Differential and Integral Equations

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

Luc Molinet

Full-text: Open access

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 $.

Article information

Source
Differential Integral Equations, Volume 13, Number 1-3 (2000), 189-216.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356124296

Mathematical Reviews number (MathSciNet)
MR1811955

Zentralblatt MATH identifier
0974.35109

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35A05 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B65: Smoothness and regularity of solutions

Citation

Molinet, Luc. The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation. Differential Integral Equations 13 (2000), no. 1-3, 189--216. https://projecteuclid.org/euclid.die/1356124296


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