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

Lu Yang; Xi Liu; Zhibo Hou

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 1, page 49-70
  • ISSN: 0011-4642

Abstract

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We consider the Keller-Segel-Navier-Stokes system n t + 𝐮 · ∇ n = Δ n - ∇ · ( n ∇ v ) , x ∈ Ω , t > 0 , v t + 𝐮 · ∇ v = Δ v - v + w , x ∈ Ω , t > 0 , w t + 𝐮 · ∇ w = Δ w - w + n , x ∈ Ω , t > 0 , 𝐮 t + ( 𝐮 · ∇ ) 𝐮 = Δ 𝐮 + ∇ P + n ∇ φ , ∇ · 𝐮 = 0 , x ∈ Ω , t > 0 , which is considered in bounded domain Ω ⊂ ℝ N ( N ∈ { 2 , 3 } ) with smooth boundary, where φ ∈ C 1 + δ ( Ω ¯ ) with δ ∈ ( 0 , 1 ) . We show that if the initial data ∥ n 0 ∥ L N / 2 ( Ω ) , ∥ ∇ v 0 ∥ L N ( Ω ) , ∥ ∇ w 0 ∥ L N ( Ω ) and ∥ 𝐮 0 ∥ L N ( Ω ) is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state ( n ¯ 0 , n ¯ 0 , n ¯ 0 , 0 ) exponentially with n ¯ 0 : = ( 1 / | Ω | ) ∫ Ω n 0 ( x ) d x .

How to cite

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Yang, Lu, Liu, Xi, and Hou, Zhibo. "Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production." Czechoslovak Mathematical Journal 73.1 (2023): 49-70. <http://eudml.org/doc/299551>.

@article{Yang2023,
abstract = {We consider the Keller-Segel-Navier-Stokes system \[ \{\left\lbrace \begin\{array\}\{ll\} n\_t+\{\bf u\}\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla v ),& x\in \Omega ,\ t>0,\\ v\_t +\{\bf u\}\cdot \nabla v=\Delta v -v+w, &x\in \Omega ,\ t>0,\\ w\_t+\{\bf u\}\cdot \nabla w=\Delta w -w+n, &x\in \Omega ,\ t>0,\\ \{\bf \{u\}\}\_t + (\{\bf \{u\}\}\cdot \nabla )\{\bf \{u\}\} = \Delta \{\bf \{u\}\} + \nabla P + n\nabla \phi ,\ \nabla \cdot \{\bf u\}=0, &x\in \Omega ,\ t>0, \end\{array\}\right.\} \] which is considered in bounded domain $\Omega \subset \mathbb \{R\}^N$$(N \in \lbrace 2,3\rbrace )$ with smooth boundary, where $\phi \in C^\{1+\delta \}(\overline\{\Omega \})$ with $\delta \in (0,1)$. We show that if the initial data $\Vert n_0\Vert _\{L^\{\{N\}/\{2\}\}(\Omega )\}$, $\Vert \nabla v_0\Vert _\{L^N(\Omega )\}$, $\Vert \nabla w_0\Vert _\{L^N(\Omega )\}$ and $\Vert \{\bf u\}_0\Vert _\{L^N(\Omega )\}$ 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 $\{\bar\{n\}\}_0:=(1/|\Omega |)\int _\{\Omega \}n_0(x)\{\rm d\}x$.},
author = {Yang, Lu, Liu, Xi, Hou, Zhibo},
journal = {Czechoslovak Mathematical Journal},
keywords = {Keller-Segel-Navier-Stokes; global solution; decay estimate; indirect process},
language = {eng},
number = {1},
pages = {49-70},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production},
url = {http://eudml.org/doc/299551},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Yang, Lu
AU - Liu, Xi
AU - Hou, Zhibo
TI - Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 1
SP - 49
EP - 70
AB - We consider the Keller-Segel-Navier-Stokes system \[ {\left\lbrace \begin{array}{ll} n_t+{\bf u}\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla v ),& x\in \Omega ,\ t>0,\\ v_t +{\bf u}\cdot \nabla v=\Delta v -v+w, &x\in \Omega ,\ t>0,\\ w_t+{\bf u}\cdot \nabla w=\Delta w -w+n, &x\in \Omega ,\ t>0,\\ {\bf {u}}_t + ({\bf {u}}\cdot \nabla ){\bf {u}} = \Delta {\bf {u}} + \nabla P + n\nabla \phi ,\ \nabla \cdot {\bf u}=0, &x\in \Omega ,\ t>0, \end{array}\right.} \] which is considered in bounded domain $\Omega \subset \mathbb {R}^N$$(N \in \lbrace 2,3\rbrace )$ with smooth boundary, where $\phi \in C^{1+\delta }(\overline{\Omega })$ with $\delta \in (0,1)$. We show that if the initial data $\Vert n_0\Vert _{L^{{N}/{2}}(\Omega )}$, $\Vert \nabla v_0\Vert _{L^N(\Omega )}$, $\Vert \nabla w_0\Vert _{L^N(\Omega )}$ and $\Vert {\bf u}_0\Vert _{L^N(\Omega )}$ 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 ${\bar{n}}_0:=(1/|\Omega |)\int _{\Omega }n_0(x){\rm d}x$.
LA - eng
KW - Keller-Segel-Navier-Stokes; global solution; decay estimate; indirect process
UR - http://eudml.org/doc/299551
ER -

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