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Admissible solutions of the Euler equations
Abstract
In this paper, we prove the existence of an extensive class of initial data--characterized by intermittency dimension and roughness--for which the d-dimensional Euler equations (d \ge 2) admit a decreasing energy channel with infinitely many admissible solutions arising from the same initial data. This class includes initial data in C^{1-κ}(\mathbb{T}^d) and W^{1,p}(\mathbb{T}^d) for any 0 < κ< 1 and 2 < p < \infty, thereby demonstrating the sharpness of the Lions and Yudovich uniqueness principles. Our proof is based on a new backward convex integration scheme employing approximate Λ-Euler equations to reduce the initial error and on uncovering a quantitative relationship between regularity, energy, and intermittency.
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BibTeX
@misc{cheskidov2026admissible,
title = {{Admissible solutions of the Euler equations}},
author = {Alexey Cheskidov and Zirong Zeng and Deng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22491},
eprint = {2610.22491},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22491},
}
RIS
TY - UNPB
TI - Admissible solutions of the Euler equations
AU - Alexey Cheskidov
AU - Zirong Zeng
AU - Deng Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22491
UR - https://arxiv.org/abs/2610.22491
AB - In this paper, we prove the existence of an extensive class of initial data--characterized by intermittency dimension and roughness--for which the d-dimensional Euler equations (d \ge 2) admit a decreasing energy channel with infinitely many admissible solutions arising from the same initial data. This class includes initial data in C^{1-κ}(\mathbb{T}^d) and W^{1,p}(\mathbb{T}^d) for any 0 < κ< 1 and 2 < p < \infty, thereby demonstrating the sharpness of the Lions and Yudovich uniqueness principles. Our proof is based on a new backward convex integration scheme employing approximate Λ-Euler equations to reduce the initial error and on uncovering a quantitative relationship between regularity, energy, and intermittency.
ER -