Geometric optics expansions for hyperbolic corner problems II : from weak stability to violent instability

Antoine Benoit (1, 2)

1 MEPHYSTO - Quantitative methods for stochastic models in physics
2 ULB - Université libre de Bruxelles

In this article we are interested in the rigorous construction of geometric optics expansions for weakly well-posed hyperbolic corner problems. More precisely we focus on the case where selfinteracting phases occur and are exactly the phases where the uniform Kreiss-Lopatinskii condition fails. We then show that the associated WKB expansion suffers arbitrarily many amplifications in a fixed finite time. As a consequence, we show that this corner problem can not be well-posed even at the price of a huge loss of derivatives. The new result, in that framework, is that the violent instability does not come from the failure of the weak Kreiss-Lopatinskii condition, but of the accumulation of arbitrarily many weak instabilities.

Domaines: Equations aux dérivées partielles [math.AP]

Antoine Benoit. Geometric optics expansions for hyperbolic corner problems II : from weak stability to violent instability. 2016. ⟨hal-01242899v3⟩