ANNALES DE L'INSTITUT FOURIER

On the ${L}^{2}$ well posedness of Hyperbolic Initial Boundary Value Problems
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 1809-1863
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.
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.
Revised : 2016-04-11
Accepted : 2016-10-21
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3123
Classification:  35L04
Keywords: Hyperbolic first order systems, Cauchy problems, boundary value problems, Lopatinski conditions, semi-group estimates
@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},
publisher = {Association des Annales de l'institut Fourier},
volume = {67},
number = {5},
year = {2017},
pages = {1809-1863},
doi = {10.5802/aif.3123},
language = {en},
url = {https://aif.centre-mersenne.org/item/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, Volume 67 (2017) no. 5, pp. 1809-1863. doi : 10.5802/aif.3123. https://aif.centre-mersenne.org/item/AIF_2017__67_5_1809_0/

[1] Audiard, Corentin On mixed initial-boundary value problems for systems that are not strictly hyperbolic, Appl. Math. Lett., Tome 24 (2011) no. 5, pp. 757-761 | Article | Zbl 1213.35294

[2] Benoit, Antoine Finite speed of propagation for mixed problems in the $WR$ class, Commun. Pure Appl. Anal., Tome 13 (2014) no. 6, pp. 2351-2358 | Article | Zbl 1304.35405

[3] Benzoni-Gavage, Sylvie; Serre, Denis Multi-dimensional hyperbolic partial differential equations. First-order systems and applications, Oxford University Press, Oxford Mathematical Monographs (2007), xxv+508 pages | Zbl 1113.35001

[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 0446.35001

[5] Friedrichs, Kurt Otto Symmetric hyperbolic linear differential equations, Commun. Pure Appl. Math., Tome 7 (1954), pp. 345-392 | Article | Zbl 0059.08902

[6] Friedrichs, Kurt Otto Symmetric positive linear differential equations, Commun. Pure Appl. Math., Tome 11 (1958), pp. 333-418 | Article | Zbl 0083.31802

[7] Friedrichs, Kurt Otto; Lax, Peter David Boundary Value Problems for First Order Operators, Commun. Pure Appl. Math., Tome 18 (1965), pp. 355-388 | Article | Zbl 0178.11403

[8] Friedrichs, Kurt Otto; Lax, Peter David On Symmetrizable Differential Operators, Proc. Sympos. Pure Math., Tome 10 (1967), pp. 128-137 | Article | Zbl 0184.36603

[9] Gårding, Lars Linear hyperbolic partial differential equation with constant coefficients, Acta Math., Tome 85 (1951), pp. 1-62 | Article | Zbl 0045.20202

[10] Gués, Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin Uniform stability estimates for constant-coefficient symmetric hyperbolic boundary value problems, Commun. Partial Differ. Equations, Tome 32 (2007) no. 4, pp. 579-590 | Article | Zbl 1133.35404

[11] Hersh, Reuben Mixed problems in several variables, J. Math. Mech., Tome 12 (1963), pp. 317-334 | Zbl 0149.06602

[12] Hörmander, Lars The Analysis of Partial Differential Operators I–IV, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 256, 257, 274 and 275 (1983, 1984 and 1985) | Zbl 0521.35001; 0521.35002 ; 0601.35001 ; 0612.35001

[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, Tome 29 (1974) no. 5, pp. 3-70 (in russian. English version in Russ. Math. Surv. 29 (1974), no. 5, p. 1-70) | Zbl 0312.35049

[14] Kashiwara, Masaki; Kawai, Takahiro Micro-hyperbolic pseudo-differential operators. I., J. Math. Soc. Japan, Tome 27 (1975), pp. 359-404 | Article | Zbl 0305.35066

[15] Kreiss, Heinz-Otto Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., Tome 23 (1970), pp. 227-298 | Article | Zbl 0193.06902

[16] Majda, Andrew The stability of multi-dimensional shock fronts, Mem. Am. Math. Soc., Tome 275 (1983) | Zbl 0506.76075

[17] Majda, Andrew; Osher, Stanley Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Commun. Pure Appl. Math., Tome 28 (1975), pp. 607-675 | Article | Zbl 0314.35061

[18] Métivier, Guy The block structure condition for symmetric hyperbolic systems, Bull. Lond. Math. Soc., Tome 32 (2000) no. 6, pp. 689-702 | Article | Zbl 1073.35525

[19] Métivier, Guy Small viscosity and boundary layer methods. Theory, stability analysis, and applications, Birkhäuser Boston, Modeling and Simulation in Science, Engineering and Technology (2004), xii+194 pages | Zbl 1133.35001

[20] Métivier, Guy Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Edizioni della Normale, Centro di Ricerca Matematica Ennio De Giorgi Series, Tome 5 (2008), xi+140 pages | Zbl 1156.35002

[21] Métivier, Guy ${L}^{2}$ well-posed Cauchy Problems and symmetrizability of first order systems, J. Éc. Polytech., Math., Tome 1 (2014), pp. 39-70 | Article | Zbl 1332.35205

[22] Métivier, Guy; Zumbrun, Kevin Symmetrizers and Continuity of stable subspaces for parabolic-hyperbolic boundary value problems, Discrete Contin. Dyn. Syst., Tome 11 (2004) no. 1, pp. 205-220 | Article | Zbl 1102.35332

[23] Métivier, Guy; Zumbrun, Kevin Hyperbolic Boundary Value Problems for Symmetric Systems with Variable Multiplicities, J. Differ. Equations, Tome 211 (2005) no. 1, pp. 61-134 | Article | Zbl 1073.35155

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

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