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

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

2007-07-10

arxiv.org/abs/0707.1447v1

Mathematics

Analysis of PDEs

Scientific paper

10.1007/s00222-008-0124-z

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 $s\geq 1/4$ in the case of a boundary less manifold and $s\geq 8/21$ in the case of a manifold with boundary.

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