Regularity of a Weak Solution to the Navier-Stokes Equation in Dependence on Eigenvalues and Eigenvectors of the Rate of Deformation Tensor

2005 ◽  
pp. 197-212 ◽  
Author(s):  
Jiří Neustupa ◽  
Patrick Penel
Keyword(s):  
Weak Solution ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Deformation Tensor ◽  
Eigenvalues And Eigenvectors ◽  
Navier Stokes Equation

scholarly journals On Regularity of a Weak Solution to the Navier–Stokes Equations with the Generalized Navier Slip Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Jiří Neustupa ◽  
Patrick Penel
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Regularity Criterion ◽  
Navier Stokes ◽  
Deformation Tensor ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equation ◽  
Navier Slip ◽  
Slip Boundary

The paper shows that the regularity up to the boundary of a weak solution v of the Navier–Stokes equation with generalized Navier’s slip boundary conditions follows from certain rate of integrability of at least one of the functions ζ1, (ζ2)+ (the positive part of ζ2), and ζ3, where ζ1≤ζ2≤ζ3 are the eigenvalues of the rate of deformation tensor D(v). A regularity criterion in terms of the principal invariants of tensor D(v) is also formulated.

scholarly journals On the Convergence of Solutions of Globally Modified Navier-Stokes Equations with Delays to Solutions of Navier-Stokes Equations with Delays

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Pedro Marín-Rubio ◽  
José Real ◽  
Antonio M. Márquez-Durán
Keyword(s):  
Weak Solution ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Additional Case ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Convergence Of Solutions

AbstractWe prove that under suitable assumptions, from a sequence of solutions of Globally Modified Navier-Stokes equations with delays we can extract a subsequence which converges in an adequate sense to a weak solution of a three-dimensional Navier-Stokes equation with delays. An additional case with a family of different delays involved in the approximating problems is also discussed.

scholarly journals TAMED 3D NAVIER–STOKES EQUATION: EXISTENCE, UNIQUENESS AND REGULARITY

2009 ◽  
Vol 12 (04) ◽  
pp. 525-549 ◽  
Author(s):  
MICHAEL RÖCKNER ◽  
XICHENG ZHANG
Keyword(s):  
Weak Solution ◽  
Stokes Equation ◽  
Existence And Uniqueness ◽  
Smooth Solution ◽  
Navier Stokes ◽  
Suitable Weak Solution ◽  
Navier Stokes Equation ◽  
Whole Space ◽  
New Construction ◽  
Bounded Smooth

In this paper, we prove the existence and uniqueness of a smooth solution to a tamed 3D Navier–Stokes equation in the whole space. In particular, if there exists a bounded smooth solution to the classical 3D Navier–Stokes equation, then this solution satisfies our tamed equation. Moreover, using this tamed equation we can give a new construction for a suitable weak solution of the classical 3D Navier–Stokes equation introduced in Refs. 16 and 2.

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