Minimal initial data for potential Navier–Stokes singularities

Abstract

Assuming some initial data u 0 ∈Ḣ 1/2 (R 3 ) lead to a singularity for the 3d Navier-Stokes equations, we show that there are also initial data with the minimalḢ 1/2 -norm which will produce a singularity.

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

“…This was shown in [3] for the whole-space case using the 'persistence of singularities' established by Rusin andŠverák in [38]. Subsequently, this was established for the halfspace case by Seregin in [40].…”

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

“…This is enough to pass to the limit in (12). Notice that the a priori estimate forces div u L 2 t L 2…”

Global Weak Besov Solutions of the Navier–Stokes Equations and Applications

“…Proof. In order to prove the above theorem, we utilise the weak- * stability properties of global Besov solutions, along with arguments related to those contained in [45] and [44]. Assume the hypotheses of the theorem.…”

“…One might interpret the situation |x n | → ∞ and inf n∈N t n > ε (4. 45) as meaning that u 0 has certain nice properties which cause the singularities to disappear at spatial infinity as the initial data approaches the sphere of radius ρ u 0 X centered on u 0 , and similarly for t n → ∞. Since ρ X ≤ ρ u 0 X , one is tempted to say that, in terms of its ability to "prevent" the blowup of nearby solutions, u 0 is at least as good as zero initial data.…”