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| In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equations belonging to the H\"older space with H\"older exponent greater than $1/3$ conserve energy; conversely, he conjectured the existence of solutions belonging to any H\"older space with exponent less than $1/3$ which dissipate energy.
The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). In this talk we discuss recent work related to resolving the second component of Onsager's conjecture. In particular, we present a proof of a weak version of the conjecture: there exists weak non-conservative solutions to the Euler equations whose $1/3-\epsilon$ H\"older norm in space is Lebesgue integrable in time. |
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