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arXiv · 2507.02338

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

Motofumi Aoki·Yasunori Maekawa
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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.

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Motofumi Aoki, Yasunori Maekawa. 2025-07-03. Self-similar vorticity around the boundary and non-uniqueness of solutions to the two-dimensional Navier-Stokes equations in the half space. https://arxiv.org/abs/2507.02338

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