Dispersive estimates for the Schr{\"o}dinger equation in a strictly convex domain and applications

TL;DR

This paper establishes dispersive and Strichartz estimates for the semi-classical Schrödinger equation in a strictly convex domain, revealing a boundary-induced loss and applying results to nonlinear equations.

Contribution

It provides the first optimal dispersive estimates with boundary effects for convex domains and applies these to solve the cubic nonlinear Schrödinger equation.

Findings

Dispersive decay rate with boundary loss of (h/t)^{1/4} proven optimal.

Strichartz estimates established for convex domains.

Application to cubic nonlinear Schrödinger equation in 3D domain.

Abstract

We consider an anisotropic model case for a strictly convex domain of dimension $d\geq 2$ with smoothboundary and we describe dispersion forthe semi-classical Schr{\"o}dinger 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.