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Global Existence of Strong Solutions to the Kinetic Cucker--Smale Model Coupled with the Stokes Equations

Chunyin Jin

Abstract

In this paper, we investigate existence of global-in-time strong solutions to the kinetic Cucker--Smale model coupled with the Stokes equations in the whole space. By introducing a weighted Sobolev space and using space-time estimates for the linear non-stationary Stokes equations, we present a complete analysis on existence of global-in-time strong solutions to the coupled model, without any smallness requirements on initial data.

Global Existence of Strong Solutions to the Kinetic Cucker--Smale Model Coupled with the Stokes Equations

Abstract

In this paper, we investigate existence of global-in-time strong solutions to the kinetic Cucker--Smale model coupled with the Stokes equations in the whole space. By introducing a weighted Sobolev space and using space-time estimates for the linear non-stationary Stokes equations, we present a complete analysis on existence of global-in-time strong solutions to the coupled model, without any smallness requirements on initial data.

Paper Structure

This paper contains 8 sections, 9 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The kinetic Cucker--Smale model
  4. Linear non-stationary Stokes equations
  5. The Helmholtz decomposition in $\mathbb R^3$
  6. Local Existence of Strong Solutions to the Coupled System
  7. A Priori Estimates
  8. Global Existence of Strong Solutions

Key Result

Theorem \oldthetheorem

Let $0<R_0<\infty$. Assume the initial data $f_0(\hbox{\boldmath $x$},\hbox{\boldmath $v$})\ge 0$, $f_0(\hbox{\boldmath $x$},\hbox{\boldmath $v$}) \in H_{\omega}^1(\mathbb R^3 \times \mathbb R^3)\cap L^{\infty}(\mathbb R^3 \times \mathbb R^3)$, and $\hbox{\boldmath $u$}_0(\hbox{\boldmath $x$}) \in H Then the Cauchy problem eq-cs-s-eq-sys-inidata admits a unique global-in-time strong solution in th

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem \oldthetheorem
  • Remark 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.1: The Helmholtz decomposition
  • Proposition 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 5 more