Trend to equilibrium for flows with random diffusion

TL;DR

This paper proves that solutions to certain active scalar equations with random diffusion in a periodic setting exponentially converge to a uniform distribution over time, indicating a trend to equilibrium.

Contribution

It establishes the exponential convergence to equilibrium for a broad class of active scalar equations with random diffusion, extending previous results on global existence.

Findings

Solutions converge exponentially fast to uniform distribution

Random diffusion prevents finite-time blowup in these equations

Results apply to Hamiltonian and gradient flow models

Abstract

Motivated by the possibility of noise to cure equations of finite-time blowup, recent work arXiv:2109.09892 by the second and third named authors showed that with quantifiable high probability, random diffusion restores global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as the Patlak-Keller-Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from arXiv:2109.09892 in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow\infty$.