Symmetrizers and Continuity of Stable Subspaces for Parabolic-Hyperbolic Boundary Value Problems

In this paper we prove the continuity of stable subspaces associated to parabolic-hyperbolic boundary value problems, for limiting values of parameters. The analysis is based on the construction performed in a previous paper of Kreiss' type symmetrizers.


Introduction
This paper is motivated by the analysis of multidimensional small viscosity boundary layers or shock profiles (see [MZ], [GMWZ1] [GMWZ2]). The stability criterium for layers or profiles is described through the nonvanishing of an Evans function. After freezing the tangential coefficients, performing a tangential Fourier-Laplace transform and using a reduction to a normal form, the problem is reduced to an ordinary N × N constant coefficient differential system in the normal variable: where p are parameters and ζ denote the tangential frequency variables (see [MZ]). Typically, ζ = (τ − iγ, η) ∈ C × R d−1 where τ − iγ is the Fourier-Laplace frequency dual to time t and η = (η 1 , . . . , η d−1 ) are the Fourier frequencies dual to the spatial tangential variables y. The equation is supplemented by boundary conditions (1.2) Γ(p, ζ)U |x=0 = 0 .
For ρ = |ζ| > 0 and γ ≥ 0, G(p, ζ) has no purely imaginary eigenvalues and the bounded solutions of the homogeneous equation (1.1) are where E − (p, ζ) is the stable subspace at +∞ of G(p, ζ), that is the space generated by the generalized eigenvectors associated to eigenvalues belonging the the half space {Re µ < 0}. The stability condition, that is the well posedness of the boundary value problem (1.1) (1.2) in L p spaces, reads where, for linear subspaces E and F of C N , det(E, F ) is zero if dim E + dim F = N and equal to the determinant obtained by taking orthonormal basis in E and F when dim E + dim F = N ; this is necessary for multidimensional stability. The function D is commonly known as an Evans function.
The uniform stability condition requires that D (or a rescaled version for high frequencies) be uniformly bounded from below by a positive constant when ρ > 0 and γ ≥ 0. This has been shown in several contexts to be sufficient for multidimensional stability; see [Z], [MZ], .
Of particular importance is the analysis for low frequencies where the hyperbolic character becomes crucial. There one uses polar coordinates (1.5) ζ = ρζ , ρ = |ζ|.
This result was known under additional assumptions such as strict hyperbolicity of the hyperbolic part or constant rank of branching points, see [Rou], [ZS], or [Z]. The Assumptions 3.1 are the somewhat minimal conditions of hyperbolicity and parabolicity as in [MZ]. An important corollary is that (1.6) D(p, ρζ) = β∆(p,ζ) + o(1) , as ρ → 0 uniformly for |ζ| = 1 withγ ≥ 0, where ∆(p,ζ) is the Lopatinski determinant of the appropriate limiting hyperbolic boundary value problem, and β is a transversality coefficient for the inner layer ordinary differential equation (the viscous shock or boundary profile problem); indeed, the two statements are essentially equivalent. This reduces the somewhat abstract uniform stability condition, at least in the crucial low frequency regime, to a pair of checkable conditions, associated respectively with outer (hyperbolic) and inner (stationary parabolic) problems in the formal matched asymptotic approximation of the boundary layer, and is one of the central conclusions of [ZS], [Z], [Rou].
In the original planar shock stability analysis of [Z], which was carried out by explicit estimates on the Laplace-Fourier inversion formula, continuous extension of stable subspaces was also used in an essential way in obtaining the linearized decay estimates from which nonlinear stability ultimately follows. In the more recent curved boundary layer analysis of [MZ], carried out by Kreiss symmetrizer techniques, it was shown that the uniform stability condition alone suffices, whether or not continuity holds, yielding applications to much more general situations than considered in [Z]; this was possible because of the flexibility of the symmetrizer method. This abstract result, however, sidestepped the issue of checkability of the uniform stability condition/validity of the fundamental relation (1.6).
