Memory effects in measure transport equations

TL;DR

This paper studies a nonlinear transport equation incorporating fractional time derivatives to model memory effects in complex systems, establishing well-posedness and a generalized solution representation.

Contribution

It introduces a well-posedness theory for weak measure solutions of nonlinear transport equations with fractional time derivatives and generalizes the classical push-forward formula.

Findings

Established existence and uniqueness of solutions

Derived an integral formula generalizing classical push-forward

Applicable to systems with memory effects in physics, chemistry, biology

Abstract

Transport equations with a nonlocal velocity field have been introduced as a continuum model for interacting particle systems arising in physics, chemistry and biology. Fractional time derivatives, given by convolution integrals of the time-derivative with power-law kernels, are typical for memory effects in complex systems. In this paper we consider a nonlinear transport equation with a fractional time-derivative. We provide a well-posedness theory for weak measure solutions of the problem and an integral formula which generalizes the classical push-forward representation formula to this setting.