About local continuity with respect to $$L_{2}$$ initial data for energy solutions of the Navier–Stokes equations

Abstract

In this paper we consider classes of initial data that ensure local-in-time Hadamard well-posedness of the associated weak Leray–Hopf solutions of the three-dimensional Navier–Stokes equations. In particular, for any solenodial $$L_{2}$$ initial data $$u_{0}$$ belonging to certain subsets of $$VMO^{-1}(\mathbb {R}^3)$$ , we show that weak Leray–Hopf solutions depend continuously with respect to small divergence-free $$L_{2}$$ perturbations of the initial data $$u_{0}$$ (on some finite-time interval). Our main result is inspired by and improves upon previous work of the author (Barker in J Math Fluid Mech 20(1):133–160, 2018) and work of Jean–Yves Chemin (Commun Pure Appl Math 64(12):1587–1598, 2011). Our method builds upon [4, 9]. In particular our method hinges on decomposition results for the initial data inspired by Calderon (Trans Am Math Soc 318(1):179–200, 1990) together with use of persistence of regularity results. The persistence of regularity statement presented may be of independent interest, since it does not rely upon the solution or the initial data being in the perturbative regime.