Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2020-05-21 , DOI: 10.1007/s00033-020-01310-y Myowin Htwe , Peter Y. H. Pang , Yifu Wang
We are concerned with the Keller–Segel–Navier–Stokes system
$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \rho _t+u\cdot \nabla \rho =\Delta \rho -\nabla \cdot (\rho \mathcal {S}(x,\rho ,c)\nabla c)-\rho m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ m_t+u\cdot \nabla m=\Delta m-\rho m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ c_t+u\cdot \nabla c=\Delta c-c+m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ u_t+ (u\cdot \nabla ) u=\Delta u-\nabla P+(\rho +m)\nabla \phi ,\quad \nabla \cdot u=0, &{}\quad (x,t)\in \varOmega \times (0,T) \end{array}\right. \end{aligned}$$subject to the boundary condition \((\nabla \rho -\rho \mathcal {S}(x,\rho ,c)\nabla c)\cdot \nu =\nabla m\cdot \nu =\nabla c\cdot \nu =0, u=0\) in a bounded smooth domain \(\varOmega \subset \mathbb {R}^3\). It is shown that this problem admits a global classical solution with exponential decay properties when \(\mathcal {S}\in C^2(\overline{\varOmega }\times [0,\infty )^2)^{3\times 3}\) satisfies \(|\mathcal {S}(x,\rho ,c)|\le C_S \) for some \(C_S>0\), and the initial data satisfy certain smallness conditions.
中文翻译：
三维Keller-Segel-Navier-Stokes系统模拟珊瑚施肥的经典解的渐近行为
我们关注的是Keller–Segel–Navier–Stokes系统
$$ \ begin {aligned} \ left \ {\ begin {array} {l @ {\ quad} l} \ rho _t + u \ cdot \ nabla \ rho = \ Delta \ rho-\ nabla \ cdot（\ rho \数学{S}（x，\ rho，c）\ nabla c）-\ rho m，＆{} \ quad（x，t）\ in \ varOmega \ times（0，T），\\ m_t + u \ cdot \ nabla m = \ Delta m- \ rho m，＆{} \ quad（x，t）\ in \ varOmega \ times（0，T），\\ c_t + u \ cdot \ nabla c = \ Delta c-c + m，＆{} \ quad（x，t）\ in \ varOmega \ times（0，T），\\ u_t +（u \ cdot \ nabla）u = \ Delta u- \ nabla P +（\ rho + m） \ nabla \ phi，\ quad \ nabla \ cdot u = 0，＆{} \ quad（x，t）\ in \ varOmega \ times（0，T）\ end {array} \ right。\ end {aligned} $$服从边界条件\（（\ nabla \ rho-\ rho \ mathcal {S}（x，\ rho，c）\ nabla c）\ cdot \ nu = \ nabla m \ cdot \ nu = \ nabla c \ cdot \ nu = 0，u = 0 \）在有界光滑域\（\ varOmega \ subset \ mathbb {R} ^ 3 \）中。结果表明，当\（\ mathcal {S} \ in C ^ 2（\ overline {\ varOmega} \ times [0，\ infty）^ 2）^ {3 \时，该问题接受具有指数衰减性质的全局经典解。 3} \）满足\（| mathcal {S}（x，\ rho，c）| \ le C_S \）对于某些\（C_S> 0 \），并且初始数据满足某些较小性条件。
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