Dimension of the singular set of wild Hölder solutions of the incompressible Euler equations

Abstract For ? &lt; 1 3 , we consider ${C}^{\beta }\left({\mathbb{T}}^{3}\times [0,T]\right)$?> C T × [ 0 , T stretchy="false">] weak solutions of the incompressible Euler equations that do not conserve kinetic energy. We prove for such closed and non-empty set singular times $\mathcal{B}$?> mathvariant="script">B satisfies ${\text{dim}}_{\mathcal{H}}(\mathcal{B})\geqslant \frac{2\beta }{1-\beta }$?> dim mathvariant="script">H stretchy="false">( stretchy="false">) ? 2 ? . This lower bound on Hausdorff dimension in time is intrinsically linked to Hölder regularity energy conjecture it be sharp. As a first step this direction, every {\beta }^{\prime }< ? are able construct, via convex integration scheme, non-conservative system ${\text{dim}}_{\mathcal{H}}(\mathcal{B})\leqslant \frac{1}{2}+\frac{1}{2}\frac{2{\beta }}{1-{\beta }}$?> ? + The structure wild build allows moreover deduce non-uniqueness Cauchy problem from smooth initial datum.