On the L 2 well posedness of Hyperbolic Initial Boundary Value Problems
[Sur le problème mixte hyperbolique dans L 2 ]
Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 1809-1863.

On montre que le problème mixte est bien posé dans L 2 , sous l’hypothèse nécessaire de Lopatinski uniforme, pour une classe de systèmes hyperboliques qui contient les systèmes de multiplicités constantes, mais significativement plus large. On montre en outre que la vitesse de propagation du problème aux limites est la même que la vitesse de propagation à l’intérieur du domaine. Au contraire, on montre sur un exemple que, même pour les systèmes symétriques au sens de Friedrichs mais à multiplicités variables, le problème mixte peut être mal posé sous la seule condition de Lopatinski uniforme.

In this paper we give a class of hyperbolic systems, which includes systems with constant multiplicity but significantly wider, for which the initial boundary value problem (IBVP) with source term and initial and boundary data in L 2 , is well posed in L 2 , provided that the necessary uniform Lopatinski condition is satisfied. Moreover, the speed of propagation is the speed of the interior problem. In the opposite direction, we show on an example that, even for symmetric systems in the sense of Friedrichs, with variable coefficients and variable multiplicities, the uniform Lopatinski condition is not sufficient to ensure the well posedness of the IBVP in Sobolev spaces.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3123
Classification : 35L04
Keywords: Hyperbolic first order systems, Cauchy problems, boundary value problems, Lopatinski conditions, semi-group estimates
Mot clés : Systèmes hyperboliques du premier ordre, problèmes de Cauchy, problèmes aux limites, conditions de Lopatinski, estimations de semi-groupe
Métivier, Guy 1

1 Université de Bordeaux - CNRS Institut de Mathématiques de Bordeaux 351 Cours de la Libération 33405 Talence Cedex (France)
@article{AIF_2017__67_5_1809_0,
     author = {M\'etivier, Guy},
     title = {On the $L^2$ well posedness of {Hyperbolic} {Initial} {Boundary} {Value} {Problems}},
     journal = {Annales de l'Institut Fourier},
     pages = {1809--1863},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     doi = {10.5802/aif.3123},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.3123/}
}
TY  - JOUR
AU  - Métivier, Guy
TI  - On the $L^2$ well posedness of Hyperbolic Initial Boundary Value Problems
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 1809
EP  - 1863
VL  - 67
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.3123/
DO  - 10.5802/aif.3123
LA  - en
ID  - AIF_2017__67_5_1809_0
ER  - 
%0 Journal Article
%A Métivier, Guy
%T On the $L^2$ well posedness of Hyperbolic Initial Boundary Value Problems
%J Annales de l'Institut Fourier
%D 2017
%P 1809-1863
%V 67
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.3123/
%R 10.5802/aif.3123
%G en
%F AIF_2017__67_5_1809_0
Métivier, Guy. On the $L^2$ well posedness of Hyperbolic Initial Boundary Value Problems. Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 1809-1863. doi : 10.5802/aif.3123. http://www.numdam.org/articles/10.5802/aif.3123/

[1] Audiard, Corentin On mixed initial-boundary value problems for systems that are not strictly hyperbolic, Appl. Math. Lett., Volume 24 (2011) no. 5, pp. 757-761 | DOI | Zbl

[2] Benoit, Antoine Finite speed of propagation for mixed problems in the WR class, Commun. Pure Appl. Anal., Volume 13 (2014) no. 6, pp. 2351-2358 | DOI | Zbl

[3] Benzoni-Gavage, Sylvie; Serre, Denis Multi-dimensional hyperbolic partial differential equations. First-order systems and applications, Oxford Mathematical Monographs, Oxford University Press, 2007, xxv+508 pages | Zbl

[4] Chazarain, Jacques; Piriou, Alain Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier-Villars, 1981, viii+466 pages (Ouvrage publie avec le concours du C.N.R.S.) | Zbl

[5] Friedrichs, Kurt Otto Symmetric hyperbolic linear differential equations, Commun. Pure Appl. Math., Volume 7 (1954), pp. 345-392 | DOI | Zbl

