  We consider logarithmic spiral vortex sheets consisting of two branches.
Based on some simple assumptions that appear true by numerical computations, we
fully classify their long-time behavior and asymptotics, where in all cases
each branch decays to $0$ or blows up in finite time. Furthermore, we present
illustrations determining which range of initial data corresponds to each case.
We also determine the asymptotic stability of the symmetric and asymmetric
self-similar spirals.
