Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Biharmonic wave maps into spheres
HTML articles powered by AMS MathViewer

by Sebastian Herr, Tobias Lamm and Roland Schnaubelt PDF
Proc. Amer. Math. Soc. 148 (2020), 787-796 Request permission

Abstract:

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau-type approximation.
References
  • Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
  • Jishan Fan and Tohru Ozawa, On regularity criterion for the 2D wave maps and the 4D biharmonic wave maps, Current advances in nonlinear analysis and related topics, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 32, Gakk\B{o}tosho, Tokyo, 2010, pp. 69–83. MR 2668271
  • A. Freire, Global weak solutions of the wave map system to compact homogeneous spaces, Manuscripta Math. 91 (1996), no. 4, 525–533. MR 1421290, DOI 10.1007/BF02567971
  • Andreas Gastel, The extrinsic polyharmonic map heat flow in the critical dimension, Adv. Geom. 6 (2006), no. 4, 501–521. MR 2267035, DOI 10.1515/ADVGEOM.2006.031
  • Emmanuel Hebey and Benoit Pausader, An introduction to fourth order nonlinear wave equations, http://www.math.univ-paris13.fr/ pausader/HebPausSurvey.pdf.
  • Sebastian Herr, Tobias Lamm, Tobias Schmid, and Roland Schnaubelt, Biharmonic wave maps: Local wellposedness in high regularity, arXiv e-prints (2019), arXiv:1903.01813.
  • Tobias Lamm, Heat flow for extrinsic biharmonic maps with small initial energy, Ann. Global Anal. Geom. 26 (2004), no. 4, 369–384. MR 2103406, DOI 10.1023/B:AGAG.0000047526.21237.04
  • Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR 1329547, DOI 10.1007/978-3-0348-9234-6
  • Roger Moser, Weak solutions of a biharmonic map heat flow, Adv. Calc. Var. 2 (2009), no. 1, 73–92. MR 2494507, DOI 10.1515/ACV.2009.004
  • Benoit Pausader, Scattering for the defocusing beam equation in low dimensions, Indiana Univ. Math. J. 59 (2010), no. 3, 791–822. MR 2779061, DOI 10.1512/iumj.2010.59.3966
  • A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
  • Tobias Schmid, Energy bounds for biharmonic wave maps in low dimensions, CRC 1173-Preprint 2018/51, Karlsruhe Institute of Technology, 2018.
  • Jalal Shatah, Weak solutions and development of singularities of the $\textrm {SU}(2)$ $\sigma$-model, Comm. Pure Appl. Math. 41 (1988), no. 4, 459–469. MR 933231, DOI 10.1002/cpa.3160410405
  • Jalal Shatah and Michael Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics, vol. 2, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. MR 1674843
  • Changyou Wang, Heat flow of biharmonic maps in dimensions four and its application, Pure Appl. Math. Q. 3 (2007), no. 2, Special Issue: In honor of Leon Simon., 595–613. MR 2340056, DOI 10.4310/PAMQ.2007.v3.n2.a9
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35L75, 58J45
  • Retrieve articles in all journals with MSC (2010): 35L75, 58J45
Additional Information
  • Sebastian Herr
  • Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
  • MR Author ID: 785145
  • Email: herr@math.uni-bielefeld.de
  • Tobias Lamm
  • Affiliation: Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
  • MR Author ID: 748462
  • Email: tobias.lamm@kit.edu
  • Roland Schnaubelt
  • Affiliation: Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
  • MR Author ID: 603222
  • Email: schnaubelt@kit.edu
  • Received by editor(s): December 10, 2018
  • Received by editor(s) in revised form: June 24, 2019
  • Published electronically: August 7, 2019
  • Additional Notes: The second and third author gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.
  • Communicated by: Joachim Krieger
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 787-796
  • MSC (2010): Primary 35L75, 58J45
  • DOI: https://doi.org/10.1090/proc/14744
  • MathSciNet review: 4052215