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 Lt1(Cx1/3−ε); namely, x ↦ v (x,t) is ⅓−ε‐Hölder continuous in space at a.e. time t and the integral ∫[ υ(⋅,t) ]1/3−εdt is finite. A well‐known open conjecture of L. Onsager claims that such solutions exist even in the class Lt∞(Cx1/3−ε).© 2016 Wiley Periodicals, Inc.