Proceedings of the St. Petersburg Mathematical Society, Volume II (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].

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.

Regularity for the stationary Navier-Stokes equations in bounded domain

Archive for Rational Mechanics and Analysis, 1994

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain f~ _~ IR N, 5 ~< N < oo. If u, p satisfy the additional conditions (B) fVuV(ujdx<__f(~+p)u.Vydx+ff.uTdx V7 ~ C~(a), y __> 0, they become regular. Moreover, it is proved that every weak solution u, p satisfying (A) with q = oe is regular. The existence of such solutions for N = 5 has been established in a former paper [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.

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.

Local regularity of the Navier–Stokes equations near the curved boundary

Journal of Mathematical Analysis and Applications, 2010

We present some regularity conditions for suitable weak solutions of the Navier-Stokes equations near the curved boundary of a sufficiently smooth domain. Our extend the work that was results established near a flat boundary by Gustafson, Kang and Tsai (2006) [6].