The incompressible limits of compressible Navier-Stokes equations in the whole space with general initial data

2009 ◽  
Vol 30 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Ling Hsiao ◽  
Qiangchang Ju ◽  
Fucai Li
Keyword(s):  
Initial Data ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
General Initial Data ◽  
Whole Space

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.

A simple proof of existence of weak and statistical solutions of Navier–Stokes equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 1-11 ◽  
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Natural Number ◽  
Simple Proof ◽  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Existence Proofs ◽  
Statistical Solutions

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

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