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

Abstract

This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation $$\{\begin{array}{l}{n}_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c),x\in \Omega ,t>0,c_t+u\cdot \nabla c=\Delta c-c+n,x\in \Omega ,t>0,\\ u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+n\nabla \varphi ,x\in \Omega ,t>0,\\ \nabla \cdot u=0,x\in \Omega ,t>0,\end{array}$$ which describes chemotaxis-fluid interaction in cases when the evolution of the chemoattractant is essentially dominated by production through cells, 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 $$|S(x,n,c)|\le C_S(1+n{)}^{-\alpha }$$ with some $C_S>0$ and $\alpha \ge 0.$ If $\alpha >0$ and $\Omega \subseteq {\mathbb{R}}^2$ 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 classical solution which is bounded on $\Omega \times (0,\infty )$. This extends a recent result by Wang-Winkler-Xiang (Annali della Scuola Normale Superiore di Pisa-Classe di Scienze. XVIII, (2018), 2036--2145) which asserts global existence of bounded solutions under the constraint $\Omega \subseteq {\mathbb{R}}^2$ is a bounded {\bf convex domain} with smooth boundary. Moreover, we shall improve the result of Wang-Xiang (J. Diff. Eqns., 259(2015), 7578--7609), who proved the possibility of global and bounded, in the case that $\kappa \equiv 0$ and $\alpha >0$. In comparison to the result for the corresponding fluid-free system, the {\bf optimal condition} on the parameter $\alpha$ for both {\bf global existence} and {\bf boundedness} are obtained.

37

References in corpus (1)

• New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models

Cited by in corpus (1)

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