[6] Friedrichs, Kurt Otto Symmetric positive linear differential equations, Commun. Pure Appl. Math., Volume 11 (1958), pp. 333-418 | DOI | Zbl

[7] Friedrichs, Kurt Otto; Lax, Peter David Boundary Value Problems for First Order Operators, Commun. Pure Appl. Math., Volume 18 (1965), pp. 355-388 | DOI | Zbl

[8] Friedrichs, Kurt Otto; Lax, Peter David On Symmetrizable Differential Operators, Proc. Sympos. Pure Math., Volume 10 (1967), pp. 128-137 | DOI | Zbl

[9] Gårding, Lars Linear hyperbolic partial differential equation with constant coefficients, Acta Math., Volume 85 (1951), pp. 1-62 | DOI | Zbl

[10] Gués, Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin Uniform stability estimates for constant-coefficient symmetric hyperbolic boundary value problems, Commun. Partial Differ. Equations, Volume 32 (2007) no. 4, pp. 579-590 | DOI | Zbl

[11] Hersh, Reuben Mixed problems in several variables, J. Math. Mech., Volume 12 (1963), pp. 317-334 | Zbl

[12] Hörmander, Lars The Analysis of Partial Differential Operators I–IV, Grundlehren der Mathematischen Wissenschaften, 256, 257, 274 and 275, Springer, 1983, 1984 and 1985 | Zbl

[13] Ivrii, V.Ja.; Petkov, Vesselin M. Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed, Usp. Mat. Nauk, Volume 29 (1974) no. 5, pp. 3-70 in russian. English version in Russ. Math. Surv. 29 (1974), no. 5, p. 1-70 | Zbl

[14] Kashiwara, Masaki; Kawai, Takahiro Micro-hyperbolic pseudo-differential operators. I., J. Math. Soc. Japan, Volume 27 (1975), pp. 359-404 | DOI | Zbl

[15] Kreiss, Heinz-Otto Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., Volume 23 (1970), pp. 227-298 | DOI | Zbl

[16] Majda, Andrew The stability of multi-dimensional shock fronts, Mem. Am. Math. Soc., Volume 275 (1983) | Zbl

[17] Majda, Andrew; Osher, Stanley Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Commun. Pure Appl. Math., Volume 28 (1975), pp. 607-675 | DOI | Zbl

[18] Métivier, Guy The block structure condition for symmetric hyperbolic systems, Bull. Lond. Math. Soc., Volume 32 (2000) no. 6, pp. 689-702 | DOI | Zbl

[19] Métivier, Guy Small viscosity and boundary layer methods. Theory, stability analysis, and applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2004, xii+194 pages | Zbl

[20] Métivier, Guy Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Centro di Ricerca Matematica Ennio De Giorgi Series, 5, Edizioni della Normale, 2008, xi+140 pages | Zbl

[21] Métivier, Guy L 2 well-posed Cauchy Problems and symmetrizability of first order systems, J. Éc. Polytech., Math., Volume 1 (2014), pp. 39-70 | DOI | Zbl

[22] Métivier, Guy; Zumbrun, Kevin Symmetrizers and Continuity of stable subspaces for parabolic-hyperbolic boundary value problems, Discrete Contin. Dyn. Syst., Volume 11 (2004) no. 1, pp. 205-220 | DOI | Zbl

[23] Métivier, Guy; Zumbrun, Kevin Hyperbolic Boundary Value Problems for Symmetric Systems with Variable Multiplicities, J. Differ. Equations, Volume 211 (2005) no. 1, pp. 61-134 | DOI | Zbl

[24] Rauch, Jeffrey B. L 2 is a continuable initial condition for Kreiss’ mixed problems, Commun. Pure Appl. Math., Volume 25 (1972), pp. 26-285 | DOI | Zbl

[25] Rauch, Jeffrey B.; Massey, Frank J.III Differentiability of Solutions to Hyperbolic Initial Boundary Value Problems, Trans. Am. Math. Soc., Volume 189 (1974), pp. 303-318 | Zbl

Cité par Sources :