Conservation Laws for Fourth Order Systems in Four Dimensions

Abstract

Following an approach of the second author (Rivière, 2007 Rivière , T. ( 2007 ). Conservation laws for conformally invariant variational problems . Invent. Math. 168 : 1 – 22 . [Crossref], [Web of Science ®] [Google Scholar] ) for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuity of weak solutions of this system. Moreover we use the conservation law to derive the existence of a unique global weak solution of the extrinsic biharmonic map flow in the energy space.