With the establishment of Theorem 1.1, we now resolve this issue completely, under the same minimal hypotheses considered in [MZ]. The matrix perturbation problem associated with determination of stable subspaces is a rather complicated one involving many parameters, and would be difficult if not impossible to describe completely in this generality. In the inviscid case, this problem can be sidestepped thanks to the block structure condition enjoyed by systems with constant multiplicity characteristics; see [Mé3]. However, the additional parameter ρ generically destroys this block structure, and so the issue apparently arises again. A notable aspect of [MZ] was the construction of Kreiss symmetrizers without the use of block structure in either γ or ρ: that is, the identification of different roles played by Laplace and Fourier frequencies in the analysis. Here, we show that, somewhat surprisingly, the same symmetrizer construction used in [MZ] to avoid the need for continuity of subspaces in fact establishes that continuity! That is, the structure of stable subspaces is already hidden in the details of the nonstandard symmetrizer construction.
As the proof of the main theorem relies on the use of symmetrizers for G(p, ζ), we first review the construction performed in [MZ] of Kreiss' type symmetrizers for parabolic-hyperbolic boundary value problems and we prove the new continuity theorem next.

Symmetrizers
Recall first the essence of the "method of symmetrizers" as it applies to general boundary value problems on the half line {x ≥ 0}: Here, u and f are functions on [0, ∞[ values in some Hilbert space H, and G(x) is a C 1 family of (possibly unbounded) operators defined on D, dense subspace of H. A symmetrizer is a family of C 1 functions x → S(x) with values in the space of operators in H such that there are C 0 , λ > 0, δ > 0 and C 1 such that In (2.2), the norm of S(x) is the norm in the space of bounded operators in H. Similarly S * (x) is the adjoint operator of S(x). The notation Re T = 1 2 (T + T * ) is used in (2.3) for the real part of an operator T . When T is unbounded, the meaning of Re T ≥ λ, is that all u ∈ D belongs to the domain of T and satisfies (2.5) Re T u, u ≥ λ|u| 2 .
The property (2.3) has to be understood in this sense.
Lemma 2.1. If there is a symmetrizer S, then for all Here, | · | is the norm in H and · the norm in L 2 ([0, ∞[; H).
In applications to evolution boundary value problems, G(x) is a tangential differential system G(t, y, x, ∂ t , ∂ y ). One usually studies (2.1) in weighted spaces e γt L 2 , with γ > 0 for forward propagation. With u γ = e −γt u and f γ = e −γt f , the equation (2.1) is transformed into with G γ (t, y, x, ∂ t , ∂ y ) = G(t, y, x, ∂ t + γ, ∂ y ), which is studied in L 2 . When G has constant coefficients in (t, y), the equation (2.8) transformed into a system of o.d.e. in x using a tangential space time Fourier transform. For "slowly" varying coefficients, one first freeze the coefficients and then perform the Fourier transform, yielding a family of systems depending on the parameters (t, z) and the frequencies ζ = (τ, η, γ): where G(t, y, x, ζ) = G(t, y, x, iτ + γ, iη).
In this framework, one can look for symmetrizers for the system (2.9). This is a matrix S(t, y, x, ζ) such that When G and S do not depend on (t, y), one immediately gets symmetrizers for (2.8), taking S as Fourier multiplier. When S depends on (t, y), one can try to take as a symmetrizer the pseudo-differential operator or a para-differential version of it. The problem is then to convert the symbolic estimates (2.10) into estimates for the operators S(x). This strategy has been shown to be successful for hyperbolic problems ( [Kr], [CP]) and small viscosity parabolic-hyperbolic problems ( [MZ], [GMWZ1], [GMWZ2]).
A further reduction to constant coefficients in x can be performed when G converges at an exponential rate to a limit when x tend to infinity: for some θ > 0 and ζ in a neighborhood of ζ. Then: Lemma 2.2 ( [MZ]). There is a neighborhood ω of ζ and there is a matrix All this motivates the construction of symmetrizers for constant coefficients systems (1.1) with boundary conditions (1.2). Note that the reduction to constant coefficients above leads to frequency dependent boundary conditions. In the sequel, we concentrate on hyperbolic-parabolic systems arising in the study of boundary layers or shock profiles.

