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

TL;DR

This paper introduces a fourth-order wave map equation for low-dimensional manifolds and establishes energy bounds that guarantee global existence and uniqueness of smooth solutions in dimensions one and two.

Contribution

It develops a new fourth-order wave map model and proves energy estimates ensuring global well-posedness in low dimensions, extending previous results for second-order wave maps.

Findings

Energy estimates prevent blow-up in low dimensions

Global existence of smooth solutions in dimensions 1 and 2

Uniqueness of solutions bounded in Sobolev norms

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 $\textit{a priori}$ 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.