Analytic regularity of uniformly stable shock fronts with analytic data

+ C l19~w(z, 9x)llp-j is,

for all ZE Coo@, @&), XE Cm(E, @), qaqp. Proposition 1 shows that (4.2), holds and obtain (4.2), we proceed by induction. So assume that (4.2), has been proved for all q< k and let us show that (4.2)k+, holds.

352

PAUL

Since cY,Z=Y~Z-~~?-~’ that

GODIN

zz’i’ 5$18,2--q

d,‘Z

(where J& =I),

it follows

N-l $‘2

Writing

Illa,Zlllk

<

c(?1’2

lll~zlllk

+

#‘2

1 I=0

lllalZlllk

+

t13’2

lllzlllk).

(4-3)

.

(4.4)

q = go - a,, we obtain from (4.3) that

q”’

lllaNZlllk

6

c

~-1’2(ll/90~qzlilk

+

IIk?qzlllk+l)

1 N-l

+ c2

(

,;. Ill~,4llk f Ill~o4ll,

On the other hand, we may apply (4.2), first with Z, go’ox, then with Z, X replaced by a,Z, 8,X, 0 6 1~ Nright-hand sides by standard commutator arguments. with (4.4), (4.2), + 1 follows easily for q large enough. proof of Proposition 8.

)I

X replaced by goZ, 1, and estimate the If we combine this This completes the

Remark. If 8;~ 1,= o = 0 and ajx 1,=o = 0 for 0 < j < p, a global result (that is, without a domain of influence) corresponding to (4.1) has been proved in [7, Propostion 5.11. (Actually normal derivatives of z on xN = 0 are also estimated in [7, Proposition 5.11.) Let v be as in Sections 2, 3. It follows from (3.1), (3.2) and our construction of D, E that 9% = 0 in a neighborhood of D in 0 x R +, while 9?(djv,V~jcp)=OinaneighborhoodofEinox(O}xW+ ifO
F,, j =

N-l

a; - 2 a,B,a,v + C (Bka; akajv-ag(B,a,aj4) >

k=O

in D, while the condition

(4.5)

k=O

9(aju, V ajq) = 0 in E implies that

N-l &,j

=

*go

ta;fajkd

+ 4,v,,(a;

bk(V)-a;(ajk@k(v)))

ad - a;Wcu,v,,(ajv))

(4.6)

in E. When proving estimates for FE’,.j, g, jr it is convenient to use a more symmetric notation. So let w’ be the vector with components {Q,, Ial = 1, 16 I< 2m}, and let w” be the vector with components {a;cp, (al = 2). We also define vectors G’, 3 by ci, = ‘(0, Vcp), 6 = ‘(v, w’, Vcp). The components

ANALYTIC

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353

of w’, w”, I? will be denoted by w:, wf, GS. Notice that it follows from (4.5) that F,,j = T, + T2 in D, with G,(6) + c w:G,,(+) s T2 = c (8; $w:H#)-

,

a;(8jw:Hk(6))),

(4.7) (4.8)

and that it follows from (4.6) that g,

j

= C (Ia;

ajGsJs(G)

-

aF(ajssJs(a)))

(4.9)

in E, where the vector valued functions Gr, G2,s, Hk, J, are analytic. For bounding F,,j and g,j, we shall need the following lemma. LEMMA 1. There exists a constant C > 0 such that for all j E Z + with O< j<,u, all qEZ+, all jl,..., j,EZ+ with j, + ... + j, = j, and all fi, .... fq E C”(d, @), the estimate

holds. Proof of Lemma 1. We may of course assume that q 32. Choose an extension operator 8: P(b, C) + Hp(RN+‘, C) such that for every integer 1 with 0 < I< p, d is continuous if C”(iT, C) is equipped with the H’(D, @) topology and H”( RN+ I, C) is equipped with the H’(RN+‘, C) topology. Using &‘, one sees that (4.10), is a consequence of the analogous inequality

C) if s = 1, .... q. Notice that in for all F,, .... F, with F, E Hp’-iS(RN+l, (4.11 )4 one has j, # 0 for at most j 6 ~1values of s; hence it suffices to prove (4.11 )4 for q < ~1only. Now for fixed j < p and fixed q ,< JL, there is only a finite number of ways of writing j = j, + . . . + j, with j, > 0. Furthermore from the inequality p > [(N + 1)/2], it follows that for all q E Z + and all j, there exists C’>O such that the . ..+j., II, ...f .i, E .z+ with j=j,+ inequality

SC’ If lllFsIII~r-j~~w~+I,c, H@-qRN+‘,C)

(4.12)

s=l

holds (see, e.g., [7, pp. 8 l-821). Hence (4.11 )4 follows readily from (4.12). The proof of Lemma 1 is complete.

