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

TL;DR

This paper investigates the well-posedness of the fifth order KP-I equation in a new energy space, establishing local solutions for low regularity initial data using advanced harmonic analysis techniques.

Contribution

It introduces an interpolated energy space and proves local well-posedness for the fifth order KP-I equation at low regularity levels, employing novel bilinear estimates in Bourgain spaces.

Findings

Established local well-posedness in $E_s$ for $0<s extless=1$

Developed bilinear estimates using dyadic Strichartz and dispersive smoothing

Demonstrated the effectiveness of the interpolated energy space approach

Abstract

We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as $\partial_tu+\alpha\partial_x^3u+\partial^5_xu+\partial_x^{-1}\partial_y^2u+uu_x=0,$ while $\alpha\in \mathbb{R}$. We introduce an interpolated energy space $E_s$ to consider the well-posedeness 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 $0<s\leq1$. 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. Key words: The fifth order KP-I equation, Bourgain space, Dyadic decomposed Strichartz estimate, Dispersive smoothing effect, Maximal estimate.