I will discuss a proof of Onsager’s conjecture that 1/3-Hölder is the threshold regularity for energy conservation for the 3D incompressible Euler equations. The construction is based on the method of convex integration, which was first developed for the incompressible Euler equations by De Lellis and Székelyhidi to build Hölder-continuous Euler flows that fail to conserve energy, and was later improved by Isett and by Buckmaster-De Lellis-Székelyhidi to obtain further partial results towards Onsager's conjecture. The proof to be discussed of the full conjecture combines the use of “Mikado flows” due to Daneri-Székelyhidi with a new "gluing approximation" technique. The latter technique exploits a special structure in the incompressible Euler equations and Euler-Reynolds equations.
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