Scalar conservation laws with multiple rough fluxes

TL;DR

This paper extends the theory of scalar conservation laws to include multiple rough fluxes with inhomogeneous and path-dependent features, using kinetic formulations inspired by stochastic viscosity solutions.

Contribution

It introduces a novel approach for pathwise entropy solutions with multiple rough fluxes, expanding previous work to more complex flux dependencies.

Findings

Develops a framework for scalar conservation laws with multiple rough fluxes.

Extends stochastic viscosity solution techniques to kinetic formulations.

Provides theoretical foundations for handling inhomogeneous and path-dependent fluxes.

Abstract

We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame and Souganidis who considered spatially independent and inhomogeneous fluxes with multiple paths and a single driving singular path respectively. The approach is motivated by the theory of stochastic viscosity solutions which relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws this is best implemented at the level of the kinetic formulation which we follow here.