Viscous spreading of a narrow planetary ring

In a local Keplerian sheet, slow viscous drift gives ${\partial }_t\Sigma =3{\partial }_x^2(\nu \Sigma )$. For $\nu =A{\Sigma }^2$, the compact profile $\Sigma =\sigma \sqrt{1-x^2/w^2}$ preserves its shape, conserves $\sigma w$, and spreads with $w\propto t^{1/4}$ at late times.