  We derive continuous dependence estimates for weak entropy solutions of
degenerate parabolic equations with nonlinear fractional diffusion. The
diffusion term involves the fractional Laplace operator, $\Delta^{\alpha/2}$
for $\alpha \in (0,2)$. Our results are quantitative and we exhibit an example
for which they are optimal. We cover the dependence on the nonlinearities, and
for the first time, the Lipschitz dependence on $\alpha$ in the $BV$-framework.
The former estimate (dependence on nonlinearity) is robust in the sense that it
is stable in the limits $\alpha \downarrow 0$ and $\alpha \uparrow 2$. In the
limit $\alpha \uparrow 2$, $\Delta^{\alpha/2}$ converges to the usual
Laplacian, and we show rigorously that we recover the optimal continuous
dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231-251,
1999) for local degenerate parabolic equations (thus providing an alternative
proof).
