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.

Original languageEnglish (US)
Pages (from-to)1151-1162
Number of pages12
JournalDiscrete and Continuous Dynamical Systems - Series S
Issue number5
StatePublished - Oct 2013


  • Coherent structures
  • Mathematical geosciences
  • Time-averages
  • Waves

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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