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Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations


Author: Martin Hadac
Journal: Trans. Amer. Math. Soc. 360 (2008), 6555-6572
MSC (2000): Primary 35Q53; Secondary 35B30
DOI: https://doi.org/10.1090/S0002-9947-08-04515-7
Published electronically: July 22, 2008
MathSciNet review: 2434299
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Abstract: We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space $ H^{s_1,s_2}(\mathbb{R}^2)$ with $ s_1>-\frac12$ and $ s_2\geq 0$. On the $ H^{s_1,0}(\mathbb{R}^2)$ scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for the following generalisation of the KP II equation:

$\displaystyle (u_t - \vert D_x\vert^\alpha u_x + (u^2)_x)_x + u_{yy} = 0, \quad u(0) = u_0, $

for $ \frac43<\alpha\leq 6$, $ s_1>\max(1-\frac34 \alpha,\frac14-\frac38 \alpha)$, $ s_2\geq 0$ and $ u_0\in H^{s_1,s_2}(\mathbb{R}^2)$. We deduce global well-posedness for $ s_1\geq 0$, $ s_2=0$ and real valued initial data.


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Additional Information

Martin Hadac
Affiliation: Mathematical Institute of the University of Bonn, Beringstraße 1, D-53115 Bonn, Germany
Email: hadac@math.uni-bonn.de

DOI: https://doi.org/10.1090/S0002-9947-08-04515-7
Keywords: Kadomtsev-Petviashvili II equation, Cauchy problem, local well-posedness.
Received by editor(s): January 22, 2007
Published electronically: July 22, 2008
Additional Notes: The research for this work was mainly carried out while the author was employed at the Department of Mathematics of the University of Dortmund.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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