354

PAUL

GODIN

In order to bound F,, j and g,j, we shall need to estimate derivatives of composite functions, and we shall do it in the spirit of [ 1,2]. For qE E+\(O), put my = cq!/(q + 1)2, where c is so small that c?

%I

= (P) 0
u c (8) 0
MIS1

mlu-pl Gmlal,

mla-BI+

1 G

14

for all

%I

CLE(iZ+)N\{O}.

(Here /I < a means that /Ij < aj for all j and that /I # a). If E> 0, put M, = Eleqrn If ZE C”(b, lFP) for some n~Z+\{0}, define lzlP = 8;~) where 19 > 0 will be chosen later, and [z], = sup,or,=“; Qe,,- I,(

SUP0 O, C>O, which depend only on 4, lll4lp+l (actually lljzlll, for (i) and (ii)), y, such that for alIp> 1, all 0>0, and all E > 0, the condition [z], < V/E implies the following:

(i) C&)1, d CCzl,; (ii) (iii)

Mz)l,+ I d C(lzl,+ 1+ M,+ ICzl,); for Jal=p+l,j=O

,..., N-l,

ands=l,...,

n

Qep(aF(ajzs#(z))- 8; ajzsd(z)) Proof of Lemma 2. It follows the lines of the proof of Lemmas 3.3.1 and 3.3.2 of [ 11, or of Lemma 3.1.1 of [2], so we just indicate briefly how to modify those proofs. To prove (i) we write for 1aI= p and 0 G j< p, e-ecp-')r

aj i3$cj(z))

=

C da,, oL,+ ... +lr,=cr

where c(aI, .... a,) =

JL1

a,)=~e-ec9-lw @(a1, ee.3

.... a,) @(a,,

-., 4,

a,, .... a,, E (Z+)N, I<;<“ak9and . . a/

I-j. V((W . . . %%Nz)) O (4.13)

ANALYTIC

REGULARITY

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FRONTS

355

In (4.13), $s,h = ($;) e-e(lv~hl-l)~ a$* ayz,, with Js,h =CsSrGqh jr,h + C h
also use an analogous result for functions on E. For put rnb = c’q!/(q + 1)5/2, where c’ is so small that

for all aE(Z+)“‘\(O}. For e>O, write Mq=el-qmb. If ZEY($ IJY) for some ncZ+\{O}, define lzl; = SUP~,~zP Q&t- ,,@p)> L-4; = sup0
v > 0, C > 0, which depend only on 4,

Ml Ir+ 1, y, such that for all p 2 1, all 0 > 0 and all E > 0, the condition [z];

6 V/E implies the following:

0) C&)1; S CCzl;; (ii) (iii)

I&)I;+l

GCc(l~l;+~ +M;+,Czl;);

for (a( =p+

l,j=O,

.... N-

Qbp(aF(ajzs4(z)) - 8; GCc(p+ w4;+,

1, and s= 1, .... n, djZsb(z))

+q+I(l+

CqJ’>~

The proof of Lemma 3 is similar to that of Lemma 2, so we may omit it. Remark. The choice of rnb is not fortuitous. It is imposed to make mp and pli2rnb of the same size, for purposes which will become apparent in the proof of Proposition 9 below.

356

PAUL

GODIN

In the next two lemmas we collect estimates for Ti, T2 (defined in (4.7), (4.8)). We put )a) =p.

LEMMA 4. There exist constants v > 0, C > 0, depending only on G, , G2,*, III9)(( p, y, such that for all p 2 2, all 0 > 0, and all E > 0, the condition [$I,1 d V/E implies that

C&q,-,,V,KC(l~l,

+~,Wl,-,(I

+&b”lp-A+

W’IJ

We have T, = T,, + T12, with T,, =a;(G,(O)), Lemma 2(ii) gives that Qe~p-l~U'll)G C(IO(, + MJE],i). On the other hand we can write iZ;(~tG~,,(a))= S, + S, + S,, with S1 = B;wtG,,J%), S, = COcs~c,,,,.(~) Iwl:IIsl I%(~h-,,~ which can be bounded by sM,Ji+&,- 1 [w”lp- i if we use Lemma 2(i) (for G&6)) and the inequality C 0 < B< c((;) M,,, M,, _ BI < EM,%,. The proof of Lemma 4 is complete. Proof of Lemma 4. T,, = Cs a;WG,,sbW.

