Well-posedness and stationary states for a crowded active Brownian system with size-exclusion

TL;DR

This paper establishes the existence and uniqueness of solutions and stationary states for a complex non-linear, non-local PDE modeling a crowded active Brownian particle system with size-exclusion, using gradient flow and entropy methods.

Contribution

It extends the boundedness-by-entropy method to a non-linear, non-local PDE with degenerate diffusion, providing new existence and uniqueness results.

Findings

Existence of solutions to the PDE was proven.

Uniqueness established when the non-local drift is zero.

Stationary solutions were characterized and shown to be unique under certain conditions.

Abstract

We prove the existence of solutions to a non-linear, non-local, degenerate equation which was previously derived as the formal hydrodynamic limit of an active Brownian particle system, where the particles are endowed with a position and an orientation. This equation incorporates diffusion in both the spatial and angular coordinates, as well as a non-linear non-local drift term, which depends on the angle-independent density. The spatial diffusion is non-linear degenerate and also comprises diffusion of the angle-independent density, which one may interpret as cross-diffusion with infinitely many species. Our proof relies on interpreting the equation as the perturbation of a gradient flow in a Wasserstein-type space. It generalizes the boundedness-by-entropy method to this setting and makes use of a gain of integrability due to the angular diffusion. For this latter step, we adapt a classical interpolation lemma for function spaces depending on time. We also prove uniqueness in the particular case where the non-local drift term is null, and provide existence and uniqueness results for stationary equilibrium solutions.