Global existence to a $3D$ chemotaxis-Navier-stokes system with nonlinear diffusion and rotation

Abstract

This paper is concerned with the following quasilinear chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-nc,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.\eqno(CNF) $$ is considered under the no-flux boundary conditions for $n,c$ and the Dirichlet boundary condition for $u$ in a three-dimensional convex domain $\Omega \subseteq {\mathbb{R}}^3$ with smooth boundary, which describes the motion of oxygen-driven bacteria in a fluid. Here % $\Omega \subseteq {\mathbb{R}}^3$ is a , $\kappa \in \mathbb{R}$ and $S$ denotes the strength of nonlinear fluid convection and a given tensor-valued function, respectively. Assume $m>\frac{10}{9}$ and $S$ fulfills $|S(x,n,c)|\le S_0(c)$ for all $(x,n,c)\in \bar{\Omega }\times [0,\infty )\times [0,\infty )$ with $S_0(c)$ nondecreasing on $[0,\infty )$, then for any reasonably regular initial data, the corresponding initial-boundary problem $(CNF)$ admits at least one global weak solution.

References in corpus (1)

• A new approach toward locally bounded global solutions to a $3D$ chemotaxis-stokes system with nonlinear diffusion and rotation