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Lecture Series by Sara Daneri

Time

15:00-17:00, Dec 3-Dec 5
16:00-18:00, Dec 6

Venue

Room 306, No. 5 Science Building, Minhang Campus, Shanghai Jiao Tong University

Program

Convex integration and applications to the Onsager’s conjecture

Abstract
In 1949, L. Onsager formulated the following conjecture: all solutions of the incompressible Euler equations which are H"older continuous with exponent bigger than $1/3$ conserve the total kinetic energy, while there are solutions which dissipate the total kinetic energy in the space of H"older continuous functions with exponent smaller than $1/3$. The conjecture was formulated in accordance with Kolmogorov’ s ‘41 theory of turbulence. The first part of the conjecture, trivial for $C^1$ solutions, was proved by Constantin, E and Titi in ‘94, using mollification and commutator estimates. The second part of the conjecture challenged the mathematical community for a long time. The first non-conservative solutions of the incompressible Euler equations where found by Scheffer and Shnirelman in the ’90s. Then in 2007 De Lellis and Szekelyhidi discovered that the method of convex integration, used for the first time by Nash in ‘54 to prove the existence of infinitely many $C^1$ isometric embeddings of $n$-dimensional Riemannian manifolds in $\R^{n+2}$, could be developed to produce infinitely many bounded weak solutions with non-constant energy. The regularity of such dissipative solutions was improved up to the critical exponents $1/3$ due to the contribution of various authors, among which De Lellis, Szekelyhidi, Isett, Buckmaster and myself, culminating into the proof of the second part of the Onsager’s conjecture by Isett in 2016. In this series of lectures, after giving an overview of the results which led to the proof of the conjecture, I will explain the main ideas of the proof, underlying the technical issues and the strategies to overcome it. In particular, I will focus on the application of convex integration, making also a parallel with the problem of isometric embeddings.

Prerequisite knowledge: notion of Riemannian manifold and embedding, classical Schauder estimates for elliptic operators, classical mollification estimates, Gronwall’s inequality.

References

“Non-uniqueness and h-principle for H"older-continuous weak solutions of the Euler equations”, Sara Daneri and Szekelyhidi, Arch. Rat. Mech. Anal., 2016
“From isometric embeddings to turbulence” lecture notes by L. Szekelyhidi,
“Onsager’s conjecture for admissible weak solutions”, Buckmaster, De lellis, Szekelyhidi and Vicol, published on Comm. Pure Appl. Math, 2018.