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

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