Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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: \begin{equation*} \left\{
\begin{array}{ll}
\rho_t+u\cdot\nabla\rho=\Delta\rho-\nabla\cdot(\rho\mathcal{S}(x,\rho,c)\nabla
c)-\rho m, & \quad (x,t)\in \Omega\times (0,T), \\ m_t+u\cdot\nabla m=\Delta
m-\rho m, & \quad (x,t)\in \Omega\times (0,T), \\ c_t+u\cdot\nabla c=\Delta
c-c+m, & \quad (x,t)\in \Omega\times (0,T), \\ u_t=\Delta u-\nabla
P+(\rho+m)\nabla\phi,\quad \nabla\cdot u=0, & \quad (x,t)\in \Omega\times (0,T)
\end{array}\right. \end{equation*} subject to the boundary conditions $\nabla
c\cdot \nu=\nabla m\cdot \nu=(\nabla\rho-\rho \mathcal{S}(x,\rho,c)\nabla
c)\cdot \nu=0$ and $u=0$, and suitably regular initial data $(\rho_0(x),m_0(x),
c_0(x),u_0(x))$, where $T\in (0,\infty]$, $\Omega\subset\mathbb R^3$ is a
bounded domain with smooth boundary $\partial\Omega$. This system describes the
spatio-temporal dynamics of the population densities of sperm $\rho$ 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 $\mathcal{S}\in C^2(\overline\Omega\times [0,\infty)^2)^{3\times 3}$
satisfies $|\mathcal{S}(x,\rho,c)|\leq C_S(1+\rho)^{-\alpha}$ with some $C_S>0$
and $\alpha\geq 0$. We will show that for $\alpha\geq \frac 13$, 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 $\alpha\geq 0$, a similar result is valid when the initial
data satisfy a certain smallness condition.

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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci