On a three-dimensional chemotaxis-Stokes system with nonlinear sensitivity modeling coral fertilization

This paper is concerned with the effects of nonlinear sensitivity on boundedness of solutions for the following chemotaxis-Stokes system {n(t) + u . del n = del . (del n - S(n)del c) - nm, (x, t) is an element of Omega x (0, infinity), c(t) + u . del c = Delta c - c + m, (x, t) is an element of Omega x (0, infinity), m(t) + u . del m = Delta m - mn, (x, t) is an element of Omega x (0, infinity) u(t) = Delta u + del P + (n + m)del phi, (x, t) is an element of Omega x (0, infinity), del . u = 0, (x, t) is an element of Omega x (0, infinity), in a smoothly bounded domain Omega subset of R-3 under no-flux boundary conditions for n, c, m and no-slip boundary conditions for u, where n and m denote the densities of unfertilized sperms and eggs, respectively, c stands for the concentration of the signal, u represents the velocity of fluid, P is the pressure within the fluid and phi is the gravitational potential. This system describes the process of coral fertilization occurring in ocean flow. Based on the novel conditional estimates for c and u, it is proved that for all appropriately regular nonnegative initial data, this system possesses a unique globally hounded solution provided that S is an element of C-2 ([0, infinity)) satisfies S(n) <= chi n(n + 1)(alpha-1) with chi > 0 and alpha < 1, which improves the known subcritical exponent alpha < 2/3 under fluid-free case.