,
shock front to be weakly ~
define a planar
w
r
(3.7) is imposed, thus, the assertion
ready been defined
M
+ k
to
,
of (l.ll) with
and belonging
to
F == g
has al-
Hersh, we
+ dim E (s, w)
=
E+(s, w)
We note that it follows from Hersh's
,
lemma
0 with the for any
([4])
s
that
and
for (B)
We prove bility
n >
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 bO
(3.18)
for
xl
o ,
(w~, w{, p')
.1:...£L,
dW~
__1 + (W+ - (J) __1 + + _1_+ w 1 2 dX dt dX. 2 1
i
for
0
+ dX. 1
xl > 0
I
g
t>O
where
(3.19)
2 + [p] 2
[p]w
[p]w
2
b
O
[pw ] 1 [p]
b
1
[pw ]w2 1 [p]w
2
is given
qy
ANDREW MAJDA
36
and
(3.20)
+ 2
w
+
0
P
"2 c
AD
+ w2
+
0
P
0
0
2 c 1 2 c
+ w2 2 (-) c
-s
(3.21)
. + s + lWW 2
=
~
v
1
~
for the transform direction corresponding
~I
w
2
v
2
/2
( )-1 ~') ('" P w1 + p+c
" v 3
/21
( ~I -w1 +
1
(
)
p+c
-1 ~I
P
)
A tedious
calculation
the ordinary
using
differential
(3.22)
-s + l
l2(w+ 1
-+
E (s, w) ,
2 _
d
=
Ml(S, w)
discussed
c
2
0
+
2
0)
h(w~- 0)
-s + c)
+ (w l
-
0
+ c)
0
in (3.22), we can calculate
in the introduction
- (wl - 0)
satisfies
-ciw
/2(w~ -
-ciw + I2(Wl - 0 - c)
With the form of
v(s, w, Xl)
-ciw 0)
-ciw
r
that
equation
(w
dV dx
(3.18), establishes
the boundary
space,
and needed to check the uniform
W(w+l - 0),
Thus, with
{e~, e;}
1812
+
IwI2
span = 1,
E+(s, w.) Re
A
(3.27)
>0
,
except at the single 0 .
iw yClv3
A
V
l
v
2
where
8 ~
Re 8
(3~3
Xl
0
point
defined
in (3'.2;
Next, we explicitly ditions
s
calculate
the Lopatinski
in (3.27) using the basis
+ A2(w~ - 0)
=
0
Lopatinski
determinant
Therefore,
at points where
[+(s, w)
for
so we anticipate
..J
T
0
and
of the boundary
con-
from (3.24); from (3.25),
this factor.
of (3.27) using
s
determinant
The result is that the
(3.24) is given by
-s..JT W (+w
l
- 0) ,
LEMMA 3.1.
Assume that
a
2
>0
(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+
the first condition
•
satisfy
(3.:
one can d which
es-
Liu's entro-
in (1.17) is violated
£Q.
+ div(pw)
aat
(pE) +
at
where w
=
+
3
L
j=l
f-
(pw.E + pw.)
xj
J
0
J
E = ~ Iwl2 (w3'
state,
w2'
wl) 1
p(T, S) ,
T = p
and the entropy,
(1 - shocks) propagating
along the
hanical
from (3.15) together
shock conditions
xl
e ( T + , P +) - e (-T , P -)
direction
are determined
with the additional
1 (+ + 2 T
- T -)( P + + P -)
by the mecjump con-
dp(p,
s)
>0
dp
given in (1.16)
P = exp ( -
S)
C
V
T
-r '
2
( 3)
When ( 3.4)1
is violated but
(t - 1) + M
the estimate in (1.18) is valid.
(W'
+ iw
x w~, .) 3
I
and defines
/'\,
W
W
X2 2
At
+ W
W
x3 3
Iwl
I
3' w2' wl'
p', s/)
Q,
>
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)
<0
when
a
l
<0
and
f
is increasing when
only happen when
<
f(.l...)
i
1
(0':2) , the proof of Lemma 3.1 follows easily. d
a
2
~ 0,
this can
IIvII ~. n
<
00,
vanishing
for
t
<0
mulae in (2.18) and (2.19) that
+ + a
Lv
IF+,n 120
+
IF-,n I~
+ (g}20 ,n
is finite.
