Indexed on: 10 Mar '16Published on: 05 Jun '15Published in: Communications on Pure and Applied Mathematics
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 (13−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). © 2015 Wiley Periodicals, Inc.