Boundary epsilon regularity for incompressible Navier--Stokes equations via weak-strong uniqueness

TL;DR

This paper proves boundary regularity for weak solutions of 3D incompressible Navier--Stokes equations under small boundary norm conditions, using a novel slicing technique.

Contribution

It establishes boundary regularity criteria based on the $L^4_tL^4_x$-norm, addressing a previously open problem with a new slicing method.

Findings

Weak solutions are regular up to the boundary if the boundary norm is sufficiently small.

Introduces a new slicing construction near the boundary for analysis.

Answers an open problem in boundary regularity for Navier--Stokes equations.

Abstract

We show that finite-energy weak solutions to the incompressible Navier--Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the $L^4_tL^4_x$-norm of the solution is smaller than a constant depending only on the domain. This answers a problem raised in [D. Albritton, T. Barker, and C. Prange, J. Math. Fluid Mech. 25 (2023), Paper No. 49]. Our proof relies on a new slicing construction near the boundary of the domain.