Abstract

The Euler equations govern the flow of incompressible inviscid fluids. In two dimensions the vorticity (the curl of the velocity vector field) is preserved along particle trajectories. The talk will focus on a special class of weak solutions, called vortex patches, where the initial vorticity is the indicator function of some bounded simply connected region. Since vorticity is preserved, the solution will remain an indicator of some bounded simply connected region, with the region evolving in time.

After reviewing some basic notions such as particle trajectory maps and the vorticity stream formulation, we define a suitable weak formulation in order to be able to discuss such solutions. We will then sketch an argument to show that if the boundary of the vortex patch is initially sufficiently smooth, it will remain smooth globally in time.