scholarly journals Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces

2020 ◽  
Vol 70 (4) ◽  
pp. 877-892
Author(s):  
Jamel Benameur ◽  
Lotfi Jlali
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Well Posedness ◽  
Time Decay ◽  
Navier Stokes Equation ◽  
Gevrey Spaces ◽  
Gevrey Space ◽  
Long Time ◽  
Incompressible Navier Stokes Equation

AbstractIn this paper, we prove a global well-posedness of the three-dimensional incompressible Navier-Stokes equation under initial data, which belongs to the Lei-Lin-Gevrey space $\begin{array}{} Z^{-1}_{a,\sigma} \end{array}$(ℝ3) and if the norm of the initial data in the Lei-Lin space 𝓧−1 is controlled by the viscosity. Moreover, we will show that the norm of this global solution in the Lei-Lin-Gevrey space decays to zero as time approaches to infinity.

Modelling of Laminar Canonical Flows: Revisit

2012 ◽  
Author(s):  
Kofi Freeman K. Adane ◽  
Mark F. Tachie
Keyword(s):  
Three Dimensional ◽  
Shear Thinning ◽  
Secondary Flows ◽  
Newtonian Fluids ◽  
Navier Stokes ◽  
Jet Flows ◽  
Wall Jet ◽  
Navier Stokes Equation ◽  
Incompressible Navier Stokes Equation ◽  
Insight Into

Three-dimensional laminar lid-driven and wall jet flows of various shear-thinning non-Newtonian and Newtonian fluids were numerically investigated. The complete nonlinear incompressible Navier-Stokes equation was solved using a collocated finite-volume based in-house CFD code. From the results, velocity profiles at several locations, jet spread rates, secondary flows and vorticity distributions were used to provide insight into the characteristics of three-dimensional laminar canonical flows of shear-thinning non-Newtonian and Newtonian fluids.

scholarly journals On three-dimensional incompressible Navier-Stokes fluid on cantor sets in spherical Cantor type co-ordinate system

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 853-858
Author(s):  
Zhi-Jun Meng ◽  
Yao-Ming Zhou ◽  
Dong-Mu Mei
Keyword(s):  
Heat Conduction ◽  
Stokes Equations ◽  
Heat Conduction Problem ◽  
Three Dimensional ◽  
Cantor Sets ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stokes Fluid ◽  
Incompressible Navier Stokes Equation

This paper addresses the systems of the incompressible Navier-Stokes equations on Cantor sets without the external force involving the fractal heat-conduction problem vial local fractional derivative. The spherical Cantor type co-ordinate method is used to transfer the incompressible Navier-Stokes equation from the Cantorian co-ordinate system into the spherical Cantor type co-ordinate system.

scholarly journals Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows

2015 ◽  
Vol 783 ◽  
pp. 412-447 ◽  
Author(s):  
Basile Gallet
Keyword(s):  
Reynolds Number ◽  
Stokes Equation ◽  
Three Dimensional ◽  
Threshold Value ◽  
Navier Stokes ◽  
Rossby Number ◽  
Navier Stokes Equation ◽  
Rotating Turbulence ◽  
2D Flow ◽  
Long Time

We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body force. This system admits vertically invariant solutions that satisfy the 2D Navier–Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations. We first consider the 3D rotating Navier–Stokes equation linearized around a statistically steady 2D flow solution. We show that this base flow is linearly stable to vertically dependent perturbations when the global rotation is fast enough: under a Reynolds-number-dependent threshold value$Ro_{c}(Re)$of the Rossby number, the flow becomes exactly 2D in the long-time limit, provided that the initial 3D perturbations are small. We call this property linear two-dimensionalization. We compute explicit lower bounds on$Ro_{c}(Re)$and therefore determine regions of the parameter space$(Re,Ro)$where such exact two-dimensionalization takes place. We present similar results in terms of the forcing strength instead of the root-mean-square velocity: the global attractor of the 2D Navier–Stokes equation is linearly stable to vertically dependent perturbations when the forcing-based Rossby number$Ro^{(f)}$is lower than a Grashof-number-dependent threshold value$Ro_{c}^{(f)}(Gr)$. We then consider the fully nonlinear 3D rotating Navier–Stokes equation and prove absolute two-dimensionalization: we show that, below some threshold value$Ro_{\mathit{abs}}^{(f)}(Gr)$of the forcing-based Rossby number, the flow becomes two-dimensional in the long-time limit, regardless of the initial condition (including initial 3D perturbations of arbitrarily large amplitude). These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number$Re$and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone–anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.

scholarly journals Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces

2019 ◽  
Vol 150 (2) ◽  
pp. 569-606 ◽  
Author(s):  
Dat Cao ◽  
Luan Hoang
Keyword(s):  
Asymptotic Expansion ◽  
Body Force ◽  
Stokes Equations ◽  
Three Dimensional ◽  
The Body ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Gevrey Space ◽  
Long Time ◽  
The Individual

AbstractThe Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

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.

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