Boundedness and Asymptotic Behavior in a 3D Keller-Segel-Stokes System Modeling Coral Fertilization with Nonlinear Diffusion and Rotation

This paper dealswith the four-component Keller-Segel-Stokesmodel of coral fertilization {nt(+) u .del n= Delta n(m) - del. (nS(x, n,c) .del c)-n rho, c(t)vertical bar u.del c= Delta c- c vertical bar rho, rho t+u.del rho = Delta rho - n rho, u(t)+del P= Delta u+(n+rho)del phi, del.u= 0 in a bounded and smooth domain Omega subset of R-3 with zero-flux boundary for n, c, rho and no-slip boundary for u, where m>0, phi is an element of W-2,W-infinity (Omega), and S:(Omega) over bar x[0, infinity)(2) -> R-3x3 is given sufficiently smooth function such that vertical bar S(x, n, c)vertical bar <= S-0(c)(n+ 1)(-alpha) for all (x,n, c). is an element of(Omega) over bar x [0, infinity)(2) with alpha >= 0 and some nondecreasing function S-0 : [0, infinity) ->[0, infinity). It is shown that if m > 1-alpha for 0 <=alpha <= 2/3, or m >= 1/3 for alpha>2/3, then for any reasonably regular initial data, the corresponding initial-boundary value problemadmits at least one globally boundedweak solution which stabilizes to the spatially homogeneous equilibrium (n(infinity), rho(infinity), rho(infinity),0) in an appropriate sense, where n(infinity) := 1/vertical bar Omega vertical bar {integral(Omega)n(0)- integral(Omega)rho(0)}(+) and rho(infinity): -1/vertical bar Omega vertical bar{integral(Omega)rho(0)- integral(Omega)n(0)}(+). These results improve and extend previously known ones.