Renormalized stochastic entropy solution for degenerate parabolic-hyperbolic equations with Levy noise

TL;DR

This paper develops a well-posedness theory for renormalized entropy solutions of degenerate parabolic-hyperbolic PDEs with Levy noise, using approximation and doubling variables techniques.

Contribution

It introduces a novel framework for existence and uniqueness of solutions to stochastic PDEs with Levy noise and unbounded domains.

Findings

Existence of renormalized entropy solutions established.

Uniqueness proved using adapted Kruzhkov's doubling variables method.

Applicable to PDEs with general L1-data on unbounded domains.

Abstract

In this article, we establish the well-posedness theory for renormalized entropy solutions of a degenerate parabolic-hyperbolic PDE perturbed by a multiplicative Levy noise with general L1-data on the unbounded domain. By using a suitable approximation procedure based on the vanishing viscosity technique and bounded data, we prove the existence of a renormalized entropy solution to the underlying problem. The uniqueness of the solution is settled by adapting Kru\v{z}kov's doubling the variables technique in the presence of noise.