Generic types and transitions in hyperbolic initial-boundary value ...

Generic types and transitions in hyperbolic initial-boundary value problems Sylvie Benzoni-Gavage, Frederic Rousset, Denis Serre, Kevin Zumbrun November 16, 2000



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Contents 1 Introduction 2 Weakly stable IBVPs of real type 2.1 2.2 2.3 2.4

Reality of E (i; ) . . . . . . . The set R . . . . . . . . . . . Analytical structure near @ R The class WR . . . . . . . . .

3 Generic transitions 3.1 3.2 3.3 3.4

The case with \surface waves" The case with a singularity . . The case when D leaves R . . The case when
(1; 0) = 0 . .

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4 The wave equation 5 Linear elasticity

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CNRS S. B.-G., F. R. and D. S. : ENS Lyon, UMPA (UMR 5669 CNRS), 46, allee d'Italie, F-69364 Lyon Cedex 07. Research of S. B.-G., F. R. and D. S. was done in accomplishment of the TMR project \Hyperbolic conservation laws", contract # ERB FMRX-CT96-0033. z Department of Mathematics, Indiana University, Rawles Hall, Bloomington, IN 47405. K. Z. thanks the ENS Lyon for its hospitality along his visit on March 2000, during which this project was initiated. Research of K.Z. was supported in part by U.S. National Science Foundation Grants number DMS9107990 and DMS-0070765.  y

1

6 The stability of shock waves

22

6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.2 Gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.3 Phase boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Abstract The stability of linear initial-boundary value problems for hyperbolic systems (with constant coecients) is linked to the zeroes of the so-called Lopatinskii determinant. Depending on their location, problems may be either unstable, strongly stable or weakly stable. The rst two classes are known to be \open", in the sense that the instability or the strong stability persists under a small change of coecients in the dierential operator and/or in the boundary condition. Here we show that a third open class exists, that we call \weakly stable of real type". Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system. Applications to the wave equation, linear elasticity, shock waves and phase boundaries in uid mechanics are given.

Contents 1

Introduction

Let t (t 2 R+) be a time variable and x be a space variable, with d scalar components. We consider t-hyperbolic systems of partial dierential equations. Though our analysis is valid for arbitrary orders, we shall present it for the simpler case of rst-order systems. These read (1.1)

d @u + X @u = 0; A  @t =1 @x

where A are n  n matrices with real entries, satisfying the hyperbolicity property. This means that, for every frequency vector  2 Rd, the matrix

A() :=

d X =1

 A

is diagonalizable with real eigenvalues. In some cases, we shall make a stronger assumption : either strict hyperbolicity, that is for  6= 0, the multiplicities of the eigenvalues do not depend on  , or symmetrizability. When x is restricted to a domain dierent from the whole space Rd, with a smooth boundary, a system like (1.1) must be supplemented with boundary conditions. The boundary is said to be characteristic at a point x0 if A( ) is singular,

2

where  denotes the unit normal to @ at x0. Here, we are interested in domains whose boundary is non-characteristic everywhere. The corresponding initialboundary value problem (IBVP) has been analyzed in [16, 27, 20] (also see [9, 23] for characteristic problems), via the normal-mode analysis. A complete description can be found in [8]. This method consists, after a localization permitted by the nite propagation speed present in (1.1), in solving a similar IBVP in a half-space

0 := Rd?1  R+. Then, the non-characteristic property amounts to det Ad 6= 0. This new IBVP is attacked with a Laplace transform in t and a Fourier transform in the tangential variable y := (x1 ; : : : ; xd?1). This yields a linear ODE of the form

dv dxd = A(; )v;

(1.2)

where A(;  ) is constant (that is independent of xd ). It is given by

p A(; ) = ?A?d (In + iA(; 0)); i = ?1: The vector  2 Rd? is the space frequency while  2 C , the time frequency, has a 1

1

positive real part. A solution of (1.2) is associated to a growing solution of (1.1), of the form (1.3)

u(x; t) = et+i
y v(xd ):

We shall denote by C + the set of time frequencies : C+

:= f 2 C ; 0g:

A well-known lemma, due to Hersh [16], states that the matrix A(;  ) is of hyperbolic type in the sense of dynamical systems, for all  2 Rd?1 and  2 C + . That means that its eigenvalues have non-zero real parts. This ensures that C n is the direct sum of the stable and unstable invariant subspaces of A(;  ). The stable one is the set of initial data v 0 for which the solution of (1.2) with v (0) = v 0 tends to zero as xd ! +1. Similarly, the unstable one is the set of initial data for which the solution decays to zero as xd ! ?1. In both cases, the decay is actually exponentially fast. In the sequel, we denote the stable subspace by E (;  ). It depends analytically on (;  ) ; in particular, its dimension p is constant and equals the sum of multiplicities of positive eigenvalues of Ad . This dimension is the \number of incoming characteristics". One easily proves that the number q of independent scalar boundary conditions must be equal to p for the IBVP to be well-posed. It would be underdetermined if q < p and overdetermined if q > p. However, q = p is not sucient to ensure the well-posedness. To see this, we consider simple solutions of (1.1) of the form (1.3). Let

b1(u(y; 0; t)) =


= bp (u(y; 0; t)) = 0 be (linear homogeneous) boundary conditions, where bj are real linear forms. Then u as in (1.3) is a solution of the IBVP if and only if b1; : : : ; bp vanish on v (0). Focusing on L2-solutions, which are relevant in the functional space setting, v (0) is any vector in E (;  ). Now, let us assume that b1 ; : : : ; bp vanish on some non-zero vector of

(1.4)

3

E (; ), for some pair (; ). Then the corresponding u solves the IBVP and grows at the rate exp(t 0. This proves that a solution of the IBVP exists with the growth rate exp(t 0; @u + X A  d @t =1 @x b1(u(y; 0; t)) = : : : = bp (u(y; 0; t)) = 0; y 2 Rd?1; t > 0; u(x; 0) = u0 (x); xd > 0

(1:1)

(1.5) (1.6) satis es the Lopatinskii condition if the linear map E (; ) ! C p v 7! (b1(v); : : : ; bp(v)) is one-to-one, for every (;  ) 2 C +  Rd?1. This notion is easily extended to higher order problems and has been fruitfully used in free boundary problems, such as the propagation of shock waves in systems of conservation laws (see [7, 11, 22, 21], as well as [2, 23] for characteristic cases). It has been shown sucient for existence and uniqueness of a solution for C 1 data, assuming C 1 -compatibility between the initial and boundary data (see [27]). Unfortunately, it is still too weak to imply well-posedness in L2 spaces. There are actually two (at least) notions of L2 well-posedness, a strong and a weak one. The (non-homogeneous) IBVP (1.1,1.5,1.6) with right-hand sides (f; g; u0) is said to be strongly well-posed in L2 if it has a solution for every L2 data and the following energy estimate holds for each T > 0 and suitable constant CT : Z

