Trend to Equilibrium for Flows With Random Diffusion

Abstract

Abstract Motivated by the possibility of noise to cure equations of finite-time blowup, the recent work [ 90] 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 [ 90] in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow \infty $.