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

2007/07/10 by N. Burq, N. Tzvetkov
Mathematics · #math.AP #msc:35Q55 #msc:35BXX #msc:37K05 #msc:37L50 #msc:81Q20

paper · pdf · doi:10.1007/s00222-008-0124-z

arxiv created 2007/07/10 · arxiv updated 2009/12/01

Abstract

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.

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