Global well-posedness in the energy space for a modified KP II equation via the Miura transform

Abstract

We prove global well-posedness of the initial value problem for a modified Kadomtsev-Petviashvili II (mKP II) equation in the energy space. The proof proceeds in three main steps and involves several different techniques. In the first step, we make use of several linear estimates to solve a fourth-order parabolic regularization of the mKP II equation by a fixed point argument , for regular initial data (one estimate is similar to the sharp Kato smoothing effect proved for the KdV equation by Kenig, Ponce, and Vega, 1991). Then, compactness arguments (the energy method performed through the Miura transform) give the existence of a local solution of the mKP II equation for regular data. Finally, we approximate any data in the energy space by a sequence of smooth initial data. Using Bourgain's result concerning the global well-posed-ness of the KP II equation in L 2 and the Miura transformation, we obtain convergence of the sequence of smooth solutions to a solution of mKP II in the energy space.