Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global well-posedness of the 3D Patlak-Keller-Segel system near a straight line

Bowei Tu

Abstract

We consider the Cauchy problem of the three-dimensional parabolic-elliptic Patlak-Keller-Segel chemotactic model. The initial data is almost a Dirac measure supported on a straight line with mass less than $8π$. We prove that if the data is sufficiently close to the straight line, then global well-posedness holds. This result is parallel to the work on vortex filament solutions of the Navier-Stokes equations by Bedrossian, Germain and Harrop-Griffiths \cite{filament}.

Global well-posedness of the 3D Patlak-Keller-Segel system near a straight line

Abstract

We consider the Cauchy problem of the three-dimensional parabolic-elliptic Patlak-Keller-Segel chemotactic model. The initial data is almost a Dirac measure supported on a straight line with mass less than . We prove that if the data is sufficiently close to the straight line, then global well-posedness holds. This result is parallel to the work on vortex filament solutions of the Navier-Stokes equations by Bedrossian, Germain and Harrop-Griffiths \cite{filament}.
Paper Structure (12 sections, 150 equations)

This paper contains 12 sections, 150 equations.

Table of Contents

  1. Introduction
  2. Statement of Results and Outline of the Proof
  3. Function Spaces
  4. Mild Solution
  5. Main Results
  6. Linear Estimates for the Core Part
  7. Statement of the Estimates
  8. Long Time Estimates
  9. Short Time Estimates
  10. Proof of Theorem 3.1
  11. Linear Estimates for the Background Part
  12. Nonlinear Estimates

Theorems & Definitions (9)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof