Well-posedness of an integro-differential model for active Brownian particles

TL;DR

This paper establishes the well-posedness of a nonlinear integro-differential model for active Brownian particles, providing a rigorous mathematical foundation for analyzing such systems with nonlocal interactions and low-regularity initial data.

Contribution

It introduces a novel existence and uniqueness proof for a class of nonlinear integro-differential equations modeling active Brownian particles, including cases with distributional initial data.

Findings

Proved global existence and uniqueness of weak solutions.

Extended well-posedness to very weak initial data.

Included finite systems with a finite number of directions.

Abstract

We propose a general strategy for solving nonlinear integro-differential evolution problems with periodic boundary conditions, where no direct maximum/minimum principle is available. This is motivated by the study of recent macroscopic models for active Brownian particles with repulsive interactions, consisting of advection-diffusion processes in the space of particle position and orientation. We focus on one of such models, namely a semilinear parabolic equation with a nonlinear active drift term, whereby the velocity depends on the particle orientation and angle-independent overall particle density (leading to a nonlocal term by integrating out the angular variable). The main idea of the existence analysis is to exploit a-priori estimates from (approximate) entropy dissipation. The global existence and uniqueness of weak solutions is shown using a two-step Galerkin approximation with appropriate cutoff in order to obtain nonnegativity, an upper bound on the overall density and preserve a-priori estimates. Our anyalysis naturally includes the case of finite systems, corresponding to the case of a finite number of directions. The Duhamel principle is then used to obtain additional regularity of the solution, namely continuity in time-space. Motivated by the class of initial data relevant for the application, which includes perfectly aligned particles (same orientation), we extend the well-posedness result to very weak solutions allowing distributional initial data with low regularity.