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, the local energy inequality, and belong to $C_t^0(W^{1/3-,3}\cap L^{\infty-})$. More precisely, for every $\beta < 1/3$, we can construct such solutions in the space $C_t^0(B_{3,\infty}^\beta \cap L^{\frac{1}{1-3\beta}})$.