Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion

TL;DR

This paper proves the existence of global weak solutions for a three-dimensional chemotaxis-Navier-Stokes system with slow p-Laplacian diffusion, under certain conditions on parameters and initial data.

Contribution

It establishes the first global weak solution existence result for this complex chemotaxis-Navier-Stokes model with slow diffusion in three dimensions.

Findings

Global weak solutions exist for p > 32/15.

Solutions are valid for smooth initial data.

The results depend on structural assumptions on f and χ.

Abstract

This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion \begin{eqnarray} \left\{\begin{array}{lll} n_t+u\cdot\nabla n=\nabla\cdot(|\nabla n|^{p-2}\nabla n)-\nabla\cdot(n\chi(c)\nabla c),& x\in\Omega,\ t>0, c_t+u\cdot\nabla c=\Delta c-nf(c),& x\in\Omega,\ t>0, u_t+(u\cdot\nabla) u=\Delta u+\nabla P+n\nabla\Phi,& x\in\Omega,\ t>0, \nabla\cdot u=0,& x\in\Omega,\ t>0 \end{array}\right. \end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $\Omega\subset \mathbb{R}^3$ with smooth boundary. Here, $\Phi\in W^{1,\infty}(\Omega)$, $0<\chi\in C^2([0,\infty))$ and $0\leq f\in C^1([0,\infty))$ with $f(0)=0$. It is proved that if $p>\frac{32}{15}$ and under appropriate structural assumptions on $f$ and $\chi$, for all sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least one global weak solution.