On scalar nonlinear balance laws with singular nonlocal sources

TL;DR

This paper studies scalar nonlinear balance laws with singular nonlocal sources, proving global existence of solutions, partial uniqueness, and criteria for smoothness or wave breaking, covering notable equations like Burger-Poisson.

Contribution

It establishes existence, partial uniqueness, and smoothness criteria for solutions to balance laws with singular convolution sources, extending understanding of these complex equations.

Findings

Proved global existence of entropy weak solutions in L^2(R).

Established partial uniqueness in periodic and non-periodic settings.

Derived criteria for local smoothness and wave breaking.

Abstract

We investigate one-dimensional scalar balance laws with singular convolution-type source terms. Under appropriate convexity and kernel assumptions, we establish the global existence of entropy weak solutions in ${\bf L}^2(\mathbb{R})$, together with two partial uniqueness results, in the ${\bf L}^2$-periodic setting and non-periodic setting with ${\bf L}^1(\mathbb{R})$ kernel. In the ${\bf L}^1$-kernel case, the characteristic speed satisfies an Oleinik-type estimate, and entropy weak solutions possess locally bounded fractional variation for all positive times. Furthermore, we derive a simple criterion characterizing local smoothness and wave breaking of solutions, which, in particular, includes both the Burger-Poisson and the Burgers-Hilbert equation as special cases.