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arXiv · 2102.12866

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

Tobias Schmid
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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.

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Tobias Schmid. 2022-09-18. Energy bounds for a fourth-order equation in low dimensions related to wave maps. https://doi.org/10.1090/proc%2F16100

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