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

Oana Ivanovici (1)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

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.

Domaines: Equations aux dérivées partielles [math.AP]

Oana Ivanovici. Dispersive estimates for the Schrödinger equation in a strictly convex domain and applications. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, inPress. ⟨hal-02950431v3⟩