A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

TL;DR

This paper studies a complex stochastic fractional degenerate parabolic-hyperbolic equation, establishing existence, uniqueness, and stability results using advanced mathematical techniques like entropy solutions and BV estimates.

Contribution

It introduces a new framework for proving uniqueness of stochastic fractional degenerate equations and provides explicit error estimates for the vanishing viscosity method.

Findings

Established existence of stochastic entropy solutions.

Proved uniqueness using a new technical framework.

Derived explicit continuous dependence and error estimates.

Abstract

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method "a la Kruzkov" for more general equations where the noise coefficient may depends explicitly on the spatial variable.