Some further progress for existence and boundedness of solutions to a two-dimensional chemotaxis-(Navier-)Stokes system modeling coral fertilization

TL;DR

This paper establishes conditions under which a complex two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization admits globally existing and bounded solutions, advancing understanding of such coupled biological-fluid systems.

Contribution

It proves global existence and boundedness of solutions for a chemotaxis-Navier-Stokes system under new parameter conditions, extending prior results in the field.

Findings

Global classical solutions exist under specified conditions.

Solutions are unique and uniformly bounded for stronger assumptions.

The results apply to models of coral fertilization in ocean flow.

Abstract

In this paper, we investigate the effects exerted by the interplay among Laplacian diffusion, chemotaxis cross diffusion and the fluid dynamic mechanism on global existence and boundedness of the solutions. The mathematical model considered herein appears as \begin{align}\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot( nS(n)\nabla c)-nm,\quad x\in \Omega, t>0, \disp{ c_{ t}+u\cdot\nabla c=\Delta c-c+w},\quad x\in \Omega, t>0, \disp{w_{t}+u\cdot\nabla w=\Delta w-nw},\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0,\\ \end{array}\right.\eqno(KSNF) \end{align} in a bounded domain $\Omega\subset \mathbb{R}^2$ with a smooth boundary, which describes the process of coral fertilization occurring in ocean flow. Here $\kappa\in \mathbb{R}$ is a given constant, $\phi\in W^{2,\infty}(\Omega)$and $S(n) $ is a scalar function satisfies $|S(n)|\leq C_S(1+n)^{-\alpha}$ {for all} $n\geq 0$ with some $C_S>0$ and $\alpha\in\mathbb{R}$. It is proved that if either $\alpha>-1,\kappa=0$ or $\alpha\geq-\frac{1}{2},\kappa\in\mathbb{R}$ is satisfied,then for any reasonably smooth initial data, the corresponding Neumann-Neumann-Neumann-Dirichlet initial-boundary problem $(KSNF)$ possesses a globally classical solution. In case of the stronger assumption $\alpha>-1,\kappa = 0$ or $\alpha>-\frac{1}{2},\kappa \in\mathbb{R},$ we moreover show that the corresponding initial-boundary problem admits a unique global classical solution which is uniformly bounded on $\Omega\times(0,\infty)$.