Commentary on local and boundary regularity of weak solutions to Navier-Stokes equations (original) (raw)

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.

Regularity of Pressure in the Neighbourhood of Regular Points of Weak Solutions of the Navier-Stokes Equations

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 48 (2003)

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 .

Regularity criterion for solutions to the Navier–Stokes equations in the whole 3D space based on two vorticity components

Journal of Mathematical Analysis and Applications, 2018

We prove, among others, the following regularity criterion for the solutions to the Navier-Stokes equations: If u is a global weak solution satisfying the energy inequality and ω = ∇ × u, then u is regular on (0, T), T > 0, if two components of ω belong to the space L q (0, T ; Ḃ −3/p ∞,∞) for p ∈ (3, ∞) and 2/q + 3/p = 2. This result is an improvement of the results presented by Chae and Choe (1999) [7] or Zhang and Chen (2005) [38]. Our method of the proof uses a suitable application of the Bony decomposition and can also be used for the proofs of some other kin criteria. One such example is presented in Appendix.