On the low regularity of the fifth order Kadomtsev–Petviashvili I equation

Abstract

We consider the fifth order Kadomtsev–Petviashvili I (KP-I) equation as ∂ t u + α ∂ x 3 u + ∂ x 5 u + ∂ x −1 ∂ y 2 u + u u x = 0 , while α ∈ R . We introduce an interpolated energy space E s to consider the well-posedness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in E s for 0 < s ⩽ 1 . To obtain the local well-posedness, we present a bilinear estimate in the Bourgain space in the framework of the interpolated energy space. It crucially depends on the dyadic decomposed Strichartz estimate, the fifth order dispersive smoothing effect and maximal estimate.