Preprint

Inserted: 5 sep 2024
Last Updated: 5 sep 2024

Year: 2024

Abstract:

We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $L^\infty L^2$ weak solutions with vorticity in $L^\infty L^p$ for some $p>1$, surpassing for the first time the critical scaling of the standard convex integration technique.

To achieve this, we introduce several new ideas, including: \begin{itemize} \item(i) A new family of building blocks built from the Lamb-Chaplygin dipole. \item(ii) A new method to cancel the error based on time averages and non-periodic, spatially-anisotropic perturbations. \end{itemize}

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• Nonuniqueness_of_Euler_Solutions @13@.pdf