CT

TZ

0

Z

TZ

2

xd >0

TZ

Z

ku(x; t)k dx dt +

0

kf (x; t)k dx dt +

Z

2

TZ

R

d?1

ku(y; 0; t)k dy dt  2

Z

kg(y; t)k dy dt + d?1 2



ku (x)k dx : 2

xd >0 R This kind of stability is suitable for perturbation analysis in nonlinear problems, as done in [21]. However, it is not satis ed by reversible problems, such as the wave equation with Neumann boundary condition. We say that the IBVP is weakly wellposed in L2 if it has a unique solution whenever g  0, and the following energy estimate holds 0

Z

xd >0

TZ

0

ku(x; t)k dx dt  CT

0

Z

TZ

2

xd >0

0

kf (x; t)k dx dt +

0



Z

2

xd >0

xd >0

ku (x)k dx : 0

2

A reinforced statement, called the uniform Lopatinskii condition, characterizes the strongly well-posed IBVPs in L2 . The analysis uses Laplace-Fourier transform,

4

algebraic geometry, energy estimates and the Paley-Wiener theorem. To describe this stronger condition, we recall that E (;  ) admits a unique limit as  ! i, with (;  ) 2 Rd n f0g, (see for example [8]). We feel free to denote this limit by E (i; ). Let us point out that, though (;  ) varies in Rd n f0g, E (i;  ) is not fully homogeneous, but only positively homogeneous. Homogeneity is useful since it allows to restrict the analysis on compact sets, either projective spaces (for full homogeneity) or spheres (for positive homogeneity). From now on, the words \homogeneous", \homogeneity", will refer only to positive homogeneity. De nition 2 The IBVP (1.1,1.5,1.6) satis es the uniform Lopatinskii condition if the linear map E (; ) ! C p v 7! (b1(v); : : : ; bp(v)) =: Bv is one-to-one, for every (;  ) 2 (C +  Rd?1) n (0; 0). By homogeneity, this de nition may be rewritten for the pairs (;  ) 2 K + := f 0. Let F be the generalized eigenspace associated to . As  varies around i, F ( ) varies continuously as an invariant subspace associated to eigenvalues of positive imaginary parts. Therefore F ( ) is contained in E (; ) for 0. Passing to the limit, we obtain that F  E (i; ). When E (i;  ) is real, we have a contradiction. Therefore B (;  ) has only real eigenvalues. The converse is trivial since a real matrix with a real spectrum may only have real generalized eigenspaces, therefore only real invariant subspaces.



When jj >> k k, the real matrix ?1 B (;  ) is close to ?A?d 1 . If Ad has simple

eigenvalues (which are real since system (1.1) is hyperbolic), then the eigenvalues of B (;  ) are real and simple. Therefore, E (i; ) is real. A more accurate result actually holds. Let us say that the system (1.1) is strictly hyperbolic if the multiplicities of the eigenvalues of A( ) do not depend on  2 R as long as  6= 0. We denote by C the characteristic cone of the system, de ned by

C := f(; ) 2 R d ; det(In + A()) = 0g: 1+

We now denote by ?+ the connected component of the vector (1; 0) in the complement of C ; this is the set of temporal vectors. Also, we de ne ? = ?+ [ (??+ ). Let nally G + and G denote the projections of ?+ and of ? on the d rst components (; 1; : : : ; n?1 ). Proposition 1 Let the system (1.1) be strictly hyperbolic. Then, for (; ) 2 G , the matrix A(;  ) is diagonalisable, with pure imaginary eigenvalues associated to real eigenvectors.

Proof.

For the sake of simplicity, we give the proof in the simplest case where A( ) has simple real eigenvalues for  2 R,  6= 0. Let B (;  ) = ?A?d 1 (In + A(; 0)), which is a real matrix. We denote by U the connected component of (1; 0) in the set of pairs (;  ) 2 G + such that B (;  ) has simple eigenvalues. By continuation, these eigenvalues remain real. Let (0; 0) be a boundary point of U . If moreover (0; 0) 2 G + then B (0 ; 0) must have a non-simple eigenvalue 0. In other words, the polynomial P (; ; ) := det(In + A(; )) satis es P (0 ; 0; 0) = @P @ (0; 0; 0) = 0:

7

On the other hand, the Euler identity for the homogeneous polynomial P implies

 @P @P 0 @ + 0
r P + 0 @ (0; 0; 0) = 0: We conclude that, for every real number , the vector (0; 0; ) is orthogonal to r; P (0; 0; 0). This means that the operator @t + A(rx) is not strictly hyperbolic in the direction (0; 0; ). >From theorem 3.13 in [8] (page 305), it cannot belong to ?. Therefore (0; 0) does not belong to G + , a contradiction. We have proved that U is closed in G + . Since it is open (from the implicit 

function theorem), and since G + is connected (it is known to be convex, see [8]), we conclude that U = G .



For practical applications, we also may use the following result. Proposition 2 Let the system (1.1) be symmetrisable. Then, for (; ) 2 G , the matrix A(;  ) is diagonalisable, with only pure imaginary eigenvalues associated to real eigenvectors.

Proof.

P

Let S0@t u +  S @ u = 0 be a symmetric form of the system. Then it is known that ?+ consists in those pairs (;  ) such that the symmetric matrix S0 + S ( ) is positive de nite. Let (;  ) be a vector in G + . There exists a real number  such that (; ; ) 2 ?+ . Therefore, S0 + S (; ) is positive de nite. Its product by the symmetric matrix (S d )?1 will thus be diagonalisable with real eigenvalues. This means that In ? B (;  ) is diagonalisable with real eigenvalues.



2.2 The set R

We shall denote by R the set of real parameters (;  ) for which E (;  ) is of real type. In the literature, this set is called the hyperbolic domain of @t + A(rx ) at the boundary. From proposition 1, R contains G and therefore is a neighborhood of the line  = 0,  6= 0. It is closed in C  Rn n f0g. Trivial examples, where all matrices A are diagonal, or close to diagonal, show that R and G may dier signi cantly. However, under generic assumptions, the closure of G is just a connected component of R. To see this, it is sucient to prove that (generically), the boundary of G is also a boundary of R. We therefore consider a point (0; 0) of @ G + . Then there exists a real number 0 such that (0; 0; 0) 2 @ ?+ , but since (0; 0; ) does not belong to ?+ , for every  2 R. Then P (0; 0; 0) = @P=@(0; 0; 0) = 0. Under the generic assumption that @ 2P=@2 and r; P do not vanish at (0; 0; 0), the equation of the characteristic cone has a normal form ( ? 0)2 = `( ? 0;  ? 0) + O(( ? 0)2 + ( ? 0)2); where ` is a non-zero linear form. This also is the equation giving eigenvalues of B(; ) close to 0 when (; ) is close to (0; 0). Due to the branching, these

