Dissipative Euler Flows with Onsager‐Critical Spatial Regularity

For any ɛ > 0 we show the existence of continuous periodic weak solutions v of the Euler equations that do not conserve the kinetic energy and belong to the space ; namely, x ↦ v ( x , t ) is ⅓−ε‐Hölder continuous in space at a.e. time t and the integral is finite. A well‐known open conjecture of L. Onsager claims that such solutions exist even in the class .