The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation

Abstract

We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers (KP-Burgers) equation in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts which makes it quite particular. After showing local existence by contraction principle for initial data $ \varphi\in H^s(\mathbb R^2) $ such that $ {\mathcal F}^{-1} (\frac{k_2}{k_1} \widehat{\varphi})\in H^r(\mathbb R^2) $, $ 0{\leqslant} r {\leqslant} s- 1 $, we try to extend the solutions for all positive times. Whereas for $ {\varepsilon}=-1 $ and $ 1{\leqslant} p < 4/3 $ this will be done without any assumption on the the initial data, we will require a smallness condition on the initial data otherwise. In a last part we prove a local smoothing effect in the transverse direction, which enables us to establish the existence of weak global solutions in $ L^2(\mathbb R^2) $ when $ {\varepsilon}=-1 $ and $ 1{\leqslant} p < 4/3 $.