8

eigenvalues cannot be real, for some pairs (;  ). From lemma 1, we conclude that (0; 0) does not belong to R. Let us focus on the strictly hyperbolic case. Then P splits into

P (; ; !) =

Y

l

( ? l (; ! ))ml;

where l are distinct eigenvalues of A(; ! ) and ml are their (constant) multiplicites. Moreover, the kernel of A(; ! ) + lIn has dimension ml . The functions l are real analytic on Rd n f0g, homogeneous of degree one and extend holomorphically to a conical neighborhood in C d n f0g. Denoting the eigenvalues of Ad by al 's, we have l(0; ! )  ?al ! . Since al 6= 0 (by the non-characteristic assumption), we have @l=@!(0; !) 6= 0. Therefore, each of these functions may be inverted locally, and give rise to new functions !l (; ). These satisfy !l (0; ) = ?=al. The numbers i!l(?i; ) are the eigenvalues of A(; ). When 0, the stable eigenvalues of A(; 0) are the ?=al where we retain only the ones for which al > 0. When  6= 0, there is no reason why the range of l(;
) would be a half line  > 0.p Let us take as an example the wave equation2 utt =
u. Then 1 = ?2 = kk2 + !2. The range of 1 is only [kk; +1[. This shows that l(;
) is not globally invertible in general. On @ R, at least one of the derivatives @l=@! vanishes from the implicit function theorem. Let us assume rst that ml = 1 (the eigenvalue l is simple), and let us denote by `l and rl left and right eigenvectors (for symmetric systems, we should have `l = rlT ) :

`l(lIn + A(; !)) = 0; (lIn + A(; !))rl = 0: Normalizing with `l rl = 1, we obtain

@l =@! = ?`lAd rl : In particular, this derivative vanishes when `l Ad rl does. Since the kernel and the range of A(i;  )?i!In are C rl and (`lAd )? respectively, we see that @l =@! vanishes if and only if i! is a non-semi-simple eigenvalue of A(i;  ). Proposition 3 Let (; !) 2 Rd n f0g, such that @l=@!(; !) 6= 0, and let  := l(; !). Then rl(; !) belongs to E (i; ) if and only if

Proof.

@l (; !) < 0: @!

Let us denote by  this derivative. We just observe that as  = i enters C + , keeping  constant, the real part of i! varies in the same way as the real part of  . But, when  2 C + , rl belongs to E (;  ) if and only if i! 2 C ? . Since E depends continuously on  , this proves the statement. 2

As mentioned before, this analysis applies to higher order equations or systems.

9



For an eigenvalue of constant multiplicity, the result is similar : choosing left and right eigenbases `1 ; : : : ; `ml ; r1; : : : ; rml with `j rk = jk , the matrix `Ad r is a multiple of the identity. The derivative vanishes when this matrix does, that is when i! is a non-semi-simple eigenvalue of A(i; ). In that case, the multiplicity of this eigenvalue is at least 2ml. Let (0; !0) 2 Rd n f0g be such that

@l ( ; ! ) = 0: @! 0 0

We make the generic assumption that Then, locally, the equation

@ 2l ( ; ! ) 6= 0: @! 2 0 0

@l (; !) = 0 @! de nes a graph ! = l ( ), with l (0) = !0 . This projects locally on the (;  )-space as a graph  = ^l ( ) := l(; l( )). We easily have (restricting the calculations to the case ml = 1) @ ^l @l @ = @ = ?`l Arl : 



Hereabove, ^l is homogeneous of degree one. Because of homogeneity, all these calculations have a counterpart in the sphere S d?1 . The functions l, l and ^l have analytical extensions to pairs (!; ), when ! is not any more real, but still close to !0 . We complete the description with an analytical choice of eigenvectors rl(; ! ), which belong to the bases B(il(; ! );  ). The condition 0; l = 1; : : : ; p: l @!

Such pairs (;  ) are therefore interior points of R. The maps (;  ) 7! E (i;  ); B(i; ) are analytic on this interior. In particular,
is analytic on IntR [



C+

10



 Rd? : 1

Thus the boundary @ R consists of the pairs (; ! ) at which at least one of the derivatives @l=@! vanishes. The generic case is that only one vanishes (let us call l the corresponding index), and the corresponding second derivative

@ 2l (; ! ) l @! 2

does not. Near such a boundary point (0; 0), the domain R may be viewed as a closed set in a Riemannian manifold, that is the upper sheet of a folding. We use local coordinates (;  := ! ? l ( )), so that the boundary (locally denoted by ?l ) is represented by  = 0. A given point (;  ) near ?l admits two representations of the form  = l(;  + l ( )). The corresponding values of  have opposite signs, which means that one point (; ! ) belongs to the lower sheet and the other one to the upper sheet. However, the sign of @l=@! changes accross f = 0g. From proposition 3, it follows that one sheet is relevant for the construction of E (i;  ) and the other is not. This proves that (;  ) 7! (; ! ) is a local change of variable between a half-ball bounded by ?l and a half-ball bounded by f = 0g. The Riemannian structure that we shall consider is the one given by coordinates (; ! ). It is not equivalent to the one de ned by (;  ), since the change of variables is singular along ?l . In order to be as clear as possible, we shall denote by R! this new structure. It extends to an open neighborhood V in C d . Let rj (; ! ) be an analytical basis of ker(j (; ! )In + A(; ! )). Since 0 is a regular value of j (0;
) for j 6= l, we may de ne locally Vj (;  ) by (here,  is close to !j ) Vj (j (; ); ) := rj (; ); j 6= l: We then de ne R~j (; !) := Vj (l(; ); ); j 6= l: Finally, we choose r~l  rl . The set fr~1; : : : ; r~pg is a basis B(i;  ), which is analytic on R! .

2.4 The class WR

Since a basis B = fR1; : : : ; Rpg may be chosen holomorphically in V (see above subsection), we may de ne a holomorphic function D : V ! C by

D := det(BR1 ; : : : ; BRp ): In IntR, we use coordinates ( = ?i;  ) and have
(i;  ) = D(;  ). In a neighborhood of ?l , we use coordinates (; ! ) and have
(il(; ! ); ) = D(; ! ). We shall consider weakly stable IBVPs :
does not vanish on C + but does at some boundary points 0: An elimination gives the criteria a > 0, a2 + kbk2 = c2. We denote by ?cr the corresponding hypersurface. In order to characterize the parameters for which D leaves R, we work in variables (; ! ). The boundary @ R is given by ! = 0, We write that D vanishes at some point (; 0), where dD is parallel to d! , that is r D = 0. This reads as

 + b
 = 0;  = ck k; c2  + b = 0: Eliminating, we obtain kbk = c. We denote by ?l the corresponding hypersurface. Finally,
(1; 0) = 0 reads 1 ? a! = 0; ! = 1=c; that is a = c. We denote by ?0 the corresponding hypersurface.

