Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

Abstract

This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space L 3 , then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia andŠverák, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if (0, T * ) is a singular point thenThis result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely L 3,∞ and the Besov spaceḂ −1+ 3 p p,∞ , p ∈ (3, ∞).

Short Time Regularity of Navier–Stokes Flows with Locally L3 Initial Data and Applications

“…We thank Professors Barker and Prange who kindly sent us their preprint [1] while we are finishing this paper. The preprint [1] contains a result similar to our Corollary 1.2 for local energy weak solutions.…”

Scale-Invariant Estimates and Vorticity Alignment for Navier–Stokes in the Half-Space with No-Slip Boundary Conditions

“…In this case, we have the following lighter notation: (p, r; 0, 0). 7 Technically, (28) must be assumed instead of (16). For the purpose of this discussion we overlook this point.…”

“…This point is discussed in more details in Section 1.3 below. (3) In the paper [7] on concentration phenomena for Type I blow-up solutions of the Navier-Stokes equations, a scale-critical condition such as (28) was used to provide a control in L 2,uloc (R 3 ) of a rescaled solution.…”

“…In particular, in [20] it was shown that if η : [0, ∞) → R is a nondecreasing continuous function with η(0) = 0 , ξ = ω |ω| and Ω d := {(x, t) ∈ R 3 + × (0, T ) : |ω(x, t)| > d} then the following holds true. Namely, the assumption (7) |ξ(x, t) − ξ(y, t)| ≤ η(|x − y|) for all (x, t), (y, t) ∈ Ω d implies that u is bounded up to t = T .…”

Quantitative Regularity for the Navier–Stokes Equations Via Spatial Concentration

“…The contradiction is achieved by showing that the limit function must be zero by applying a Liouville type theorem based on backward uniqueness for parabolic operators satisfying certain differential inequalities. By now there are many generalizations of (6) to cases of other critical norms. See, for example, [12,16,35,50].…”