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Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming

Nilasis Chaudhuri, Young-Pil Choi, Oliver Tse, Ewelina Zatorska

Abstract

We consider a one-dimensional hydrodynamic model featuring nonlocal attraction-repulsion interactions and singular velocity alignment. We introduce a two-velocity reformulation and the corresponding energy-type inequality, in the spirit of the Bresch-Desjardins estimate. We identify a dependence between the communication weight and interaction kernel and between the pressure and viscosity term allowing for this inequality to be uniform in time. It is then used to study long-time asymptotics of solutions.

Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming

Abstract

We consider a one-dimensional hydrodynamic model featuring nonlocal attraction-repulsion interactions and singular velocity alignment. We introduce a two-velocity reformulation and the corresponding energy-type inequality, in the spirit of the Bresch-Desjardins estimate. We identify a dependence between the communication weight and interaction kernel and between the pressure and viscosity term allowing for this inequality to be uniform in time. It is then used to study long-time asymptotics of solutions.
Paper Structure (32 sections, 16 theorems, 268 equations)

This paper contains 32 sections, 16 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. A priori energy and modulated energy estimate
  4. Existence of regular solutions to the approximate system
  5. The approximate system
  6. The approximate initial data
  7. Formulation of the main result
  8. Total energy and modulated energy esimates
  9. Uniform estimates for the density
  10. Conclusion of the proof of Theorem \ref{['Th:3']}
  11. Compactness results for approximate solutions
  12. Derivation of the Mellet-Vasseur inequality
  13. Convergence of the density
  14. Strong convergence of $(\rho^\varepsilon)^\gamma$
  15. Strong convergence result
  16. ...and 17 more sections

Key Result

Theorem 2.3

Let $\tau\ge 0$ and the conditions $(\mathcal{A})$ hold. Then there exist a weak solution to system main_eq in the sense of Definition Def:1, and $\lambda>0$ such that the modulated energy inequality holds with constants $\ell_{\tau,\lambda}\in\mathbb R$ and $c_\lambda > 0$. If either $\tau>0$ or $\phi$ is uniformly bounded from below, i.e. $\phi\ge \underline{\phi}$ for some constant $\underline

Theorems & Definitions (33)

  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • Remark 3.2
  • proof : Proof of Lemma \ref{['lem:bd-estimate']}
  • Proposition 4.1
  • ...and 23 more