# Onsager's Conjecture

# Onsager's Conjecture

Now rightfully called Onsager’s Theorem, we discuss the prediction published by Lars Onsager in 1949 which described a sharp regularity threshold for energy conservation in ideal fluid turbulence. Seemingly unknown to the PDE community before the 1990s — and practically unmentioned again after Onsager’s announcement until 1975 — the conjecture and the developments leading to its full resolution by 2017 (due to the works of De Lellis, Szekelyhidi Jr., Buckmaster, Vicol, Isett, and many more) have become a hot topic in the field. The use of the word “conjecture” in the title of this talk is because the focus will be on the mathematical analysis and physical phenomenology which led Onsager to form his prediction and the first works to resolve the “rigid” half of the statement published in the 1990s. The counterintuitive theorems proven in the past decade and a half which led to the proof of the “flexible” side of the conjecture have surprising connections to the embedding theorems of John Nash, but we will merely attempt to state these results.