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Keller-Segel-Navier-Stokes systems involving general sensitivities with Signal-Dependent Power-Law Decay

Jaewook Ahn, Sukjung Hwang

Abstract

This paper investigates a two-dimensional Keller--Segel--Navier--Stokes system with a tensor-valued chemotactic sensitivity $S(x,n,c)$. Under a signal-dependent power-decay condition $|S(x,n,c)| \le s_0 (s_1+c)^{-γ}$, we establish the global existence and uniform-in-time boundedness of classical solutions for both fluid-coupled ($γ> 1/2$) and fluid-free ($γ> 0$) systems. The proof relies on a sequence of localized energy estimates, including the $L^{2}_{\rm loc}$-smallness of the weighted gradient of the signal concentration, to overcome the mathematical difficulties arising from signal production and fluid transport. Furthermore, under specific structural assumptions on the sensitivity tensor, we prove that solutions of the fluid-free system converge exponentially to the spatially homogeneous steady state. To this end, we establish an interpolation inequality involving the Hölder norm, which is of independent interest and seems to have broad applications.

Keller-Segel-Navier-Stokes systems involving general sensitivities with Signal-Dependent Power-Law Decay

Abstract

This paper investigates a two-dimensional Keller--Segel--Navier--Stokes system with a tensor-valued chemotactic sensitivity . Under a signal-dependent power-decay condition , we establish the global existence and uniform-in-time boundedness of classical solutions for both fluid-coupled () and fluid-free () systems. The proof relies on a sequence of localized energy estimates, including the -smallness of the weighted gradient of the signal concentration, to overcome the mathematical difficulties arising from signal production and fluid transport. Furthermore, under specific structural assumptions on the sensitivity tensor, we prove that solutions of the fluid-free system converge exponentially to the spatially homogeneous steady state. To this end, we establish an interpolation inequality involving the Hölder norm, which is of independent interest and seems to have broad applications.
Paper Structure (4 sections, 28 theorems, 241 equations)

This paper contains 4 sections, 28 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Global Existence and Boundedness: Proof of Theorem \ref{['THM1']}
  4. Large--Time Behavior: Proof of Theorem \ref{['THM3']}

Key Result

Theorem 1.1

Let $\Omega\subset{ \mathbb{R} }^2$ be a bounded smooth domain. (i) Fluid-coupled case. Assume that $\gamma>\frac{1}{2}$ and $s_1>0$. Then KS01--KS02 admits a unique global classical solution $(n,c,u,\pi)$ such that and moreover In particular, there exists $C=C (\Omega,\alpha,s_0,s_1,\gamma,\int_\Omega n_0, \|n_0\|_{L^{\infty}(\Omega)},\|u_0\|_{L^2(\Omega)},\|A^\alpha u_0\|_{L^2(\Omega)},\|\nab

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 36 more