On the Global Existence for the Kuramoto-Sivashinsky Equation

We address the global existence of solutions for the 2D Kuramoto-Sivashinsky equations in a periodic domain [0,L1]×[0,L2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,L_1]\times [0,L_2]$$\end{document} with initial data satisfying ‖u0‖L2≤C-1L2-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert u_0\Vert _{L^2}\le C^{-1}L_2^{-2}$$\end{document}, where C is a constant. We prove that the global solution exists under the condition L2≤1/CL13/5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2\le 1/C L_1^{3/5}$$\end{document}, improving earlier results. The solutions are smooth and decrease energy until they are dominated by CL13/2L21/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C L_1^{3/2}L_2^{1/2}$$\end{document}, implying the existence of an absorbing ball in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}.