abstract:
We present a result concerning the stochastic heat equation with trace class noise and zero Dirichlet boundary condition on a two-dimensional bounded polygonal domain. 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 $r<-1/2$ in time.
The regular part $u_R$ admits spatial Sobolev regularity of order $s=2$, while the spatial Sobolev regularity of $u_S$ is restricted by $s<1+\gamma/\pi$, where $\gamma$ is the largest interior angle at the boundary of the domain. 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.
Our considerations are motivated by the question whether adaptive approximation methods for the solutions to SPDEs pay off in the sense that they admit better convergence rates than nonadaptive (uniform) approximation methods. It is known that the approximation rate that can be achieved by uniform methods is determined by the Sobolev regularity of the exact solution, whereas the approximation rate of adaptive methods is determined by the regularity in a specific scale of Besov spaces.
The presented result complements recent results on the Besov regularity of the solutions to SPDEs. |