LEMMA 5. There exist constants

v > 0, C> 0, depending only on Hk, ll1~lllp+ 1, y, such that for all p L 2, all 8 > 0, and all E > 0, the condition [$I,1 < V/E implies that Q,,,-,,(W=(P Proof.

Iw’lp + Mp +~p!N,-Al+

PIN,-I)}.

Since the proof is similar to that of Lemma 3.3.l(iii)

of [l],

we

omit it. From Lemmas 4 and 5, we obtain the estimate

Qscp-,,(f’a,j)G C(P IW’Ip+ Iw”Ip + l@lp+ Mp[+]p- I(1 +&Cw”Ip- 1) (4.14)

+ P~pc~l,-1cw’l,-A7 provided obtain

[%],-1


Q&p- l,(Sa,j)

G CP{

Using (4.9) and Lemma 3(iii), we

IaIL + Mp(l+ Cal;- ,)‘>,

(4.15)

provided [$I;- i < V/E and (a( = p > 2. As mentioned already, we plan to apply Proposition 8 with z, x replaced by 8; aiu, a; a,(p, respectively (Ial = p, 0 < j < N - 1). We take care of the terms arising from Do and E0 in the following lemma (in which derivatives in the x,,, direction could also have been considered).

351

ANALYTICREGULARITYOFSHOCKFRONTS

LEMMA 6. One can find constants C, > 0, C, > 0 such that the following estimates hold for all a E (Z ’ )“, all /?E (Z + )” such that Ifi/ d p, and all j E

(0, 1, .... N(i) (ii) (iii)

11: sup,, la;+s f?,ol G C, Ckl (al!, sup, la;+s J,ul < C, Ckl [al!, sup, la;+fi a,V,,cp( 6 C, Cyl Icrl!.

Proof of Lemma 6. Recall that in Section 3 we have defined /?j( y), for 0 < j< N- 1 and y = (x’, t) E E, as the C-linear form vanishing on X,‘, such that (fij( y), bk(aA(x’, 0, t))) = 1 if j = k and 0 if j # k. It is easy to check that /3’ must be of the form pj( y) = bi(ul(x’, 0, t)) with bj analytic. Hence (3.2) implies that

(4.16) in E, where a](u) = (b’(u), bN(u)) is an analytic function of u. Write 6(x, t) = u(x’, 0, t). Since dpu =0 in D, it follows from (4.16) and the definition of 2 that

2 = @(u,a,u,5)

(4.17)

in D, with Q, analytic. But u(~, is analytic up to 8D,; hence using (4.17) and the classical Cauchy-Kowalewski computation (see, e.g., Lemma 3.6.1 of Cl]), we obtain (i). (ii) is an immediate consequence of (i) and (iii) follows at once from (ii) and (4.16). The proof of Lemma 6 is complete. If H > 0 and p 2 2, let us say that condition (p, H) holds if [G], _, < H, ~[w”]p*-~ ,
PROPOSITION 9. One can find strictly positive numbers E, 0, H such that condition (p, H) holdsfor all p > 2.

Proof of Proposition 9.

We proceed by induction.

Condition

(2, H) will

hold provided

if Ho is large enough

HaHo

(4.18)

E
(4.19)

(independent

of

(E,

0)) and .sO is small enough

358

PAUL

GODIN

(independent of H, 0). Assume now that condition (q, H) holds for 2 < q < p and let us manage that (p + 1, H) also holds. We apply Proposition 8 with z replaced by 8; aju, x replaced by a; a,q, [cl1= p > 2, O
eal,

with qr given in Proposition 8. Henceforth we denote by C, K or C1 various strictly positive constants independent of H, E, 8, p. (4.1) implies that (W-

l))“*

ch,,- ,,(a;

G Ci(Q-

11)-l’*

ajo)

+

da;

QQ+

Qo(,-I)(~~-,,~>+

ajv)

+

Q&

Q&p-l)(ga,j)+

l)(a; ajVcp) S,>,

(4.21)

where s, =e-e(P-l)Y

1 ((qpWIQU

l))“-lPl

x (Iv;+@ aj41D, + (W-

I))-“’

ll~~+a~j~,dI~o).

