On scalar nonlinear balance laws with singular nonlocal sources

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 ${\mathbf{L}}^2(\mathbb{R})$, together with two partial uniqueness results, in the ${\mathbf{L}}^2$-periodic setting and non-periodic setting with ${\mathbf{L}}^1(\mathbb{R})$ kernel. In the ${\mathbf{L}}^1$-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.

45pages