The convergence rate of the fast signal diffusion limit for a Keller–Segel–Stokes system with large initial data

2020 ◽  
pp. 1-41
Author(s):  
Min Li ◽  
Zhaoyin Xiang
Keyword(s):  
Initial Data ◽  
Convergence Rate ◽  
Stokes System ◽  
Smooth Boundary ◽  
Three Dimensional ◽  
Diffusion Limit ◽  
Large Initial Data ◽  
Volume Filling ◽  
Algebraic Convergence ◽  
Current Setting

In this paper, we investigate the fast signal diffusion limit of solutions of the fully parabolic Keller–Segel–Stokes system to solution of the parabolic–elliptic-fluid counterpart in a two-dimensional or three-dimensional bounded domain with smooth boundary. Under the natural volume-filling assumption, we establish an algebraic convergence rate of the fast signal diffusion limit for general large initial data by developing a series of subtle bootstrap arguments for combinational functionals and using some maximal regularities. In our current setting, in particular, we can remove the restriction to asserting convergence only along some subsequence in Wang–Winkler and the second author (Cal. Var., 2019).

scholarly journals Linear stability of blowup solution of incompressible Keller–Segel–Navier–Stokes system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yan Yan ◽  
Hengyan Li
Keyword(s):  
Initial Data ◽  
Linear Stability ◽  
Stokes System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Blowup Solution ◽  
Existence Time ◽  
Whole Space

AbstractIn this paper, we consider the linear stability of blowup solution for incompressible Keller–Segel–Navier–Stokes system in whole space $\mathbb{R}^{3}$ R 3 . More precisely, we show that, if the initial data of the three dimensional Keller–Segel–Navier–Stokes system is close to the smooth initial function $(0,0,\textbf{u}_{s}(0,x) )^{T}$ ( 0 , 0 , u s ( 0 , x ) ) T , then there exists a blowup solution of the three dimensional linear Keller–Segel–Navier–Stokes system satisfying the decomposition $$ \bigl(n(t,x),c(t,x),\textbf{u}(t,x) \bigr)^{T}= \bigl(0,0, \textbf{u}_{s}(t,x) \bigr)^{T}+\mathcal{O}(\varepsilon ), \quad \forall (t,x)\in \bigl(0,T^{*}\bigr) \times \mathbb{R}^{3}, $$ ( n ( t , x ) , c ( t , x ) , u ( t , x ) ) T = ( 0 , 0 , u s ( t , x ) ) T + O ( ε ) , ∀ ( t , x ) ∈ ( 0 , T ∗ ) × R 3 , in Sobolev space $H^{s}(\mathbb{R}^{3})$ H s ( R 3 ) with $s=\frac{3}{2}-5a$ s = 3 2 − 5 a and constant $0< a\ll 1$ 0 < a ≪ 1 , where $T^{*}$ T ∗ is the maximal existence time, and $\textbf{u}_{s}(t,x)$ u s ( t , x ) given in (Yan 2018) is the explicit blowup solution admitted smooth initial data for three dimensional incompressible Navier–Stokes equations.

scholarly journals Analysis of long-term solution of chemotactic model with indirect signal consumption in three-dimensional case

2021 ◽  
Author(s):  
Qiaoling Hu Hu
Keyword(s):  
Weak Solution ◽  
Large Time ◽  
Smooth Boundary ◽  
Three Dimensional ◽  
Neumann Boundary ◽  
Global Weak Solution ◽  
Large Initial Data ◽  
Chemotaxis Model ◽  
Large Time Limit

In this paper, we consider the chemotaxis model u_t&=\Delta u-\nabla\cdot(u\nabla v),& \qquad x\in\Omega,\,t>0,v_t&=\Delta v-vw,& \qquad x\in\Omega,\,t>0,w_t&=-\delta w+u,& \qquad x\in\Omega,\,t>0,under homogeneous Neumann boundary conditions in a bounded and convex domain $\Om\subset \mathbb{R}^3$ with smooth boundary, where $\delta>0$ is a given parameter. It is shown that for arbitrarily large initial data, this problem admits at least one global weak solution for which there exists $T>0$ such that the solution $(u,v,w)$ is bounded and smooth in $\Om\times(T,\infty)$. Furthermore, it is asserted that such solutions approach spatially constant equilibria in the large time limit.

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