To what extent is cross-diffusion controllable in a two-dimensional chemotaxis-(Navier–)Stokes system modeling coral fertilization

Abstract

We study the chemotaxis-(Navier–)Stokes system modeling coral fertilization: $n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c)-nm$, $c_t+u\cdot \nabla c=\Delta c-c+m$, $m_t+u\cdot \nabla m=\Delta m-nm$, $u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \phi$ and $\nabla \cdot u=0$ in a bounded and smooth domain $\Omega \subset \mathbb {R}^2$, where $\kappa \in \mathbb {R}$, $\phi \in W^{2,\infty }(\Omega )$, and $S\in C^2({\bar{\Omega }}\times [0,\infty )^2;\mathbb {R}^{2\times 2})$ satisfies $|S(x,n,c)|\le S_0(c)(1+n)^{-\alpha }$ for all $(x,n,c)\in {\bar{\Omega }}\times [0,\infty )^2$ with $\alpha \in \mathbb {R}$ and the function $S_0:[0,\infty )\rightarrow [0,\infty )$ nondecreasing. Under the relatively weak destabilizing action of cross-diffusion for $\alpha \ge 0$, the global boundedness of classical solutions was obtained in Espejo and Winkler (Nonlinearity 31:1227–1259, 2018) and Li (Differ Equ 267:6290–6315, 2019). In this paper, we show that even if n|S| with $-\frac{1}{2}<\alpha <0$ bears a superlinear growth of n, the corresponding initial-boundary value problem (with any $\kappa \in \mathbb {R}$) still possesses a global classical solution emanating from any suitably smooth initial data. Moreover, when $\kappa =0$, this solution is globally bounded.