A regularity result for quasilinear stochastic partial differential equations of parabolic type
For mathematicians studying regularity of solutions to quasilinear SPDEs, this provides a theoretical regularity result under classical a priori estimates.
The paper establishes Hölder continuity in time and optimal spatial regularity for weak solutions of quasilinear parabolic SPDEs with multiplicative noise, under conditions on coefficients and initial data. The proof uses a decomposition into a linear SPDE and a linear PDE with random coefficients.
We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is Hölder continuous in time and possesses spatial regularity that is only limited by the regularity of the given data. Our proof is based on an efficient method of increasing regularity: the solution is rewritten as the sum of two processes, one solves a linear parabolic SPDE with the same noise term as the original model problem whereas the other solves a linear parabolic PDE with random coefficients. This way, the required regularity can be achieved by repeatedly making use of known techniques for stochastic convolutions and deterministic PDEs.