New Sufficient Conditions of Local Regularity for Solutions to the 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.

Conditions Implying Regularity of the Three Dimensional Navier-Stokes Equation

Applications of Mathematics, 2005

We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac like inequalities. As part of the our methods, we give a different approach to a priori estimates of Foiaş, Guillopé and Temam.

Regularity for Suitable Weak Solutions to the Navier-Stokes Equations in Critical Morrey Spaces

2006

A class of sufficient conditions of local regularity for suitable weak solutions to the nonstationary three-dimensional Navier-Stokes equations are discussed. The corresponding results are formulated in terms of functionals which are invariant with respect to the Navier-Stokes equations scaling. The famous Caffarelli-Kohn-Nirenberg condition is contained in that class as a particular case.

Regularity criteria for the three-dimensional Navier-Stokes equations

Indiana University Mathematics Journal, 2008

In this paper we consider the three-dimensional Navier-Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one component of the velocity field, or alternatively in terms of one component of the pressure gradient, for the regularity of strong solutions to the three-dimensional Navier-Stokes equations.

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].

Regularity Criteria in Weak L3 for 3D Incompressible Navier-Stokes Equations

Funkcialaj Ekvacioj, 2015

We study the regularity of a distributional solution (u, p) of the 3D incompressible evolution Navier-Stokes equations. Let B r denote concentric balls in R 3 with radius r. We will show that if p ∈ L m (0, 1; L 1 (B 2)), m > 2, and if u is sufficiently small in L ∞ (0, 1; L 3,∞ (B 2)), without any assumption on its gradient, then u is bounded in B 1 × (1 10 , 1). It is an endpoint case of the usual Serrin-type regularity criteria, and extends the steady-state result of Kim-Kozono to the time dependent setting. In the appendix we also show some nonendpoint borderline regularity criteria.

The Regularity of Weak Solutions of the 3D Navier―Stokes Equations in B―1∞,∞

Archive For Rational Mechanics and Analysis, 2010

We show that if a Leray-Hopf solution u to the 3D Navier-Stokes equation belongs to C((0, T ]; B −1 ∞,∞ ) or its jumps in the B −1 ∞,∞norm do not exceed a constant multiple of viscosity, then u is regular on (0, T ]. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya-Prodi-Serrin criterion.