The $L^3$-based strong Onsager theorem

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 $C^0_t (W^{\frac 13-, 3} \cap L^{\infty-})$. More precisely, for every $\beta