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On the space-time analyticity of the Keller-Segel-Navier-Stokes system

Elie Abdo, Zhongtian Hu

Abstract

In this paper, we study the coupled Keller-Segel-Navier-Stokes system, which models chemotaxis occuring in ambient viscous fluid. We consider this nonlinear, nonlocal system on a periodic strip, equipped with homogeneous Neumann boundary conditions for the Keller-Segel part and no-slip boundary condition for the fluid part. We prove the simultaneous space-time analyticity of the solution up to the boundary based on energy methods.

On the space-time analyticity of the Keller-Segel-Navier-Stokes system

Abstract

In this paper, we study the coupled Keller-Segel-Navier-Stokes system, which models chemotaxis occuring in ambient viscous fluid. We consider this nonlinear, nonlocal system on a periodic strip, equipped with homogeneous Neumann boundary conditions for the Keller-Segel part and no-slip boundary condition for the fluid part. We prove the simultaneous space-time analyticity of the solution up to the boundary based on energy methods.
Paper Structure (11 sections, 9 theorems, 137 equations)

This paper contains 11 sections, 9 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Functional Setting.
  3. Main Result
  4. Outline of the proof of Theorem \ref{['maintheorem']}.
  5. Local Regularity in Lebesgue Spaces
  6. Gevrey Elliptic Regularity
  7. Gevrey Boundedness of Solutions to the Keller-Segel-Navier-Stokes System
  8. Interpolation Inequalities
  9. Combinatorial Inequalities
  10. Estimates of $M_i$
  11. Estimates of $N_i$

Key Result

Theorem 1.1

Let $r \ge 3$ be an integer. There exists $\epsilon, \tilde{\epsilon}, \bar{\epsilon} \in (0,1]$ depending only on $r$ such that for any initial scalar $\rho_0 \in H^{2r}(\Omega)$ and any initial divergence-free velocity field $u_0 \in H^{2r}(\Omega)$ satisfying the compatibility conditions, there e holds.

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • Remark 3.1
  • Corollary 3.1
  • proof
  • Lemma 3.1
  • proof
  • ...and 10 more