Onsager-type conjecture for active scalar equations

Let $\theta$ be a weak solution of the active scalar equation, let $\delta$ be the parameter appearing in the equation, and set

$$\Lambda=(-\Delta)^{\frac12}.$$

The Hamiltonian is

$$\mathcal{H}(t)=\frac12\int_{\mathbb{T}^2}\theta T_0[\theta](x,t)\,dx,$$

with $T_0$ the Fourier multiplier defined in the preceding setup. Onsager-type conjecture. All weak solutions $\theta$ satisfying

$$\Lambda^{-1}\theta\in C_{t,x}^{1+\frac{2\delta}{3}}$$

conserve the Hamiltonian. Conversely, for every

$$\frac{1+\delta}{2}\leq\gamma<1+\frac{2\delta}{3},$$

there exist weak solutions satisfying

$$\Lambda^{-1}\theta\in C_t^0C_x^\gamma$$

that do not conserve the Hamiltonian. This conjecture proposes a sharp Onsager-type regularity threshold for Hamiltonian conservation in the active scalar equations; the positive and negative parts respectively assert conservation above the threshold and anomalous dissipation below it.