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Global well-posedness in the energy space for a modified KP II equation via the Miura transform


Authors: Carlos E. Kenig and Yvan Martel
Journal: Trans. Amer. Math. Soc. 358 (2006), 2447-2488
MSC (2000): Primary 35Q53; Secondary 35G25
DOI: https://doi.org/10.1090/S0002-9947-06-04072-4
Published electronically: January 24, 2006
MathSciNet review: 2204040
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Abstract: We prove global well-posedness of the initial value problem for a modified Kadomtsev-Petviashvili II (mKP II) equation in the energy space. The proof proceeds in three main steps and involves several different techniques.

In the first step, we make use of several linear estimates to solve a fourth-order parabolic regularization of the mKP II equation by a fixed point argument, for regular initial data (one estimate is similar to the sharp Kato smoothing effect proved for the KdV equation by Kenig, Ponce, and Vega, 1991).

Then, compactness arguments (the energy method performed through the Miura transform) give the existence of a local solution of the mKP II equation for regular data.

Finally, we approximate any data in the energy space by a sequence of smooth initial data. Using Bourgain's result concerning the global well-posedness of the KP II equation in $ L^2$ and the Miura transformation, we obtain convergence of the sequence of smooth solutions to a solution of mKP II in the energy space.


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Additional Information

Carlos E. Kenig
Affiliation: Institute for Advanced Study and Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Yvan Martel
Affiliation: Centre de Mathématiques, Ecole Polytechnique 91128 Palaiseau Cedex, France
Address at time of publication: Department of Mathematics, Université de Versailles–Saint-Quentin-en- Yvelines, 78035 Versailles, France

DOI: https://doi.org/10.1090/S0002-9947-06-04072-4
Keywords: Modified KP II equation, initial value problem, Miura transform, energy method
Received by editor(s): April 22, 2004
Published electronically: January 24, 2006
Additional Notes: The first author was supported in part by the NSF and at IAS by The von Neumann Fund, The Oswald Veblen Fund and the Bell Companies Fellowship.
Part of this work was done while the second author was a Member of the IAS. He was partially supported by the NSF, under agreement No. DMS-0111298
Article copyright: © Copyright 2006 American Mathematical Society

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