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

Abstract

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 $\text{a priori}$ 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 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 $n=1,2$. We also give a proof of the uniqueness of solutions that are bounded in these Sobolev norms.

v2: typos fixed, introductory section updated and title changed according to request of referee. To appear Proc. Amer. Math. Soc

References in corpus (2)

• Biharmonic wave maps into spheres
• Biharmonic wave maps: Local wellposedness in high regularity

Cited by in corpus (2)

• Numerical approximation of bi-harmonic wave maps into spheres
• Global results for a Cauchy problem related to biharmonic wave maps