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 $$\begin{align}\begin{cases} &n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(nS(x, n, c)\nabla c),\quad &x\in \Omega, t>0, \\ &c_t+u\cdot\nabla c=\Delta c-c+n,\quad &x\in \Omega, t>0, \\ &u_t+\kappa (u\cdot\nabla)u+\nabla P=\Delta u+n\nabla \phi,\quad &x\in \Omega, t>0, \\ &\nabla\cdot u=0,\quad &x\in \Omega, t>0, \end{cases}\end{align}$$ where $\kappa\in \mathbb{R},\phi\in W^{2,\infty}(\Omega)$ and $S$ is a given function with values in $\mathbb{R}^{2\times2}$ which fulfills $$ |S(x,n,c)| \leq C_S $$ 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 $$\int_{\Omega}(n_{\varepsilon} +\varepsilon)^{m}+\int_{\Omega}|\nabla c_\varepsilon|^{2},$$ where $n_\varepsilon$ and $c_\varepsilon$ are components of the solutions to (2.1) below.