Probabilistic well-posedness for supercritical wave equation on $\mathbb{T}^3$

TL;DR

This paper proves that supercritical wave equations on the three-dimensional torus are almost surely globally well-posed for initial data in a broad range of Sobolev spaces, extending previous deterministic results.

Contribution

It establishes probabilistic well-posedness for supercritical wave equations on 3, extending the known range of initial data regularity beyond deterministic limits.

Findings

Almost sure global well-posedness for supercritical wave equations.

The regularity threshold is significantly lower than the critical index.

Method follows strategies from Burq and Oh-Po, adapted to supercritical cases.

Abstract

In this article, we follow the strategies, listed in \cite{Burq2011} and \cite{OhPo}, in dealing with supercritical cubic and quintic wave equations, we obtain that, the equation \begin{equation*} \left\{ \begin{split} &(\partial^2_t-\Delta)u+|u|^{p-1}u=0,\ \ 3<p<5 &\big(u,\partial_tu\big)|_{t=0}=(u_0,u_1)\in H^{s}\times H^{s-1}=:\mathcal{H}^s, \end{split} \right. \end{equation*} is almost surely global well-posed in the sense of Burq and Tzvetkov\cite{Burq2011} for any $s\in (\frac{p-3}{p-1},1)$. The key point here is that $\frac{p-3}{p-1}$ is much smaller than the critical index $\frac{3}{2}-\frac{2}{p-1}$ for $3<p<5$.