The main result
Consider a first order N × N system The variables are t ∈ R, y ∈ R d−1 and x ∈ R + :=]0, +∞[. Moreover, In addition, p ∈ R M is a place holder for the variables themselves, the unknown in case of quasilinear systems and all the other parameters such as source terms or control. Next, we consider a parabolic viscous perturbation of (3.1) This kind of systems arises in the study of boundary layers or shock profiles, as linearized form of viscous perturbations of hyperbolic conservation laws, see e.g. [MZ] [GMWZ1], [GMWZ2]. In [MZ] [GMWZ2] we considered rather the small viscosity problem L 1 − εL 2 . Rescaling the variables as t = εt, y = εỹ, x = εx reduces to ε = 1 and the rescaled variables are much more adapted to the geometrical analysis of the symbol. (H2) There is c > 0 such that for all p ∈ O and ξ ∈ R d the eigenvalues We perform a Laplace-Fourier transform with respect to t and a Fourier transform with respect to the variables y. This amounts to performing the substitution In applications, to study the forward propagation, dictated by the sign of L 2 , one restricts γ to be nonnegative. After this Laplace-Fourier transform, L becomes a second order differential operator in x depending on the parameters p and ζ = (τ, η, γ) ∈ R d+1 . We reduce this system to first order introducing the variable U = (u, ∂ x u) with values in R 2N . This leads to the following 2N × 2N system and the matrices A j and B j,k are evaluated at p. The Assumption 3.1 immediately implies the following result (see [MZ] for a detailed proof).
ii) When ζ = 0, G(p, 0) has 0 as a semi-simple eigenvalue, of multiplicity N . The nonvanishing eigenvalues are those of (B d,d ) −1 A d .
For ζ = 0 and γ ≥ 0 we denote by E − (p, ζ) [resp. E + (p, ζ)] the spectral subspace of G(p, ζ) associated to the spectrum in the left hand half space {Re µ < 0} [resp. {Re µ > 0}]. Both have dimension N and depend smoothly on p and ζ as long as ζ = 0 and γ ≥ 0. To analyze the behavior of this spaces when ζ tends to zero, we introduce polar coordinates ζ = ρζ with ρ = |ζ| and set We can now state the main result of this paper: Remark 3.4. As already said, this result was already known under additional hypothesis [Rou], [ZS], [Z]. In [MZ] we used that theĚ − (p,γ, ρ) is continuous up to ρ = 0 whenγ > 0 and thatĚ(p,ζ, 0) has the form see section 4 below. Moreover, E hyp − is the negative subspace corresponding to the hyperbolic system L 1 . Following Kreiss [Kr] (see also [CP] and [Mé3] for the extension to hyperbolic systems of constant multiplicity) E hyp − (p,ζ) can be extended continuously up toγ = 0. Therefore, in [MZ] we used the existence of the double limit lim γ→0 lim ρ→0Ě − (p,ζ, ρ) to establish desired linear decay estimates. The Theorem 3.3 asserts the existence of the limit asγ and ρ tend to zero, independently.

Strategy of the proof
By Lemma 3.2, for ζ small the eigenvalues of G split into a group of N eigenvalues close to 0 and a group of N eigenvalues away from the imaginary axis. More precisely, given a point p, there is a C ∞ invertible matrix V defined for (p, ζ) in neighborhood ω of (p, 0) such that G 1 := V −1 GV has the block diagonal form with H(p, 0) = 0 and P (p, 0) = (B d,d ) −1 A d . The eigenvalues of P (p, ζ) satisfy |Re µ| ≥ c for some c > 0 and Therefore, for (p, ζ) ∈ ω, with ζ = 0 and γ ≥ 0: where E H − [resp. E P − ] is the spectral space associated to eigenvalues of H [resp. P ] in {Re µ < 0}.
