scholarly journals On the continuity of the solutions to the Navier–Stokes equations with initial data in critical Besov spaces

2019 ◽  
Vol 198 (5) ◽  
pp. 1495-1511 ◽  
Author(s):  
Reinhard Farwig ◽  
Yoshikazu Giga ◽  
Pen-Yuan Hsu
Keyword(s):  
Initial Data ◽  
Besov Spaces ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Critical Besov Spaces

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 Algebra Properties in Fourier-Besov Spaces and Their Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Xuhuan Zhou ◽  
Weiliang Xiao
Keyword(s):  
Initial Data ◽  
Besov Spaces ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Well Posedness ◽  
Gevrey Regularity ◽  
Navier Stokes Equations ◽  
Small Initial Data ◽  
Scale Functions

We estimate the norm of the product of two scale functions in Fourier-Besov spaces. As applications of these algebra properties, we establish the global well-posedness for small initial data and local well-posedness for large initial data of the generalized Navier-Stokes equations. Particularly, we give a blow-up criterion of the solutions in Fourier-Besov spaces as well as a space analyticity of Gevrey regularity.

scholarly journals Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces

2018 ◽  
Vol 23 (8) ◽  
pp. 3427-3460 ◽  
Author(s):  
Minghua Yang ◽  
◽  
Zunwei Fu ◽  
Jinyi Sun ◽  
◽  
...  
Keyword(s):  
Besov Spaces ◽  
Stokes Equations ◽  
Global Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Critical Besov Spaces

scholarly journals On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces

2015 ◽  
Vol 31 (4) ◽  
pp. 1375-1402 ◽  
Author(s):  
Qionglei Chen ◽  
Changxing Miao ◽  
Zhifei Zhang
Keyword(s):  
Besov Spaces ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Critical Besov Spaces
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