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For any $\gamma<1/3$, we construct a nontrivial weak solution $u$ to thetwo-dimensional, incompressible Euler equations, which has compact support intime and satisfies $u\in C^\gamma(\mathbb R_t \times \mathbb T^2_x)$. Inparticular, the constructed solution does not conserve energy and, thus,settles the flexible part of the Onsager conjecture in two dimensions. Theproof involves combining the Nash iteration technique with a new linear Newtoniteration.

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