Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3D ball

TL;DR

This paper proves new global well-posedness results for the radial nonlinear wave equation on a 3D ball using Gibbs measure evolution, employing two different analytical approaches.

Contribution

It introduces two novel methods for establishing global solutions: one based on contraction mapping for certain nonlinearities, and another analyzing convergence for the full admissible range.

Findings

Established global well-posedness via Gibbs measure evolution.

Developed a general approach applicable to nonlinear Schrödinger equations.

Provided convergence analysis for solution sequences.

Abstract

We establish new global well-posedness results along Gibbs measure evolution for the nonlinear wave equation posed on the unit ball in $\mathbb{R}^3$ via two distinct approaches. The first approach invokes the method established in the works \cite{B1,B2,B3} based on a contraction-mapping principle and applies to a certain range of nonlinearities. The second approach allows to cover the full range of nonlinearities admissible to treatment by Gibbs measure, working instead with a delicate analysis of convergence properties of solutions. The method of the second approach is quite general, and we shall give applications to the nonlinear Schr\"odinger equation on the unit ball in subsequent works \cite{BB1,BB2}.