We also require that the coef-
There is an S
A E H
S
loc '
with
B E H
s >2[~J
,
and
+ 7 ,
>
EO
0
so that if
lal T + (b) T <00, s, 0 s, 0
the function
V
lal + Ibl
<
EO '
for some fixed integer
satisfies the basic estimate
2
II vII 0
,11
IA
and
1
(All of the quantities
+
als
T
R
, 0' 0
(A + a, B + b)
I
s
in (4.5) are assumed to be finite.) V
as
V
= I
V
j,k
where j,k
We choose the partition ordinate to a local cover by coordinate
in (2.13) on the coefficients
guarantees
of unity
systems which flatten the boundary.
that for
It
I > TO
and
Ixl
' O
for a fixed finite number of functions, we map the boundary (1.11), by utilizing
~k'
locally to a half-space a standard partition
where
supp
¢~
[-2, 2J.
If
and use the folding map above
of unity argument, we only need to
2
IIv.J , kilO , T)
~
C (
2
IF+
j,k
1
O,T)
1
Provided that the constants,
Cl
+
I
2
-
Fj,k
1
O,T)
T)
and
C , 2
depend only upon the quantities
appearing in (l~. 5) to deduce the estimate in Proposition 4.1. Vj,k = (vj,k'
¢j,k)
==
(v;,k(x,
x',
t),
vj,k'
¢j,k(x/,
Here
t))
fies the half-space problem
-
iav.J,k
(1.11) built from
L+ a
F.
J,k
and
La
from (2.18) and
(M +
Vj,k
vanishes for
m)v.J, k
Ixl > RO'
F. k
J,
+ (bO + bO)(~·
J,
t
< 0,
and
k)t
1 > 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
sl ;;;. s[~] + 6, and
provided
that
C n) E HS s, "comp
Ix I
< RO
supp v T)
s {xl1xl
> 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<
a )
in
HSSO,n
N-l
L (A
_d_ + N dJ
+ j
a.) J
boundary
depends
_d_ + dd t dXj
conditions
continuously
on
La
satisfies
that
Ial
stants
the block structure
± IK
01'
homogeneous
I <
EO
°> ° 2
where
assumption
EO is sufficiently
and a matrix-valued
-1
+
aN)
is Hermitian
*
-
01 KKKK + R(~
+ ~)
small.
function
of degree zero with repsect to
R(~
form Appendix
B.
Also, suppose
Then there are con-
K),
R(xN, x', t, ~, w, n, a,
(~, w, n)
such that the following
•
~ 011 ,
N-l (3)
L
Re(R((i~+n)I+
iWj(f:'j+aj)))~02rn,
Ixl+ltl";;;TO+RO
j=l
(4)
Fix
s
with
are finite.
IIKK" S,O,S function
s ~ [n+lJ + 1 2
of
~
and assume that
Then the function
with values in
Hs-l So,n comp
IA
R(x , N and a
+ al
.
)
s,TO,RO
and
is a continuous l
C
function
of
We define
the boundary
e+( b x , t, ~, w, n) I
condtions
Lemma 4.2 provided that
in (4.20) with all of the properties le~1 + Iml
< EO
guaranteed
where we fix the integer,
s,
in with
+a
-)-1(-L
a
+llw )
e-llt-g
for xN
I I -;;;
V
so that
0,
t
>0
dx' dt
(w, R(~)v)
The first steps in the proof of the estimate in (4.7) involve a quantized version of the argument in [6J. take the
L2
inner product with
We apply w,
R(~) to the equation in (4.21),
and integrate over
x
N
to derive
I
O O
Re
dW
A
0 (W,
R(~)
dX
~ N
N-l
OO
+ Re
Re
J:
I
0
(W,
(W, R((AN
R
L A.
