Energy bounds for biharmonic wave maps in low dimensions

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