Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

Abstract

We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^{1}$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^{1}$-contraction property for the solutions, generalizing the results obtained in [Ann. Probab. 44 (2016) 1916–1955].