Conservation laws driven by L\'{e}vy white noise

TL;DR

This paper studies multidimensional conservation laws influenced by Lévy white noise, proving existence, uniqueness, and stability of entropy solutions using advanced stochastic calculus and measure-theoretic techniques.

Contribution

It introduces a framework for entropy solutions under Lévy noise, extending classical results to stochastic, multidimensional settings with generalized entropy inequalities.

Findings

Existence of entropy solutions via vanishing viscosity method

Uniqueness established through $L^1$-contraction principle

Generalized entropy inequalities accommodate Young measure-valued solutions

Abstract

We consider multidimensional conservation laws perturbed by multiplicative L\'{e}vy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the It\^{o}-L\'{e}vy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first establish the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the $L^1$-contraction principle. Finally, the $L^1$ contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.