Global boundedness and decay property of a three-dimensional Keller-Segel-Stokes system modeling coral fertilization

This paper is concerned with the four-component Keller-Segel-Stokes system modelling the fertilization process of corals: {ρ_t + u · ∇ρ = Δρ - ∇ · (ρS(x, ρ, c)∇c) - ρm, (x, t) ∈ Ω × (0, T), m_t + u · ∇m = Δm - ρm, (x, t) ∈ Ω × (0, T), c_t + u · ∇c = Δc - c + m, (x, t) ∈ Ω × (0, T), u_t = Δu-∇P + (ρ + m)∇,φ, ∇ · u = 0, (x, t) ∈ Ω × (0, T) subject to the boundary conditions ∇c · _ = ∇m · = (∇ρ - ρS(x, ρ, c)∇c) · = 0 and u = 0, and suitably regular initial data (ρ0(x),m0(x), c0(x), u0(x)), where T ∈ (0,∞], Ω ∈ ℝ³ is a bounded domain with smooth boundary Ω. This system describes the spatio-temporal dynamics of the population densities of sperm ρ and egg m under a chemotactic process facilitated by a chemical signal released by the egg with concentration c in a fluid-flow environment u modeled by the incompressible Stokes equation. In this model, the chemotactic sensitivity tensor S ∈ C²(Ω × [0,∈)²)³×³ satisfies |S(x, ρ, c)| ≤ C_S (1 + ρ)^-α with some C_S > 0 and α ≥ 0. We will show that for α ≥ 1/3 , the solutions to the system are globally bounded and decay to a spatially homogeneous equilibrium exponentially as time goes to infinity. In addition, we will also show that, for any α ≥ 0, a similar result is valid when the initial data satisfy a certain smallness condition.