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Well-posedness for the fifth-order KdV equation in the energy space


Authors: Carlos E. Kenig and Didier Pilod
Journal: Trans. Amer. Math. Soc. 367 (2015), 2551-2612
MSC (2010): Primary 35Q53, 35Q35, 35A01; Secondary 37K05, 76B15
DOI: https://doi.org/10.1090/S0002-9947-2014-05982-5
Published electronically: December 4, 2014
MathSciNet review: 3301874
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Abstract: We prove that the initial value problem (IVP) associated to the fifth-order KdV equation

$\displaystyle \partial _tu-\partial ^5_x u=c_1\partial _xu\partial _x^2u+c_2\partial _x(u\partial _x^2u)+c_3\partial _x(u^3),$ (0.1)

where $ x \in \mathbb{R}$, $ t \in \mathbb{R}$, $ u=u(x,t)$ is a real-valued function and $ \alpha , \ c_1, \ c_2, \ c_3$ are real constants with $ \alpha \neq 0$, is locally well-posed in $ H^s(\mathbb{R})$ for $ s \ge 2$. In the Hamiltonian case (i.e. when $ c_1=c_2$), the IVP associated to (0.1) is then globally well-posed in the energy space $ H^2(\mathbb{R})$.

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Additional Information

Carlos E. Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: cek@math.uchicago.edu

Didier Pilod
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP: 21945-970, Rio de Janeiro, RJ, Brazil
Email: didier@im.ufrj.br

DOI: https://doi.org/10.1090/S0002-9947-2014-05982-5
Keywords: Fifth-order KdV equation, fifth-order water-waves models, initial value problem
Received by editor(s): May 3, 2012
Received by editor(s) in revised form: June 23, 2012, and October 6, 2012
Published electronically: December 4, 2014
Additional Notes: The first author was partially supported by NSF Grant DMS-0968472
The second author was partially supported by CNPq/Brazil, Grant 200001/2011-6
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.