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 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 Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving the Velocity Partial Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
TianLi Li ◽  
Wen Wang
Keyword(s):  
Weak Solution ◽  
Partial Derivative ◽  
Weak Solutions ◽  
Global Regularity ◽  
Stokes Equations ◽  
Regularity Criterion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Embedded Technology

In this paper, we study the regularity of the weak solutions for the incompressible 3D Navier–Stokes equations with the partial derivative of the velocity. By the embedded technology, we prove that the weak solution u is regular on (0, T] if ∂ 3 u ∈ L p 0 , T ; L q R 3 with 2 / p + 3 / q = 70 / 37 + 15 / 37 q , 15 / 4 ≤ q ≤ ∞ , or 2 / p + 3 / q = 34 / 19 + 9 / 19 q , 9 / 4 ≤ q ≤ ∞ .

REGULARITY OF WEAK SOLUTIONS FOR THE NAVIER–STOKES EQUATIONS IN THE CLASS L∞(BMO-1)

2012 ◽  
Vol 14 (03) ◽  
pp. 1250020 ◽  
Author(s):  
WENDONG WANG ◽  
ZHIFEI ZHANG
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Removable Singularity ◽  
Navier Stokes ◽  
Singularity Theorem ◽  
Navier Stokes Equations ◽  
Regularity Of Weak Solutions ◽  
Mild Assumption

We study the regularity of weak solution for the Navier–Stokes equations in the class L∞( BMO-1). It is proved that the weak solution in L∞( BMO-1) is regular if it satisfies a mild assumption on the vorticity direction, or it is axisymmetric. A removable singularity theorem in ∈ L∞( VMO-1) is also proved.

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