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

par Benoit, Antoine
Référence SIAM journal on mathematical analysis, 49, 5, page (3335-3395)
Publication Publié, 2017

Article révisé par les pairs

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Résumé :

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 self-interacting phases occur and where one of them is exactly the phase where the uniform Kreiss-Lopatinskii condition fails. We show that the associated WKB expansion suffers arbitrarily many amplifications before a fixed finite time. As a consequence, we show that such a corner problem cannot be weakly well-posed even at the price of a huge loss of derivatives. The new result, in that framework, is that the violent instability (or Hadamard instability) does not come from the degeneracy of the weak Kreiss-Lopatinskii condition but from the accumulation of arbitrarily many weak instabilities.