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Suppression of blow-up in 3-D Keller-Segel model via Couette flow in whole space

Shijin Deng, Binbin Shi, Weike Wang

Abstract

In this paper, we study the 3-D parabolic-parabolic and parabolic-elliptic Keller-Segel models with Couette flow in $\mathbb{R}^3$. We prove that the blow-up phenomenon of solution can be suppressed by enhanced dissipation of large Couette flows. Here we develop Green's function method to describe the enhanced dissipation via a more precise space-time structure and obtain the global existence together with pointwise estimates of the solutions. The result of this paper shows that the enhanced dissipation exists for all frequencies in the case of whole space and it is reason that we obtain global existence for 3-D Keller-Segel models here. It is totally different from the case with the periodic spatial variable $x$ in [2,10]. This paper provides a new methodology to capture dissipation enhancement and also a surprising result which shows a totally new mechanism.

Suppression of blow-up in 3-D Keller-Segel model via Couette flow in whole space

Abstract

In this paper, we study the 3-D parabolic-parabolic and parabolic-elliptic Keller-Segel models with Couette flow in . We prove that the blow-up phenomenon of solution can be suppressed by enhanced dissipation of large Couette flows. Here we develop Green's function method to describe the enhanced dissipation via a more precise space-time structure and obtain the global existence together with pointwise estimates of the solutions. The result of this paper shows that the enhanced dissipation exists for all frequencies in the case of whole space and it is reason that we obtain global existence for 3-D Keller-Segel models here. It is totally different from the case with the periodic spatial variable in [2,10]. This paper provides a new methodology to capture dissipation enhancement and also a surprising result which shows a totally new mechanism.
Paper Structure (13 sections, 12 theorems, 127 equations)

This paper contains 13 sections, 12 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Green's functions and computational lemmas
  3. Green's Functions
  4. Green's function for $n(x,y,z,t)$
  5. Green's function for $c(x,y,z,t)$
  6. Computational Lemmas: Nonlinear Wave Interactions
  7. Computational Lemmas: Tools for Pointwise Estimates of $c(x,y,z,t)$
  8. Global existence, nonlinear stability and pointwise structure
  9. Local well-posedness and regularity criterion
  10. Representations of solutions and linear estimates
  11. Proof of Theorem \ref{['main']}: Parabolic-Parabolic Case
  12. Proof of Theorem \ref{['main0']}: Parabolic-Elliptic Case
  13. Appendix

Key Result

Theorem 1.1

(Parabolic-Parabolic case.) Suppose $\epsilon=1$ in pro and the initial functions $n_0(x,y,z), c_0(x,y,z)$ satisfy and for constants $C_0, C_*>1$ and $0<C_0^*\ll 1$. Then there exists a positive constant $C_0\ll A< C_0/C_0^*$ such that the solution $(n(x,y,z,t),c(x,y,z,t))$ of pro exists globally in time and satisfies Here, the time decay structure function $\mathscr{A}(t;A,\theta,\gamma)$ and

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 15 more