Dissipative Euler flows with Onsager-critical spatial regularity

For any $ϵ>0$ we show the existence of continuous periodic weak solutions $v$ of the Euler equations which do not conserve the kinetic energy and belong to the space $L_t^1(C_x^{\frac{1}{3}-ϵ})$, namely $x\mapsto v (x,t)$is$(\frac{1}{3}-\epsilon)$-H\ddot{o}ldercontinuousinspaceata\mathrm{.}e\mathrm{.}time$t$andtheintegral$\int [v(\cdot, t)]_{\frac{1}{3}-\epsilon} dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L_t^{\mathrm{\infty }}(C_x^{\frac{1}{3}-ϵ})$.