Differential and Integral Equations

Local well posedness for modified Kadomstev-Petviashvili equations

C. E. Kenig and S. N. Ziesler

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Abstract

In this paper we consider the Kadomstev-Petivashvili equation and also the modified Kadomstev-Petviashvili equation, with nonlinearity $\partial_x(u^3).$ We improve on previous results of Iório and Nunes [5], and also on previous work of the authors, [13]. For the modified $(KP-II)$ equation we give optimal (up to endpoint) maximal function type estimates for the solution of the associated linear initial-value problem. These estimates enable us to obtain a local well-posedness result via the contraction mapping principle. For modified $(KP-I)$ we use methods developed by Kenig in [9], which use an energy estimate together with Strichartz estimates and "interpolation inequalities." We give some counterexamples to well posedness via the contraction mapping principle, for both the Kadomstev-Petviashvili equation and the modified equation.

Article information

Source
Differential Integral Equations, Volume 18, Number 10 (2005), 1111-1146.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060108

Mathematical Reviews number (MathSciNet)
MR2162626

Zentralblatt MATH identifier
1212.35419

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

Citation

Kenig, C. E.; Ziesler, S. N. Local well posedness for modified Kadomstev-Petviashvili equations. Differential Integral Equations 18 (2005), no. 10, 1111--1146. https://projecteuclid.org/euclid.die/1356060108


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