Regularity of Pressure in the Neighbourhood of Regular Points of Weak Solutions of the Navier-Stokes Equations (original) (raw)

Commentary on local and boundary regularity of weak solutions to Navier-Stokes equations

Electronic Journal of Differential Equations, 2004

We present results on local and boundary regularity for weak solutions to the Navier-Stokes equations. Beginning with the regularity criterion proved recently in [14] for the Cauchy problem, we show that this criterion holds also locally. This is also the case for some other results: We present two examples concerning the regularity of weak solutions stemming from the regularity of two components of the vorticity ([2]) or from the regularity of the pressure ([3]). We conclude by presenting regularity criteria near the boundary based on the results in [10] and [16].

On the regularity of the solutions of the Navier–Stokes equations via one velocity component

Nonlinearity, 2010

We consider the regularity criteria for the incompressible Navier-Stokes equations connected with one velocity component. Based on the method from [4] we prove that the weak solution is regular, provided u 3 ∈ L t (0, T ; L s (R 3)), 2 t + 3 s ≤ 3 4 + 1 2s , s > 10 3 or provided ∇u 3 ∈ L t (0, T ; L s (R 3)), 2 t + 3 s ≤ 19 12 + 1 2s if s ∈ (30 19 , 3] or 2 t + 3 s ≤ 3 2 + 3 4s if s ∈ (3, ∞]. As a corollary, we also improve the regularity criteria expressed by the regularity of ∂p ∂x 3 or ∂u 3 ∂x 3 .

A new regularity criterion for weak solutions to the Navier–Stokes equations

Journal de Mathématiques Pures et Appliquées, 2005

In this paper we obtain a new regularity criterion for weak solutions to the 3-D Navier-Stokes equations. We show that if any one component of the velocity field belongs to L α ([0, T); L γ (R 3)) with 2 α + 3 γ ≤ 1 2 , 6 < γ ≤ ∞, then the weak solution actually is regular and unique. Titre. Un nouveau critère de régularité pour les solutions faibles deséquations de Navier-Stokes Resumé. Dans cet article, on obtient un nouveau critère de régularité pour les solutions faibles deséquations de Navier-Stokes en dimension 3. On démontre que si une conposante quelconque du champ de vitesse appartientà L α ([0, T ]; L γ (R 3)) avec 2 α + 3 γ ≤ 1 2 , 6 < γ ≤ ∞, alors la solution faible est régulière et unique.

Some New Regularity Criteria for the Navier-Stokes Equations Containing Gradient of the Velocity

Applications of Mathematics, 2000

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz 49 (2004) APPLICATIONS OF MATHEMATICS No. 5, 483-493 SOME NEW REGULARITY CRITERIA FOR THE NAVIER-STOKES EQUATIONS CONTAINING GRADIENT OF THE VELOCITY*