On the regularity of entropy solutions to stochastic degenerate parabolic 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.