Global well-posedness of the KP-I initial-value problem in the energy space

We prove that the KP-I initial-value problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{cases} \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 \,\text{ on }\,\mathbb{R}^2_{x,y}\times\mathbb{R}_t;\\ u(0)=\phi, \end{cases}$$\end{document} is globally well-posed in the energy space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{E}^1(\mathbb{R}^2)=\big\{\phi:\mathbb{R}^2\to\mathbb{R}: \|\phi\|_{\mathbf{E}^1(\mathbb{R}^2)}\approx\|\phi\|_{L^2}+\|\partial_x\phi\|_{L^2}+\big\|\partial_x^{-1}\partial_y\phi\big\|_{L^2}<\infty\big\}.$$\end{document}