16

The above calculations result in the gure 1, with two distinct cases : d = 2 and d  3. We now discuss the location of classes WR, SS, SU, using the curves ?j , j = S; cr; l; 0. Their union splits the plane into connected components on which the class is constant. We rst recall that the IBVP is strongly well-posed in L2 when b = 0 and a < 0; the energy estimate5

d Z (u2 + c2kruk2)dx = ac2 Z  @u 2 dy dt xd>0 t xd =0 @xd gives rst a priori estimates of ut ; ru in L1 (R+; L2(xd > 0)) and of the trace of ut ; @u=@xd in L2 (R+  Rd?1). Then a multiplication of (4.7) by @u=@x ( < d) and an integration by parts gives also an estimate of the trace of @u=@x in the same space. One also holds easily non-zero data 9f; g ). It follows that the connected component de ned by a < 0 and kbk < c consists only of SS problems. Admitting that genericity holds accross each ?j , we have a transition accross ?l , so that the IBVP is WR when a < 0 and kbk > c. We now must consider separately the cases d = 2 and d  3. When d = 2, the IBVP is WR for a < c and jbj > c, since there is no transition accross a = 0 for jbj > c. However, there is a SS-SU transition accross a = 0 for jbj < c, so that the IBVP is SU for a > 0 and a2 + b2 < c2 . Let now consider the lens between this zone and the lines a = c, b = c (or b = ?c as well). One reaches it from the half-disk when crossing ?cr ; this shows that IBVPs are either WR or SU in the lens. We also reach it from a region of class WR when crossing ?l ; this shows that IBVPs are either SS or WR in the lens. Finally, this lens lies within the class WR. Now, the class WR extends to all parameters such that a > c, since ?0 cannot be a genuine transition in space dimension d = 2. When d  3, the whole axis is a transition towards SU. Therefore, the IBVP is SU in the half-ball a > 0, a2 + kbk2 = c2 , and also if 0 < a < c with kbk > c. Since ?l cannot serve as a transition from SU, the SU class extend in the lens from kbk > c. Finally, all the problems with 0 < a < c are SU. There remain two connected components for which a > c and either kbk < c or kbk > c. One reaches then from SU components, when crossing ?0 . Therefore, they belong to either SU or WR. Also, since ?l cannot serve as a transition WR-SU, both components belong to the same class. We easily check that this class is WR, by considering the case b = 0 :
= 0 reads  = a, so that a2 2 + k k2 = c22 . This shows that  2 iR, which forbids 0. We summarize the above discussion in the well-known gure 2.

5

Linear elasticity

In this section, we consider 0 = R2  R+, as an isotropic elastic medium, and we study its deformations. Denoting by z 2 R3 the displacement vector, the simplest 5

One may as well check this case by a direct computation of
.

17

model is the second order linear system 

(5.10)

ztt = div(Tz); x 2 0; Tz = (rz + rz t ) + ( ? )div(z)Id;

where  > 0, > 0 are Lame coecients. We consider a stressed boundary condition on xd = 0

Tz
 = zt;

(5.11)

where  is the outgoing unit normal and 2 R is a parameter. The aim of this section is to study the nature of the IBVP, according to the parameter , using main results of this paper about generic transitions and MAPLEr software when complexity of computations makes it necessary. More precisely, we will show Proposition 5 The IBVP is  SS for < 0  SU for > 0. For = 0, the IBVP is weakly stable, the transition being due to a surface wave.

Remark:

Note that when = 0 (free boundary condition) the well known surface wave (cf. for example [3], [31], [29]) called a Rayleigh wave is of nite energy. Proof. We rst compute the Lopatinskii determinant. Set v = ?zt, w = rz,the second order system (5.10) becomes a rst order system with unknowns vw 

(5.12)

@tv + div(w) + r(trace(w)) = 0; @tw + rv = 0:

The boundary condition (5.11) becomes

B

(5.13)



v w



:= (w + wt ) + ( ? )trace(w) + v = 0:

To determine E (;  ), we look for solutions of (5.12) 

v w



= et ei
y ei!xd



V W



;

where 0,  2 Rd?1, =m ! > 0. We get a linear system 

where  =



V + iW + i trace(W ) = 0; W + iV  t = 0;



 : 

18

Solving this system, we nd

E (; ) = F s (; )  F p (; ): Here, F s (;  ) and F p (;  ) are eigenspaces associated to eigenvalues

p



1



2 !s = i c2 + jj2 s

1

2 and !p = i c2 + j j2 p

2

p

2

;

where cs =  and cp =  + : The square roots for complex numbers are associated to the principal determination of the argument. More precisely,

F s (; ) =







;

st V

= 0; W



= ?iV st ; s =



 !s







; 

 ; k2C : ; V = kp ; W !p We notice that the dimensions of F s (;  ) and F p (;  ) are respectively d ? 1 and 1. Taking a basis of E (;  ), and using boundary condition (5.13), we can compute F p (; ) =

V W

V W

= ?iV pt ; p =

the Lopatinskii determinant. We nd


(; ; ) =
1 (; ; )
2(; ; );

(5.14) where (5.15)


1 (; ; ) =  + i!s ;

(5.16)


2(; ; ) =  2 2 j j2 + !s !p ? i 3 (!s + !p )  ?
?  2 + 2c2s jj2 2 + 4c4s jj2!s !p :

?




Looking at the de nition of !s , !p and F s (;  ), F p (;  ), we nd that  2  R = (; ); jj ? c2  0 ; p 

2

In this domain, we have (5.17)



   2 2 2 2   2 2 !s = ?sgn() c2 ? j j ; !p = ?sgn() c2 ? j j s p 1

1

In the complement of R; there are two subdomains. In the rst one, where j j2? c2s > 0; and 2

(5.18)

   2 2 2 2   2 !s = i jj ? c2 ; !p = i j j ? c2 ; s p 

1

1

2

19

surface waves are of nite energy. In the second one waves are not necessary of nite energy and we have (5.19)



1

2 !s = ?sgn() c2 ? j j2 s

2



2 ; !p = i j j ? c2 p 2

1

2

:

