Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

Abstract

We prove two results concerning local regularity theory of the Navier-Stokes equations near a curved portion Γ ⊂ ∂ Ω of the boundary. Suppose that u is a boundary suitable weak solution with singularity (x * , T * ), where x * ∈ Ω ∪ Γ. Then (under a weak background assumption) the L 3 norm of u tends to infinity in every ball centered at x * :Furthermore, u gives rise to a non-trivial mild bounded ancient solution in R 3 or R 3 + through a rescaling procedure that zooms in on the singularity. This generalizes results of Seregin et al. into the local setting with curved boundary. Our proof relies on a truncation procedure for boundary suitable weak solutions. The former result is based on energy estimates for L 3 initial data and a Liouville theorem. For the latter result, we apply perturbation theory for L ∞ initial data due to K. Abe and Y. Giga. The curved boundary is treated in two different ways.

Quantitative regularity for the Navier-Stokes equations via spatial concentration

“…This result has been further refined to other wider critical spaces and to domains other than R 3 . See, for example, [2], [3]- [4], [8] and [28]. All these arguments are qualitative and achieved by contradiction and compactness arguments.…”

“…As a consequence of these quantitative estimates, Tao 3 We say (X, • X ) ⊂ S (R 3 ) is critical if u0 ∈ X ⇒ u 0λ (x) := λu(λx) ∈ X with X norm equal to that of u0. 4 The argument in [5] is in turn taken from the talk given by G . Seregin.…”

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

“…In [11], Constantin and Fefferman provided a geometric regularity criteria for the vorticity ω = ∇ × u of solutions on the whole-space, which remarkably does not depend on scale-invariant quantities. 1 Specifically, they showed that if 2 (2) | sin(∠(ω(x + y, t), ω(x, t))| ≤ C|y| for (x, t) ∈ Ω d := {(y, s) ∈ R 3 × (0, T ) : |Ω(y, s)| > d} Date: June 20, 2019. 1 By scale-invariant quantities, we mean quantities which are invariant with respect to the Navier-Stokes rescaling (u λ (y, s), p λ (y, s)) = (λu(λy, λ 2 s), λ 2 p(λy, λ 2 s)).…”

“…The vast majority of regularity criteria for the Navier-Stokes equations are stated in terms of scale-invariant quantities since, heuristically at least, the diffusive effects and non-linear effects are 'balanced'. 2 Here ∠(a, b) denotes the angle between the vectors a and b.…”

“…The following lemma is a crucial tool. It is taken from the first author's Thesis [6, Proposition 5.5]; see also [2,Proposition A.5] and [37,Proposition 5.21] for subsequent generalisations. The stability of singular points was first proven in the interior case by Rusin and Šverák [38].…”