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 $\mathbf E = \{u\in L^2,~{ \partial_x}^2 u \in L^2, ~{ \partial_x}^{-1} { \partial_y} u \in L^2\}$ which is the natural energy space associated with this equation.

中文翻译：

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关于周期五阶KP-I方程的柯西问题

本文的目的是研究周期性五阶KP-I方程的Cauchy问题$$ {\ partial_t} u-{\ partial_x} ^ 5 u-{\ partial_x} ^ {-1} {\ partial_y} ^ 2u + u {\ partial_x} u = 0，〜（t，x，y）\ in \ mathbb R \ times \ mathbb T ^ 2。$$我们证明了空间$ \ mathbf E = \ {u \ in L ^ 2，〜{\ \ partial_x} ^ 2 u \ in L ^ 2，〜{ \ partial_x} ^ {-1} {\ partial_y} u \ inL ^ 2 \} $，它是与该方程关联的自然能量空间。