Finite-time breakdown conjecture for the b-equations hierarchy

Let $A$ be a formally symmetric nonnegative pseudodifferential operator, and consider the one-dimensional Euler–Arnold equation

$$\omega_t+u\omega_r+2u_r\omega=0,\qquad \omega=Au.$$

For $b\in\mathbb{R}$, its $b$-version is obtained by replacing the coefficient $2$ in the transport law by $b$.

Finite-time breakdown conjecture for the b-equations. If, for some operator $A$, the Euler–Arnold equation has $C^1$ solutions that break down in finite time, then for every $b>1$ the corresponding $b$-version also has $C^1$ solutions that break down in finite time.

This conjecture predicts that finite-time breakdown persists throughout the range $b>1$, independently of the particular operator once breakdown is known for the original transport law. The source emphasizes that global behavior depends strongly on $b$, and gives no proof of this assertion.