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Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains

Matthias Hieber, Hideo Kozono, Sylvie Monniaux, Patrick Tolksdorf

TL;DR

The authors analyze coupled chemotaxis-fluid systems (Keller-Segel-Navier-Stokes and a chemotaxis-consumption variant) in bounded Lipschitz domains, focusing on strong solutions in critical Besov-type spaces and equilibria stability. They develop a time-weighted maximal regularity framework for the linearized problem and use a fixed-point argument to obtain local well-posedness in $E_{p,μ}^T$, with $μ = n/(2q) + 1/p - 1/2$, and then prove global existence for small data with exponential decay toward a stationary state. For smoother data, they establish global boundedness and positivity preservation, and the approach extends to the chemotaxis-consumption model with analogous results. The work advances the analysis of chemotaxis-fluid interactions in non-smooth geometries and provides robust tools for studying diffusion, reaction, and buoyancy in 3D.

Abstract

Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local strong as well as global strong solutions for small data in the setting of critical Besov spaces. Moreover, non-trivial equilibria are shown to be exponentially stable. For smoother data, these solutions are shown to be globally bounded and to preserve positivity properties. The approach presented is based on optimal $\mathrm{L}^q$-regularity properties of the Neumann Laplacian and the Stokes operator in bounded Lipschitz domains.

Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains

TL;DR

The authors analyze coupled chemotaxis-fluid systems (Keller-Segel-Navier-Stokes and a chemotaxis-consumption variant) in bounded Lipschitz domains, focusing on strong solutions in critical Besov-type spaces and equilibria stability. They develop a time-weighted maximal regularity framework for the linearized problem and use a fixed-point argument to obtain local well-posedness in , with , and then prove global existence for small data with exponential decay toward a stationary state. For smoother data, they establish global boundedness and positivity preservation, and the approach extends to the chemotaxis-consumption model with analogous results. The work advances the analysis of chemotaxis-fluid interactions in non-smooth geometries and provides robust tools for studying diffusion, reaction, and buoyancy in 3D.

Abstract

Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local strong as well as global strong solutions for small data in the setting of critical Besov spaces. Moreover, non-trivial equilibria are shown to be exponentially stable. For smoother data, these solutions are shown to be globally bounded and to preserve positivity properties. The approach presented is based on optimal -regularity properties of the Neumann Laplacian and the Stokes operator in bounded Lipschitz domains.
Paper Structure (8 sections, 16 theorems, 184 equations)

This paper contains 8 sections, 16 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Results
  3. Proof of Local solvability
  4. Global existence for small initial data
  5. Positivity
  6. Regularity and boundedness
  7. The Chemotaxis-Consumption Model
  8. Appendix: Real interpolation of solenoidal function spaces

Key Result

Theorem 2.1

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain and $n=2$ or $n=3$. Then there exists $\varepsilon= \varepsilon(\Omega) >0$, depending on the Lipschitz geometry of $\Omega$, such that for all $p,q \in (1,\infty)$ satisfying and all and all $f \in \mathrm{L}^n (\Omega;\mathbb{R}^n)$ there exists $T>0$ such that the Keller-Segel-Navier-Stokes system Eq: KSNS admits a unique strong

Theorems & Definitions (32)

  • Theorem 2.1: Local existence
  • Theorem 2.3: Global existence and stability
  • Theorem 2.4: Boundedness and positivity
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Lemma 3.1: Fixed point theorem
  • Remark 3.2
  • Lemma 3.3
  • ...and 22 more