Energy bounds for a fourth-order equation in low dimensions related to wave maps

Abstract

For compact, isometrically embedded Riemannian manifolds N ↪ R L N \hookrightarrow \mathbb {R}^L , we introduce a fourth-order version of the wave maps equation. By energy estimates, we prove an a priori estimate for smooth local solutions in the energy subcritical dimension n = 1 n = 1 , 2 2 . The estimate excludes blow-up of a Sobolev norm in finite existence times. In particular, combining this with recent work of local well-posedness of the Cauchy problem, it follows that for smooth initial data with compact support, there exists a (smooth) unique global solution in dimension n = 1 n=1 , 2 2 . We also give a proof of the uniqueness of solutions that are bounded in these Sobolev norms.