scholarly journals Density properties for solenoidal vector fields, with applications to the Navier-Stokes equations in exterior domains

1992 ◽  
Vol 44 (2) ◽  
pp. 307-330 ◽  
Author(s):  
Hideo KOZONO ◽  
Hermann SOHR
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Exterior Domains ◽  
Solenoidal Vector ◽  
Navier Stokes Equations

scholarly journals The 3D Navier–Stokes Equations: Invariants, Local and Global Solutions

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 41 ◽  
Author(s):  
Vladimir Semenov
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Uniform Boundedness ◽  
Global Solutions ◽  
Navier Stokes ◽  
Solenoidal Vector ◽  
Navier Stokes Equations ◽  
Scaling Procedure ◽  
Sufficient And Necessary Conditions ◽  
Special Invariant

In this article, I consider local solutions of the 3D Navier–Stokes equations and its properties such as an existence of global and smooth solution, uniform boundedness. The basic role is assigned to a special invariant class of solenoidal vector fields and three parameters that are invariant with respect to the scaling procedure. Since in spaces of even dimensions the scaling procedure is a conformal mapping on the Heisenberg group, then an application of invariant parameters can be considered as the application of conformal invariants. It gives the possibility to prove the sufficient and necessary conditions for existence of a global regular solution. This is the main result and one among some new statements. With some compliments, the rest improves well-known classical results.

Some Elements of Functional Analysis

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Functional Analysis ◽  
Weak Solutions ◽  
Vector Fields ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Dual Spaces ◽  
Type Spaces ◽  
Definition Of

Before introducing the concept of Leray’s weak solutions to the incompressible Navier–Stokes equations, classical definitions of Sobolev spaces are required. In particular, when it comes to the analysis of the Stokes operator, suitable functional spaces of incompressible vector fields have to be defined. Several issues regarding the associated dual spaces, embedding properties, and the mathematical way of considering the pressure field are also discussed. Let us first recall the definition of some functional spaces that we shall use throughout this book. In the framework of weak solutions of the Navier– Stokes equations, incompressible vector fields with finite viscous dissipation and the no-slip property on the boundary are considered. Such H1-type spaces of incompressible vector fields, and the corresponding dual spaces, are important ingredients in the analysis of the Stokes operator.

scholarly journals On the steady Navier–Stokes equations in 2D exterior domains

2020 ◽  
Vol 269 (3) ◽  
pp. 1796-1828 ◽  
Author(s):  
Mikhail V. Korobkov ◽  
Konstantin Pileckas ◽  
Remigio Russo
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Exterior Domains ◽  
Navier Stokes Equations

scholarly journals Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 2. The Stokes equations

2005 ◽  
Vol 47 (1) ◽  
pp. 39-50 ◽  
Author(s):  
G. D. McBain
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Momentum Equation ◽  
Integral Representations ◽  
Navier Stokes ◽  
Field Equations ◽  
Navier Stokes Equations ◽  
Time Discretisation ◽  
Mass Constraint ◽  
Conservation Of Momentum

AbstractWe continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.

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