+ ~~)-l ( j=l
(w,
(w,
R(w))
I ~=O
d~. + ~ J
dd;N) dJ)j
II:
(4.27)
J
(;~)w)
~I
+ (n +
B)w))
d~
(w,
Rw) A
I
;;;. -O~ 1(w,
* K0CKN)
0
+
1 "2
(w)O2
~=O
*
KKPb(x
,
, t, ~, w, n) ,
o
*
-1 ( 3 ( K P ) b
to the boundary
2
0
(M + ffi)w)O
+ (w,
conditions
A
Rw)1 ~=O
in (4.22) to obtain
provided
Tl
> C2
From the composition
OO
Re
J
0
and adjoint formulae
( w, R( (A N
in Lemma 4.2, Proposition
N-l + ~
)-l(
\' dW dW (n + B)w) j~l Aj dX + + j
at
'I
A-l,
where
p(XN' x'
Ixl + It perties
I
t)
~RO.
is a smooth function
of compact
From Lemma 4.1 and standard
in Appendix
8obolev
estimates
P
=
1
for
using the pro-
(B-7) and (B-8), we deduce that
Q( x,,
t , '0, r- w,
Q(x', t, D ' D , n) x t
ply the operator
support with
use the composition
formula
E HS
n)
8-1,
n
comp
to the boundary
in Proposition
4.2 with
m
conditions l
=
-1
and
in (4.22), m2
=
1 ,
and also (4.17) to obtain that
provided
> C2.
n
that
By combining
(4.36) and again choosing we have derived 1/1
= e-nt-¢j,k
n
sufficiently
the main estimate
and
w
=
e
-nt(-+
the estimate
large with
in (4.7) provided
--)
vj,k' vj,k
.
in (4.39) with the one in
n
> C2
'
we see that
that we recall that
where
F+, F_,
g
vanish for
2
I
v+ - v+ 1 + n O,T
12
++ L v n - F+, 0 T +
1
1-L v
t
1
<0
v- - v n
1
and the coefficients of
-\2
O,T
+
(~
n
2 «(+ n - F -, 0 T + B vn ,
- ~ )2
1,T
----+
L+
0
the square root of the positive coordinates,
this operator
REMARK.
The boundary
is pseudo-local
self-adjoint
has the principal
value problem
when the unusual
norm,
operator, symbol
(-"'If + 1) 1 0
(I gijwiWj
+ 1)1/2
in (5.1) does have an adjoint which 12([0, T], Hl/2(M )) O
,
is used for
satisfYing
the condition
value problems
in (2.13), it follows that the regularized
from (5.3) are uniformly
There are constants, of
E
for
solution
mating tangential tion, applying
Cl
well-posed
and
C2,
so that if
E ~ EO
yE
provided
that
boundary E
satisfies
independent is a smooth
of (5.3), then
derivatives
the estimate
M
E [vt
an
of in
'
Y
along
(5.4)
(M -vM
MO
to these tangential
+ 1)1/2]~ 0
by differentiating
'J'
the solu-
derivatives,
and
.;;; c
l,s
(
IF+ 2
IF
+
1
S, T)
2 s, n + (
1 -
T)
REMARK. Here the dependence of
C l,s
and
C 2,s
)2
g s,
on the coefficients is
of no importance; however, for the nonlinear problem treated in [10], such
PROPOSITION 5.1.
r any
0
< E .;;; EO'
(5.3) has a unique strong solution, VE long to
HS(Q+
x
[0, TJ) , HS(Q
x
the boundary value problem in
Furthermore, if
[0, TJ) , and HS(M
o
x
(F+, F_, g) [0, TJ)
be-
respec-
HS(n_ x [0, T]) ~ HS+l(M x [0, T]) .
o
Next, we use Proposition 5.1 and the estimates in (5.4) and (5.5) to con-
as
E. ~ J
O.
differentiability
By definition,
this means that
V
is a strong solution.
of this strong solution follows V
immediately
The
from the esti-
E
j
of Theorem
I, we only need to establish
N
L
A(n)
Proposition
+
A.n.