Using Lemma 6, we get S, < CKpp!.

(4.22)

(W”JP
(4.23)

Note that

Also

IA, G I$ + l%lq G GWl,-,

+ WlqA

(4.24)

where CB is independent of H, E, p and tends to 0 as 8 tends to cc. Because of the first and third part of condition (q, H), the right-hand side of (4.24) may be bounded by CoHM4-1(l+(q-1)1/2/~) if 2
(4.25 )o.

if 2
< CH.

(4.26)

On the other hand, (p, H) also implies that [~],-,(l+~Cw~l~~,)~H(l+H(~-l)~‘*).

(4.27)

ANALYTIC

REGULARITY

OF SHOCK

359

FRONTS

Assume that (4.28)

EH
with v as in (4.14) and (4.15): Using (4.23), (4.25),, (4.27), (4.26), and (p, H), to bound respectively the second, third, fourth, and fifth term of the right-hand side of (4.14), we iset

Qoc,-,,(f'qjKC(~

Iw’lp + Iw”II, +M,H+M,HU

+H(P-

l)“*)

+ ~43’) d C(p (w’lp + Iw”I;

+ pM,H(H+

1)).

(4.29)

On the other hand (4.28) (4.15) and (p, H) imply that

Q&p- l,(ga,j) G CPML(H+ Hence if B0 is large enough (independent

112*

(4.30)

of p, E, H) and (4.31)

tee,, it follows from (4.21), (4.29), (4.30) that

(epp* lwyp + lwy; + lwy; ~C{p”2~-“2A4~H(H+1)+pM;(H+1)2+KPp!}.

Notice that ltilL+, in particular that

uw/*

< C( Iw’IL + I ,“I;);

(4.32)

hence, if we use (4.23), (4.32) implies

b+ + I+I;+~ + ba dC{p”20-“2M,H(H+1)+pM;(H+1)2+KPp!).

(4.33)

Let us assume that ell*E ,> 1.

(4.34)

If we multiply (4.33) by E(P”~M~)-’ and use the inequalities “*IV; < CM,, it follows that c&M;+,? P

lw’lp I IWb+l+ -< E W’lp M,

M; + 1

p112M,

CE(H+

l)* + C, P~‘~(EK)~.

pl’*MP

<

(4.35)

We ask that 4Cc(H+

l)*<

H,

(4.36)

360

PAUL

GODIN

so that the first term in the right-hand side of (4.35) is not larger than H/4. To make the second term in the right-hand side of (4.35) not larger than H/4, we impose that 2EK<

(4.37)

1

and that 4C, sup (2F’p3’*)

< H.

(4.38)

pa2

It follows from (4.35) (4.36), (4.37), (4.38) that (4.39)

Finally if 8i is large enough (independent (4.25), that

of E, H, p), we obtain from

l+lP<-H

(4.40)

MP ’ 2

provided

e>e,.

(4.41)

Inequalities (4.39) and (4.40) imply at once (p + 1, H). So it suffices to choose H so large that (4.18) and (4.38) hold, then E so small that (4.19), (4.28), (4.36), (4.37) are satisfied, and finally 6 so large that (4.20), (4.31), (4.34), (4.41) hold. The proof of Proposition 9 is complete. Using Proposition 9 and the Sobolev estimates of the form sup la”cp( 6 C1 Cyl Y E

imbedding

theorem,

we get (4.42)

Ia)’ .,

(4.43)

sup la%1 Y < C, Cyl(a(!. D

Inequality

(4.42) shows that cp is analytic in E. Since u satisfies the equation

-=au

-A"~‘(U,Vql) >

in D, it follows from (4.42), (4.43), and the classical Cauchy-Kowalewski computation (see, e.g., Proposition 3.6.1 of [ 11) that u is analytic in 4.