On one hand, because the spectrum of P does not intersect the imaginary axis, the definition of the spaces E P − extends smoothly to (p, ζ) ∈ ω. On the other hand, the eigenvalues of H are small, and we now investigate how they split in the complex complex domain when ζ is small. In polar coordinates, there holds where EȞ − is the negative space associated toȞ. The hyperbolicity assumption (H1) implies that H 0 (p,ζ) has no eigenvalues on the imaginary axis whenγ = 0. This remains true forȞ(p,ζ, ρ) for ρ small enough (depending onγ). In particular, for allζ ∈ R d+1 witȟ γ > 0, the definition of the bundle EȞ(p,ζ, ρ) extends smoothly to a neighborhood of (p,ζ, 0) in R M × R d+1 × R. This remains true for E(p, ρζ) since E P − (p, ρζ) and V(p, ρζ) are smooth in ρζ. Thus, it remains to prove Theorem 3.3 on a neighborhood of a point (p,ζ, 0) whereγ = 0 andζ = 0.
The strategy of the proof is as follows. We first construct explicitly spaces E − and E + such that Indeed, extending Kreiss' analysis to hyperbolic systems of constant multiplicity as in [Mé3], one can show that the bundle E H 0 − (p,ζ) extends continuously toγ = 0 and we take E − to be equal to the value of this extension at (p,ζ), but we will not use this result, see below. Next the main point is to construct symmetrizers. In the next statement we denote by Π ± the projectors on E ± associated to the decomposition (4.7).

Proof of Theorem 4.1
Though not stated explicitly, the result of Theorem 4.1 is given in Appendix A, section 3, of [MZ]. For the convenience of the reader, we give here the main steps of the proof. a) We first remark that it is sufficient to prove the theorem for κ large enough. Next, we can replace (4.9) by the weaker condition for S = S(p,ζ, 0). Indeed, if (5.1) holds, then, by continuity of S, on a possibly smaller neighborhood of (p,ζ, 0), there holds and therefore 1 2 S satisfies (4.8) for κ/2 and also (4.8) and (4.10) with the constant c/2. b) Recall that G 1 = V −1 GV has the form (4.1). We construct spaces E H ± and E P ± such that Denote by Π H ± and Π P ± the corresponding projectors. Suppose that for all κ large enough there are symmetrizers S H and S P defined on some neighborhood of (p,ζ) and such that Here, we have denoted by S H and S P the value of S H and S P respectively at the base point (p,ζ, 0).
Therefore, choosing δ = 1/C, (5.4) and (5.7) imply (5.1) with κ replaced by κ/C and it is sufficient to construct the symmetrizers S H and S P .
c) The construction of S P is easy. Because the eigenvalues of P do not belong to the imaginary axis, one can perform a further reduction (or choose V in (4.1)) such that for (p, ζ) in a neighborhood of (p, 0): where the spectrum of P ± is contained in {±Re µ > 0}. In this basis, the spaces E ± (p, ζ) are constant. we call them E ± . They are generated by the first and last vectors of the basis corresponding to the blocks P + and P − respectively. There are self-adjoint matrices S P ± and c > 0 such that For instance, one can choose with appropriate positive constants C ± Then the symmetrizers satisfy (5.6) to (5.8).
Remark 5.1. The symmetrizers S P constructed above are smooth functions of (p, ζ). This was important in [MZ] in the quantization of the symbolic calculus.

d)
We now turn to the construction of S H . As already mentioned in (4.4),Ȟ is a perturbation of H 0 . We denote by µ k = iξ k the distinct eigenvalues of H 0 := H 0 (p,ζ) located on the imaginary axis. There is δ > 0, a neighborhood Ω 0 of (p,ζ, 0) and a smooth matrix V (p,ζ, ρ) such that (5.10) such that each block H k has its spectrum either in {|Re µ| ≥ 2δ} or in the ball of radius δ centered at µ k . Moreover, we can assume that the balls of radius 2δ centered at the µ k do not intersect each other.
