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

Abstract

This paper is concerned with the four-component Keller-Segel-Stokes system modelling the fertilization process of corals:subject to the boundary conditions ∇c · ν = ∇m · ν = (∇ρ − ρS(x, ρ, c)∇c) · ν = 0 and u = 0, and suitably regular initial data⊂ R 3 is a bounded domain with smooth boundary ∂Ω. This system describes the spatiotemporal dynamics of the population densities of sperm ρ and egg m under a chemotactic 1 process facilitated by a chemical signal released by the egg with concentration c in a fluidflow environment u modeled by the incompressible Stokes equation. In this model, the chemotactic sensitivity tensor S ∈ C 2 (Ω × [0, ∞) 2 ) 3×3 satisfies |S(x, ρ, c)| ≤ C S (1 + ρ) −αwith 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.

Global classical small-data solutions for a three-dimensional Keller--Segel--Navier--Stokes system modeling coral fertilization

“…It is noted that a similar result was proved in [18] for the three-dimensional Stoke variant of (1.4). However, as is well-known, the nonlinear convection (u • ∇)u in the three-dimensional Navier-Stokes equation may enforce the spontaneous emergence of singularities in the sense of blow-up with respect to the norm in L ∞ (Ω), we thereby subject the study of classical solutions of (1.4) to small initial data by an essentially one-step contradiction argument, unlike that in the two-dimensional case ( [9]).…”

“…Although the proof of Lemma 3.2 and Lemma 3.3 below is very similar to that of Lemma 3.11 and Lemma 3.12 in [18], respectively, we give their proofs for the convenience of the interested reader.…”

Existence and uniqueness of mild solutions to the chemotaxis-fluid system modeling coral fertilization

“…For example, Chae, Kyungkeun, and Lee [9] studied the global well-posedness of coral fertilization models. Li, Pang, and Wang [10] explored the global boundedness and decay property. Zheng [11] showed the global weak solution of this system in a three-dimensional space.…”

Large time behavior in a three-dimensional degenerate chemotaxis-Stokes system modeling coral fertilization

“…Specially, in spatially two-dimensional setting, it is proved that system (1.1) with κ = 1 possesses a unique classical solution which is globally bounded and approaches to some constant equilibrium as time goes to infinity ( [14]). While for physically most relevant three-dimensional case, the corresponding global solvability of (1.1) in the context of Stokes-fluid or Navier-Stokes-fluid requires some smallness hypothesis for initial data ( [22,27,7,18]), or necessary aids of some type of nonlinear mechanism, such as porous medium diffusion ( [26]), or signal-dependent sensitivity ( [21]), or saturation effects of cells ( [22,18,24,38]), or p-Laplace diffusion of cells ( [25]). Recently, from some refined models proposed in [1,5,6], it can be observed that the migration of cells rests with some gradient-dependent limitations, which inspires investigations of global dynamics on associated initial-boundary problems.…”