ANALYTICREGLJLARITYOFSHOCKFRONTS

Theorem 1 is now an immediate uf (x’, Ik b, - dx’, t)), t).

consequence of the equality

361 u&(x, t) =

5. PROOFOF THEOREM 2 The proof of Theorem 2 is similar to that of Theorem 1 so we shall describe it only briefly. After a translation of time we may assume that to = 0. We change variables as in Sections 2, 3 and obtain Eqs. (3.1) and (3.2) which now are valid for (x, t) E G8 x ] -6, S[, (x’, t) E og x ] -6, S[, respectively. If t*O, put D*=lJlaclcTDt, E*=UrwctxTEt, where D,, E, are defined as in Section 3. One can find 1, > 0, 6, > 0, independent of t* ~0, such that for each I 0} by the set {t E R, t > t*} and if one checks the fact that b and 2 can be chosen independently of t* < 0 throughout the proofs. (In the proof of Proposition 4, one should take CL(X,t) = (i- t*) cr,(b Ix-%IJ(tt*)) q(bt/(it*)) now.) So now fix 2 O, R>O, so that bR < T- t* and a6 x [ -8, S] is a neighborhood of B* in D x 58, and such that v is analytic in a neighborhood of B,. while cp is analytic in a neighborhood of E,. . The results of Section 4 apply with obvious modifications (i.e., one has to consider Cauchy data on t = t* and take 5 = t + y with y > It*1 now) to yield analyticity of u in 6* and of cp in E*. This immediately implies Theorem 2.

APPENDIX

In this appendix we have collected the proofs of several results (Propositions 2, 3, 4) used in Section 3. The proofs of Propositions 2 and 3 are close to the sketched proof of a corresponding result in Section B.3 of [9]. (In [9] only the case m = N= 2 is considered, which leads to some simplifications.) However we give indications about the proofs of Propositions 2 and 3 for the sake of completeness. First we check that, at the origin of w x (0) x R, the boundary value problem (9, @) (resp. (p*, %*)) satisfies the uniform Lopatinski condition (resp. the backward uniform Lopatinski condition). If Q is a differential operator in o x (01 x R, we denote by c(Q)(O, <‘, r) its principal symbol at ((0, O,O), (r’, r))~ (o x (0) x R) x (RN-’ x C).

362

PAUL

LEMMA A.l.

[<‘I’+

OnecanJindC>Osuch

GODIN

thatforall(5’,z)E[WN~‘x@

with

)712= 1 and I m z < 0, and all z E E+ (0, t’, z), the inequality

holds. Proof of Lemma A.l. The uniform stability condition (2.6)(0,0,0j implies that r/IO(O) + ~~=-r’ [$Ij(0) and .M(O)z form a uniformly strictly positive anglewhenImr
2 Cl IY(O) Jw)zl 2 c, I(1 -P(O)) mow

(A.11

uniformly when (<‘j2+l~)2=1, Imz 0,

I~‘(O)zl a c, I4

(A.2)

uniformly when l~‘12+1r12=1, Imr
t, D,, D,) = 9*(x, -t, D,, -D,), W**(x’, t, D,., D,) = D,., -D,) (where D = i-’ a) and denote by E**(O, <‘, 7) the space associated to P’** in the same way as E+(O, t’, 7) is associated to 9. %9*(x’, -t,

LEMMA A.2. l<‘12+17\2=1

There exists C> 0 such that for all (r’, 7) E RN-’ and1 m 7 < 0, and all z E E**(O, t’, T), the estimate

I(aw**w,

x @ with

t-3 7))zl 2 c Id

holds. Proof of Lemma A.2.

d%(O) is an isomorphism

of @“” onto itself. We

have dX(O)(Ker

a(‘X**)(O,

g’, 7)) = (Ker a(V)(O, t’, -7))‘.

(A.3)

In fact, it is not hard to check that the left-hand side of (A.3) is contained in the right-hand side; since a simple computation shows that both sides have dimension equal to m - 1, (A.3) follows.

ANALYTIC

REGULARITY

OF SHOCK

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363

On the other hand, it is shown in [3, pp. 333-3341 that d;(o)

E**(O, (‘, 1) = (E+(O, 5’, -?))I.

