Very mild diffusion enhancement and singular sensitivity: Existence of bounded weak solutions in a two-dimensional chemotaxis-Navier--Stokes system

TL;DR

This paper proves the existence of global bounded weak solutions for a two-dimensional chemotaxis-Navier--Stokes system with mild diffusion enhancement and singular sensitivity, under specific conditions on the diffusion coefficient and initial data.

Contribution

It establishes the existence of global bounded weak solutions in 2D for a chemotaxis-Navier--Stokes system with mild diffusion and singular sensitivity, extending previous results.

Findings

Existence of global bounded weak solutions under certain diffusion conditions.

Solutions are classical if diffusion coefficient is positive at zero.

Results apply to a range of initial data and sensitivity functions.

Abstract

We consider an initial-boundary value problem for the chemotaxis-Navier--Stokes system \begin{align*} \left\{ \begin{array}{c@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\nabla n=\nabla\cdot\big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c\big),\ &x\in\Omega,& t>0,\\ c_{t}+u\cdot\nabla c=\Delta c-cn,\ &x\in\Omega,& t>0,\\ u_{t}+(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\Phi,\quad \nabla\cdot u=0,\ &x\in\Omega,& t>0,\\ \big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c)\cdot\nu=\nabla c\cdot\nu=0,\ u=0,\ &x\in\partial\Omega,& t>0,\\ n(\cdot,0)=n_0,\ c(\cdot,0)=c_0,\ u(\cdot,0)=u_0,\ &x\in\Omega. \end{array}\right. \end{align*} in a smoothly bounded domain $\Omega\subset\mathbb{R}^2$. Assuming $S:\overline{\Omega}\times[0,\infty)\times(0,\infty)\rightarrow \mathbb{R}^{2\times 2}$ to be sufficiently regular and such that with $\gamma\in[0,\frac56]$ and some non-decreasing $S_0:(0,\infty)\to(0,\infty)$, we have \begin{align*} \big|S(x,n,c)\big|\leq \frac{S_0(c)}{c^\gamma}\quad\text{for all }(x,n,c)\in\overline{\Omega}\times[0,\infty)\times(0,\infty), \end{align*} we show that if $D:[0,\infty)\to[0,\infty)$ is suitably regular and positive throughout $(0,\infty)$, then for all $M>0$ one can find $L(M)>0$ such that whenever $$\liminf_{n\to\infty} D(n)>L\quad\text{and}\quad \liminf_{n\searrow0}\frac{D(n)}{n}>0$$ are satisfied and the initial data $(n_0,c_0,u_0)$ are suitably regular and satisfy $\|c_0\|_{L^{\infty}(\Omega)}\leq M$ there is a global and bounded weak solution for the initial-boundary value problem above. Under the additional assumption of $D(0)>0$ this solution is moreover a classical solution of the same problem.