5.1, for a fixed
0
J J
j=l
N 0
L
j=l
A:n. J J
E
< EO
.
Vrr)
denote the projections on the positive (negative) eigenspace of
where
f
a-V
+
belongs to
E(-6
M0
tan any fixed
t
12([0, T] x
+ 1)1/2
and fixed
MO)
and
n ~ O.
The operator,
is an elliptic operator of order one on E
>
0
A(n)
with principal symbol satisfying
12(MO)
for
From (5.8)-(5.10) and standard arguments we deduce the following properties of the soltion in (5.7).
(Below, we set
(Rn(f) E
1 T
s+,
the mappingf
f -
e-ntf
and
denote
< CS (E)(f)s,T l----->
Rn(f)
is given by an
elliptic pseudodifferential operator with principal symbol in
S-l,n
given by
~
by
well-posed for
+ L,
pseudo-differential L
-
boundary
conditions
2 1FT) 1
- O,T
+ IFT)12 + (gT»2 +
O,T
O,T
finite, the boundary value problem
in (5.12) has a unique strong solution. (2)
For
T) sufficiently large and
!FT)12 + IFT)12 + (gT»2 +
S,T
- S,T
O,T
F~, gT) with
finite and
and
wi
E HS(M
Mox[O,T]
(5.15)(a)
T)* + (1+) u
F
-+
in
n+ x [0, T]
(16)*u-
F
in
n x [0, T]
(u+, u-)
vanish at
t
T
o
x [0, T]).
(8 + B(V
RnT))*, t an £
L2
inner product on
M
O
x [0, T]
is computed relative to the
and from (5.ll) it follows that this oper-
ator of order zero (with norm depending on
£)
:
also,
A+, A
are defined
that for smooth solutions of (5.12) and (5.15), respectively, with gn
= g =
0 , the Green's formulp
(w_,
(L ) *u
tions as in [14] can be repeated "weak
=
strong."
Once
L2
"weak
with only changes
=
in notation
strong" is established,
to establish
we can use the
PSEUDO-DIFFERENTIAL
The assertion smooth symbols
provided tiation
that
in
s ~
t" c"
WITH SOBOLEV
SPACE COEFFICIENTS:
in (l)(a) follows
the standard
L2-continuity
SO,n.
By Plancherel's
So = [%]
under the integral
with the remark that a ( x,
OPERATORS
+ 1.
for
theorem,
The result
in (l)(b) follows
of the representation
na(x,~,
proof
n) E HS SO,n comp
in
(4.14)
provided
from differen-
and (l)(a) together
that
S
n ) E H comp S-l,n.
To verify the assertions and adjoint
a(x,
formulae
in (2), we follow the proofs
given in [5].
sl ml,n ~, n) E H S comp
Therefore,
we consider
for the composition a multiple
symbol,
8
and
b E H
+ 2
and
2
camp
m2,n S
with
m +m (R 1
2 ) v 0
8
m +m (R 1
n
2 )2 + v 1
m +m
n
2( R 1
n
2 )2 v 0
~
1
:> [~] 2
8
2
:> 2[~] 2
+ 6
e
J
-iy·e
J d(x,
d(x,
E"
y, n) dy
E, + e, e, n) de
(A-7) A
d(x, E, + e, e, n) -
la [""
RN(X,
ensure continuity the appropriate
Y
E"
a ea (jD;<:)d(x, [" e, n) a! >
n) -
of the associated function
space.
J
~(x,
E"
e,
operator with
From the explicit
n)
de
s = So -
(m
l
+ m
2
- 1)
form in (A-l), the defini-
A
tion of
RN
in (A-7), and the Taylor remainder
on
formula
we derive,
with
.,;; c e N+S 1
I
L
sup N +N =N 0
1";;5
1
rals over the two separate the region
I e I .,;; ~ I~ I ,
regions,
lel";;~
1~1
and
Ie I > ~ I~ I .
For
the inequality
sup <1
o
provided a • )
that
s2 ~ N +
if
s2
>N
+
+ ~.
