View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Applied Mathematics Letters 24 (2011) 757–761 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On mixed initial–boundary value problems for systems that are not strictly hyperbolic Corentin Audiard ∗ Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, Rhone, France article info a b s t r a c t Article history: The classical theory of strictly hyperbolic boundary value problems has received several Received 25 June 2010 extensions since the 70s. One of the most noticeable is the result of Metivier establishing Received in revised form 23 December 2010 Majda's ``block structure condition'' for constantly hyperbolic operators, which implies Accepted 28 December 2010 well-posedness for the initial–boundary value problem (IBVP) with zero initial data. The well-posedness of the IBVP with non-zero initial data requires that ``L2 is a continuable Keywords: initial condition''. For strictly hyperbolic systems, this result was proven by Rauch. We Boundary value problem prove here, by using classical matrix theory, that his fundamental a priori estimates are Hyperbolicity Multiple characteristics valid for constantly hyperbolic IBVPs. ' 2011 Elsevier Ltd. All rights reserved. 1. Introduction In his seminal paper [1] on hyperbolic initial–boundary value problems, H.O. Kreiss performed the algebraic construction of a tool, now called the Kreiss symmetrizer, that leads to a priori estimates. Namely, if u is a solution of 8 d X C >@ u C A .x; t/@ u D f ;.t; x/ 2 × Ω; <> t j xj R jD1 (1) C >Bu D g;.t; x/ 2 @ × @Ω; :> R ujtD0 D 0; C Pd where the operator @t jD1 Aj@xj is assumed to be strictly hyperbolic and B satisfies the uniform Lopatinski˘ı condition, there is some γ0 > 0 such that u satisfies the a priori estimate p γ kuk 2 C C kuk 2 C ≤ C kf k 2 C C kgk 2 C ; (2) Lγ .R ×Ω/ Lγ .R ×@Ω/ Lγ .R ×Ω/ Lγ .R ×@Ω/ 2 2 −γ t for γ ≥ γ0. Here above, Lγ is the usual L space with a weight e : Z 2 C −2γ t 2 L .R × O/ D u V e juj dxdt < 1 : C R ×O (Ralston [2] then extended this result to the case of complex coefficients.) ∗ Tel.: +33 6 87 65 44 13. E-mail address: [email protected]. 0893-9659/$ – see front matter ' 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.12.028 758 C. Audiard / Applied Mathematics Letters 24 (2011) 757–761 J. Rauch proved that the initial–boundary value problem is in fact well-posed for arbitrary L2 initial data. More precisely, for u0 the initial datum, he obtained the fundamental a priori estimate p −γ T e ku.T /k 2 C γ kuk 2 C kuk 2 ≤ C ku k 2 C kf k 2 C C kgk 2 C (3) L .Ω/ Lγ .T0;T ]×Ω/ Lγ .T0;T ]×@Ω/ 0 L .Ω/ Lγ .R ×Ω/ Lγ .R ×@Ω/ for Friedrichs symmetrizable systems (in his thesis) and soon after for strictly hyperbolic systems [3]. Motivated by physical systems that are not strictly hyperbolic and by characteristic IBVPs, Majda and Osher [4] pointed out that the construction of Kreiss symmetrizers can be performed as soon as the system of equations satisfies the so called `block structure condition' (see also [5]). More recently, Metivier [6] thoroughly investigated algebraic properties of the symbols of constantly hyperbolic operators. Definition 1. Let L be a first-order operator d D C X L @t Aj.x; t/@xj ; (4) jD1 with Aj V .x; t/ ! Aj.x; t/ 2 Mr .C/. D Pd It is said to be constantly hyperbolic if the symbol A(η/ jD1 Ajηj is diagonalizable with real eigenvalues, and the d multiplicity of the eigenvalues remains constant for η 2 R n f0g. The main result of Metivier in [6] is that, if L is a constantly hyperbolic differential operator, then it satisfies the block structure condition. P The proof relies on a factorization of the determinant of the symbol τ C Ajηj as in the Weierstrass preparation theorem. Here we will need a slightly different result proved in [7] that we shall state in the third part. The aim of our second part is to rapidly explain the scheme of proof of Rauch's theorem. In particular we emphasize where the strict hyperbolicity assumption is necessary. In the third part we describe a modification of Rauch's proof that adapts it to constantly hyperbolic IBVPs. 