Singular Behavior of the Solution to the Stochastic Heat Equation on a Polygonal Domain

TL;DR

This paper analyzes the singular behavior of solutions to the stochastic heat equation on polygonal domains, revealing how corner angles influence regularity and providing estimates useful for numerical methods.

Contribution

It introduces a decomposition of the solution into regular and singular parts, characterizing their regularity and dependence on boundary angles, with estimates for Sobolev norms.

Findings

Solution decomposes into regular and singular parts.

Regularity of the solution is limited by boundary angles.

Provides Sobolev norm estimates for solution components.

Abstract

We study the stochastic heat equation with trace class noise and zero Dirichlet boundary condition on a bounded polygonal domain O in R^2. It is shown that the solution u can be decomposed into a regular part u_R and a singular part u_S which incorporates the corner singularity functions for the Poisson problem. Due to the temporal irregularity of the noise, both u_R and u_S have negative L_2-Sobolev regularity of order s<-1/2 in time. The regular part u_R admits spatial Sobolev regularity of order r=2, while the spatial Sobolev regularity of u_S is restricted by r<1+\pi/\gamma, where \gamma is the largest interior angle at the boundary of O. We obtain estimates for the Sobolev norm of u_R and the Sobolev norms of the coefficients of the singularity functions. The proof is based on a Laplace transform argument w.r.t. the time variable. The result is of interest in the context of numerical methods for stochastic PDEs.