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

Abstract

This paper is concerned with geometric regularity criteria for the Navier-Stokes equations in R 3 + × (0, T ) with no-slip boundary condition, with the assumption that the solution satisfies the 'ODE blow-up rate' Type I condition. More precisely, we prove that if the vorticity direction is uniformly continuous on subsets ofwhere the vorticity has large magnitude, then (0, T ) is a regular point. This result is inspired by and improves the regularity criteria given by Giga, Hsu and Maekawa in [20].We also obtain new local versions for suitable weak solutions near the flat boundary. Our method hinges on new scaled Morrey estimates, blow-up and compactness arguments and 'persistence of singularites' on the flat boundary. The scaled Morrey estimates seem to be of independent interest.

Localized smoothing and concentration for the Navier-Stokes equations in the half space

“…As a consequence, as of now there is still no proof of a Constantin and Fefferman-type geometric nonlinearity depletion criterion [CF93] outside the critical setting. In the critical case, i.e., under a Type I assumption, such a result was obtained in [GHM14] thanks to the proof of a complicated Liouville theorem, and in [BP20b] via a new strategy based on the stability of Type I singularities. The relationship between boundary effects and potential singularity formation is discussed in [LT16].…”

“…Hence in short, p ≃ u ⊗ u for the half space, contrary to the whole space. This issue brought about considerable difficulties in the work [BP20b] when clarifying the relationship between different notions of Type I singularities in the half space; see also the discussion on page 5 above Theorem 1.4. A way to circumvent the difficulty was to rely on 'fractional pressure estimates' pioneered in the work [CK18] and reproved in [BP20b] via the formulas of [MMP19].…”

“…This issue brought about considerable difficulties in the work [BP20b] when clarifying the relationship between different notions of Type I singularities in the half space; see also the discussion on page 5 above Theorem 1.4. A way to circumvent the difficulty was to rely on 'fractional pressure estimates' pioneered in the work [CK18] and reproved in [BP20b] via the formulas of [MMP19]. (4) The pressure associated to the Stokes resolvent problem in a bounded domain Ω with no-slip boundary condition and source term f satisfies the following bound:…”

“…The assumption (1.7) is a weak form of Type I, adapted to the scaling-invariant energy. For Leray-Hopf solutions, (1.7) is implied by the L ∞ and L 3,∞ conditions, though this is not trivial, see [BP20b,Theorem 2] for the precise statement with boundary and [AB19] for a discussion without boundary.…”