Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35Q53, 35B30

Retrieve articles in all journals with MSC (2000): 35Q53, 35B30


Additional Information

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

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.