Boundedness and Large Time Behavior in a Two-Dimensional Keller-Segel-Navier-Stokes System with Nonlinear Diffusion Modeling Coral Fertilization

Abstract

This paper deals with a two-dimensional Keller-Segel-Navier-Stokes system of coral fertilization with porous medium diffusion. When the nonlinear diffusion exponents of sperms and eggs \(m>\frac{1}{2}\), \(l>0\) respectively, as well as the strength of fertilization \(\mu >0\), the system possesses a global bounded weak solution. Furthermore, if \(0< l<1\), the corresponding global weak solution stabilizes to the spatially homogeneous equilibrium \((n_{\infty },\rho _{\infty },\rho _{\infty },0)\) in an appropriate sense, where \(n_{\infty }:=\frac{1}{|\Omega |}(\int _{\Omega }n_{0}-\int _{\Omega } \rho _{0})_{+}\) and \(\rho _{\infty }:=\frac{1}{|\Omega |}(\int _{\Omega }\rho _{0}-\int _{ \Omega }n_{0})_{+}\).