Random data Cauchy problem for the wave equation on compact manifold

TL;DR

This paper develops local strong solutions for cubic and quintic nonlinear wave equations with random initial data on compact manifolds, improving previous results and covering various manifold dimensions and boundary conditions.

Contribution

It extends the existence results for nonlinear wave equations with random data to broader settings and lower regularity thresholds, improving prior work by Burq and Tzvetkov.

Findings

Constructed solutions for cubic wave equations on 3D manifolds with boundary.

Established solutions for quintic wave equations on 2D boundaryless manifolds.

Achieved lower regularity thresholds for initial data in solution existence.

Abstract

Inspired by the work of Burq and Tzvetkov (Invent. math. 173(2008), 449-475.), firstly, we construct the local strong solution to the cubic nonlinear wave equation with random data for a large set of initial data in $H^{s}(M)$ with $s\geq \frac{5}{14}$, where M is a three dimensional compact manifold with boundary, moreover, our result improves the result of Theorem 2 in (Invent. math. 173(2008), 449-475.); secondly, we construct the local strong solution to the quintic nonlinear wave equation with random data for a large set of initial data in $H^{s}(M)$ with $s\geq\frac{1}{6}$, where M is a two dimensional compact boundaryless manifold; finally, we construct the local strong solution to the quintic nonlinear wave equation with random data for a large set of initial data in $H^{s}(M)$ with $s\geq \frac{23}{90}$, where M is a two dimensional compact manifold with boundary.