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

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Bruna, Maria 
Burger, Martin 
Esposito, Antonio 
Schulz, Simon 

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.

Description

Keywords

35K20, 35K58, 35Q70, 35Q92, math.AP, math.AP

Journal Title

SIAM Journal on Mathematical Analysis

Conference Name

Journal ISSN

0036-1410
1095-7154

Volume Title

Publisher

Society for Industrial and Applied Mathematics