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

Yang, Lu ; Liu, Xi ; Hou, Zhibo

Czechoslovak Mathematical Journal, Tome 73 (2023) no. 1, pp. 49-70.

Résumé

We consider the Keller-Segel-Navier-Stokes system $$ \begin{cases} n_t+{\bf u}\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla v ), x\in \Omega ,\ t>0,\\ v_t +{\bf u}\cdot \nabla v=\Delta v -v+w, \in \Omega ,\ t>0,\\ w_t+{\bf u}\cdot \nabla w=\Delta w -w+n, \in \Omega ,\ t>0,\\ {\bf {u}}_t + ({\bf {u}}\cdot \nabla ){\bf {u}} = \Delta {\bf {u}} + \nabla P + n\nabla \phi ,\ \nabla \cdot {\bf u}=0, \in \Omega ,\ t>0, \end{cases} $$ 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 ${\bar n}_0:=(1/|\Omega |)\int _{\Omega }n_0(x){\rm d}x$.

Détail

DOI : 10.21136/CMJ.2022.0399-21

Classification : 35B35, 35B40, 35K55, 35Q35, 92C17
Keywords: Keller-Segel-Navier-Stokes; global solution; decay estimate; indirect process