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Global well-posedness for a two-dimensional Keller-Segel-Euler system of consumption type

Jungkyoung Na

TL;DR

The paper analyzes the 2D Keller-Segel system of consumption type coupled with the incompressible Euler equations in $\mathbb{R}^2$, proving local existence of smooth solutions for arbitrary smooth initial data and global existence under a smallness condition on the initial oxygen concentration. The authors develop a $W^{1,q}$ energy framework with $q>2$ and utilize the sharp Brézis-Wainger inequality to control $L^{\infty}$ norms in the absence of velocity diffusion, drawing on insights from the partially inviscid 2D Boussinesq system. This approach extends known 2D global well-posedness results from KS coupled with Navier-Stokes to the Euler-coupled setting, demonstrating global regularity under small initial oxygen and highlighting how low-viscosity fluid dynamics interact with chemotaxis-driven consumption. The work contributes to the understanding of chemotaxis-fluid coupling in low-dissipation regimes and provides a rigorous framework for global well-posedness in 2D KS-E systems. The techniques may inform analyses of related active-fluid models and inspire further study of inviscid-fluid effects on chemotactic aggregation. $

Abstract

We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in $\mathbb{R}^2$. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the $W^{1,q}$-energy estimate $(q>2)$ motivated by the partially inviscid two-dimensional Boussinesq system in \cite{C06}. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations.

Global well-posedness for a two-dimensional Keller-Segel-Euler system of consumption type

TL;DR

The paper analyzes the 2D Keller-Segel system of consumption type coupled with the incompressible Euler equations in , proving local existence of smooth solutions for arbitrary smooth initial data and global existence under a smallness condition on the initial oxygen concentration. The authors develop a energy framework with and utilize the sharp Brézis-Wainger inequality to control norms in the absence of velocity diffusion, drawing on insights from the partially inviscid 2D Boussinesq system. This approach extends known 2D global well-posedness results from KS coupled with Navier-Stokes to the Euler-coupled setting, demonstrating global regularity under small initial oxygen and highlighting how low-viscosity fluid dynamics interact with chemotaxis-driven consumption. The work contributes to the understanding of chemotaxis-fluid coupling in low-dissipation regimes and provides a rigorous framework for global well-posedness in 2D KS-E systems. The techniques may inform analyses of related active-fluid models and inspire further study of inviscid-fluid effects on chemotactic aggregation. $

Abstract

We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in . This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the -energy estimate motivated by the partially inviscid two-dimensional Boussinesq system in \cite{C06}. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations.
Paper Structure (5 sections, 8 theorems, 127 equations)

This paper contains 5 sections, 8 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Previous works
  3. Main results and Discussion
  4. Local well-posedness
  5. Global well-posedness

Key Result

Theorem 1.1

Let $m>3$. Assume that $\chi$ and $k$ are defined by intro, and initial data $\left (\rho_{0}, c_{0}, u_{0}\right)$ and $\phi$ satisfy Then there exist some $T_{loc}>0$ and a unique solution $\left (\rho,c,u\right)$ with initial data $\left (\rho_{0}, c_{0}, u_{0}\right)$ for KS-E satisfying

Theorems & Definitions (13)

  • Theorem 1.1: Local well-posedness
  • Theorem 1.2: Global well-posedness
  • Proposition 2.1
  • Proposition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 3 more