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Global Existence For A Nonlocal Multi-Species Aggregation-Diffusion Equation

Elaine Cozzi, Zachary Radke

TL;DR

The paper addresses the global-in-time regularity of smooth solutions to a multi-species, nonlocal aggregation-diffusion system with singular kernels. It develops a small-data framework, proving $L^2$-decay and bootstrapped global existence under precise viscosity and kernel-norm conditions, and proves preservation of pointwise density inequalities via two comparison theorems for ideal kernels. A single-species analogue is provided, and a detailed discussion clarifies how the results extend beyond balanced kernel assumptions. Collectively, these results advance understanding of global regularity in nonlocal cross-diffusion systems and offer tools applicable to chemotaxis- and ecology-inspired models without requiring detailed balance on kernels.

Abstract

We consider the question of global existence of smooth solutions to a multi-species aggregation-diffusion equation for a class of singular interaction kernels. We establish a smallness condition on the initial data which yields global existence of smooth solutions. We also give conditions on the species interaction which ensure that pointwise inequalities comparing species densities are preserved by the evolution.

Global Existence For A Nonlocal Multi-Species Aggregation-Diffusion Equation

TL;DR

The paper addresses the global-in-time regularity of smooth solutions to a multi-species, nonlocal aggregation-diffusion system with singular kernels. It develops a small-data framework, proving -decay and bootstrapped global existence under precise viscosity and kernel-norm conditions, and proves preservation of pointwise density inequalities via two comparison theorems for ideal kernels. A single-species analogue is provided, and a detailed discussion clarifies how the results extend beyond balanced kernel assumptions. Collectively, these results advance understanding of global regularity in nonlocal cross-diffusion systems and offer tools applicable to chemotaxis- and ecology-inspired models without requiring detailed balance on kernels.

Abstract

We consider the question of global existence of smooth solutions to a multi-species aggregation-diffusion equation for a class of singular interaction kernels. We establish a smallness condition on the initial data which yields global existence of smooth solutions. We also give conditions on the species interaction which ensure that pointwise inequalities comparing species densities are preserved by the evolution.
Paper Structure (14 sections, 22 theorems, 167 equations)

This paper contains 14 sections, 22 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Definitions and Preliminary Lemmas
  3. Notational Conventions and Definitions
  4. Interaction Kernels
  5. Useful Inequalities
  6. Mild solutions of (\ref{['aggregation equation']}) and (\ref{['PDE']})
  7. Proof of Theorem \ref{['smallHexistence']}
  8. Proofs of Theorems \ref{['comparison1']} and \ref{['comparison2']}
  9. Concluding Remarks
  10. A Priori Estimates
  11. Positivity of Solution
  12. Conservation of Mass
  13. The Mild Solution
  14. The Classical Solution

Key Result

Theorem 1.1

Let $k\geq 3$, $r>2$, and $N\geq 2$. Assume that $\rho$ is a non-negative solution to (PDE) on $[0,T]$ belonging to $Lip([0,T]; W^{k,1}\cap W^{k,\infty}(\mathbb{R}^d))$. Then there exists a constant $C>0$, depending on $N$, $r$, and when $d=2$, the support of $\mathcal{K}$, such that if either $\mat or $\mathcal{K}$ is allowable and $\nu$ satisfies then $\rho$ can be extended to a unique non-nega

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 37 more