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

TL;DR

This paper proves a local-in-space smoothing effect for the Navier-Stokes equations in the half space, addressing the challenges posed by non-local pressure effects and applying it to analyze blow-up behavior.

Contribution

It establishes a new local smoothing result for Navier-Stokes in the half space, extending previous whole space results and handling non-local pressure effects.

Findings

Proves local smoothing effect in half space for Navier-Stokes

Shows critical $L^3_x$ norm concentration at specific scales during blow-up

Addresses non-local pressure effects in half space setting

Abstract

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 \v{S}ver\'ak [J\v{S}14], is a central tool in two of the authors' recent work on quantitative $L^3_x$ blow-up criteria [BP21]. 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 $L^3_x$ norm must concentrate at scales $\sim \sqrt{T^* - t}$ in the presence of a Type I blow-up.