(A-4)

From Lemma A.1 we know that Ker cr(9?)(0, <‘, -Z) and E+(O, r’, -t) form a uniformly positive angle. Furthermore their (direct) sum is equal to c2m, since dim E+ (0, t’, -t) = m - 1. Hence Lemma 2 follows from (A.3) and (A.4). Put %7*(x’, t, D,,, D,)=%$(x’, -t, D,., -D,) forj= 1, 2 and for q>O define the operators A,, = (q’ + ID,. I* + Of)‘/*, 9, = edV’ 9eV’, $P** = e -V ?Z**t?, (WI), =e-“* (&1e”‘, (VT*), = eerlrq*e”‘. Using Lemma; A.l, A.2, and assuming & (defined in (3.3)) small enough, one can construct Kreiss symmetrizers (just as in [6, Lemma 4.33, see also [5, lo]) for the “inner” operators YV, Y;* with corresponding boundary operators (“qJq),

(w;*)$

respectively. Hence arguing as in Section 4 of [6], we end up with the following result: there exist constants C > 0, q0 > 0, such that for all ‘12 q,,, the following inequalities hold.

v”* llle-q’41~. R+ Ile-W ox tojx RG C(rl-‘I* IIle~q’Flll~x R + Il~~‘e-~‘~~ Ilox foj x w + IP’h2

Ilox 10) x w),

(A.51

if z E eVrH’(Q x R, @2m), 9z = F in 52 x R, and Vjz = hi in o x (0) x R for j= 1,2;

y1”* lll~“‘411Qxw + lle’Vlwx fojx w Q C(V”* Ill~“rf’lllox R + IM;‘e% llwx io) x w + Ileq’h2IL lo) x d,

(A.6)

if zEe- ~‘H1(QxlR,@*“),8*z=Fin~x!R,and%?,~z=hjinox{0}xlR for j= 1, 2. We may now prove Proposition 2. This will be done in three steps. Proof of Proposition 2. Step 1. If q > 0, write 9: = erlr$p*e-Vr, U: = F,, = evfF. We are going to show that the following holds.

e’%?*e-“I,

If rl is large

enough,

there exists f, E L*(Q x R, C*“) n (w x R, @‘“))) such that Y;fq = F,, in (l7je *+ Q x R, %;yoTqNl 0 in.o x (0) x !R in the distribution sense, if y. means trace on 0 x (0) x R. (A.71 cj(Rf

H-‘/*-i

364

PAUL

GODIN

To see this, put W,, = e-‘%‘eV’ and write Y = {w E C,$)(G x R, CZm), %?,, w= 0 in ox (0) x R}. It follows easily from (AS) that the linear map @: gV Y + @ which sends yq w to (w, Fq)nX R is well defined and that l@~=+4l G c, lIlQ4lI*. Iwfor some C, > 0 (depending on q) if q 2 Q, > 0. Hence there exists fq E L2(52 x R, Czm) such that

for all w E Y. Since distribution sense. cj(R,+, H-U-i (w other hand, writing

C;(Q x R, C2”) c Y, it follows that g:fq = F,, in the Since det J& never vanishes, it follows that fq E x R, Czm)) for alljE h+ (see, e.g., [4, Chap. 21). On the (D,) = (1 + JD, l*)i/* (with y = (x’, t)), one has

To prove (A.9), let x E C,“(R”‘) satisfy j x(y) dy = 1; put x,(y) = E-“‘x(y/s) if E> 0. Define xE dr as the measure with density x,(y) on o x (0) x R and write f, = f* (x, dy). Integration by parts shows that (A.9) holds with f replaced by fE. If E tends to 0, (A.9) follows easily. Now write u = e-‘“y, f,,, yl=(uEC~(Wx{O)x[W,~2m), Vu=0 in ox (0) x R}. Let us denote by (T, ~p)~ the action of the Cj valued distribution T in o x (0) x R on the Cj valued test function cp. If j= 2m, we simply write (T, cp). Comparison of (A.8) and (A.9) easily shows that

(Yod%U, v > = 0

(A.10)

for all VE Y’. If ~EC~~(OX(O}X(W,@) and v=‘(v+,v~)E CF(o x (0) x R, C2m) satisfy A!+ v + + dt- v- = ~j?s-o’ (~W/ay,) /Ii ox (0) x OB,it follows from (A.lO) that

in

(A.ll)

(A:)-‘d+u=(.d*_)--1db,

(A-12) Notice that (A.1 1) means that V:u = 0 in o x (0) x R; hence, writing (WT), for eq’%?eKqr, we get J re),

rof,

=

0

in

ox(O}xR.

