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Construction of weak solutions to a pressureless viscous model driven by nonlocal attraction-repulsion

Piotr B. Mucha, Maja Szlenk, Ewelina Zatorska

Abstract

We analyze the pressureless Navier-Stokes system with nonlocal attraction-repulsion forces. Such systems appear in the context of models of collective behavior. We prove the existence of weak solutions on the whole space $\mathbb{R}^3$ in the case of density-dependent degenerate viscosity. For the nonlocal term it is assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. Under these assumptions, we derive the analog of the Bresch-Desjardins and Mellet-Vasseur estimates for the nonlocal system. In particular, we are able to adapt the approach of Vasseur and Yu to construct a weak solution.

Construction of weak solutions to a pressureless viscous model driven by nonlocal attraction-repulsion

Abstract

We analyze the pressureless Navier-Stokes system with nonlocal attraction-repulsion forces. Such systems appear in the context of models of collective behavior. We prove the existence of weak solutions on the whole space in the case of density-dependent degenerate viscosity. For the nonlocal term it is assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. Under these assumptions, we derive the analog of the Bresch-Desjardins and Mellet-Vasseur estimates for the nonlocal system. In particular, we are able to adapt the approach of Vasseur and Yu to construct a weak solution.
Paper Structure (21 sections, 24 theorems, 334 equations, 1 figure, 1 table)

This paper contains 21 sections, 24 theorems, 334 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The main result
  3. Fundamental level of approximation
  4. Truncation to periodic domain
  5. The Galerkin approximation
  6. Energy estimate
  7. The BD inequality and limit passages with $\varepsilon,\nu,\delta,\eta\to 0$.
  8. Convergence lemmas
  9. Limit passage with $\nu,\varepsilon\to 0$
  10. Limit passage with $\eta,\delta\to 0$.
  11. Mellet -- Vasseur estimates
  12. Preparation of initial data.
  13. Preparation of the test function.
  14. Limit passage with $m\to\infty$
  15. Limit passage with $\kappa\to 0$ and $k\to \infty$
  16. ...and 6 more sections

Key Result

Theorem 2.3

Let $(\varrho_0,m_0)$ satisfy (A1-A2). Then there exists a global in time weak solution $(\varrho,u)$ to (main) in the sense of Definition Def:main. In addition, this solution satisfies: (i) the energy estimate (ii) the Bresch-Desjardins estimate (iii) the Mellet-Vasseur estimate

Figures (1)

  • Figure 1: Attractive-repulsive potential $k(|x|)=K(x)$.

Theorems & Definitions (49)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 39 more