Vortex dynamics on a domain with holes

We explore the relationship between the hydrodynamic Green’s function and the Dirichlet Green’s function in a bounded domain with holes. This relationship is expressed using the harmonic measures on the domain, following the work of Flucher and Gustafsson (Vortex motion in two-dimensional hydrodynamics, energy renormalization and stability of vortex pairs, TRITA preprint series, 1997). The explicit form of this relation expresses velocity in terms of vorticity in a way which turns out to be very convenient, especially for analysis. We explain the advantages and describe some applications.