(A.13)

Now we may decompose Z = (A: ) - ’ &+ u as Z = Z’ + Cj”=-ol *fljZ,, where (Z’, *), = (Z, *-~~m, l = CzY XSj>mv

ANALYTIC

REGULARITY

OF SHOCK

FRONTS

365

<‘Bjzj9 *>m = Czj9

x { 0} x R, V’) and all x E (B'IcI>>I f or all $ E C;(o C;(UI x (0) x R, @). From (A.12) it follows easily that (A.14)

Since &Y&,’ Zj dyj = (9*)-l (I- P)Z, (A.14) o x (0) x R; hence (+?S:), y,,f, = 0 there. Together that %;yOfV = 0 in w x (0) x 0% Hence the proof Step 2. Let us prove that the distribution f, the following regularity property:

shows that %‘:u = 0 in with (A.13), this implies of (A.7) is complete. constructed in (A.7) has

For any s > 1, there exists q(s) > 0, independent of F, such that the following holds when q > q(s): if f, E ZZ-‘(0 x R, c2”) and y0fv E H”-‘(co x (0) x R, @2m), then f,, E H”(f2 x R, czrn) and (A.15) y,f, E H”(o x (0) x R, C”“). To prove (A.15), we use the standard mollification technique (see [4, Sect. 2.41). Let x E C,“(RN) satisfy the following conditions, if 6 means Fourier transform: (i) (ii)

for some k > s, i(c) = O((clk) as i --t 0 in IV, if CEUV” and j(XJ=O for all AEIW, then c=O.

Let us define as before x,(y) = ~-~x(y/e), E> 0, and fv,E= f, * (x, dy), if xE dy is the measure with density x, on w x (0) x R. We apply (A.5) with z replaced by e - “‘j&, square and integrate from 0 to 1 with respect to the measure E-2”-1(1+B2/~2)-1d&.Normsofthetypes(~~lllu*x,III~.R~-2”-’ (1 +a2/c2)-’ dc)1’2, cj: IIYo~*XEll~x{O}xW~-2s- ‘( 1 + d2/t2)-’ dc)lj2 are studied systematically in Section 2.4 of [4]; in particular it is shown there that they are essentially equivalent to 111(Z-d2 dY)-1/2 (Z-d,,)“” u(ljnX u, ll(Z-d2 dY)-‘12 (I- dY)‘12 you(JoX (o) X u, respectively. Standard commutation arguments using the estimates of Section 2.4 of [4] easily yield (A.15). Step 3. If q is large enough, then fq E H’(Q x R, cZm), so ed9’f, must vanish for large t in view of (A.5), since F does. Assume now that q, tf are large and that ii> q. If w is equal either to fq or to e-‘V-‘r”f,, then w E H’(l2 x R, @2m) and 9’;~ = F, in IR x R, %‘;w = 0 in ox (0) x R. Hence it follows from (A.5) that e-cq-‘r)‘jq = f4, since rj is large. Therefore if we put f = e-*‘f? for large q, f is independent of q and f E %+. Furthermore 9’*f=FinQxRand%?*f=Oinox{O}xR. To solve Problem (3.10), choose first 2 E C$)(fi x R, c2”) such that g = g such that Y*f = F- LT*g in Q x R, W*f = 0 inwx(O}xR, thenfEX+ in o x {0) x R. Then z = f + g is a solution of Problem (3.10).

366

PAUL

GODIN

To complete the proof of Proposition 2, it remains to show that (3.11) holds. To obtain (3.11) we use a well known trick (also used at the end of the proof of Proposition 6). Let J,(t) E Cm( R, [0, 1 I), 0 < E< 1, be equal to 1 ift>&andtoOift<--E.Putg,(y)=J,(t)g(y),F,(x,t)=J,(t)F(x,t).It follows from the proof of the existence part of Proposition 2 just above that one can find y10>O independent of E and z, E (n,,,, e-“‘H’(SZ x [w,@*“)) A X+ such that Z*z, = F, in 4 x Iw, V*(z, - g,) = 0 in w x (0) x R. Hence z: =z-zz, Elf+, and applying (A.6) to zk, we obtain that zh = 0 if t > E. So for q 2 y10(with v0 independent of E), we have vl’* IIIeq’411Q.]E,+~[

+

Ile~t41wxj~~xlE,+mC

6 #‘* IlIe% IllRxR + lle%IIw.~o~xR
v1j2 IlWzlll Qx R+ + Ile’l’zlloxtoix R+ < C(rl-“* lIleq’f’lIl~. R+ + Ileqtglloxioi x R+).