(The constant
lei >~ I~I ,
For the region
however,
s
s
+
%'
C
the inequality
we derive by similar
depends
upon the support of
in (A-ll) is not useful; estimates
as in (A-12)
(A-14) CN([A2(-N+l)I~I]
provided that
s2
>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\ <
Ej(O, zo' 0)
where
=
0,
Kj
is real scalar,
and
Cj
is the nilpotent
matrix
~ax 2
2
n +1; +w
,;;;; C B ( z 0 '
(V) Is
I1jJD~n,l;,w)
IA.J
+ a
j
I s, T 0R 0 )
=1
Ren;;;'O
C
B
1jJ
is a constant
is a fixed nonnegative
supp 1jJ
s {zl Iz -
independent
of
a
provided
smooth function of
zol < El(ZO)}
and
1jJ
=
(B-6)
always exists provided
1
2ni
These projections
V ,
l Iz - zol <-2 with the smoothness
that the operator
J
(M
!K.-zl=r J
are well-defined
with E
for
1
Next, we verify that a transformation, ties in
z
that
and
I(M
a
a
- zI)-l
L
is strictly hyper-
dz
- ZI)-ll > cO>
proper-
0
for
IKj
- zl
E (ZO) 3
=
r
provided
that
r
is fixed appropriately
Her and below we will use the following S
Ifgl
s
If
So =
(B-9)
max Res >0 v+ E E+ (x' , t ,s ,w )
1
2
So
C
.;;;Co
and
Co
and
n+l [-2-J + 1
IK(x', t,
n, ~,
w)v+1
Iz -
~
Yllv+1
zol
properties
as regards multiplication
F(x, t, f)
depends only on
lal +
two well-known
.;;; C If I Igi s s s
Ifj
where
:>-. _ [n+ lJ +
and
and
<
max
n 222 +~ +w = 1 Ren ;;;'0
n+l So ;;;. [-2-]
+ 1 .
(1) R
is Hermitian;
°
R ;;;. 11 ;;;. c2nI ,
( 2)
C~~;KK +
(3)
Re(RM ) a
z E S z E S
,
(B-13)
max 222
n
=1
+~ +w
Re;;' 0
where
EO
is sufficiently max (Ial + Es
cients satisfy
S
small,
is fixed, and the perturbation
coeffi-
IKI) < EO
Z
Kreiss builds the symmetrizer
=
n
with
0
in a neighborhood
of any point
Zo E S
in the form
D.(zO) + B.(z', a) + iF.(zO) J
where DjC
j
Dj(ZO)
*j =
+ CjD
Fj(ZO) in Lemma
is an appropriate 0
of
[6].
J
constant
and has the explicit
is an appropriate
4.4
J
symmetric matrix with
form from Lemma 4.1 of
[6].
The matrix
constant anti symmetric matrix with the properties
In the construction,
hat they can be chosen independent
of
a
j
first
and
K
D
j
is chosen, then
for sufficiently
Fj ,
small
fixed neighborhoods. The size of shows that
Dj(zO)
Dj(ZO)
depends on the constant
Yl
in (B-9) and Kreiss
can be chosen in the form of his Lemma 4.1 and obeying the
[Here we have abused notation
shows that the entries of
and called
Dl
=
Rl
and set
F
j
dE.
atl
for
(zo'O)
n
>0
and
n
< E5
The construction
of
B.(z', a)
involves the implicit function theorem.
J
Consider the bilinear
F(B, M)
matrix function
defined by
* + M* )(D. + B) (D . + B)(C. + M) + (C. J
By construction, this map at
D
j
(0, 0)
J
J
was chosen so that with respect to
B
F(O, 0)
J
=
is given by
0
B
j
from (B-14) as given by
G(E.(z',
a))
J
Since
Ej(o,
< S6(zo).
0)
=a
,
Furthermore, from the smoothness properties of the transformation
V in (B-6) and (B-S), it follows that
where
~
is a fixed nonnegative smooth function with
lli-DS(n,e"w
max 222
n
uniformly for
t"
)(B.)!
