scholarly journals Global solvability of the rotating Navier-Stokes equations with fractional Laplacian in a periodic domain

2017 ◽  
Vol 372 (1-2) ◽  
pp. 743-779
Author(s):  
Nobu Kishimoto ◽  
Tsuyoshi Yoneda
Keyword(s):  
Fractional Laplacian ◽  
Stokes Equations ◽  
Global Solvability ◽  
Navier Stokes ◽  
Periodic Domain ◽  
Navier Stokes Equations

scholarly journals Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data

2008 ◽  
Vol 57 (6) ◽  
pp. 2775-2792 ◽  
Author(s):  
Yoshikazu Giga ◽  
Katsuya Inui ◽  
Alex Mahalov ◽  
Juergen Saal
Keyword(s):  
Initial Data ◽  
Stokes Equations ◽  
Global Solvability ◽  
Navier Stokes ◽  
Navier Stokes Equations

scholarly journals Limits of the Stokes and Navier–Stokes equations in a punctured periodic domain

2019 ◽  
Vol 18 (02) ◽  
pp. 211-235
Author(s):  
Michel Chipot ◽  
Jérôme Droniou ◽  
Gabriela Planas ◽  
James C. Robinson ◽  
Wei Xue
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Stokes Equations ◽  
Periodic Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Periodic Domain ◽  
Navier Stokes Equations ◽  
Forcing Function ◽  
Connected Set

We treat three problems on a two-dimensional “punctured periodic domain”: we take [Formula: see text], where [Formula: see text] and [Formula: see text] is the closure of an open connected set that is star-shaped with respect to [Formula: see text] and has a [Formula: see text] boundary. We impose periodic boundary conditions on the boundary of [Formula: see text], and Dirichlet boundary conditions on [Formula: see text]. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier–Stokes equations, all with a fixed forcing function [Formula: see text], and examine the behavior of solutions as [Formula: see text]. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of [Formula: see text] with periodic boundary conditions.

Hydrodynamics at the smallest scales: a solvability criterion for Navier–Stokes equations in high dimensions

2011 ◽  
Vol 369 (1935) ◽  
pp. 359-370 ◽  
Author(s):  
T. M. Viswanathan ◽  
G. M. Viswanathan
Keyword(s):  
Stokes Equations ◽  
Global Solvability ◽  
Navier Stokes ◽  
High Dimensional ◽  
High Dimensions ◽  
Complex Interplay ◽  
Navier Stokes Equations ◽  
Solvability Criterion ◽  
Scale Transformations ◽  
Hydrodynamic Systems

Strong global solvability is difficult to prove for high-dimensional hydrodynamic systems because of the complex interplay between nonlinearity and scale invariance. We define the Ladyzhenskaya–Lions exponent α l ( n )=(2+ n )/4 for Navier–Stokes equations with dissipation −(− Δ ) α in , for all n ≥2. We review the proof of strong global solvability when α ≥ α l ( n ), given smooth initial data. If the corresponding Euler equations for n >2 were to allow uncontrolled growth of the enstrophy , then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier–Stokes equations for α < α l ( n ). The energy is critical under scale transformations only for α = α l ( n ).

Model of a viscous layer deformation by thermocapillary forces

2002 ◽  
Vol 13 (2) ◽  
pp. 205-224 ◽  
Author(s):  
V. V. PUKHNACHOV
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Global Solvability ◽  
Navier Stokes ◽  
Free Boundaries ◽  
Qualitative Properties ◽  
Navier Stokes Equations ◽  
Opposite Case ◽  
Thermocapillary Forces ◽  
Layer Deformation

Three-dimensional nonstationary flow of a viscous incompressible liquid is investigated in a layer, driven by a nonuniform distribution of temperature on its free boundaries. If the temperature given on the layer boundaries is quadratically dependent on horizontal coordinates, external mass forces are absent, and the motion starts from rest then the free boundary problem for the Navier–Stokes equations has an ‘exact’ solution in terms of two independent variables. Here the free boundaries of the layer remain parallel planes and the distance between them must be also determined. In present paper, we formulate conditions for both the unique solvability of the reduced problem globally in time and the collapse of the solution in finite time. We further study qualitative properties of the solution such as its behaviour for large time (in the case of global solvability of the problem), and the asymptotics of the solution near the collapse moment in the opposite case.

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