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For $\beta<\frac13$, we consider $C^\beta(\mathbb{T}^3\times [0,T])$ weaksolutions of the incompressible Euler equations that do not conserve thekinetic energy. We prove that for such solutions the closed and non-empty setof singular times $\mathcal{B}$ satisfies $\dim_{\mathcal{H}}(\mathcal{B})\geq\frac{2\beta}{1-\beta}$. This lower bound on the Hausdorff dimension of thesingular set in time is intrinsically linked to the H\"older regularity of thekinetic energy and we conjecture it to be sharp. As a first step in thisdirection, for every $\beta<\beta'<\frac{1}{3}$ we are able to construct, via aconvex integration scheme, non-conservative $C^\beta(\mathbb{T}^3\times [0,T])$weak solutions of the incompressible Euler system such that$\dim_{\mathcal{H}}(\mathcal{B})\leq\frac{1}{2}+\frac{1}{2}\frac{2\beta'}{1-\beta'}$. The structure of the wildsolutions that we build allows moreover to deduce non-uniqueness of$C^\beta(\mathbb{T}^3\times [0,T])$ weak solutions of the Cauchy problem forEuler from every smooth initial datum.

Dimension of the singular set of wild H\"older solutions of the incompressible Euler equations