Invariance properties of the solution operator for measure-valued semilinear transport equations

TL;DR

This paper establishes conditions ensuring measure-valued semilinear transport equations preserve positivity and regularity properties of initial conditions, including absolute continuity and $L^p$ regularity, in the space of finite signed Radon measures.

Contribution

It provides new invariance results for measure-valued solutions of nonlinear transport equations, extending understanding of their positivity and regularity preservation.

Findings

Positivity is preserved under certain conditions.

Absolute continuity with respect to Lebesgue measure is maintained.

Solutions retain initial $L^p$ regularity if present initially.

Abstract

We provide conditions under which we prove for measure-valued transport equations with non-linear reaction term in the space of finite signed Radon measures, that positivity is preserved, as well as absolute continuity with respect to Lebesgue measure, if the initial condition has that property. Moreover, if the initial condition has $L^p$ regular density, then the solution has the same property.