Global existence, smooth and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux

Abstract

We consider the spatially $3$-D version of the following Keller-Segel-Navier-Stokes system with rotational flux $$\{\begin{array}{l}{n}_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c),x\in \Omega ,t>0,c_t+u\cdot \nabla c=\Delta c-c+n,x\in \Omega ,t>0,\\ u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+n\nabla \varphi ,x\in \Omega ,t>0,\\ \nabla \cdot u=0,x\in \Omega ,t>0\end{array}(\ast )$$ under no-flux boundary conditions in a bounded domain $\Omega \subseteq {\mathbb{R}}^3$ with smooth boundary, where $\varphi \in W^{2,\infty }(\Omega )$ and $\kappa \in \mathbb{R}$ represent the prescribed gravitational potential and the strength of nonlinear fluid convection, respectively. Here the matrix-valued function $S(x,n,c)\in C^2(\bar{\Omega }\times [0,\infty {)}^2;{\mathbb{R}}^{3\times 3})$ denotes the rotational effect which satisfies $|S(x,n,c)|\le C_S(1+n{)}^{-\alpha }$ with some $C_S>0$ and $\alpha \ge 0$. In this paper, by seeking some new functionals and using the bootstrap arguments on system $(\ast )$, we establish the existence of global weak solutions to system $(\ast )$ for arbitrarily large initial data under the assumption $\alpha \ge 1$. Moreover, under an explicit condition on the size of $C_S$ relative to $C_N$, we can secondly prove that in fact any such {\bf weak} solution $(n,c,u)$ becomes smooth ultimately, and that it approaches the unique spatially homogeneous steady state $({\bar{n}}_0,{\bar{n}}_0,0)$, where ${\bar{n}}_0=\frac{1}{|\Omega |}{\int }_{\Omega }{n}_0$ and $C_N$ is the best Poincaré constant. To the best of our knowledge, there are the first results on asymptotic behavior of the system.

59. arXiv admin note: text overlap with arXiv:1907.11823; text overlap with arXiv:1506.05592 by other authors

References in corpus (1)

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