(A.16)

We are now going to prove the following estimate, which together with (A.16) clearly suffices to complete the proof of (3.11). There exist q1 > 0, C > 0 such that for all q > vi and all z E eC”‘H’(52 x R, C2m)

GC(V”*

Illeq’~*zllI,. R+ + Ileq’41,,{o~.R+).

(A.17)

When proving (A.17), we may and shall assume that z E C;,(d x IX, @2m). Since do is positive definite and ZZJ~ZZ$ is real symmetric ifj= 1, .... N, (A.17) is easily obtained if one integrates by parts in Re(eqLz, ZZ*&oeq’z)Q, n+ (actually this is the classical energy integral method, see, e.g., [8], p. 140). The proof of Proposition 2 is complete. Proof of Proposition 3. Proposition 3 can be proved by arguments similar to those used in the proof of Proposition 2 (Steps 1, 2 and beginning of Step 3). Hence we may omit its proof. (At any rate, Proposition 3 is also a consequence of the results of [6]. )

Finally we prove Proposition

4.

Proof of Proposition 4. Choose tli, a, E C,“(BB, R) such that al(s) = s if 1.~1< 1, ai =0 if IsJ >2, a*(s)= 1 if IsI < 1, a,(s)=0 if IsI >2. Define a(x, t) = &,(b (x-X1,/t) or,(bt/t). Proposition 2 shows that for q large

ANALYTIC

REGULARITY

OF SHOCK

367

FRONTS

enough, there exists h E X+ such that .Y*h= H in D x R, V*h =0 in o x (0) x R. If one lets q tend to co, (3.11) implies that h = 0 for t large. Below we shall prove that One can find b, >O> 1, >O such that when O 0, C > 0, such that the estimate q ‘I2 111 eq(’ + “(“~t))h 111 R x R + 11 e”(’ + a(X,t))hIIw X Iol X R

(A.18)

< Cq - ‘I2 111 eq(’ + acX,t))HIIIR x w

holds provided q > ylO. In (A.18), the dependence of ‘lo and C on b, 1, E does not matter; the important thing is that when b, 1, E are fixed, C can be chosen independently of q when q is large. Assume for a moment that (A.18) is proved. It is no restriction to suppose that b,, < 1. Then a(x, t) = b )x-X(, in G&X, t), so there exists p > 0 such that t + a(x, t) < i- ,u on the support of H. Letting q tend to cc in (A.18), we obtain that h = 0 if t + CL(X,t) 2 i-p; this of course implies that h = 0 in Sb,JX, t) and completes the proof of Proposition 4. So it remains to prove (A.18). Write j(x, t) = t - a(x, -t), h’(x, t) = h(x, -t). Then V**h’ = 0 in o x (0) x R! and the estimate in (A.18) is a consequence of an estimate of the type v”* Ille-“jh’lllax

R + lle-4ih’ll~x

< Cq-1’2

l)le-4iY**h’111nX

foIx R R

(A.19)

with C independent of q for large q, To prove (A.19), we make the transformation (2, ?) = @(x, t) = (x, j(x, 1)). We assume that 2b sup ltll I . sup l&,/d\1 d 1; then 0 is a diffeomorphism of 52 x R. Let 8, 9, K be the images of Z**, %?**, h’ by Cp. Notice that FEZ?. Denote by Y$*, %$* the differential operators with constant coefficients obtained when freezing the coefficients of T**, %** at (0,O) EQ x R. An explicit computation shows that each coeflicient in the principal part of g or @ is a perturbation with compact support and sup norm less that c(b +A) of the corresponding coefficient in Y$*, %?$* (here c is a constant independent of 6, I, E, X, ~3. Hence if b, and 1, are small enough, we may proceed just as in the proof of (A.5) above to obtain, for 1 large, the estimate rl”* llle-qi~lll~f. < Cq-‘I*

~~+ Ile-tli~llwi.x iolx wi llle-q’Billln,,

wi

which holds since @E=O in We, x (0) x [wi. (A.19) is an immediate sequence of (A.20). The proof of Proposition 4 is complete.

(A.20) con-

368

PAUL GODIN REFERENCES

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