Long-time behavior of logarithmic spiral vortex sheets with two branches

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.