On the regularity of entropy solutions to stochastic degenerate parabolic equations
Marko Erceg, Kenneth H. Karlsen, Darko Mitrović
TL;DR
This work establishes space-time fractional regularity for entropy solutions to stochastic degenerate parabolic equations with anisotropic diffusion by developing a kinetic formulation featuring two transport equations (one second-order, one first-order). A quantitative non-degeneracy condition on the symbol is used, notably without requiring bounds on the derivative of the kinetic symbol, which broadens applicability beyond prior theory. The main result shows that the velocity-averaged kinetic function gains fractional Sobolev regularity in space-time, with exponents explicitly tied to the non-degeneracy parameter and the spatial dimension; in the deterministic case the regularity improves further. The approach relies on velocity averaging in the stochastic kinetic setting and leverages a parabolic regularity mechanism within the kinetic framework, providing a robust tool for stochastic compactness at the entropy level across a broad class of degenerate equations.
Abstract
We study the regularity of entropy solutions for quasilinear parabolic equations with anisotropic degeneracy and stochastic forcing. Building on previous works, we establish space-time regularity under a non-degeneracy condition that does not require an assumption on the derivative of the symbol of the corresponding kinetic equation, a restriction imposed in earlier studies. This allows us to obtain regularity results for certain equations not accounted for by prior theory, albeit with reduced regularity exponents. Our approach uses a kinetic formulation with two transport equations, one of second order and one of first order, leveraging a form of "parabolic regularity" inherent in these equations that was not utilized in previous studies.