Weighted regularity criteria for the three-dimensional Navier—Stokes equations

2009 ◽  
Vol 139 (3) ◽  
pp. 661-671 ◽  
Author(s):  
Yong Zhou
Keyword(s):  
Weak Solution ◽  
Velocity Field ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Regularity Criteria ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Image Position

We establish some regularity criteria in terms of the velocity field with weight for the Navier—Stokes equations in ℝ3. It is proved that if the weak solution satisfiesorthen the weak solution actually is regular up to time T.

scholarly journals A Geometrical Regularity Criterion in Terms of Velocity Profiles for the Three-Dimensional Navier–Stokes Equations

2019 ◽  
Vol 72 (4) ◽  
pp. 545-562 ◽  
Author(s):  
C V Tran ◽  
X Yu
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Regularity Criterion ◽  
Velocity Profiles ◽  
Regularity Criteria ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Large Velocity ◽  
Whole Space ◽  
Velocity Region

Summary In this article, we present a new kind of regularity criteria for the global well-posedness problem of the three-dimensional Navier–Stokes equations in the whole space. The novelty of the new results is that they involve only the profiles of the magnitude of the velocity. One particular consequence of our theorem is as follows. If for every fixed $t\in (0,T)$, the ‘large velocity’ region $\Omega:=\{(x,t)\mid |u(x,t)|>C(q)\left|\mkern-2mu\left|{u}\right|\mkern-2mu\right|_{L^{3q-6}}\}$, for some $C(q)$ appropriately defined, shrinks fast enough as $q\nearrow \infty$, then the solution remains regular beyond $T$. We examine and discuss velocity profiles satisfying our criterion. It remains to be seen whether these profiles are typical of general Navier–Stokes flows.

scholarly journals Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear Damping

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Weihua Wang ◽  
Guopeng Zhou
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Regularity Criterion ◽  
Nonlinear Damping ◽  
Navier Stokes ◽  
Navier Stokes Equations

This paper is concerned with the regularity criterion of weak solutions to the three-dimensional Navier-Stokes equations with nonlinear damping in critical weakLqspaces. It is proved that if the weak solution satisfies∫0T∇u1Lq,∞2q/2q-3+∇u2Lq,∞2q/2q-3/1+ln⁡e+∇uL22ds<∞,  q>3/2, then the weak solution of Navier-Stokes equations with nonlinear damping is regular on(0,T].

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.

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