Global well-posedness for the incompressible Navier–Stokes equations in the critical Besov space under the Lagrangian coordinates

2021 ◽  
Vol 274 ◽  
pp. 613-651
Author(s):  
Takayoshi Ogawa ◽  
Senjo Shimizu
Keyword(s):  
Besov Space ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Well Posedness ◽  
Lagrangian Coordinates ◽  
Navier Stokes Equations ◽  
Critical Besov Space

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.

scholarly journals NAVIER-STOKES EQUATIONS IN THE BESOV SPACE NEAR L∞ AND BMO

2003 ◽  
Vol 57 (2) ◽  
pp. 303-324 ◽  
Author(s):  
Hideo KOZONO ◽  
Takayoshi OGAWA ◽  
Yasushi TANIUCHI
Keyword(s):  
Besov Space ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

Stability of Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Well Posedness ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Three Space

In this chapter we intend to investigate the stability of the Leray solutions constructed in the previous chapter. It is useful to start by analyzing the linearized version of the Navier–Stokes equations, so the first section of the chapter is devoted to the proof of the well-posedness of the time-dependent Stokes system. The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes equations, and the more delicate case of three space dimensions will be dealt with in Sections 3.3–3.5.

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