Homogeneous Dirichlet problem for degenerate parabolic-hyperbolic PDE driven by Levy noise

TL;DR

This paper establishes the well-posedness of entropy solutions for a degenerate parabolic-hyperbolic PDE with Levy noise, extending previous Brownian noise results with a simpler approach to global inequalities.

Contribution

It introduces a new, simplified method for proving the global Kato's inequality in the context of Levy noise, advancing the theory of entropy solutions for such PDEs.

Findings

Proved existence and uniqueness of entropy solutions

Developed a simplified approach to global Kato's inequality

Extended well-posedness theory to Levy noise perturbations

Abstract

In this article, we study the homogeneous Dirichlet problem for a degenerate parabolic-hyperbolic PDE perturbed by Levy noise. In particular, we develop the well-posedness theory of entropy solution based on the Kru\v{z}kov's semi-entropy formulation. In comparison to the pioneered work by Bauzet et al. (J. Funct. Anal. 266, (2014), 2503-2545), concerning the existence and uniqueness of entropy solution for the Dirichlet problem for conservation laws driven by Brownian noise, our present analysis involves a simpler approach to obtain the global Kato's inequality.