Regularity and Uniqueness for a Model of Active Particles with Angle-Averaged Diffusions

TL;DR

This paper proves regularity and uniqueness of solutions for a nonlocal degenerate parabolic equation modeling active particles, using De Giorgi's method, Galerkin approximation, and heat kernel estimates.

Contribution

It introduces a novel approach to establish regularity and uniqueness for a nonlocal, degenerate parabolic equation derived from active particle models.

Findings

Weak solutions are smooth for positive times.

The equation is shown to be uniformly parabolic under certain conditions.

Uniqueness of solutions is established with stronger initial data constraints.

Abstract

We study the regularity and uniqueness of weak solutions of a degenerate parabolic equation, arising as the limit of a stochastic lattice model of self-propelled particles. The angle-average of the solution appears as a coefficient in the diffusive and drift terms, making the equation nonlocal. We prove that, under unrestrictive non-degeneracy assumptions on the initial data, weak solutions are smooth for positive times. Our method rests on deriving a drift-diffusion equation for a particular function of the angle-averaged density and applying De Giorgi's method to show that the original equation is uniformly parabolic for positive times. We employ a Galerkin approximation to justify rigorously the passage from divergence to non-divergence form of the equation, which yields improved estimates by exploiting a cancellation. By imposing stronger constraints on the initial data, we prove the uniqueness of the weak solution, which relies on Duhamel's principle and gradient estimates for the periodic heat kernel to derive $L^\infty$ estimates for the angle-averaged density.