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Influence of Flux Limitation on Large Time Behavior in a Three-Dimensional Chemotaxis-Stokes System Modeling Coral Fertilization
Acta Applicandae Mathematicae ( IF 1.6 ) Pub Date : 2021-08-06 , DOI: 10.1007/s10440-021-00427-0
Ji Liu 1
Affiliation  

In this paper, we consider the following system

$$ \left \{ \textstyle\begin{array}{ll} n_{t}+u\cdot \nabla n&=\Delta n-\nabla \cdot (n\mathcal{S}(|\nabla c|^{2}) \nabla c)-nm, \\ c_{t}+u\cdot \nabla c&=\Delta c-c+m, \\ m_{t}+u\cdot \nabla m&=\Delta m-mn, \\ u_{t}&=\Delta u+\nabla P+(n+m)\nabla \Phi ,\qquad \nabla \cdot u=0 \end{array}\displaystyle \right . $$

which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where \(\mathcal{S}\) is a given function fulfilling

$$ |\mathcal{S}(\sigma )|\leq K_{\mathcal{S}}(1+\sigma )^{- \frac{\theta }{2}},\qquad \sigma \geq 0 $$

with some \(K_{\mathcal{S}}>0\). Based on conditional estimates of the quantity \(c\) and the gradients thereof, a relatively compressed argument as compared to that proceeding in related precedents shows that if

$$ \theta >0, $$

then for any initial data with proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded, and which also enjoys the stabilization features in the sense that

$$\begin{aligned} &\|n(\cdot ,t)-n_{\infty }\|_{L^{\infty }(\Omega )}+\|c(\cdot ,t)-m_{ \infty }\|_{W^{1,\infty }(\Omega )} +\|m(\cdot ,t)-m_{\infty }\|_{W^{1, \infty }(\Omega )}\\ &\quad{}+\|u(\cdot ,t)\|_{L^{\infty }(\Omega )}\rightarrow 0 \quad \text{as}~t\rightarrow \infty \end{aligned}$$

with \(n_{\infty }:=\frac{1}{|\Omega |}\left \{ \int _{\Omega }n_{0}-\int _{ \Omega }m_{0}\right \} _{+}\) and \(m_{\infty }:=\frac{1}{|\Omega |}\left \{ \int _{\Omega }m_{0}-\int _{ \Omega }n_{0}\right \} _{+}\).



中文翻译:

在三维趋化-斯托克斯系统建模珊瑚施肥中通量限制对长时间行为的影响

在本文中,我们考虑以下系统

$$ \left \{ \textstyle\begin{array}{ll} n_{t}+u\cdot \nabla n&=\Delta n-\nabla \cdot (n\mathcal{S}(|\nabla c|^ {2}) \nabla c)-nm, \\ c_{t}+u\cdot \nabla c&=\Delta c-c+m, \\ m_{t}+u\cdot \nabla m&=\Delta m -mn, \\ u_{t}&=\Delta u+\nabla P+(n+m)\nabla \Phi ,\qquad \nabla \cdot u=0 \end{array}\displaystyle \right . $$

它模拟了珊瑚受精过程,在一个平滑的三维有界域中,其中\(\mathcal{S}\)是一个给定的函数,满足

$$ |\mathcal{S}(\sigma )|\leq K_{\mathcal{S}}(1+\sigma )^{- \frac{\theta }{2}},\qquad \sigma \geq 0 $$

一些\(K_{\mathcal{S}}>0\)。基于对数量\(c\)及其梯度的条件估计,与相关先例中的程序相比,相对压缩的论证表明,如果

$$ \theta >0, $$

那么对于任何具有适当规律性的初始数据,在无通量/无通量/无通量/狄利克雷边界条件下的相关初始边界问题承认一个独特的全局有界的经典解,并且在某种意义上也享有稳定特征那

$$\begin{aligned} &\|n(\cdot ,t)-n_{\infty }\|_{L^{\infty }(\Omega )}+\|c(\cdot ,t)-m_ { \infty }\|_{W^{1,\infty }(\Omega )} +\|m(\cdot ,t)-m_{\infty }\|_{W^{1, \infty }( \Omega )}\\ &\quad{}+\|u(\cdot ,t)\|_{L^{\infty }(\Omega )}\rightarrow 0 \quad \text{as}~t\rightarrow \infty \end{对齐}$$

\(n_{\infty }:=\frac{1}{|\Omega |}\left \{ \int _{\Omega }n_{0}-\int _{ \Omega }m_{0}\right \} _{+}\)\(m_{\infty }:=\frac{1}{|\Omega |}\left \{ \int _{\Omega }m_{0}-\int _{ \欧米茄 }n_{0}\right \} _{+}\) .

更新日期:2021-08-10
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