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| The study of incompressible Euler flows with Holder regularity is motivated by the theory of hydrodynamic turbulence. In connection with this theory, L. Onsager conjectured that solutions to incompressible Euler with Holder regularity less than $1/3$ may fail to satisfy the conservation of energy. C. DeLellis and L. Sz\'{e}kelyhidi, Jr. have pioneered an approach to constructing such irregular flows based on an iteration scheme known as convex integration. 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 an improved framework which leads to solutions with Holder regularity as much as $1/5-$. |
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