Analysis and simulation of nonlinear and nonlocal transport equations

Abstract

This article is devoted to the analysis of some nonlinear conservative transport equations, including the so-called aggregation equation with pointy potential, and numerical method devoted to its numerical simulation. Such a model describes the collective motion of individuals submitted to an attractive potential and can be written as a continuity transport equation with a velocity field computed through a self-consistent interaction potential. In the strongly attractive setting, Lp solutions may blow up in finite time, then a theory of existence of weak measure solutions has been defined. In this approach, we show the existence of Filippov characteristics allowing to define solutions of the aggregation initial value problem as a pushforward measure. Then numerical analysis of an upwind type scheme is proposed allowing to recover the dynamics of aggregates beyond the blowup time.