Regularity for Suitable Weak Solutions to the Navier-Stokes Equations in Critical Morrey Spaces (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 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.
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.
Journal of Mathematical Fluid Mechanics, 2019
In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, ∂ 3 u 3. We show that if ∂ 3 u 3 satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor ∂ 1 u 3 are also established, which covers some previous literature.
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].