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

TL;DR

This paper investigates the well-posedness and regularity of quasilinear degenerate parabolic-hyperbolic SPDEs with multiplicative noise, establishing new regularity results and proving existence, uniqueness, and stability of solutions in an $L^1$ framework.

Contribution

It introduces novel regularity results using averaging techniques and proves existence and uniqueness of solutions without growth restrictions on nonlinearities.

Findings

Established new regularity results for solutions.

Proved existence and uniqueness in full $L^1$ setting.

Demonstrated comparison and $L^1$-contraction properties.

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.