Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production

Abstract

We consider the Keller-Segel-Navier-Stokes system

$$\left\{ {\begin{array}{*{20}{c}} {{n_t} + {\bf{u}} \cdot \nabla n = \Delta n - \nabla \cdot (n\nabla v),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in \Omega ,\;t > 0,} \\ {{v_t} + {\bf{u}} \cdot \nabla v = \Delta v - v + w,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in \Omega ,\;t > 0,} \\ {{w_t} + {\bf{u}} \cdot \nabla w = \Delta w - w + n,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in \Omega ,\;t > 0,} \\ {{{\bf{u}}_t} + ({\bf{u}} \cdot \nabla ){\bf{u}} = \Delta {\bf{u}} + \nabla P + n\nabla \phi ,\;\nabla \cdot {\bf{u}} = 0,}&{x \in \Omega ,\;t > 0,} \end{array}} \right.$$

which is considered in bounded domain Ω ⊂ ℝ^N (N ∈ {2, 3}) with smooth boundary, where $\phi \in {C^{1 + \delta }}\left( {\overline \Omega } \right)$ with δ ∈ (0, 1). We show that if the initial data ${\left\| {{n_0}} \right\|_{{L^{N/2}}\left( \Omega \right)}}$, ${\left\| {\nabla {v_0}} \right\|_{{L^N}\left( \Omega \right)}}$, ${\left\| {\nabla {w_0}} \right\|_{{L^N}\left( \Omega \right)}}$ and ${\left\| {{{\bf{u}}_0}} \right\|_{{L^N}\left( \Omega \right)}}$ 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 ${\bar n_0}: = \left( {1/\left| \Omega \right|} \right)\int_\Omega {{n_0}\left( x \right){\rm{d}}x}$.