The Onsager conjecture in 2D: a Newton-Nash iteration

For any $\gamma <1/3$ $\gamma <1/3$ , we construct a nontrivial weak solution $u$ $u$ to the two-dimensional, incompressible Euler equations, which has compact support in time and satisfies $u\in C^{\gamma }({\mathbb{R}}_t\times {\mathbb{T}}_x^2)$ $u\in C^{\gamma }(\mathbb{R}_{t} \times \mathbb{T}^{2}_{x})$ . In particular, the constructed solution does not conserve energy and, thus, settles the flexible part of the Onsager conjecture in two dimensions. The proof involves combining the Nash iteration technique with a new linear Newton iteration.