Self-similar vorticity around the boundary and non-uniqueness of solutions to the two-dimensional Navier-Stokes equations in the half space

TL;DR

This paper demonstrates the non-uniqueness of solutions to 2D Navier-Stokes equations in a half space by analyzing boundary-bound vorticity instability at high Reynolds numbers, highlighting boundary layer effects.

Contribution

It introduces a novel construction of non-unique solutions based on boundary-focused self-similar vorticity instability, extending prior work on solution uniqueness.

Findings

Non-uniqueness of mild solutions in half space

Boundary layer effects are crucial for vorticity instability

Self-similar vorticity concentrates near boundary at initial time

Abstract

In this paper we show the non-uniqueness of mild solutions to the two-dimensional forced Navier-Stokes equations in the half space under the noslip boundary condition, following the program established by Albritton, Bru{\'e}, and Colombo in 2022. Our construction of non-unique solutions is based on the instability of self-similar vorticity at high Reynolds numbers which concentrates around the boundary at the initial time. In our construction, therefore, a kind of boundary layer has to be taken into account in the analysis, contrasting to the known results where the unstable self-similar vorticity is located away from the boundary with $O(1)$ distance around the initial time.