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Global existence and asymptotic stability for the Toner-Tu model of flocking

Young-Pil Choi, Kyungkeun Kang, Woojae Lee

Abstract

This paper deals with the Toner-Tu (TT) model, which is a hydrodynamic model describing the collective motion of numerous self-propelled agents. We analytically study the global-in-time well-posedness of the TT model near the steady-state solution in the ordered phase. We also show the large-time behavior of solutions showing that the steady-state solution is polynomially stable in a Sobolev space in the sense that solutions that are initially close to that steady state converge to that at least polynomially fast as time tends to infinity. Moreover, we investigate the variant of the TT model which describes the dynamics of the actin filament.

Global existence and asymptotic stability for the Toner-Tu model of flocking

Abstract

This paper deals with the Toner-Tu (TT) model, which is a hydrodynamic model describing the collective motion of numerous self-propelled agents. We analytically study the global-in-time well-posedness of the TT model near the steady-state solution in the ordered phase. We also show the large-time behavior of solutions showing that the steady-state solution is polynomially stable in a Sobolev space in the sense that solutions that are initially close to that steady state converge to that at least polynomially fast as time tends to infinity. Moreover, we investigate the variant of the TT model which describes the dynamics of the actin filament.
Paper Structure (11 sections, 17 theorems, 121 equations)

This paper contains 11 sections, 17 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Useful inequalities
  4. Reformulation
  5. Local well-posedness
  6. Cauchy problem for the Parabolic-Parabolic Toner--Tu model
  7. Global-in-time existence of solutions
  8. Large-time behavior
  9. Cauchy problem for the Toner--Tu model
  10. Global-in-time existence of solutions
  11. Large-time behavior

Key Result

Theorem 1.1

The steady-state solution $(\rho_s, v_s)$ of the TT model original tt model1 given by steady is stable in $H^m(\mathbb R^d)$-norm for any $m \geq 3$. Moreover, it is almost polynomially stable in $(\dot{H}^{-l}\cap H^m)(\mathbb R^d) \times(\dot{H}^{-l}\cap H^m)(\mathbb R^d)$ for some $l(m,d)>0$, in

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • ...and 27 more