# Nonlinear Anisotropic Degenerate Parabolic-Hyperbolic Equations with Stochastic Forcing

## TL;DR

This paper develops a comprehensive framework for analyzing the well-posedness and stability of nonlinear stochastic degenerate parabolic-hyperbolic equations with anisotropic and heterogeneous features, incorporating fractional BV regularity and continuous dependence on data.

## Contribution

It introduces a unified approach to establish well-posedness, regularity, and continuous dependence estimates for stochastic entropy solutions of complex nonlinear equations with heterogeneity.

## Key findings

- Established well-posedness in $L^p \cap N^{\kappa,1}$ spaces.
- Proved $L^1$ continuous dependence on initial data and coefficients.
- Developed fractional BV regularity estimates for solutions.

## Abstract

We are concerned with nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing, which are heterogeneous (i.e., not space-translational invariant). A unified framework is established for the continuous dependence estimates, fractional BV regularity estimates, and well-posedness for stochastic entropy solutions of the nonlinear stochastic degenerate parabolic-hyperbolic equation. In particular, we establish the well-posedness of the nonlinear stochastic equation in $L^p \cap N^{\kappa,1}$ for $p\in (1,\infty)$ and the $\kappa$--Nikolskii space $N^{\kappa,1}$ with $\kappa>0$, and the $L^1$ continuous dependence of the stochastic entropy solutions not only on the initial data, but also on the degenerate diffusion matrix function, the flux function, and the multiplicative noise function involving in the nonlinear equation.