An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion

Abstract

This paper investigates the following quasilinear Keller-Segel-Navier-Stokes system $$\{\begin{array}{l}{n}_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (n\nabla c),x\in \Omega ,t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+n,x\in \Omega ,t>0,\\ u_t+\nabla P=\Delta u+n\nabla \varphi ,x\in \Omega ,t>0,\\ \nabla \cdot u=0,x\in \Omega ,t>0\end{array}$$ under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a three-dimensional bounded domains $\Omega \subseteq {\mathbb{R}}^3$ with smooth boundary, where $\varphi \in W^{1,\infty }(\Omega ),m>0$. It is proved that if $m>\frac{4}{3}$, then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system $(KSF)$, which in view of the known results for the fluid-free system mentioned below (see Introduction) is an optimal restriction on $m$.

arXiv admin note: text overlap with arXiv:1806.07067

References in corpus (2)

• Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions
• New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models

Cited by in corpus (1)

• An optimal result for global classical and bounded solutions in a two-dimensional Keller-Segel-Navier-Stokes system with sensitivity