Global bifurcation for vortex desingularization and internal bores

Bifurcation theory offers a robust strategy for finding nontrivial, parameter-dependent families of solutions, and has proven to be very successful in many areas of applications. The existence of families of perturbation of the trivial solutions is addressed by means of local bifurcation theory. Global bifurcation theory employs topological methods to deal with extending the local solutions as far as possible to a connected set of solutions. Since global bifurcation is not a perturbative approach, one expects that this global continuum provides solutions that are not small disturbances of the trivial ones. 

In this talk we will give a quick review of the analytic global bifurcation theory due to Dancer and Buffoni-Toland, and introduce a new machinery we recently developed with an emphasis to treat the problems on non-compact domains. As two applications in hydrodynamics, we will report results on desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices, and the existence of families of large-amplitude internal hydrodynamic bores. This is a joint work with Samuel Walsh and Miles Wheeler.