Random data Cauchy theory for supercritical wave equations I: local theory

摘要

We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H s (M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in H s (M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary.