Navier–Stokes equations, stability and minimal perturbations of global solutions

In a recent paper, we proved that assuming some initial data ϕ ∈ H ˙ 1 / 2 ( R 3 ) lead to a singularity for the 3D Navier–Stokes equations, there are also initial data with the minimal H ˙ 1 / 2 -norm which produce a singularity and the set of such data is compact up to translations and the natural scaling of the equation. The purpose of this paper is to analyze a more general case where the set of initial data potentially leading to a singularity is on a sphere centered at non-zero initial data leading to a global solution.