Invariant measure for the cubic wave equation on the unit ball of $R^3$

TL;DR

This paper proves the invariance of a measure on Sobolev spaces for the cubic nonlinear wave equation on the unit ball in three dimensions, focusing on spherical symmetry and combining analytic and probabilistic methods.

Contribution

It introduces a novel approach to establish measure invariance for the cubic wave equation on the unit ball under spherical symmetry, addressing both local flow properties and global measure invariance.

Findings

Measure is invariant under the flow of the cubic wave equation

Combines analytic and probabilistic techniques for flow analysis

Extends the flow globally while maintaining measure invariance

Abstract

This paper deals with the invariance of a measure on Sobolev spaces of low regularity under the flow of the cubic non linear wave equation on the unit ball of 3 under the assumption of spherical symmetry. It presents two aspects, an analytic one which includes the treatment of local properties of the flow, and a probabilistic one, which is mainly related to the global extension of the flow and the invariance of the measure.