scholarly journals Global Well-Posedness and Long Time Decay of Fractional Navier-Stokes Equations in Fourier-Besov Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Weiliang Xiao ◽  
Jiecheng Chen ◽  
Dashan Fan ◽  
Xuhuan Zhou
Keyword(s):  
Besov Spaces ◽  
Critical Case ◽  
Stokes Equations ◽  
Global Solutions ◽  
Navier Stokes ◽  
Well Posedness ◽  
Time Decay ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Long Time

We study the Cauchy problem of the fractional Navier-Stokes equations in critical Fourier-Besov spacesFB˙p,q1-2β+3/p′. Some properties of Fourier-Besov spaces have been discussed, and we prove a general global well-posedness result which covers some recent works in classical Navier-Stokes equations. Particularly, our result is suitable for the critical caseβ=1/2. Moreover, we prove the long time decay of the global solutions in Fourier-Besov spaces.

scholarly journals Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces

2021 ◽  
Vol 19 (1) ◽  
pp. 898-908
Author(s):  
Lotfi Jlali
Keyword(s):  
Fourier Analysis ◽  
Global Solution ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Time Decay ◽  
Navier Stokes Equations ◽  
Long Time

Abstract In this paper, we study the long time decay of global solution to the 3D incompressible Navier-Stokes equations. We prove that if u ∈ C ( R + , X − 1 , σ ( R 3 ) ) u\in {\mathcal{C}}\left({{\mathbb{R}}}^{+},{{\mathcal{X}}}^{-1,\sigma }\left({{\mathbb{R}}}^{3})) is a global solution to the considered equation, where X i , σ ( R 3 ) {{\mathcal{X}}}^{i,\sigma }\left({{\mathbb{R}}}^{3}) is the Fourier-Lei-Lin space with parameters i = − 1 i=-1 and σ ≥ − 1 \sigma \ge -1 , then ‖ u ( t ) ‖ X − 1 , σ \Vert u\left(t){\Vert }_{{{\mathcal{X}}}^{-1,\sigma }} decays to zero as time goes to infinity. The used techniques are based on Fourier analysis.

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