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Regularity and trend to equilibrium for a non-local advection-diffusion model of active particles

Luca Alasio, Jessica Guerand, Simon Schulz

Abstract

We establish regularity and, under suitable assumptions, convergence to stationary states for weak solutions of a parabolic equation with a non-linear non-local drift term; this equation was derived from a model of active Brownian particles with repulsive interactions in a previous work, which incorporates advection-diffusion processes both in particle position and orientation. We apply De Giorgi's method and differentiate the equation with respect to the time variable iteratively to show that weak solutions become smooth away from the initial time. This strategy requires that we obtain improved integrability estimates in order to cater for the presence of the non-local drift. The instantaneous smoothing effect observed for weak solutions is shown to also hold for very weak solutions arising from distributional initial data; the proof of this result relies on a uniqueness theorem in the style of M.~Pierre for low-regularity solutions. The convergence to stationary states is proved under a smallness assumption on the drift term.

Regularity and trend to equilibrium for a non-local advection-diffusion model of active particles

Abstract

We establish regularity and, under suitable assumptions, convergence to stationary states for weak solutions of a parabolic equation with a non-linear non-local drift term; this equation was derived from a model of active Brownian particles with repulsive interactions in a previous work, which incorporates advection-diffusion processes both in particle position and orientation. We apply De Giorgi's method and differentiate the equation with respect to the time variable iteratively to show that weak solutions become smooth away from the initial time. This strategy requires that we obtain improved integrability estimates in order to cater for the presence of the non-local drift. The instantaneous smoothing effect observed for weak solutions is shown to also hold for very weak solutions arising from distributional initial data; the proof of this result relies on a uniqueness theorem in the style of M.~Pierre for low-regularity solutions. The convergence to stationary states is proved under a smallness assumption on the drift term.
Paper Structure (20 sections, 26 theorems, 216 equations)

This paper contains 20 sections, 26 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Definitions and problem set-up
  3. Main theorems
  4. Preliminary Notions
  5. Boundedness of Weak Solutions
  6. Boundedness away from initial time
  7. Caccioppoli inequality
  8. Interior local boundedness on subcylinders
  9. Boundedness away from initial time
  10. Global-in-time boundedness for $L^\infty$ initial data
  11. Higher Regularity of Weak Solutions
  12. Boundedness of $\dot{f}$ away from initial time
  13. Proofs of Propositions \ref{['prop:boundedness of all time derivs']} and \ref{['prop:sobolev of all time derivs']}
  14. Uniqueness and Regularity of Very Weak Solutions
  15. Stationary States
  16. ...and 5 more sections

Key Result

Theorem 1

Assume $f_0$ is non-negative and satisfies eq:initial data assumptions, $T>0$, and let $f$ be the unique weak solution of eq:main eqn with initial data $f_0$. Then, for a.e. $t \in (0,T)$, there holds $f \in C^\infty((t,T)\times\mathbb{R}^3)$.

Theorems & Definitions (54)

  • Definition 1.1: Notions of Solution
  • Theorem 1: Smoothness away from Initial Time
  • Theorem 2: Uniqueness for Very Weak Solutions
  • Theorem 3: Regularity for Very Weak Solutions
  • Theorem 4: Global-in-time Boundedness for Bounded Initial Data
  • Theorem 5: Smoothness of Stationary States
  • Theorem 6: Convergence to Constant Stationary States for Small Péclet Number
  • Remark 1.2: Stationary States
  • Lemma 1.3: DiBenedetto, Proposition 3.2 of § 1, DiBenedetto
  • Lemma 1.4: Periodic Calderón--Zygmund Inequality
  • ...and 44 more