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

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.

Phase Separation in Systems of Interacting Active Brownian Particles

“…The presence of these cross-diffusion terms in Models 3 and 4, combined with the nonlinear advection term, makes their rigorous analysis very challenging. As a first step towards this goal, in [3] we have considered the well-posedness of Model 2.…”

“…Finally, in section 4 we present some two-dimensional numerical examples of the patterns associated with MIPS, obtained from both the stochastic microscopic models and the macroscopic models. The rigorous analysis of one of the local macroscopic models, displaying a nonlinearity in the advection term but linear diffusion in space and orientation, is addressed in [3]. The analysis of the two macroscopic models that, in addition to the nonlinearity in the advection term, display nonlinear cross-diffusion-like terms, will be the subject of future work.…”

“…1 We outline the derivation of (2.12) in subsection SM1.1. A rigorous well-posedness theory for (2.11) and (2.12) is provided in [3].…”