A REGULARITY CRITERION FOR THE NAVIER–STOKES EQUATIONS IN TERMS OF ONE DIRECTIONAL DERIVATIVE OF THE VELOCITY FIELD

2012 ◽  
Vol 10 (04) ◽  
pp. 373-380 ◽  
Author(s):  
ZHENGGUANG GUO ◽  
SADEK GALA
Keyword(s):  
Weak Solution ◽  
Velocity Field ◽  
Stokes Equations ◽  
Directional Derivative ◽  
Regularity Criterion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Multiplier Space

We consider the regularity criterion for the incompressible Navier–Stokes equations. We show that the weak solution is regular, provided [Formula: see text] for some T > 0, where Ẋr is the multiplier space. This extends a result of Kukavica and Ziane [14].

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].

On the regularity criterion for the Navier–Stokes equations in terms of one directional derivative

2017 ◽  
Vol 10 (01) ◽  
pp. 1750012 ◽  
Author(s):  
Sadek Gala ◽  
Maria Alessandra Ragusa
Keyword(s):  
Recent Work ◽  
Weak Solutions ◽  
Critical Condition ◽  
Stokes Equations ◽  
Directional Derivative ◽  
Regularity Criterion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Regularity Problem ◽  
Appl Math

In this note, we consider the regularity problem for the weak solutions, under the critical condition to the Navier–Stokes equations in [Formula: see text]. We show that, if the velocity [Formula: see text] satisfies [Formula: see text] then the solution actually is smooth on [Formula: see text]. This improves a result established in a recent work by Liu [Q. Liu, A regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity, Acta Appl. Math. 140 (2015) 1–9].

scholarly journals A Note on the Regularity Criterion of Weak Solutions of Navier-Stokes Equations in Lorentz Space

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Xunwu Yin
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Lebesgue Space ◽  
Lorentz Space ◽  
Stokes Equations ◽  
Regularity Criterion ◽  
Horizontal Velocity ◽  
Navier Stokes ◽  
Navier Stokes Equations

This paper is concerned with the regularity of Leray weak solutions to the 3D Navier-Stokes equations in Lorentz space. It is proved that the weak solution is regular if the horizontal velocity denoted byũ=(u1,u2,0)satisfiesũ(x,t)∈Lq(0,T;Lp,∞(R3))  for  2/q+3/p=1,  3<p<∞.The result is obvious and improved that of Dong and Chen (2008) on the Lebesgue space.

scholarly journals Direction of vorticity and a new regularity criterion for the Navier-Stokes equations

2005 ◽  
Vol 46 (3) ◽  
pp. 309-316 ◽  
Author(s):  
Yong Zhou
Keyword(s):  
Weak Solution ◽  
Stokes Equations ◽  
Regularity Criterion ◽  
Navier Stokes ◽  
Navier Stokes Equations

AbstractIn this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations. Under the framework of Constantin and Fefferman, a more relaxed regularity criterion in terms of the direction of vorticity is established.

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