Invariance properties of the solution operator for measure-valued semilinear transport equations
Sander C. Hille, Rainey Lyons, Adrian Muntean
TL;DR
The paper addresses the problem of invariance properties for the semilinear measure-valued transport equation $\partial_t \mu_t + \mathrm{div}_x(v_t(x)\mu_t)= f_t(\mu_t)$ by developing a robust mild-solution framework and studying the solution operator as a dynamical system on measures. It introduces a general weak-limit result ensuring that $L^p$-densities are preserved under weak convergence of measures, and proves positivity preservation and local $L^p$-invariance of densities under natural decompositions of the reaction term and mild regularity assumptions. A dilation-based reformulation of mild solutions provides a versatile tool to connect the original dynamics with a dilated semilinear problem, facilitating fixed-point arguments and stability analyses. The results have broad relevance to structured population models, traffic flow, and cell migration, where measure-valued descriptions are natural, and they lay groundwork for extending to quasi-linear problems and less regular velocity fields.
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.