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

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.