Stability and Total Variation Estimates on General Scalar Balance Laws

TL;DR

This paper derives total variation bounds and stability estimates for solutions to general scalar balance laws with space-dependent fluxes and sources, including non-local sources, enhancing understanding of solution dependence and well-posedness.

Contribution

It introduces new total variation estimates for scalar balance laws with space-dependent fluxes and sources, including non-local terms, improving stability analysis.

Findings

Total variation bounds for solutions are established.

Stability estimates for solutions with respect to flux and source variations.

Well-posedness results for balance laws with non-local sources.

Abstract

Consider the general scalar balance law $\partial_t u + \Div f(t, x,u) = F(t,x,u)$ in several space dimensions. The aim of this note is to estimate the dependence of its solutions from the flow $f$ and from the source $F$. To this aim, a bound on the total variation in the space variables of the solution is obtained. This result is then applied to obtain well posedness and stability estimates for a balance law with a non local source.