Geometry and dynamics of singular vorticities in ideal fluids

Project description

The project's purpose is to study the geometry and dynamics of singular vorticities in an ideal fluid, with emphasis on those with codimension one vorticity support. The objective of this project has two sides. The theoretical side includes: to study the geometry of the reduced dynamics of axisymmetric vortex sheets in R³, i.e. surfaces of revolution fibrate by vortex lines, to deal with vortex sheets in Rⁿ (including vortex curves in R²), to tackle the more general singular vorticities, like non-closed vortex sheets (infinite ribbons or tubes), domains with non-trivial cohomology (domains with holes), vortex patches. On the applied sides of singular vorticities, we will look for geometric integrators for vortex sheet dynamics and for a fruitful relation between spaces of vortex sheets and shape spaces of decorated surfaces.