Exponential mixing for random nonlinear wave equations: weak dissipation and localized control

TL;DR

This paper introduces a new criterion for exponential mixing in random dynamical systems, specifically applied to nonlinear wave equations with localized noise, weak dissipation, and critical nonlinearity, advancing understanding of their long-term behavior.

Contribution

It develops a novel criterion for exponential mixing applicable to dispersive equations and demonstrates its use on complex nonlinear wave systems with localized noise and weak damping.

Findings

Proves exponential mixing for certain nonlinear wave equations

Establishes a link between dissipation, randomness, and stability

Provides a new approach to analyzing localized control in dynamical systems

Abstract

We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics, i.e., asymptotic compactness in dynamical systems, global stability of evolution equations, and localized control problems. As an initial application, we exploit the exponential mixing of random nonlinear wave equations with degenerate damping, critical nonlinearity, and physically localized noise. The essential challenge lies in the fact that the weak dissipation and randomness interact in the evolution.