Global existence and boundedness of weak solutions to a chemotaxis-stokes system with rotational flux term

TL;DR

This paper proves the global existence and boundedness of weak solutions to a three-dimensional chemotaxis-Stokes system with rotational flux, under specific conditions on the chemotactic sensitivity and the nonlinear diffusion exponent.

Contribution

It establishes new conditions on the diffusion exponent ensuring global bounded weak solutions for a complex chemotaxis-fluid model.

Findings

Weak solutions are global in time and bounded for certain parameter ranges.

Conditions on the diffusion exponent m relative to l ensure boundedness.

The results extend understanding of chemotaxis-fluid interactions in 3D.

Abstract

In this paper, the three-dimensional chemotaxis-stokes system \begin{eqnarray*} \left\{\begin{array}{lll} \medskip n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n S(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0, \medskip c_t+u\cdot\nabla c=\Delta c-nf(c),&x\in\Omega,\ \ t>0, \medskip u_t+\nabla P=\Delta u +n\nabla\phi,&x\in\Omega,\ \ t>0, \nabla\cdot u=0, &x\in\Omega,\ \ t>0,, \end{array}\right. \end{eqnarray*} posed in a bounded domain $\Omega\subset\mathbb{R}^3$ with smooth boundary is considered under the no-flux boundary condition for $n$, $c$ and the Dirichlect boundary condition for $u$ under the assumption that the Frobenius norm of the tensor-valued chemotactic sensitivity $S(x,n,c)$ satisfies $S(x,n,c)<n^{l-2}\widetilde{S}(c)$ with $l>2$ for some non-decreasing function $\widetilde{S}\in C^{2}((0,\infty))$. In present work, it is shown that the weak solution is global in time and bounded while $m>m^\star(l)$, where \begin{eqnarray*} m^\star(l)= \left\{\begin{array}{lll} \medskip l-\frac{5}{6},\ &\mathrm{if}\ \frac{31}{12}\geq l>2, \medskip \frac{7}{5}l-\frac{28}{15},\ &\mathrm{if}\ l>\frac{31}{12}. \end{array}\right. \end{eqnarray*}