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

2013/05/05 by Felix Lindner, Lindner, Felix
Mathematics · #35B65 #46E35 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR #msc:35B65 #msc:35R60 #msc:46E35 #msc:60H15 #secondary: 35R60

paper · pdf · doi:10.48550/arxiv.1305.0975

49 pages, 1 figure; a few typos have been corrected

arxiv created 2013/06/07 · arxiv updated 2013/06/10

Abstract

We study the stochastic heat equation with trace class noise and zero Dirichlet boundary condition on a bounded polygonal domain O in R². 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 L₂-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+π/γ, where γis 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.

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