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

We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and Sverak [21], is a central tool in two of the authors' recent work on quantitative L3x blow-up criteria [7]. The main difficulty is that the non-local effects of the pressure in the half space are much stronger than in the whole space. As an application, we demonstrate that the critical L3x norm must concentrate at scales similar to root T* - t in the presence of a Type I blow-up.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This article establishes a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space, overcoming the challenge of stronger non-local pressure effects. The application demonstrates that the critical L3x norm must concentrate at scales similar to root T* - t in the presence of a Type I blow-up.