Blow-up prevention by nonlinear diffusion in a 2D Keller-Segel-Navier-Stokes system with rotational flux

Abstract

This paper investigates the following Keller-Segel-Navier-Stokes system with nonlinear diffusion and rotational flux $$\begin{array}{r}\{\begin{array}{llc}& n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\nabla c),& x\in \Omega ,t>0,\\ & c_t+u\cdot \nabla c=\Delta c-c+n,& x\in \Omega ,t>0,\\ & u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+n\nabla \varphi ,& x\in \Omega ,t>0,\\ & \nabla \cdot u=0,& x\in \Omega ,t>0,\end{array}\end{array}$$ where $\kappa \in \mathbb{R},\varphi \in W^{2,\infty }(\Omega )$ and $S$ is a given function with values in ${\mathbb{R}}^{2\times 2}$ which fulfills $$|S(x,n,c)|\le C_S$$ with some $C_S>0$. Systems of this type describe chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells. If $m>1$ and $\Omega \subset {\mathbb{R}}^2$ is a {\bf bounded} domain with smooth boundary, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a global and bounded (weak) solution, which significantly improves previous results of several authors. Moreover, the {\bf optimal condition} on the parameter $m$ for global existence is obtained. Our approach underlying the derivation of main result is based on an entropy-like estimate involving the functional %Our main tool is consideration of the energy functional $${\int }_{\Omega }(n_{\epsilon }+\epsilon {)}^m+{\int }_{\Omega }|\nabla c_{\epsilon }{|}^2,$$ where $n_{\epsilon }$ and $c_{\epsilon }$ are components of the solutions to (2.1) below.

30 pages. arXiv admin note: text overlap with arXiv:1903.01033