Electron. J. Diff. Eqns., Vol. 2009(2009), No. 82, pp. 1-9.

On the Cauchy-problem for generalized Kadomtsev-Petviashvili-II equations

Axel Grunrock

Abstract:
The Cauchy-problem for the generalized Kadomtsev-Petviashvili-II equation
$$ u_t + u_{xxx} + \partial_x^{-1}u_{yy}= (u^l)_x, \quad l \ge 3, $$
is shown to be locally well-posed in almost critical anisotropic Sobolev spaces. The proof combines local smoothing and maximal function estimates as well as bilinear refinements of Strichartz type inequalities via multilinear interpolation in $X_{s,b}$-spaces.

Submitted April 9, 2009. Published July 10, 2009.
Math Subject Classifications: 35Q53.
Key Words: Cauchy-problem; local well-posedness; generalized KP-II equations.

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Axel Grünrock
Rheinische Friedrich-Wilhelms-Universit&uaml;t Bonn, Mathematisches Institut
Beringstrase 1, 53115 Bonn, Germany
email: gruenroc@math.uni-bonn.de

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