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

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- by Carlos E. Kenig and Didier Pilod PDF
- Trans. Amer. Math. Soc.
**367**(2015), 2551-2612 Request permission

## Abstract:

We prove that the initial value problem (IVP) associated to the fifth-order KdV equation \begin{equation*} \tag {0.1} \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), \end{equation*} 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
- MR Author ID: 100230
- 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
- MR Author ID: 837520
- Email: didier@im.ufrj.br
- 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 - © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3301874