WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: time depending loss of derivatives

Antoine Benoit (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

In this article we are interested in linear hyperbolic systems of equations defined in the strip R × ]0, 1[. More precisely the aim of this article is to described the influence of the boundary conditions on the behaviour of the solution. This question has already been adressed in [Benb] in which the author shows that in the strip geometry, the self-interaction (meaning that a wave packet regenerates itself by repeated rebounds against the sides of the strip) phenomenon becomes generic and can lead to some exponential growth in time of the solution. Here we restrict our attention to finite time problems (so that we are not interested in the possible growth in time of the solution) however the main difficulty is that we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this article is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called W R-class of [BGRSZ02] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.

Mots clés: hyperbolic boundary value problems; weak well-posedness; geometric optics expansions; strip geometry; self-interaction phenomenon; time depending loss of regularity

Domaines: Equations aux dérivées partielles [math.AP]

Antoine Benoit. WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: time depending loss of derivatives. 2019. ⟨hal-02391809⟩