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Dispersive effects and long-time asymptotics for wave equations in exterior domains

Abstract : We are concerned with Schrödinger and wave equations, both linear and non linear, in exterior domains. In particular, we are interested in the so-called Strichartz estimates, which are a family of dispersive estimates measuring decay for the linear flow. They turn out to be particularly useful in order to study the corresponding non linear equations. In non-captive geometries, where all the rays of geometrical optics go to infinity, many results show that Strichartz estimates hold with no loss with respect to the flat case. Moreover, the local smoothing estimates for the Schrödinger equation, respectively the local energy decay for the wave equation, which are another family of dispersive estimates, are known to fail in any captive geometry. In contrast, we show Strichartz estimates without loss in an unstable captive geometry: the exterior of many strictly convex obstacles verifying Ikawa's condition. The second part of this thesis is dedicated to the study of the long time asymptotics of the corresponding non linear equations. We expect that they behave linearly in large times, or scatter, when the domain they live in does not induce too much concentration effect. We show such a result for the non linear critical wave equation in the exterior of a class of obstacles generalizing star-shaped bodies. In the exterior of two strictly convex obstacles, we obtain a rigidity result concerning compact flow solutions, which is a first step toward a general result. Finally, we consider the non linear Schrödinger equation in the free space but with a potential. We prove that solutions scatter for a repulsive potential, and for a sum of two repulsive potentials with strictly convex level surfaces. This provides a scattering result in a framework similar to the exterior of two strictly convex obstacles.
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Submitted on : Thursday, March 21, 2019 - 10:43:09 AM
Last modification on : Wednesday, October 14, 2020 - 4:24:45 AM
Long-term archiving on: : Saturday, June 22, 2019 - 1:28:42 PM


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  • HAL Id : tel-02075081, version 1


David Lafontaine. Dispersive effects and long-time asymptotics for wave equations in exterior domains. Analysis of PDEs [math.AP]. Université Côte d'Azur, 2018. English. ⟨NNT : 2018AZUR4067⟩. ⟨tel-02075081⟩



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