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: $$n_t+u\\cdot \\nabla n=\\Delta n-\\nabla \\cdot (nS(x,n,c)\\nabla c)-nm$$\n , $$c_t+u\\cdot \\nabla c=\\Delta c-c+m$$\n , $$m_t+u\\cdot \\nabla m=\\Delta m-nm$$\n , $$u_t+\\kappa (u\\cdot \\nabla )u+\\nabla P=\\Delta u+(n+m)\\nabla \\phi $$\n and $$\\nabla \\cdot u=0$$\n in a bounded and smooth domain $$\\Omega \\subset \\mathbb {R}^2$$\n , where $$\\kappa \\in \\mathbb {R}$$\n , $$\\phi \\in W^{2,\\infty }(\\Omega )$$\n , and $$S\\in C^2({\\bar{\\Omega }}\\times [0,\\infty )^2;\\mathbb {R}^{2\\times 2})$$\n satisfies $$|S(x,n,c)|\\le S_0(c)(1+n)^{-\\alpha }$$\n for all $$(x,n,c)\\in {\\bar{\\Omega }}\\times [0,\\infty )^2$$\n with $$\\alpha \\in \\mathbb {R}$$\n and the function $$S_0:[0,\\infty )\\rightarrow [0,\\infty )$$\n nondecreasing. Under the relatively weak destabilizing action of cross-diffusion for $$\\alpha \\ge 0$$\n , 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 $$-\\frac{1}{2}<\\alpha <0$$\n bears a superlinear growth of n, the corresponding initial-boundary value problem (with any $$\\kappa \\in \\mathbb {R}$$\n ) still possesses a global classical solution emanating from any suitably smooth initial data. Moreover, when $$\\kappa =0$$\n , this solution is globally bounded.