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

TL;DR

This paper proves the global existence of weak solutions for a complex 3D chemotaxis-Navier-Stokes system with nonlinear diffusion and rotation, modeling oxygen-driven bacteria in fluid dynamics.

Contribution

It establishes the existence of global weak solutions for a 3D chemotaxis-Navier-Stokes system with nonlinear diffusion and rotation under certain conditions.

Findings

Global weak solutions exist for the system under specified conditions.

The results apply to reasonably regular initial data.

The analysis handles nonlinear diffusion and fluid rotation effects.

Abstract

This paper is concerned with the following quasilinear chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-nc,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+n\nabla \phi ,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array}\right.\eqno(CNF) $$ 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 $|S(x,n,c)| \leq S_0(c)$ 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.