On Nonlinear Stochastic Balance Laws

TL;DR

This paper develops a framework for analyzing multidimensional stochastic balance laws, establishing uniform bounds, existence of solutions, and error estimates for numerical methods, advancing the mathematical understanding of stochastic PDEs.

Contribution

It identifies a class of nonlinear stochastic balance laws with uniform BV bounds and provides a multidimensional existence theory for stochastic entropy solutions.

Findings

Uniform BV bounds for vanishing viscosity approximations.

Existence of stochastic entropy solutions in multiple dimensions.

Error estimates for stochastic viscosity methods.

Abstract

We are concerned with multidimensional stochastic balance laws. We identify a class of nonlinear balance laws for which uniform spatial $BV$ bounds for vanishing viscosity approximations can be achieved. Moreover, we establish temporal equicontinuity in $L^1$ of the approximations, uniformly in the viscosity coefficient. Using these estimates, we supply a multidimensional existence theory of stochastic entropy solutions. In addition, we establish an error estimate for the stochastic viscosity method, as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions. Various further generalizations of the results are discussed.