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Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system. (arXiv:1705.06131v1 [math.AP])

来源于:arXiv
We study the chemotaxis-fluid system \begin{align*} \left\{ \begin{array}{r@{\,}c@{\,}c@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}&+&u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(\frac{n}{c}\nabla c),\ &x\in\Omega,& t>0, c_{t}&+&u\cdot\!\nabla c&=\Delta c-nc,\ &x\in\Omega,& t>0, u_{t}&+&\nabla P&=\Delta u+n\nabla\phi,\ &x\in\Omega,& t>0, &&\nabla\cdot u&=0,\ &x\in\Omega,& t>0, \end{array}\right. \end{align*} under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$, where $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary and $\phi\in C^{2}\left(\bar{\Omega}\right)$. From recent results it is known that for suitable regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution. We will show that for small initial mass $\int_{\Omega}\!n_0$ these generalized solutions will eventually b 查看全文>>