2. Rauch's theorem The proof of estimate (3) is rather long in [3]. It is based on an a priori estimate for strictly hyperbolic scalar equations whose proof relies on the method of Leray and Garding˙ (see [8]). We recall that a scalar operator P.t; x;@t ;@x/ is strictly hyperbolic (with respect to the timelike direction) if its principal symbol Pm.t; x; iτ; iη/ has roots in τ that are real and d distinct for η 2 R n f0g. Lemma 1. Let P.t; x;@t ;@x/ be a scalar strictly hyperbolic differential operator of order k, with smooth coefficients constant outside a compact set. There is a constant C such that for all T > 0, any φ 2 Hk.J × Ω/ and any " > 0 small enough, k−1 ! 1 X j kφ.T /k k−1 ≤ C "kPφk 2 C kφk k−1 C k@ φk k−1−j : (5) H .Ω/ L .]−∞;T ]×Ω/ H .]−∞;T ]×Ω/ xd H .]−∞;T ]×@Ω/ " jD0 The transition from scalar equations to first-order systems is made thanks to the following property, which is only proved with the help of Lemma 1, Kreiss's estimates (2), and the Sobolev spaces theory. Proposition 1. Let u be a solution of the boundary value problem Lu D f ;.t; x/ 2 × Ω; R (6) Bu D g;.t; x/ 2 R × @Ω: Let r be the size of the system. If L is strictly hyperbolic then for γ > 0 large enough we have the pointwise estimate ! kf k r−1 ]−∞ ]× −γ T Hγ . ;T Ω/ e ku.T /k r−1 ≤ C p C kgk r−1 ; (7) H .Ω/ γ Hγ .]−∞;T ]×@Ω/ m 2 where the Hγ spaces are the spaces built over Lγ as follows: ( Z ) m 2 X µ 2 −2γ t Hγ .Rt × Ω/ D u 2 Lγ V jD uj e dxdt < 1 : C jµ|≤m Ω×R C. Audiard / Applied Mathematics Letters 24 (2011) 757–761 759 Sketch of proof. We denote by Lco the transposed comatrix of L seen as a matrix of differential operators. Then we have LcoL D det.L/I C lower order terms. Each diagonal coefficient is a strictly hyperbolic scalar operator; thus Lemma 1 may be applied to each coordinate uj, and this gives k−1 −γ T co −γ · 1 −γ · X −γ · j e ku.T /k k−1 "kL Le uk 2 C ke uk k−1 C ke @ uk k−1−j : (8) H .Ω/ . L .]−∞;T ]×Ω/ H .]−∞;T ]×Ω/ xd H .]−∞;T ]×@Ω/ " jD0 Pk−1 −γ · j −1 Pd−1 k k k−1−j D − − C The trace terms jD0 e @xd u H .]−∞;T ]×@Ω/ are estimated thanks to the identity @xd u Ad @t u jD1 Aj@xj u f , the continuity of the trace Hm.Ω/ ! Hm−1.@Ω/, and the analogue of (2) on U − 1; T U (proved in [3]). Finally, using the inequality −γ T e kuk k−1 ]−∞ ]× ≤ kuk k−1 ; H . ;T Ω/ Hγ .]−∞;T ]×Ω/ we obtain (7). The derivation of (3) from this proposition is quite onerous; it is based on a series of analogous inequalities involving the boundary problem as well as a dual problem. Thankfully, this part does not use the strict hyperbolicity assumption and we shall therefore not describe it. As we see, the strict hyperbolicity assumption is only needed to apply Lemma 1 and Kreiss's estimate (2). Since the results of Métivier in [6] show that the estimate (2) is true for constantly hyperbolic boundary value problems, we are left to show how to adapt Lemma 1. 3. The case of constantly hyperbolic systems Rauch's proof of Proposition 1 is not valid for a non-strictly hyperbolic operator L. Even if L is a constantly hyperbolic operator, the diagonal coefficients of LcoL are not strictly hyperbolic, and Lemma 1 does not apply, as can be seen for the trivial example of two independent transport equations @x 0 @t u C u D 0: (9) 0 @x 2 co .@t C @x/ 0 Here L L D 2 . Even though (7) holds true for L D .@t C @x/I, it cannot be deduced from the scalar 0 .@t C @x/ equations 2 det.L/uj D .@t C @x/ uj D 0; j D 1; 2: In fact, to generalize the proof of Proposition 1, it suffices to find an operatoreL such thateLL D PI C lower order terms, where P is a strictly hyperbolic operator (of course the degree of P will not be the size r of the system, except in the case of strict hyperbolicity). We will need the Proposition 1.7 (p. 46) from [7] on the factorization of constantly hyperbolic operators: P Proposition 2. If L is constantly hyperbolic, the determinant of the symbol τ I C Ajηj factors as K Y qk Pk(τ; η/ ; kD1 where the Pk s satisfy: • Each Pk is a homogeneous polynomial of (τ; η/.