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arXiv · 2301.00804

Convex integration above the Onsager exponent for the forced Euler equations

Aynur Bulut·Manh Khang Huynh·Stan Palasek
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Abstract

We establish new non-uniqueness results for the Euler equations with external force on $\mathbb{T}^{d}$ $(d\geq3)$. By introducing a novel alternating convex integration scheme, we construct non-unique, almost-everywhere smooth, H\"older-continuous solutions with regularity $\frac{1}{2}-$, which is notably above the Onsager threshold of $\frac{1}{3}$. The solutions we construct differ significantly in nature from those which arise from the recent unstable vortex construction of Vishik; in particular, our solutions are genuinely $d$-dimensional ($d\geq3$), and give non-uniqueness results for any smooth data. To the best of our knowledge, this is the first instance of a convex integration construction above the Onsager exponent.

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Aynur Bulut, Manh Khang Huynh, Stan Palasek. 2023-01-02. Convex integration above the Onsager exponent for the forced Euler equations. https://arxiv.org/abs/2301.00804

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