We now investigate roots of the Lopatinskii determinant. First we easily see that when  = 0, there holds
(; 0; ) = ? csc5p ( ? cs )2 ( ? cp)2 ; hence we get one dimensional instability for = cs and = cp. Then, using homogeneity and the fact that
only depends on j j, we can x j j = 1 in the following. Functions f (; ; ) with j j = 1 will be written, with slight abuse of notation, as f (; ): Note that the the rst factor (5.15) of (5.14) is similar to the Lopatinskii determinant of the wave equation studied in section 4, with b = 0 and a = 1 in the boundary condition. Using the previous analysis, we nd that
1(; ) = 0 for some ; 0 if and only if > cs. Consequently the IBVP is SU when  cs. Next, thanks to the energy estimate d Z 1 jz j2 + jrz + rz tj2 + 1 ( ? )(div z)2 dx = Z jz j2dy; t dt xd >0 2 t 2 xd =0 we know that the IBVP is SS when < 0 and weakly stable when = 0. It is well known that when = 0 there is a Rayleigh wave (see [3, 31, 29]), hence the analysis of section 3 shows that in the generic case = 0 will be a SS ? SU transition. We now verify that this transition actually occurs. The only interesting factor is
2 since = 0 is not a transition for
1. Using [29], and (5.16), (5.18) we get
2 (i; 0) = 0 for some ; 2 < c2s if and only if Q(X ) = 0 for some X > c12s ; where Q is the third degree polynomial   4   Q(X ) = X ? 21c2 ? X 2 X ? c12 X ? c12 s s p since Q( 12 ) = 16c18s 2 D(; 0)g (); where g is a positive function and, as in the previous sections, D(; ) =
2 (i; ): The analysis made in [29] shows that Q has a 1 unique real root X0 > c12s . Set 0 = X02 ; we get D(0; 0) = 0; @ @ D(0; 0) = iK for some K > 0. Moreover, as Q( c12s ) > 0; and Q(?1) = ?1, we have Q0 (X0)  0: In the generic case Q0(X0) < 0, hence @@ D(0; 0) = C > 0, since g > 0. Consequently, using the implicit function theorem, we get a function ( ) de ned in a vicinity of zero such that 2 ( ) = 0 ? i K C + O( );

and D(( ); ) = 0. Since KC > 0; and  = i, we see that the IBVP is SU for every

2]0; "] for some " > 0: If Q0(X0) = 0, Q00(X0) = 0; otherwise there would be another root of Q greater than c12s , and Q(3)(X0) < 0. Using again g > 0; we get

D(; ) = iK + C3 +


;

20

where C > 0, K > 0: We can use the Newton polygon method to describe the roots ( ) of D in the vicinity of zero. We nd that there is a root ( ) such that

 = 0 + t 31 +


; where t3 = ?i KC , and =m t < 0: Consequently setting  = i, we see that the IBVP is still SU for every 2]0; "]; for some " > 0:

The point is now to determine if a transition between SU and another class occurs for 2 ["; cs[; or if the IBVP stays in the SU class. Using section 3, we see that such a transition occurs when there is a surface wave or when fD = 0g has a singular point in the real domain. The analysis of section 4 shows that there is no transition for the factor (5.15) of (5.14). Hence we only have to study (5.16). First we look for Rayleigh waves. Using (5.16) and (5.18), we get

=m
(i; ) =  2

1!

   2 2 2 2   ; 1 ? c2 + 1 ? c2 s p



3

1

hence there is no Rayleigh wave when 6= 0. Next we look for the other (i.e. in nite energy) type of surface wave. Using (5.19) and taking real and imaginary part in (5.16) with  = i, we get the system 8 > < > :





1

1

2 2 c2s2 ? 1 2 ? jj3 + 4c4s c2s2 ? 1 2 = 0;  1 ?
2 2 ? jj3 c2s2 ? 1 2 + 2 ? 2c2s 2 = 0:

Multiplying the rst line by ( c2s ? 1) 21 and adding it to the second line, we nd 2

!

  2  j  j jj c ? 2 c ? 1 + jj = 0: 2

3

2

1

2

2

s

s

Since this quadratic polynomial in does not have any real root when  is real, a transition with a surface wave cannot occur. Finally it remains to look for a transition in the case with a singularity. We have to solve the system 8 2 > < 1 ? c2p < 0 D(; ) = 0 > : @ @ D(; ) = 0; and since we know that the IBVP is SU for  cs , we may add the condition 0 < < cs : Moreover since > 0, we have to keep in mind that ccsp < 1: Setting 8 1  2 > cs ? c2s 2 ; > X = > c2p 2 < > > > :



Y = 1 ? cs2  = c s ; 2

21

1

2

;

we get a system of polynomial equations  2  (1 ? Y 2 + XY ) ? (X + Y ) + (1 ? 2Y 2)2 + 4(1 ? Y 2)XY = 0; (X + Y ) + 4Y 4 ? 16XY 3 + 12X 2Y 2 + 8XY ? 4X 2 ? 4Y 2 =  2 (X ? Y )2; with constraints 8 0 < X < 1; > > < 0 < Y < 1; (5.20) X < Y; > > : 0 <  < 1: We eliminate  between these two equations and nd (5.21)

8 >
:

R(X; Y ) = 0;

where

8 > > > > > >
> > > > 4X (Y ? X )(?X 4 + 3Y X 3 + 16Y 2 X 2 ? 53Y 3 X + 31Y 4 ) > : ?(Y ? X )(X 3 ? 27X 2Y + 42XY 2 + Y 3) + (Y + X )2: As the system (5.21) is hard to study by hand, we use MAPLEr software to plot the curves D = 0; N = 0; R = 0, and D ? N = 0 in the plane keeping in mind

constraints (5.20). We get gure 3. We see that the curve R = 0 is in the domain N ? D < 0, D < 0. Consequently, on the curve R = 0 we have  = ND > 1. Hence thanks to (5.20), we see that a transition with a singularity cannot occur when 0 < < cs . Finally the IBVP stays in the SU class for 0 < < cs , and the computer assisted proof is complete.

6

The stability of shock waves

6.1 Overview

The normal mode analysis has been used for a long time in the study of the stability of multi-dimensional shock waves for systems of conservation laws. It has been discussed by several authors, with varying degrees of generality and rigor. See e.g. [4, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 21, 26]. Given a strictly hyperbolic system of conservation laws of the form (6.22)

ut + divf (u) = 0; u : Rd  (0; T ) ! Rn

one is concerned with the well-posedness of the Cauchy problem with piecewise smooth initial data. A basic problem arises when this data is compatible (to high

