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

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 Šverák [JŠ14], is a central tool in two of the authors' recent work on quantitative $L_x^3$ 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 $L_x^3$ norm must concentrate at scales $\sim \sqrt{T^{\ast }-t}$ in the presence of a Type I blow-up.

36 pages, 0 figures