scholarly journals Local and global strong solutions to the stochastic incompressible Navier-Stokes equations in critical Besov space

2020 ◽  
Vol 481 (1) ◽  
pp. 123472 ◽  
Author(s):  
Lihuai Du ◽  
Ting Zhang
Keyword(s):  
Besov Space ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Critical Besov Space ◽  
Global Strong Solutions

ON THE ASYMPTOTIC ANALYSIS OF THE BGK MODEL TOWARD THE INCOMPRESSIBLE LINEAR NAVIER–STOKES EQUATION

2010 ◽  
Vol 20 (08) ◽  
pp. 1299-1318 ◽  
Author(s):  
A. BELLOUQUID
Keyword(s):  
Asymptotic Analysis ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Uniqueness Theorems ◽  
Bgk Model ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stokes Systems ◽  
Global Strong Solutions

This paper deals with the analysis of the asymptotic limit for BGK model to the linearized Navier–Stokes equations when the Knudsen number ε tends to zero. The uniform (in ε) existence of global strong solutions and uniqueness theorems are proved for regular initial fluctuations. As ε tends to zero, the solution of BGK model converges strongly to the solution of the linearized Navier–Stokes systems. The validity of the BGK model is critically analyzed.

ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN FRAMEWORK

2017 ◽  
Vol 18 (4) ◽  
pp. 829-854 ◽  
Author(s):  
Jiecheng Chen ◽  
Renhui Wan
Keyword(s):  
Initial Data ◽  
Besov Space ◽  
Besov Spaces ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Coupling Term ◽  
Navier Stokes Equations ◽  
Critical Besov Space ◽  
Image Position ◽  
Critical Besov Spaces

Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$$(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying $$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$ To accomplish this goal, we require a norm inflation coming from the coupling term $L(a)\unicode[STIX]{x1D6E5}u$ instead of $u\cdot \unicode[STIX]{x1D6FB}u$ and construct a new decomposition of the density.

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