Encontros IST-IME 

05/09/2016, 09:40 — 10:20 — Anfiteatro Pa2, Pavilhão de Matemática
Clodoaldo Ragazzo, IME, Universidade de São Paulo

Hydrodynamic Vortex on Surfaces

The equations of motion for a system of point vortices on an oriented Riemannian surface of finite topological type is presented.

The equations are obtained from a Green's function on the surface. The uniqueness of the Green's function is established under hydrodynamic conditions at the surface's boundaries and ends. The hydrodynamic force on a point vortex is computed using a new weak formulation of Euler's equation adapted to the point vortex context. An analogy between the hydrodynamic force on a massive point vortex and the electromagnetic force on a massive electric charge are presented as well as the equations of motion for massive vortices. Any noncompact Riemann surface admits a unique Riemannian metric such that a single vortex in the surface does not move (Steady Vortex Metric). Some examples of surfaces with steady vortex metric isometrically embedded in $\mathbb{R}^3$ are presented.