Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Abstract

Abstract. Let u be a weak solution of the Navier-Stokes equations in an exterior domain Ω ⊂ R 3 and a time interval [0, T [ , 0 < T ≤ ∞, with initial value u0, external force f = divF , and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u0 and f or on

Regularity of weak solutions to 3D incompressible Navier–Stokes equations

“…These results also hold for 2 < s < ∞, 3 < q < ∞ in [8] for bounded domains, and in [6] for exterior domains. However, there remains a critical case: s = ∞, q = 3.…”

Uniqueness Criteria and Strong Solutions of the Boussinesq Equations in Completely General Domains

“…Our first main result deals with existence of strong solutions of the Boussinesq equations in general domains. For the construction of strong solutions of the instationary Navier-Stokes system (see (8) below) in general domains we refer to [7,9] and to [21,Section V.4.2]. We denote by ∆ = ∆ 2 , A = A 2 the Laplace and Stokes operator, respectively.…”

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

“…In the three dimensional case, a large gap remains between the regularity available in the existence results and the additional regularity required in the sufficient conditions to guarantee the smoothness of weak solutions of the standard Navier-Stokes equations. This gap has been narrowed by the works of Iskauriaza-Seregin-Sverak [18], LadayzhenskayaSeregin [19], Scheffer [25], Serrin [27], Struwe [29], see also [2], [3], [4], [5], [6], [8], [9], [10], [13], [14], [15], [16], [20], [22], [23], [24], [26], [31], [32] and the references therein, which bring about a deeper understanding of the regularity. In particular, some local partial regularity results and Hausdorff dimension estimates on the possible singular set have been obtained for a class of suitable weak solutions defined and constructed in [7], where the principal tools are the so-called generalized energy inequality and a scaling argument.…”