Conservation laws for fourth order systems

In the first part of this talk I will briefly review the recent result of Tristan Rivière on the existence of a conservation law for weak solutions of the Euler-Lagrange equation of conformally invariant variational integrals in two dimensions. I will then show how we can adapt these arguments to show the existence of a conservation law for fourth order systems, including biharmonic maps into general target manifolds, in four dimensions. With the help of this conservation law I will prove the continuity of weak solutions of these systems. If time permits I will also indicate how one can use this conservation law to prove the existence of a unique weak solution of the biharmonic map flow in the energy space.

This is a joint work with Tristan Rivière (ETH Zuerich).