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

TL;DR

This paper proves the global existence, boundedness, and exponential convergence of solutions for a Keller-Segel-Navier-Stokes system with signal-dependent decay sensitivities, using localized energy estimates and interpolation inequalities.

Contribution

It introduces new analytical techniques to handle general sensitivities with power-law decay in chemotaxis-fluid models, including exponential convergence results.

Findings

Global existence and boundedness of solutions for $ ext{system}$

Exponential convergence to steady state in fluid-free case

Development of an interpolation inequality involving the Hölder norm

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 $|S(x,n,c)| \le s_0 (s_1+c)^{-\gamma}$, we establish the global existence and uniform-in-time boundedness of classical solutions for both fluid-coupled ($\gamma > 1/2$) and fluid-free ($\gamma > 0$) 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 H\"older norm, which is of independent interest and seems to have broad applications.