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Time-averages of fast oscillatory systems

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  • Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We describe a straightforward framework for studying time-averages of dynamical systems whose solutions exhibit fast oscillatory behaviors. Time integration averages out the oscillatory part of the solution that is caused by the large skew-symmetric operator. Then, the time-average of the solution stays close to the kernel of this operator. The key assumption in this framework is that the inverse of the large operator is a bounded mapping between certain Hilbert spaces modular the kernel of the operator itself. This assumption is verified for several examples of time-dependent PDEs.
    Mathematics Subject Classification: Primary: 76U05; Secondary: 74J99, 35Q86.

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