Energy bounds for biharmonic wave maps in low dimensions

Schmid, Tobias

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 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.

 Zugehörige Institution(en) am KIT Institut für Analysis (IANA)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2018 Sprache Englisch Identifikator ISSN: 2365-662X URN: urn:nbn:de:swb:90-889291 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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