22

order) with the Rankine-Hugoniot condition and entropy criteria accross their discontinuity locus. Then the Cauchy problem resembles a free boundary value problem. It is usually transformed in an IBVP, by introducing the equation of the shock front as an extra unknown. A localisation procedure leads to the preliminary study of piecewise constant planar shocks. A linearisation of the transformed system about such a shock yields a hyperbolic IBVP with constant coecients. Its unknowns are 2n functions of (x; t) and a scalar function of (y; t), the latter representing the disturbances of the front to leading order. The linearized Rankine-Hugoniot condition plays the r^ole of the boundary data. For such a linear system, the de nition of a Lopatinskii determinant is straightforward. As in the rst part of this article, it is homogeneous and enjoys the \reality" properties already described. Since we are concerned here with IBVPs depending on some parameters, we shall consider the stability class as a function of the shock data (ul ; ur ; s). Here, the unperturbed shock is assumed to hold (w.l.o.g.) along the hyperplane xd = st, and the solution takes the values ul;r according to the sign of xd ? st. The number s is the shock velocity. For the sake of simplicity, we rst consider Lax shocks (see [4, 5, 13, 14, 15] and Section 6.3 for undercompressive shocks), which means that in the linearized problem, the 2n  2n matrix Ad is invertible and has exactly n ? 1 positive eigenvalues. The space of parameters is the subset of triplets (ul; ur ; s) 2 R2n+1 satisfying the Rankine-Hugoniot condition (6.23)

fd (ur ) ? fd (ul) = s(ur ? ul );

plus the Lax shock inequalities (this excludes the trivial case ul = ur ). This in general de nes a smooth manifold of dimension n + 1. Strong instability of the linearized problem clearly means a non-existence at the nonlinear level for generic data, although it has not been rigorously proved so far. Also, strong linear stability has been shown to imply the nonlinear stability, that is the local existence for the Cauchy problem for (6.22) with such compatible piecewise smooth data ; see [22, 21, 25, 12]. The consequence of weak linear stability is less clear. See however [2] for the study of supersonic vortex sheets, which yields a characteristic IBVP of class WR.

6.2 Gas dynamics

We may apply the former classi cation in three generic types and four generic transitions to this context. We shall describe these calculations for the interesting case of isentropic gas dynamics. Denoting the density, uid velocity and pressure by ; z; p, this system reads (6.24)

t + div(z) = 0; (z )t + div(z z ) + rp = 0;

where p is given as a function of  : p = p(). The so-called ideal gas obeys the law p =  , where 2 (1; 1 + d=2) is the adiabatic constant. Since n = d + 1, the manifold of parameters has dimension d + 2. However, the

Galilean invariance of the system allows us to assume that the tangential compo-

23

nents of zl;r and the shock velocity s vanish. Then we are reduced to a 2-dimensional manifold6 . A convenient choice of parameters is (l ; r). @p > 0. We de ne the sound speed c := pp . Assuming hyperbolicity, we have @  d ? 1 The velocities of in nitesimal waves in the direction  2 S are z
 and z
 c. The rst one allows for contact discontinuities (vortex sheets). The strategy described above yields a characteristic IBVP when applied to contact discontinuities. These waves cannot be strongly stable. We shall focus on shock waves, which are associated to the other velocities ; they yield non-characteristic IBVPs under the Lax shock condition. Due to the symmetry (xd ; zd ) 7! (?xd ; ?zd ), we may restrict to the case where the left side xd < 0 is upstream. Then, denoting by w the component zd , the Lax inequalities read

cl < wl; 0 < wr < cr :

(6.25)

We shall not redo calculations which have been done in several papers by dierent means (see [22, 21, 29] for instance). Choosing bases with an homogeneity of order one, the Lopatinskii determinant may be put in form ?





(;  ) =  []  (c2 ? w2) + 2w + k k2[p](c2 ? w2): Hereabove,  [g] denotes the jump gr ? gl of any quantity g accross the shock,  in all other instances, w; c are wr ; cr,   is the unique solution of positive real part of (6.26)

( ? w )2 ?  2 c2 = c2(c2 ? w2)k k2:

From equation (6.26), we see that the domain R is made of pairs ( ;  ) such that 2  (c2 ? w2 )kk2. We also remark that
does not depend on the whole frequency vector  , but only on its norm. For this reason, many calculations of transitions simplify. For instance, to investigate the case when D leaves R, we only need to write that
vanishes on @ R, that is for 2 = (c2 ? w2 )k k2. In that case, (6.26) gives  = iw and
= kk2[p](c2 ? w2) ? 2 [](c2 + w2). Therefore
= 0 implies the following relation between parameters [p] = [](c2 + m2 ): This relation has long been recognized as the transition from strong to weak stability (see [22] for instance). Similarly, in the investigation of surface waves one only needs to search for a zero of
of the form (i ;  ) with 2 < (c2 ? w2)k k2. Writing =
= 0, we obtain ( ? w )(w + (c2 ? w2 ) ): The last inequality selects the root  = i which enters the right half-space when  does ; we shall not need it. Writing @D=@ k k2 = 0, we obtain [p](c2 ? w2) = 2w [] dkd  k2 ; where We deduce that

2 2 2 2( ? w ) dkd  k2 + c (c ? w ) = 0:

( ? w )[p] + c2w[] = 0: Eliminating ; and  of these equations, we obtain either [p] = 0 or [p] = []w2. However, this with p0 > 0 or with (6.23) gives r = l , which is irrelevant. We conclude again that this type of transition from WR to SU does not occur. Last, since it was shown in [28] that
(1; 0) never vanishes (we have
(1; 0) = [](c + w)2), we see that a transition between WR and SU cannot hold. Since we easily check that weak shock waves (r =l  1+ ) are strongly stable, we conclude that these Lax shocks are strongly stable if [p]=[] < c2r + wr2 and weakly stable of real type if [p]=[] > c2r + wr2.

Full gas dynamics :

full gas dynamics involves the previous equations plus the conservation of energy, with an extra unknown e, the speci c internal energy. The pressure is then given as a function p(; e), and the sound speed is s

c = @p + ?2p @p : @ @e A similar analysis holds, with only one main dierence, already quoted in [28] : the number
(1; 0) may vanish for some shocks. Therefore, a transition WR vs SU may hold, which was known in previous works (see [22, 21]).

6.3 Phase boundaries

In this section, we consider the same system of conservation laws (6.24), governing the motion of a compressible uid, but with a nonmonotone pressure law. In the region(s) where p is a decreasing function of the density, the system is of elliptic type. This region is thus forbidden. We call phase boundary a shock-like solution of Equation (6.24) where the left and right states belong to two dierent connected

25

components of the hyperbolic region. For instance this can be a liquid-vapor interface if the pressure law is given by a van der Waals isothermal. We are especially interested in subsonic phase boundaries, for which the motion of the uid is subsonic on both sides in the frame attached to the \shock" wave. That is, with similar notations as before, we require that (6.27)

jwlj < cl and jwrj < cr :

