We address the global existence of solutions for the 2D Kuramoto-Sivashinsky equations in a periodic domain \([0,L_1]\times [0,L_2]\) with initial data satisfying \(\Vert u_0\Vert _{L^2}\le C^{-1}L_2^{-2}\), where C is a constant. We prove that the global solution exists under the condition \(L_2\le 1/C L_1^{3/5}\), improving earlier results. The solutions are smooth and decrease energy until they are dominated by \(C L_1^{3/2}L_2^{1/2}\), implying the existence of an absorbing ball in \(L^2\).

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关于Kuramoto-Sivashinsky方程的整体存在性

我们解决了周期域\（[0，L_1] \ times [0，L_2] \）中二维Kuramoto-Sivashinsky方程解的全局存在性，其初始数据满足\（\ Vert u_0 \ Vert _ {L ^ 2 } \ le C ^ {-1} L_2 ^ {-2} \），其中C是常数。我们证明全局解在条件\（L_2 \ le 1 / C L_1 ^ {3/5} \）下存在，从而改善了早期结果。解是平滑的并降低能量，直到它们由\（CL_1 ^ {3/2} L_2 ^ {1/2} \）支配，这意味着\（L ^ 2 \）中存在吸收球。