论文标题
在汉堡方程的库普曼操作员上
On Koopman Operator for Burgers' Equation
论文作者
论文摘要
我们在$ l^2（[0,1]）$中的一组（小）函数上考虑汉堡方程式的流动。我们明确得出了汉堡流的库普曼分解。我们将这种分解的频率和系数确定为Koopman操作员的特征值和特征功能。我们证明，对于小凯奇（Cauchy）数据，$ t> 0 $的Koopman分解的融合，对于常规Cauchy数据，最多可达$ t = 0 $。收敛到$ t = 0 $}导致Koopman模式的“完整性”属性。我们构建所有模式和本本功能，包括与几何多重性有关的特征空间。这超出了（Page＆Kerswell，2018年）提供的总和公式，该公式只给出了每个特征值的一个任期。给出了Koopman分解的数字说明，与动态模式分解（DMD）的特征值相比，Koopman特征值（DMD）相比。
We consider the flow of Burgers' equation on an open set of (small) functions in $L^2([0,1])$. We derive explicitly the Koopman decomposition of the Burgers' flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for $t>0$ for small Cauchy data, and up to $t=0$ for regular Cauchy data. The convergence up to $t=0$} leads to a `completeness' property for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by (Page & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compared to the eigenvalues of a Dynamic Mode Decomposition (DMD).
