Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations

Abstract

Abstract. In this paper we consider the Cauchy problem for the n-dimensional Navier-Stokes equations and we prove a regularity criterion for weak solutions involving the summability of the pressure. Related results for the initial-boundary value problem are also presented.

The Cauchy problem for the magneto-hydrodynamic system

“…Uniqueness and regularity criterion based on fluid vorticity and electrical current in BMO and Besov spaces. Similarly to the case of the Navier-Stokes equations [2,4,19,31], we establish the uniqueness or regularity criterion for the MHD system (1.1)-(1.4).…”

“…One natural problem is to find suitable conditions which assure the uniqueness and regularity of the weak solutions for the MHD equation (1.1)- (1.4). Similarly to the cases of Navier-Stokes equations [2,4,21], Wu in [33] proved a regularity result for general classical weak solutions for the MHD equations in R 3 by energy integral estimates. More precisely, let (u 0 (x), b 0 (x)) ∈ L p (R 3 ) ∩ H 1 (R 3 ) with p > 3, and assume that (u(x, t), b(x, t)) ∈…”

“…Many authors studied the uniqueness and regularity of the Leray-Hopf weak solutions to Navier-Stokes equations in different frameworks, see [2,4,9], [14]- [22], [24,31]. Recently, Zhang and Chen in [39] applied Littlewood-Paley trichotomy to study the spacetime estimates for the Navier-Stokes equations and extended the well-known regularity criterion of weak solutions and blow-up criterion of the smooth solutions.…”

“…BMO(R n ) stands for the bounded mean oscillation space, and H 1 (R n ) stands for the Hardy space, which is the dual space of BMO(R n ). We take an arbitrary function ϕ in the Schwartz class S(R n ) and whose Fourier transformφ is such that 0 ≤φ ≤ 1 and φ(ξ) = 1, if |ξ| ≤ 3 4 , ϕ(ξ) = 0, if |ξ| ≥ 4 3 .…”

Локальные Формулы Для Гидродинамического Давления И Их Приложения

“…С другой стороны, π при малых r имеет порядок r 2 |∇u| 2 . Для 3-мерных уравнений Навье-Стокса несжимаемой жидкости хорошо известны критерии регулярности [1], [7]. Если бы давление удовлетворяло тем же оценкам, что и β, отсюда легко следовала бы регулярность решений 3-мерных уравнений Навье-Стокса.…”