On the Cauchy problem for the periodic fifth-order KP-I equation

TL;DR

This paper establishes the global well-posedness of the periodic fifth-order KP-I equation for initial data with constant mean in a natural energy space, advancing understanding of high-order dispersive PDEs.

Contribution

It proves global well-posedness for the periodic fifth-order KP-I equation in a natural energy space, a novel result for this high-order dispersive PDE.

Findings

Global well-posedness in the energy space.

Well-posedness holds for initial data with constant mean.

Advances understanding of high-order KP-I equations.

Abstract

The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation \[\partial_t u - \partial_x^5 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0,~(t,x,y)\in\mathbb{R}\times\mathbb{T}^2\] We prove global well-posedness for constant $x$ mean value initial data in the space $\mathbb{E} = \{u\in L^2,~\partial_x^2 u \in L^2,~\partial_x^{-1}\partial_y u \in L^2\}$ which is the natural energy space associated with this equation.