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.

Global well-posedness for 3D Navier–Stokes equations with ill-prepared initial data

2013 ◽  
Vol 13 (2) ◽  
pp. 395-411 ◽  
Author(s):  
Marius Paicu ◽  
Zhifei Zhang
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Stokes Equations ◽  
Vertical Direction ◽  
Navier Stokes ◽  
Well Posedness ◽  
Large Initial Data ◽  
Navier Stokes Equations

AbstractWe study the global well-posedness of 3D Navier–Stokes equations for a class of large initial data. This type of data slowly varies in the vertical direction (expressed as a function of $\varepsilon {x}_{3} $), and it is ill-prepared in the sense that its norm in ${C}^{- 1} $ will blow up at the rate ${\varepsilon }^{- \alpha } $ for $\frac{1}{2} \lt \alpha \lt 1$ as $\varepsilon $ tends to zero.

scholarly journals Fujita–Kato solution for compressible Navier–Stokes equations with axisymmetric initial data and zero Mach number limit

2019 ◽  
Vol 22 (05) ◽  
pp. 1950041
Author(s):  
Boris Haspot
Keyword(s):  
Mach Number ◽  
Initial Data ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Small Initial Data ◽  
Dimension Formula ◽  
Global Strong Solution ◽  
Rotational Part ◽  
Zero Mach Number Limit

In this paper, we investigate the question of the existence of global strong solution for the compressible Navier–Stokes equations for small initial data such that the rotational part of the velocity [Formula: see text] belongs to [Formula: see text] (in dimension [Formula: see text]). We show then an equivalent of the so-called Fujita–Kato theorem to the case of the compressible Navier–Stokes equations when we consider axisymmetric initial data. The main difficulty is linked to the fact that in this case the velocity is not Lipschitz, as a consequence we have to study carefully the coupling between the rotational and irrotational part of the velocity. In a second part, we address the question of convergence to the incompressible model (for ill-prepared initial data) when the Mach number goes to zero.

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