Local well posedness for modified Kadomstev-Petviashvili equations. (English) Zbl 1212.35419

Kenig, C. E.; Ziesler, S. N.

Differ. Integral Equ. 18, No. 10, 1111-1146 (2005).

Summary: In this paper we consider the Kadomstev-Petviashvili equation and also the modified Kadomstev-Petviashvili equation, with nonlinearity \(\partial _x(u^3)\). 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, 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.

MSC:

35Q53

KdV equations (Korteweg-de Vries equations)

Keywords:

Kadomstev-Petviashvili equations