Boundedness in a Chemotaxis-(Navier–)Stokes System Modeling Coral Fertilization with Slow p-Laplacian Diffusion

Abstract

In this paper, we investigate the effects exerted by the interplay among p-Laplacian diffusion, chemotaxis cross diffusion and the fluid dynamic mechanism on global boundedness of the solutions. The mathematical model considered herein appears as

$$\begin{aligned} \left\{ \begin{array}{llllll} \rho _t+u\cdot \nabla \rho =\nabla \cdot (|\nabla \rho |^{p-2}\nabla \rho )-\nabla \cdot (\rho \nabla c)-\rho m,&{}\quad x\in \Omega ,\quad ~t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+m,&{}\quad x\in \Omega ,\quad ~t>0,\\ m_t+u\cdot \nabla m=\Delta m-\rho m,&{}\quad x\in \Omega ,\quad ~t>0,\\ u_t+(u\cdot \nabla )u=\Delta u-\nabla P+(\rho +m)\nabla \phi ,&{}\quad x\in \Omega ,\quad ~t>0,\\ \nabla \cdot u=0,&{}\quad x\in \Omega ,\quad ~t>0, \end{array}\right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N~(N=2,3)$ is a general bounded domain with smooth boundary. It is proved that if either

$$\begin{aligned} p>2 \end{aligned}$$

for $\kappa \in {\mathbb {R}},N=2$ or

$$\begin{aligned} p>\frac{94}{45} \end{aligned}$$

for $\kappa =0,N=3$ is satisfied, then for each properly chosen initial data an associated initial-boundary problem admits a global weak solution which is bounded.