Stochastic Allen-Cahn equation with mobility

TL;DR

This paper studies a stochastic version of the Allen-Cahn equation with mobility and colored noise, proving existence and uniqueness of solutions with specific regularity properties in a multi-dimensional setting.

Contribution

It introduces a stochastic Allen-Cahn model with mobility and colored noise, establishing well-posedness results for solutions in dimensions up to three.

Findings

Existence of solutions with continuous paths in $H^1$

Uniqueness of solutions under given conditions

Solutions exhibit regularity in $C([0,T]; H^1)$ and $L^2([0,T];H^2)

Abstract

We introduce a class of stochastic Allen-Cahn equations with a mobility coefficient and colored noise. For initial data with finite free energy, we analyze the corresponding Cauchy problem on the $d$-dimensional torus in the time interval $[0,T]$. Assuming that $d\le 3$ and that the potential has quartic growth, we prove existence and uniqueness of the solution as a process $u$ in $L^2$ with continuous paths, satisfying almost surely the regularity properties $u\in C([0,T]; H^1)$ and $u\in L^2([0,T];H^2)$.