By de nition subsonic phase boundaries are not Lax shocks. In the mathematical theory of shock waves, they are termed undercompressive. It is now well known that such waves require an additional jump condition besides the standard RankineHugoniot condition. This can be understood by counting the \number of incoming characteristics" in the free boundary problem. We point out to te reader that this number equals the one of outgoing characteristics from the \shock". There are n = d + 1 such characteristics for a subsonic phase boundary, instead of (n ? 1) for a Lax shock. The need for prescribing an additional jump condition is thus clear, but there is no \canonical" form for it. A consensus seems to be that it should read as a kinetic relation [1]. In our Eulerian formulation, the ux of momemtum accross the wave, j := l (wl ? s) = r (wr ? s) plays the role of the wave speed in Lagrangian coordinates. Hence a suitable form for a kinetic relation is f (vr ) ? f (vl) + (vr ? vl) p(vr ) +2 p(vl) = '(j ) ; (6.28) where v denotes the speci c volume of the uid (v = 1=), f its speci c free energy (de ned up to a constant by df=dv = ? p) and ' is a \given" function. This function ' is actually the (in nite-dimensional) parameter that we are going to play with. We point out that in the case when p is nonconvex (as a function of v ) but still monotone, the system (6.24) also admits undercompressive shock waves. For such waves the function ' in (6.28) must be such that (6.29)

j '(j )  0

in order to satisfy the Lax entropy condition associated with the convex entropy E =  kzk2=2 +  f . Even when p is nonmonotone we consider as \physical" the choices of ' meeting (6.29). Furthermore, we can reasonably assume that ' is monotone (nonincreasing in the \physical" case) and vanishes at 0. As a matter of fact, when j = 0, we want to recover the so-called Maxwell equilibrium, satisfying the equal area rule f (vr ) ? f (vl) + (vr ? vl) p(vr) +2 p(vl ) = 0 : Such a function ' may be derived through the viscosity-capillarity criterion [30, 32]. However, it is not explicitly known except when the viscosity is neglected. For the pure capillarity criterion, we easily obtain that '  0. For general viscosities, ' involves the internal structure of the boundary. We can just show that ' is decreasing for small enough positive viscosities (see Equation (15) and Lemma 2 in [5]). Concentrating on dynamical phase boundaries, for which j 6= 0, we may

26

assume without loss of generality that j > 0, s = 0 and the tangential components of zl;r equal 0 (as for Lax shocks). In particular, (6.27) reduces to (6.30) 0 < wl < cl and 0 < wr < cr : In this framework, we can compute a Lopatinskii determinant (see [5]). We obtain


(;  ) = wr wcl [w]2 (cr  ? wr r ) ( cl cr  2 + wl wr l r ) l r ?   (c  ? w  ) ( c w  + c w  ) r r r l r r r l l ?  wr [w] (cl  ? wl l) ( c2r ? wr2 ) kk2 ; where  := ? j '0(j ) and l;r are de ned by 2 2 (6.32) l;r =  2 + ( c2l;r ? wl;r ) k k2 and 0 (which is by the way incompatible with (6.34) and c2r ? wr2  c2l ? wl2). Now, assume that [w] < 0. Using (6.34)(6.35) we can show that the factor of  in the brackets above is still positive. As a matter of fact, this factor equals q

p

 := cl wr [w] c2r ? wr2 + wl ( c2r + wr [w] ) c2r ? wr2 ? c2l + wl2 : Using c2l ? wl2  0, we can roughly bound  by below as follows p (6.37)   wr ( cl wr + wl [w] ) c2r ? wr2  q p 2 2 2 2 2 2 2 (6.38) + wl cr cr ? wr ? cl + wl ? cl wr cr ? wr : We see, using wr < wl and (6.30)(6.34), that the second term in the right hand-side is always positive, since

c4r ( c2r ? wr2 ? c2l + wl2 ) > c4r ( wl2 ? wr2 ) > c2l wr2 ( c2r ? wr2 ) : As to the rst term, if wr  wl =2 then

cl wr + wl [w]  wl cl ?2 wl > 0 ;

otherwise, using (6.35) we have

cl wr + wl [w] > wl wr > 0 : Therefore,  is positive and
cannot vanish.

28

We now consider the easier case c2r ? wr2 < c2l ? wl2. For ( ;  ) 2 @ R, we have q

l = 0 ; r = ? i k k sgn c2l ? wl2 ? c2r + wr2 ; (6.39)
(i ;  ) (6.40) (6.41)

? i kk sgn wrwcl

n

l r

[w]2 cl c2r ( c2r ? wr2 )3=2 q

+ cl cr wr ( + [w] ) ( cr ? wr ) c2l ? wl2 ? c2r + wr2  q 2 2 2 2 2 2 +  cl wr ( wr (cl ? wl ) ? wl (cr ? wr ) ) cl ? wl : 2

2

2

The rst two terms in between the brackets are clearly positive for   0 because of (6.30). The third one is also nonnegative for   0, at least if wr > wl since 0 < c2r ? wr2 < c2l ? wl2 . Actually, the latter inequalities imply that wr > wl because of (6.34). This completes the proof. 

Proposition 7 Assuming (6.30)(6.34) and   0, we have
(i ; ) 6= 0 for all ( ;  ) 2 Rn(0; 0) such that min( cl;r ? wl;r ) k k   max( cl;r ? wl;r ) k k : Proof. See point ii. in the proof of Theorem 1 in [5].  2

2

2

2

2

2

2

It remains to investigate the possibility of surface waves, that is, zeroes of 2
(i ;  ) for 2 < min( c2l;r ? wl;r ) k k2. In this case, l;r are both negative real numbers. Splitting
(i ;  ) into real and imaginary parts we are going to show that surface waves can only occur for  = 0, under some \reasonable" assumptions. As before, the case of expansive phase transitions, for which [w] > 0 is easier to deal with. For compressive phase transitions, we will need to re ne the conditions in (6.35). A suitable (but not necessary) assumption will be (6.42) wl2  21 cl wr : This amounts to requiring Ml := wl=cl  21 wr =wl . This can easily be checked to hold true on the curves provided in [5], the Mach number Ml being much smaller than the ratio wr =wl.

Proposition 8 Assuming  > 0, (6.30) and p  either [w] > 0 and (6.34) with wr =cr  1= 2  or [w] < 0 and (6.34)(6.42) we have
(i ;  ) = 6 0 for < min( cl;r ? wl;r ) k k : 2

2

29

2

2

Proof. Assume that
(i ; ) = 0. Taking the real part yields ? (6.43) wl wr l r ? cl cr = w [w]  cr (cl wr r + cr wl l )
r r + wl wr [w] (c ? w ) l k k ; r r 2

2

2

2

2

2

whereas taking the imaginary part gives (6.44)

wl wr l r ? cl cr 2 = c[wwr]2 ( ? r (cl wr r + cr wl l
) r + cl [w] (c2r ? wr2) k k2 :

If [w] > 0, Equation (6.43) shows that

wl wr l r ? cl cr 2 > 0 : But we know from [5] (Proposition 2) that wl wr l r ? cl cr 2 only vanishes at p 2 =  V0 kk, where V0 > 0 is explicitly computable and satis es V0 < wl wr (in practice, V02  pwl wr ). These zeroes being simple, it is thus easily seen that wl wr l r ? cl cr 2 > 0 if and only if 2 < V02 k k2. Rewriting Equation (6.44) as (6.45)

