Wave maps embody many of the difficulties in dealing with physical Lagrangian field theories such as Yang-Mills and Einstein equations, while being much more tractable because of their simple geometric formulation. There are however many interesting and physically-relevant applications where it is natural to consider the stability under small perturbations of some nontrivial static or stationary solution. Very little theoretical work has been done in any of these areas. The problem is further complicated by the presence of a boundary. I plan to study the most basic unsolved problems in this area which have to do with stability or instability of very simple, nonzero steady states to any of the systems mentioned above.
Vortices are lines of singularity for the velocity field of a fluid.
They appear in superfluid Helium in response to being rotated in a container, since the superfluid flow has to remain irrotational at all times. In the Gross-Pitaevskii model, the dynamics of a superfluid is governed by a nonlinear Schrodinger equation for the macroscopic wave function u , and vortices are identified with the zeros of u . The stationary states of a superfluid in a rotating container correspond to the minima of the free energy , which is a functional of the Ginzburg-Landau type. There are very few mathematical results on superfluid dynamics. I will also study the relativistic analogue of this problem.