We denote by N k the dimension of the block H k . By Assumption 3.1, the H k have no eigenvalues on the imaginary axis when ρ ≥ 0,γ ≥ 0 and ρ +γ > 0. Therefore, the number of eigenvalues of H k in {±Re µ > 0} is constant for ρ ≥ 0,γ ≥ 0 and ρ +γ > 0. We denote it by N k,± . Because the total number of eigenvalues of G in {Re µ < 0} is N , we have Arguing as in step b), it is sufficient to construct spaces E k ± satisfying (5.12) and for all κ large enough, symmetrizers S k , C ∞ on a neighborhood of (p,ζ, 0), such that In (5.14), S k = S k (p,ζ, 0) and Π k ± are the projectors associated to the decomposition (5.12). e) If the spectrum of H k lies in {Re µ > 2δ} [resp. {Re µ < −2δ}, then arguing as in step c), we set E k − = {0} [resp. E k − = C N k ] and S k = κ 2 S k [resp. S k = −S k ] where S k (p,ζ, ρ) is a self adjoint matrix, defined and C ∞ on a neighborhood of (p,ζ, 0) and such that Re S k ≥ Id and Re S k H k is positive definite [resp. Re S k ≤ Id and Re S k H k is negative definite].
f ) Consider now the case where the spectrum of H k is contained in the ball of radius δ centered at µ k = iξ k . We note that iξ k is an eigenvalue of H 0 if an only if −τ = λ(p,η, ξ k ) where λ(p, η, ξ) is one among the eigenvalues of η j A j (p) + ξA d (p). Suppose first that ξ k is a simple root of the eigenvalue equation, that is Denote by α k the multiplicity of λ (see Assumption (H1)). Then, according to [Mé3], H k (p,ζ, 0) is a scalar matrix q k (p,ζ)Id and by Lemma 2.10 of [MZ] (5.17) H k (p,ζ, ρ) = q k (p,ζ)Id + ρR k (p,ζ, ρ) .
Moreover, q k is purely imaginary whenγ = 0,q k := ∂γRe q k (p,ζ) does not vanish andq k Re R k (p,ζ, 0) is definite positive. Whenq k > 0 [resp.q k < 0], forγ > 0 small, the eigenvalue q k has a positive [resp. negative] real part. Thus we set is a self adjoint matrix, defined and C ∞ on a neighborhood of (p,ζ, 0) and such that Re S k ≥ Id and Re S k R k is positive definite [resp. Re S k ≤ Id and Re S k R k is negative definite]. g) We now come to the most difficult part of the construction, when ξ k is a multiple root of the eigenvalue equation, that is, when there is ν k ≥ 2 such that ∂ ξ λ = . . . ∂ ν k −1 ξ λ = 0 , ∂ ν k ξ = 0 at (p,η, ξ k ). (Note that, since λ is real analytic in ξ, there is always such an integer ν k ). The case ν k = 1 is (5.16)).
¿From [Kr] in the strictly hyperbolic case and from [Mé3] for the extension to constant multiplicity, we know that there is a smooth matrix V k 0 (p,ζ) such that where the subblock Q k is ν k × ν k matrices and there are α k such blocks, where α k is the multiplicity of the eigenvalue λ. In particular the dimension of H k is N k = α k ν k . Moreover, In addition, with Ralston's lemma [Ral], we can assume that only the first column of Q k does not vanish: where q k denotes the lower left hand corner of Q k , that Q k has purely imaginary coefficients whenγ = 0 and thatq k := ∂γRe q k (p,ζ) = 0.
Next from Lemma 2.10 of [MZ], we know that there is a smooth matrix V k on a neighborhood of (p,ζ, 0), which extends V 0 , such that where the matrix R k has the following decomposition in α k × α k blocks R k p,q of size ν k × ν k : Moreover, only the first column of the R k p,q does not vanish: where r k p,q denotes the lower left hand entry of R k p,q . In addition, denoting by R ♭ k the α k × α k matrix with entries r k p,q , the matrixq k Re R ♭ k (p,ζ, 0) is definite positive.
Arguing as in step a) and b), we are reduced to construct spaces and symmetrizers for each block H k 1 . h) From [Kr] [CP], or simply from a direct analysis of the model case we know that the candidate for the limit negative space for this block is the space generated by the first β k vectors of the canonical basis : where (5.26) β k = 1 2 ν k when ν k is even , 1 2 (ν k ± 1) when ν k is odd and ∓q k > 0 .
