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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations
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by Martin Hadac PDF
Trans. Amer. Math. Soc. 360 (2008), 6555-6572 Request permission

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 $s_1>-\frac 12$ 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: \[ (u_t - |D_x|^\alpha u_x + (u^2)_x)_x + u_{yy} = 0, \quad u(0) = u_0, \] for $\frac 43<\alpha \leq 6$, $s_1>\max (1-\frac 34 \alpha ,\frac 14-\frac 38 \alpha )$, $s_2\geq 0$ 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
  • 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.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2434299