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 $C^0_t (W^{\frac{1}{3}-, 3} \cap L^{\infty-})$. More precisely, for every $\beta\lt \frac{1}{3}$, we can construct such solutions in the space $C^0_t (B^{\beta}_{3,\infty} \cap L^{\frac{1}{1-3\beta}} )$.