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

Abstract

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