Piecewise regular solutions to scalar balance laws with singular nonlocal sources

TL;DR

This paper proves local well-posedness for piecewise regular solutions with a single shock in scalar balance laws involving singular nonlocal source terms, providing detailed solution descriptions near the shock curve.

Contribution

It introduces a novel well-posedness result for scalar balance laws with singular integral sources, focusing on solutions with a single shock and detailed local behavior.

Findings

Established local existence and uniqueness of solutions with a single shock.

Provided detailed descriptions of solutions near the shock curve.

Extended the understanding of scalar balance laws with singular nonlocal sources.

Abstract

The present paper establishes a local well-posed result for piecewise regular solutions with single shock of scalar balance laws with singular integral of convolution type kernels. In a neighborhood of the shock curve, a detailed description of the solution is provided for a general class of initial data.