Global Solutions of the Two-Dimensional Kuramoto–Sivashinsky Equation with a Linearly Growing Mode in Each Direction
Global Solutions of the Two-Dimensional Kuramoto–Sivashinsky Equation with a Linearly Growing Mode in Each Direction
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DOI：
10.1007/s00332-021-09748-8
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发表时间：
2021-02
影响因子：
3
通讯作者：
D. Ambrose;A. Mazzucato
中科院分区：
文献类型：
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作者：
D. Ambrose;A. Mazzucato
我们考虑二维空间中的Kuramoto-Sivashinsky方程。对于足够小的数据，我们首次证明了在两个空间方向上线性增长模态存在时解的整体存在性。为此，我们开发了一种新的方法，将波数分为低（线性增长模式），中间（作为低模式能量汇的线性衰减模式）和高（强线性衰减模式）。低阶和中阶模态由李雅普诺夫函数控制，而高阶模态由基于维纳代数的函数空间中的算子估计控制。
We consider the Kuramoto–Sivashinsky equation in two space dimensions. We establish the first proof of global existence of solutions in the presence of a linearly growing mode in both spatial directions for sufficiently small data. We develop a new method to this end, categorizing wavenumbers as low (linearly growing modes), intermediate (linearly decaying modes that serve as energy sinks for the low modes), and high (strongly linearly decaying modes). The low and intermediate modes are controlled by means of a Lyapunov function, while the high modes are controlled with operator estimates in function spaces based on the Wiener algebra.