= Compact self-similar disk with square-root density viscosity
{title2=$S=\tau^{-1}(1-x/(6\sqrt\tau))_+^2$}
For $\nu=\nu_0(\Sigma/\Sigma_0)^{1/2}(R/R_0)^{9/4}$, the variables $x=\sqrt{R/R_0}$, $S=(\Sigma/\Sigma_0)x^3$ and $\tau=3\nu_0t/(4R_0^2)$ reduce the source-free <Keplerian viscous diffusion equation> to $S_\tau=(xS^{3/2})_{xx}$. The displayed <self-similar solution> has an outer edge $R_e=36R_0\tau$ and a surface-density cusp proportional to $R^{-3/2}$ at the origin. The positive-part cutoff is essential: beyond the edge, $S^{3/2}$ is the cube of the absolute value, so continuing the squared polynomial without truncation does not solve the same equation.
The disk mass is $8\pi\Sigma_0R_0^2\tau^{-1/2}$ and its <angular momentum> is $12\pi\Sigma_0R_0^2\sqrt{GMR_0}$, independent of time. It therefore describes accretion with outward spreading and conserved disk angular momentum, under the formal zero-torque condition at the origin. The inward mass flux changes sign at $R=9R_0\tau/4$, so the outer portion moves outward while the inner portion accretes.
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