Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system

Yanjiang Li; Zhongqing Yu; Yumei Huang

Czechoslovak Mathematical Journal (2024)

  • Issue: 1, page 153-175
  • ISSN: 0011-4642

Abstract

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The self-consistent chemotaxis-fluid system n t + u · ∇ n = Δ n - ∇ · ( n ∇ c ) + ∇ · ( n ∇ φ ) , x ∈ Ω , t > 0 , c t + u · ∇ c = Δ c - n c , x ∈ Ω , t > 0 , u t + κ ( u · ∇ ) u + ∇ P = Δ u - n ∇ φ + n ∇ c , x ∈ Ω , t > 0 , ∇ · u = 0 , x ∈ Ω , t > 0 , is considered under no-flux boundary conditions for n , c and the Dirichlet boundary condition for u on a bounded smooth domain Ω ⊂ ℝ N ( N = 2 , 3 ) , κ ∈ { 0 , 1 } . The existence of global bounded classical solutions is proved under a smallness assumption on ∥ c 0 ∥ L ∞ ( Ω ) . Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid systems. The literature on self-consistent chemotaxis-fluid systems of this type so far concentrates on the nonlinear cell diffusion as an additional dissipative mechanism. To the best of our knowledge, this is the first result on the boundedness of a self-consistent chemotaxis-fluid system with linear cell diffusion.

How to cite

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Li, Yanjiang, Yu, Zhongqing, and Huang, Yumei. "Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system." Czechoslovak Mathematical Journal (2024): 153-175. <http://eudml.org/doc/299213>.

@article{Li2024,
abstract = {The self-consistent chemotaxis-fluid system \[ \{\left\lbrace \begin\{array\}\{ll\} n\_t+u\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla c )+\nabla \cdot (n\nabla \phi ), &x\in \Omega ,\ t>0,\\ c\_t +u\cdot \nabla c=\Delta c -nc,\quad &x\in \Omega ,\ t>0,\\ u\_t+\kappa (u\cdot \nabla ) u+\nabla P=\Delta u - n\nabla \phi +n \nabla c,\qquad &x\in \Omega ,\ t>0,\\ \nabla \cdot u=0,\quad &x\in \Omega ,\ t>0, \end\{array\}\right.\} \] is considered under no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $ \Omega \subset \mathbb \{R\}^N$$(N=2,3)$, $\kappa \in \lbrace 0,1 \rbrace $. The existence of global bounded classical solutions is proved under a smallness assumption on $\Vert c_\{0\}\Vert _\{L^\{\infty \}(\Omega )\}$. Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid systems. The literature on self-consistent chemotaxis-fluid systems of this type so far concentrates on the nonlinear cell diffusion as an additional dissipative mechanism. To the best of our knowledge, this is the first result on the boundedness of a self-consistent chemotaxis-fluid system with linear cell diffusion.},
author = {Li, Yanjiang, Yu, Zhongqing, Huang, Yumei},
journal = {Czechoslovak Mathematical Journal},
keywords = {chemotaxis; Navier-Stokes system; self-consistent; global existence; boundedness},
language = {eng},
number = {1},
pages = {153-175},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system},
url = {http://eudml.org/doc/299213},
year = {2024},
}

TY - JOUR
AU - Li, Yanjiang
AU - Yu, Zhongqing
AU - Huang, Yumei
TI - Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 153
EP - 175
AB - The self-consistent chemotaxis-fluid system \[ {\left\lbrace \begin{array}{ll} n_t+u\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla c )+\nabla \cdot (n\nabla \phi ), &x\in \Omega ,\ t>0,\\ c_t +u\cdot \nabla c=\Delta c -nc,\quad &x\in \Omega ,\ t>0,\\ u_t+\kappa (u\cdot \nabla ) u+\nabla P=\Delta u - n\nabla \phi +n \nabla c,\qquad &x\in \Omega ,\ t>0,\\ \nabla \cdot u=0,\quad &x\in \Omega ,\ t>0, \end{array}\right.} \] is considered under no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $ \Omega \subset \mathbb {R}^N$$(N=2,3)$, $\kappa \in \lbrace 0,1 \rbrace $. The existence of global bounded classical solutions is proved under a smallness assumption on $\Vert c_{0}\Vert _{L^{\infty }(\Omega )}$. Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid systems. The literature on self-consistent chemotaxis-fluid systems of this type so far concentrates on the nonlinear cell diffusion as an additional dissipative mechanism. To the best of our knowledge, this is the first result on the boundedness of a self-consistent chemotaxis-fluid system with linear cell diffusion.
LA - eng
KW - chemotaxis; Navier-Stokes system; self-consistent; global existence; boundedness
UR - http://eudml.org/doc/299213
ER -

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