scholarly journals Prodi–Serrin condition for 3D Navier–Stokes equations via one directional derivative of velocity

2021 ◽  
Vol 298 ◽  
pp. 500-527
Author(s):  
Chen Hui ◽  
Wenjun Le ◽  
Chenyin Qian
Keyword(s):  
Stokes Equations ◽  
Directional Derivative ◽  
Navier Stokes ◽  
Navier Stokes Equations

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].

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].

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