## 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.## References

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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