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

Abstract

This paper deals with the four-component Keller-Segel-Stokes model of coral fertilization

$ \left\{\begin{array}{l}n_{t}+u \cdot \nabla n=\Delta n^{m}-\nabla \cdot(n S(x, n, c) \cdot \nabla c)-n \rho, \\c_{t}+u \cdot \nabla c=\Delta c-c+\rho, \\\rho_{t}+u \cdot \nabla \rho=\Delta \rho-n \rho, \\u_{t}+\nabla P=\Delta u+(n+\rho) \nabla \phi, \quad \nabla \cdot u=0\end{array}\right.$

in a bounded and smooth domain $ \Omega \subset \mathbb{R}^{3}$ with zero-flux boundary for n, c, ρ and no-slip boundary for u, where m>0, ϕ∈W^2,∞(Ω), and S:$ \bar{\Omega} \times[0, \infty)^{2} \rightarrow \mathbb{R}^{3 \times 3}$ is given sufficiently smooth function such that $ |S(x, n, c)| \leq S_{0}(c)(n+1)^{-\alpha}$ for all $ (x, n, c) \in \bar{\Omega} \times[0, \infty)^{2}$ with α≥0 and some nondecreasing function $ S_{0}:[0, \infty) \mapsto[0, \infty)$. It is shown that if m>1−α for $ 0 \leq \alpha \leq \frac{2}{3}$, or $ m \geq \frac{1}{3}$ for $ \alpha>\frac{2}{3}$, then for any reasonably regular initial data, the corresponding initial-boundary value problem admits at least one globally bounded weak solution which stabilizes to the spatially homogeneous equilibrium (n∞,ρ∞,ρ∞,0) in an appropriate sense, where $ n_{\infty}:=\frac{1}{|\Omega|}\left\{\int_{\Omega} n_{0}-\int_{\Omega} \rho_{0}\right\}_{+}$ and $ \rho_{\infty}:=\frac{1}{|\Omega|}\left\{\int_{\Omega} \rho_{0}-\int_{\Omega} n_{0}\right\}_{+}$. These results improve and extend previously known ones.

Keywords

Keller-Segel-Stokes / nonlinear diffusion / tensor-valued sensitivity / boundedness / asymptotic behavior