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

TL;DR

This paper investigates the local existence of solutions for a supercritical cubic nonlinear wave equation on three-dimensional manifolds, demonstrating that randomization of initial data enables solutions in regimes where deterministic methods fail.

Contribution

It introduces a probabilistic approach to establish local solutions for supercritical wave equations in low regularity spaces, extending the understanding of well-posedness.

Findings

Local solutions exist for initial data in $H^s(M)$ with $s \\geq 1/4$ (boundaryless)

Solutions are constructed for $s \\geq 8/21$ (with boundary)

Randomization enables well-posedness in supercritical regimes

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