0 Ju l 2 00 7 RANDOM DATA CAUCHY THEORY FOR SUPERCRITICAL WAVE EQUATIONS I : LOCAL THEORY

— We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H(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(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.