A generalization of $H$-measures and application on purely fractional scalar conservation laws

TL;DR

This paper extends $H$-measures to more general manifolds and applies this to establish strong local $L^1$ precompactness of quasi-solutions in purely fractional scalar conservation laws.

Contribution

It introduces a generalized $H$-measure framework on arbitrary Lipschitz manifolds and uses it to analyze fractional conservation laws.

Findings

Extended $H$-measures to arbitrary Lipschitz manifolds.

Proved strong $L^1_{loc}$ precompactness of quasi-solutions.

Applied the framework to fractional scalar conservation laws.

Abstract

We extend the notion of $H$-measures on test functions defined on $\R^d\times P$, where $P\subset \R^d$ is an arbitrary compact simply connected Lipschitz manifold such that there exists a family of regular nonintersecting curves issuing from the manifold and fibrating $\R^d$. We introduce a concept of quasi-solutions to purely fractional scalar conservation laws and apply our extension of the $H$-measures to prove strong $L^1_{loc}$ precompactness of such quasi-solutions.