J
s
supp ~(z) ~ {zl Iz - zol
,,;;; CS(zo)
+~ +w = 1 Ren ;;;'0
Iz' - zol
+
lal < S6(zo)
Since
Ej(zo' 0)
=
0,
it follows
from (B-14), (B-16), and (B-1S) that
< S7(zo). The property in (2) of (B-12) is so guaranteed by (B-15) provided that we again impose the restriction Iz - zol
provided that
+ lal +
Iz - zol + la1
IKI < sS(zo)
. The discussion around a point
Zo with
n>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' ,
<
provided
it follows from (B-6), (B-20), and
formulae from (B-8) that
I I
II KII
[n;l] + 1,
max ( a zES
+
I K I) ,,;;; I a I
So
+
EO .
The estimates
regarding
the symbol norm of
R(~,
0)
and
stated at the end of Lemma 4.3 follow directly from the estimate
()
()~ R(~, in (B-13)
0)
[1 J
R. Courant and K.O. Friedrichs, Interscience, New York, 1949.
[2 J
B. Engquist and A. Majda, "Radiation Boundary Conditions for Acoustic and Elastic Wave Calculation," Comm.Pure Appl. Math, 32 (1979), pp. 313-358.
[ 3J
J.
[4 J
R. Hersh, "Mixed Problems in Several 12 (1963), pp. 317-334.
[5 J
L. Hormander, "Fourier pp. 79-183.
[6 J
H.O. Kreiss, "Initial Boundary Value Problems for Hyperbolic Comm.Pure and Appl. Math., 23 (1970), pp. 277-298.
[ 7]
P.D. Lax, "Hyperbolic Systems of Conservation Math., 10 (1957), pp. 537-467.
[ 8 J
P.D. Lax and R.S. Phillips, "Local Boundary Conditions for Dissipative Symmetric Linear Differential Operators," Comm.Pure Appl. Math., 13 (1960), pp. 427-456.
[9 J
Erpenbeck, pp. 604-614.
T.P. Liu,
"Stability
"The entropy
Supersonic
Flow and Shock Waves, Wiley-
of Step Shocks," Physics
Integral
Variables,"
Operators
condition
J. of Math. Anal. and Applications,
I,"
of Fluids,
Jour.
5 (1962),
Math. Mech.,
Acta Math.,
Laws II,"
127 (1971),
Systems,"
Comm.Pure Appl.
and the admissibility of shocks," 53 (1976), pp. 78-88.
[10]
A. Majda, "The Existence in this journal).
of Multi-dimensional
Shock Fronts,"
(to appear
[llJ
A. Majda and R. Rosales, "Nonlinear Transverse Reacting Shock Front s ," (to appear).
[12J
M. Nagase, "A NewProof of Sharp Garding Inequality," Ekvacioj, 20 (1977), pp. 259-272.
[13J
L. Nirenberg, "Lectures on Linear Partial No. 17, A.M.S. Providence, Rhode Island.
[14J
J.
[15]
J.
[16J
R. Richtmeyer, "Taylor Instability in Shock Acceleration of Compressible Fluids," Comm.Pure Appl. Math., 13 (1960), pp. 297-320.
Waves and Mach Stems in
Funkcialaj
Differential
Equations,"
Ralson, "Deficiency Indices of Symmetric Operators with Elliptic Boundary Conditions," Comm.Pure Appl. Math., 23 (1970), pp. 221-232. Rauch, "L2 is a Continuable Condition for Kreiss Comm.Pure Appl. Math., 25 (1972), pp. 265-285.
I
Mixed Problems,"
[17]
Tartakoff, "Regularity of Solutions to Boundary Value Problems for First Order Systems," Indiana Math. Journal, 21 (1972), pp. 1113-1130.
D.
Department of Mathematics University of California Berkeley, CA 94720
Number 275
Andrew Majda The stability of multi-dimensional shock fronts
Memoirs of the American Mathematical Society Providence· Rhode Island· USA January 1983 • Volume 41 • Number 275 (end of volume) • ISSN 0065-9266