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Dispersive estimates for the Schrödinger equation in a strictly convex domain and applications

Abstract : We consider an anisotropic model case for a strictly convex domain of dimension $d\geq 2$ with smooth boundary and we describe dispersion for the semi-classical Schrödinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semi-classical flow : a loss of $(\frac ht)^{1/4}$ occurs with respect to the boundary less case due to repeated swallowtail type singularities, and is proven optimal. Corresponding Strichartz estimates allow to solve the cubic nonlinear Sch\"odinger equation on such a 3D model convex domain, hence matching known results on generic compact boundaryless manifolds.
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https://hal.archives-ouvertes.fr/hal-02950431
Contributor : Oana Ivanovici Connect in order to contact the contributor
Submitted on : Tuesday, August 17, 2021 - 4:28:55 PM
Last modification on : Tuesday, September 28, 2021 - 5:16:44 PM

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  • HAL Id : hal-02950431, version 3
  • ARXIV : 2009.13810
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Oana Ivanovici. Dispersive estimates for the Schrödinger equation in a strictly convex domain and applications. 2021. ⟨hal-02950431v3⟩

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