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Keywords:
Keller-Segel-Navier-Stokes; global solution; decay estimate; indirect process
Keller-Segel-Navier-Stokes; global solution; decay estimate; indirect process
Summary:
We consider the Keller-Segel-Navier-Stokes system $$ \begin{cases} n_t+{\bf u}\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla v ),& x\in \Omega ,\ t>0,\\ v_t +{\bf u}\cdot \nabla v=\Delta v -v+w, &x\in \Omega ,\ t>0,\\ w_t+{\bf u}\cdot \nabla w=\Delta w -w+n, &x\in \Omega ,\ t>0,\\ {\bf {u}}_t + ({\bf {u}}\cdot \nabla ){\bf {u}} = \Delta {\bf {u}} + \nabla P + n\nabla \phi ,\ \nabla \cdot {\bf u}=0, &x\in \Omega ,\ t>0, \end{cases} $$ which is considered in bounded domain $\Omega \subset \mathbb {R}^N$ $(N \in \{2,3\})$ with smooth boundary, where $\phi \in C^{1+\delta }(\overline \Omega )$ with $\delta \in (0,1)$. We show that if the initial data $\|n_0\|_{L^{{N}/{2}}(\Omega )}$, $\|\nabla v_0\|_{L^N(\Omega )}$, $\|\nabla w_0\|_{L^N(\Omega )}$ and $\|{\bf u}_0\|_{L^N(\Omega )}$ is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state $({\bar n}_0,{\bar n}_0,{\bar n}_0,0)$ exponentially with ${\bar n}_0:=(1/|\Omega |)\int _{\Omega }n_0(x){\rm d}x$.
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