Caractérisation des problèmes mixtes hyperboliques bien posés

Abstract

Caractérisation des problèmes mixtes hyperboliques bien posés Mémoires de la S. M. F., tome 31-32 (1972), p. 83-88 <http://www.numdam.org/item?id=MSMF_1972__31-32__83_0> © Mémoires de la S. M. F., 1972, tous droits réservés. L'accès aux archives de la revue « Mémoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems

“…We recover the fact that the speed of propagation on the boundary is 1/|µ|, which is larger than the maximal speed of propagation for the Cauchy problem, see [4,10] and the discussion in [3, chapter 8].…”

“…As already shown for some scalar second order hyperbolic equations, see the works by Chazarain, Piriou and Ikawa [4,10], the speed of propagation for some ibvps may be larger than the speed of propagation in free space, see also the discussion in [3, chapter 8]. This is not in contradiction with Theorem 4.1 since energy can not be conserved in such problems.…”

“…When V boundary > V Cauchy , the speed of propagation for (4.1) is greater than the speed of propagation for the Cauchy problem. Examples of this kind already appeared in [4,10,3]. We also refer to section 5 for some examples.…”

“…For this system, an incident oscillatory wave coming from the interior of the domain and hitting the boundary cannot reach the bad frequency and excite the "singular mode" (4) . In other words, an incident oscillatory wave cannot produce a Rayleigh wave.…”

Loss of derivatives for hyperbolic boundary problems with constant coefficients

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Symmetric hyperbolic systems and constantly hyperbolic systems with constant coefficients and a boundary condition which satisfies a weakened form of the Kreiss-Sakamoto condition are considered. A well-posedness result is established which generalizes a theorem by Chazarain and Piriou for scalar strictly hyperbolic equations and non-characteristic boundaries [3]. The proof is based on an explicit solution of the boundary problem by means of the Fourier-Laplace transform.

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