The triple collision singularity in the 3-vortex problem is studied in this paper. Under the necessary condition k1-1 + k 2-1 + k3-1 = 0 for vorticities to have the triple collision, the main results are summarized as follows: (i) For k1 = k2, the triple collision singularity is topologically regularizable. (ii) For 0 < |k1 - k2| < ε with a sufficiently small ε, the triple collision singularity is not topologically regularizable. First of all, in order to prove these statements, all singularities in the 3-vortex problem are classified. Then, we introduce a dynamical system by blowing up the triple collision singularity with an appropriate time scaling. Roughly speaking, it corresponds to pasting an invariant manifold at the triple collision singularity on the original phase space. This technique is well known as McGehee's collision manifold (1974 Inventions Math. 27 191-227) in the N-body problem of celestial mechanics. Finally, by adopting the viewpoint of Easton (1971 J. Diff. Eqns 10 92-9), topological regularizations of the triple collision singularity are studied in detail.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics