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Energy bounds for biharmonic wave maps in low dimensions

Schmid, Tobias

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.

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DOI: 10.5445/IR/1000088929
Veröffentlicht am 21.12.2018
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht
Jahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000088929
Verlag KIT, Karlsruhe
Umfang 12 S.
Serie CRC 1173 ; 2018/51
Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015)
Schlagworte biharmonic, fourth-order wave equation, energy estimates, global solutions
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