The parity conjecture for one-dimensional vorticity models

Consider the one-dimensional vorticity equation on $\mathbb{S}^1$,

$$\partial_t\omega+u\partial_x\omega=\omega\partial_xu,\qquad u=\mathcal{K}(\omega),$$

where $\mathcal{K}$ is a non-trivial Fourier integral operator with bounded symbol $m$ satisfying

$$|m(k)|\leq\frac{C}{1+|k|}.$$

Parity conjecture. In general, $H^s$ solutions should develop singularities when $s<3/2$; if $m$ is odd, there exist analytic solutions that become singular in finite time; and if $m$ is even, $H^s$ solutions must be global whenever $s>3/2$. These claims summarize the conjectured dependence of regularity and blow-up on the parity of the symbol and remain open in this generality.