Gibbs measure evolution in radial nonlinear wave and Schr\"odinger equations on the ball

TL;DR

This paper proves the existence and uniqueness of dynamics for radial nonlinear wave and Schrödinger equations on a ball with random initial data, extending previous results by establishing well-defined Gibbs measure-supported solutions.

Contribution

It introduces new results on the well-posedness of radial nonlinear wave and Schrödinger equations on the ball with Gibbs measure initial data, complementing prior work.

Findings

Established well-defined dynamics on the support of the Gibbs measure.

Proved uniqueness of solutions for the equations with random initial data.

Extended previous results to new settings and dimensions.

Abstract

We establish new results for the radial nonlinear wave and Schr\"odinger equations on the ball in $\Bbb R^2$ and $\Bbb R^3$, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from \cite{B-T1,B-T2} and \cite {T1,T2}.