Motivated by the theory of hydrodynamic turbulence, Onsager conjectured in 1949 that solutions to the incompressible Euler equations with Holder regularity less than 1/3 may fail to conserve energy. DeLellis and Szkelyhidi have pioneered an approach to constructing such irregular flows based on an iteration scheme used by Nash to construct "wild" C^1 isometric embeddings. This approach involves correcting "approximate solutions" by adding rapid oscillations which are designed to reduce the error term in solving the equation. In this talk, I will discuss recent work on an improved iteration scheme using nonlinear phase functions for the corrections, which yields solutions in three dimensions with compact support in time and Holder regularity below 1/5.