On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier—Stokes equations

Abstract

We prove a criterion of local Hölder continuity for suitable weak solutions to the Navier-Stokes equations. One of the main part of the proof, based on a blow-up procedure, has quite general nature and can be applied to other problems in spaces of solenoidal vector fields. Mathematics Subject Classification (1991). 35K, 76D.In the present paper we deal with weak solutions to the three-dimensional NavierStokes equationsHopf proved the global existence at least one weak solution v to the first initial boundary value problem with boundary condition v| ∂Ω×[0,T ] = 0 under quite general assumptions on domain Ω, external force f and initial data v| t=0 = a. In [5] his results were discribed and the class of Hopf's solutions was introduced. Corresponding definition includes all main properties that essentially were proved by E. Hopf. More precisely, a velocity field v is called Hopf's solution if it belongs to T ] in the weak topology of L 2 (Ω; R 3 ) and satisfies the integral identityfor all t ∈ [0, T ] and for all w ∈ • J 1 2 (Ω). No information on the pressure p is given. Here • J(Ω) is the L 2 (Ω; R 3 )-closure of the set of all smooth solenoidal fields

Estimates of suitable weak solutions to the Navier-Stokes equations in critical Morrey spaces

“…Here, B(r) is the ball of radius r centered at the origin so that B = B (1). This definition is due to O. Ladyzhenskaya and the author, see [5], and slightly differs from the most popular one by L. Caffarelli, R.-V. Kohn, and L. Nirenberg given in their celebrated paper [1]. Our interest to the above question is motivated by the important observation of J. Leray made in his remarkable paper [7].…”

“…J. Leray proved the uniqueness of regular solutions in the class of turbulent solutions which are also called weak Leray-Hopf solutions. For exact definitions, we refer the reader to the paper [7] and, for example, to papers [5] and [10].…”

“…Here, we follow the definition of suitable weak solution in F.-H. Lin's reduction, see [8] and also [5] for discussions on other definitions. holds for all non-negative functions ϕ ∈ C ∞ 0 (R 3 × R 1 ) vanishing in a neighborhood of the parabolic boundary ∂ ′ Q of the cylinder Q.…”

“…The space L3 2 seems to be the most convenient for treating. For more details, we refer the reader to the paper [5].…”

“…Here and in what follows, we denote all positive universal constants by c. A proof of (2.1) is given, for example, in [5].…”