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Journal Articles SIAM Journal on Mathematical Analysis Year : 2019

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

Abstract

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.
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Dates and versions

hal-01242899 , version 1 (14-12-2015)
hal-01242899 , version 2 (30-12-2015)
hal-01242899 , version 3 (13-10-2016)
hal-01242899 , version 4 (09-03-2017)

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Antoine Benoit. Geometric optics expansions for hyperbolic corner problems II : from weak stability to violent instability. SIAM Journal on Mathematical Analysis, In press, 49 (5), pp.3335-3395. ⟨10.1137/16M1060145⟩. ⟨hal-01242899v4⟩
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