Journal of Dynamics and Differential Equations ( IF 1.4 ) Pub Date : 2021-04-19 , DOI: 10.1007/s10884-021-09985-1 Igor Kukavica , David Massatt
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 \(L^2\).
中文翻译:
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。