Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier–Stokes equations

Abstract

We review some results concerning the problem of global-in-time regularity for the initial boundary value problem for the Navier-Stokes equations in three-dimensional domains. In particular, we focus on sufficient conditions on the vorticity field which imply that strong (hence smooth) solutions exist on arbitrary time intervals.

Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

“…We start in the same way as in the proof of Theorem 2 until the inequality (36). Due to (17), there exists an arbitrarily small T 1 ∈ (0, 𝜒) such that u(T 1 ) ∈ L 3 𝜎 (Ω) ∩ L ∞ (Ω). Taking the inner product of (1) with P 𝜎 (u|u| 𝜆 ) on (T 1 , 𝜒) and integrating over Ω, we obtain…”

“…16 In this connection, see also the discussion concerning the power 1/2 by Beirao da Veiga. 17 As of now, the power 1/2 is the best (smallest) which has been achieved, and its possible improvement (decrease) seems to be a hard open problem. For the exclusion of the blow-ups of type I, the 1/2−Hölder continuity assumption can be replaced by a more general uniform continuity assumption, see Giga and Miura 18 and Giga and Sohr.…”

On the Serrin-type condition on one velocity component for the Navier-Stokes equations

“…[21], [13], [14], [9], [8], [34]). For conditions on the pressure see [29], [36], [11], [6], [3] and on the vorticity see [4], [5], [10] (for more discussion on the topic see [22]). Furthermore, in [1] it has been proved that the Serrin condition (1.4) imposed on two components guarantees the regularity (for the localization of this result see [2]).…”

Some Remarks on the Boundary Conditions in the Theory of Navier-Stokes Equations

“…The case where the viscosity is constant was studied by Shimada [37] for the Stokes problem with Navier boundary conditions. Berselli [14] gives some criteria concerning the vorticity field which imply the global regularity for the Navier-Stokes equations with stress-free boundary conditions (see also Chen-Osborne-Qian [20]).…”