We also introduce Note that β k is exactly the number of eigenvalues in {Re µ < 0} of the model (5.24) forγ > 0. This is still true for Q k (p,τ ,η,γ). Thus the number of eigenvalues in {Re µ < 0} of Q k is ν k β k whenγ > and therefore the number of eigenvalues of H k 1 in {Re µ < 0} when ρ ≥ 0,γ ≥ 0 and ρ +γ > 0, which is constant on a neighborhood of (p,ζ, 0), is Therefore, the condition (5.12) is satisfied. We now proceed to the construction of the symmetrizers. We construct S k in the block decomposition of H k 1 (5.27) where E k and E k are real symmetric matrices, and F k and H k are real and skew symmetric. E k is constant and has the special form Moreover E k (p,ζ) = 0 and F k and F ′ k are constant. The order of the construction is as follows. One first chooses E k , E k and F k as in [Kr] (see also [CP]) to construct symmetrizers for Q k , that is for ρ = 0. The new part lies in the choice of F ′ k .
1. Choice of E k . First choose the real coefficient e k,1 such that (5.29) e k,1qk ≥ 3.
Next, the coefficients e k,l for l > 1 are chosen successively to achieve that there is c > 0 such that where π k ± is the projection onto E k ± in the decomposition C ν k = E k + ⊕ E k − . (cf Lemma 5.6 in [CP], Chap. 7).
Next, following [CP] (cf (equation (5.5.3) in Chap. 7), (5.29) implies that there is a constant C and a neighborhood of (p,ζ) such that with w 1 the first component of w ∈ C ν k and w ′ ∈ C ν k −1 denotes the other components.
2. Choice of E k . Recall that Q k = i(ξ k Id+N k ) where N k is the Jordan matrix of size ν k (see (5.19)). The form of E k is chosen so that E k (ξ k Id+N k ) is real and symmetric . Next, the real matrix E k (p,ζ) is chosen so that such that (E k + E k )( 1 i Q k ) is real and symmetric whenγ = 0. This is achieved in [Kr] [CP] using the implicit function theorem and the property that 1 i Q k is real whenγ = 0.
where C is the constant in (5.31). As a consequence, we have (5.32) Re (E k + E k − iγF k )Q k =γD k , D k (p,ζ) = Re (E k ∂ γ Q k (p,ζ) + Re (F k N k ) ≥ Id .
4. Choice of F ′ k . This is the new part detailed in the Appendix A of [MZ]. Denote by E k the block diagonal matrix Diag(E k ). A vector w ∈ C N k , N k = ν k α k , is broken into α k blocks w p ∈ C ν k , with components denoted by w p,a . We denote by R k p,q the ν k ×ν k blocks of R k and by R p,a,q,b their entries. The entries of E k are denoted by E a,b . Since R p,a,q,b = 0 when b > 1, the special form of E k implies that Re (E k R k w, w) = Re E a,c R p,a,q,1 w q,1 w p,c = Re e k,1 r p,q w q,1 w p,1 + O(|w * ,1 | |w ′ * |) , where w * ,1 ∈ C α k is the collection of the first components w p,1 , w ′ * the remainder components and r p,q = R p,ν k ,q,1 the lower left hand corner entry of R p,q . The matrix Re (R ♭ k ) is definite, positive or negative according to the sign ofq k , which is the sign of e k,1 by (5.29). Thus, multiplying E k by some positive constant, we can achieve that in addition to (5.30) (5.32), the following inequality holds: (5.33) Re E k R k (p,ζ, 0)w, w ≥ 2|w * ,1 | 2 − C ′ |w ′ * | 2 .
By (5.32) and (5.34), the matrices D k and D ′ k are positive definite on a neighborhood of the base point (p,ζ, 0). This implies that S k and H k 1 satisfy (5.15).
This finishes the construction of symmetrizers for H k 1 and thus the proof of Theorem 4.1.