?
wl wr l r ? cl cr 2 = c[wwr]2 ? cr wl l r + cl wr 2 ? cl wl (c2r ? wr2) k k2 r

we infer that

wl wr l r ? cl cr 2 < c[wwr]2 ? cr wl l r ? cl wl (c2r ? 2 wr2) k k2 ; r p which is clearly negative if wr =cr  1= 2. Note that this assumption is not much ?




restrictive: in a van der Waals uid, the Mach numbers

Ml;r := wl;r =cl;r are typically very small (less than 10?1 in the examples of [5]). Hence
(i ;  ) = 0 is impossible in the case [w] > 0. We now assume that [w] < 0. From Equation (6.44) we have

wl wr l r ? cl cr 2 < 0 and thus 2 > V02 k k2 by the remark above. Taking the dierence between (6.43) and (6.45) and bounding 2 by below we obtain 0 > V02 cl cr (1 ? Mr2 ) + (c2r ? wr2 ) wr wl cl =cr + wl (V02 c2r =wr + [w] (c2r ? wr2) ) l=r : By (6.30) the rst terms in the right-hand side are positive. Then the last one must be negative. Since 0 < l =r < 1 (because of (6.34) and wr < wl), we nd that     2 V c w w l l l 0 2 cl 2 0 > w w (1 ? Mr ) c + w + (1 ? Mr ) c + 1 ? w : l r r r r r

30

This is clearly impossible if V02 is suciently close to wl wr , for instance if

V02  1 ? wr : wl wr wl

(6.46)

We recall from Lemma 5 in [5] that (6.35) implies the bound V02=(wl wr )  1 ? 2 wr =wl. We are going to improve it under the stronger assumption (6.42). We proceed similarly. Since V02 =(wl wr ) is the positive root of

Q(Y ) := (1 ? Ml2 Mr2) Y 2 + ( wwl Mr2 (1 ? Ml2 ) + wwr Ml2 (1 ? Mr2) ) Y r l

? (1 ? Ml ) (1 ? Mr ) ; 2

2

the inequality in (6.46) is equivalent to Q(1 ? wr =wl)  0. Denoting  = wr =wl and assuming Ml2  m, which implies Mr2  m because of (6.34), we easily nd that Q(1 ? )  (1 ? )2 + m (1 ? ) (  + 1= ) ? (1 ? m)2  m ( 2 + (1 ? ) (  + 1= )) ?  (2 ? ) : Hence we have Q(1 ? wr =wl )  0 for 2 2 (2 ? ) m  4  2 + (1 ? ) (2 + 1)

(since   1). This shows that (6.42) implies (6.46), and completes the proof.  To conclude, Propositions 6 and 8 show that the stability of \realistic" propagating phase boundaries (satisfying (6.30)(6.34) and (6.35) or (6.42)) only undergoes a transition at  = 0. This transition is due to surface waves. We already know from [5] (Theorem 1) that it belongs to the class SS-SU, the strong stability case corresponding to  > 0.

References [1] R. Abeyaratne and J. K. Knowles. Kinetic relations and the propagation of phase boundaries in solids. Arch. Rational Mech. Anal., 114:119{154, 1991. [2] Miguel Artola and Andrew Majda. Nonlinear development of instabilities in supersonic vortex sheets. I. the basic kink mode. Physica D, 28:253{281, 1987. [3] D. M. Barnett and J. Lothe. Free surface (Rayleigh) waves in anisotropic elastic half-spaces: the surface impedance method. Proc. Roy. Soc. London Ser. A, 402:135{152, 1985. [4] Sylvie Benzoni-Gavage. Stability of multi-dimensional phase transitions in a van der Waals uid. Nonlinear Anal. TMA, 31:243{263, 1998. [5] Sylvie Benzoni-Gavage. Stability of subsonic planar phase boundaries in a van der Waals uid. Arch. Rat. Mech. Anal., 150:23{55, 1999.

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[6] Sylvie Benzoni-Gavage, Denis Serre, and Kevin Zumbrun. Alternate evans functions and viscous shock waves. SIAM J. Math. Anal. [7] Hans Bethe. On the theory of shock waves for an arbitrary equation of state. US Army report (1942). In Classic papers in shock compression science, pages 421{492, New York, 1982. Springer-Verlag. [8] Jacques Chazarain and Alain Piriou. Introduction a la theorie des equations aux derivees partielles lineaires. Gauthier-Villars, Paris, 1981. [9] W. Domanski. Surface and boundary waves for linear hyperbolic systems: applications to basic equations of electrodynamics and mechanics of continuum. J. Tech. Phys., 30(3-4):283{300 (1990), 1989. [10] S. P. D'Yakov. On the stability of shock waves (Russian). Teoret. Fiz., 27:288{ 295, 1954. [11] Jerome Erpenbeck. Stability of step shocks. Phys. Fluids, 5:1181{1187, 1962. [12] Jacques Francheteau and Guy Metivier. Existence de chocs faibles pour des systemes quasi-lineaires hyperboliques multidimensionnels. C. R. Acad. Sci. Paris, Ser. I, 327:725{728, 1998. [13] Heinrich Freistuhler. The persistence of ideal shock waves. Appl. Math. Lett., 7:7{11, 1994. [14] Heinrich Freistuhler. A short note on the persistence of ideal shock waves. Arch. Math., 64:344{352, 1995. [15] Heinrich Freistuhler. Some results on the stability of non-classical shock waves. Journal of Partial Di. Eqn., 11:23{38, 1998. [16] Reuben Hersh. Mixed problems in several variables. J. Math. & Mech., 12:317{ 334, 1963. [17] Alan Jerey and T. Taniuti. Nonlinear wave propagation. With applications to physics and magnetohydrodrynamics. Academic Press, 1964. [18] Tosio Kato. Perturbation theory for linear operators, volume 132 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 1966. [19] V. M. Kontorovitch. On the interaction between small disturbances and discontinuities in magnetohydrodrynamics and on the stability of shock waves. Soviet Physics, JETP, 35:851{858, 1959. [20] Heinz.-Otto. Kreiss. Initial boundary value problems for hyperbolic systems. Comm. Pure & Appl. Math., 23:277{298, 1970. [21] Andrew Majda. The stability of multi-dimensional shock fronts, volume 275 of Memoirs. Amer. Math. Soc., Providence, 1983.

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b

b

c

?cr c

?S

?c

c

?l

?l

?cr a

c

?S

?c

?0

a

?0

Figure 1: Transition curves for the wave equation. Left : d = 2. Right : d  3.

b

b WR

WR

c

c

WR

WR SU

SS

?c

c

SS

a

SU

c

a

?c

WR WR

Figure 2: Classes of IBVPs for the wave equation. Left : d = 2. Right : d  3.

34

1

N>0 N=0 N0

R=0

0.8 D=0

D