The $L^3$-based strong Onsager theorem

Abstract

In this work, we prove the $L^3$-based strong Onsager conjecture for the three-dimensional Euler equations. Our main theorem states that there exist weak solutions which dissipate the total kinetic energy, satisfy the local energy inequality, and belong to $C_t^0(W^{\frac{1}{3}-,3}\cap L^{\infty -})$. More precisely, for every ${\hat{I}}^2<\frac{1}{3}$, we can construct such solutions in the space $C_t^0(B_{3,\infty }^2\cap L^{\frac{1}{1-3{\hat{I}}^2}})$.

final version, to appear in Annals of Mathematics