| Abstract: |
| If $u \in C^\theta(\mathbb{T}^3\times [0,T])$ is a solution of the incompressible Euler equations on the three dimensional torus, then its kinetic energy is in the space $C^{\frac{2\theta}{1 - \theta}}([0,T])$. In this talk I will present a recent result obtained in collaboration with L. De Rosa concerning the sharpness of this property for $\theta < 1/3$. I will start by discussing how the convex integration techniques introduced by C. De Lellis and L. Sz\`{e}kelyhidi can be used to generate \emph{approximate} counterexamples. In particular, this follows from a modification of a convex integration scheme introduced by T. Buckmaster, C. De Lellis, L. Sz\`{e}kelyhidi and V. Vicol. Then, I will show how, based on the existence of this approximate counterexamples, one can introduce a non-trivial complete metric space of $C^\theta(\mathbb{T}^3\times [0,T])$ solutions to the Euler equations in which the energy of the typical element (in the sense of Baire) cannot be more regular than $C^{\frac{2\theta}{1-\theta}}([0,T])$. |
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