• Corpus ID: 245385874

The number of degrees of freedom for the 2D Navier-Stokes equation: a connection with Kraichnan's theory of turbulence

@inproceedings{Cheskidov2021TheNO,
  title={The number of degrees of freedom for the 2D Navier-Stokes equation: a connection with Kraichnan's theory of turbulence},
  author={Alexey Cheskidov and Mimi Dai},
  year={2021}
}

• A. Cheskidov, Mimi Dai
• Published 22 December 2021
• Mathematics

The purpose of this paper is to estimate the number of degrees of freedom of solutions of the 2D NSE. More precisely, we prove that its mathematical analog, the number of determining modes, is bounded by the Kraichnan number squared κη, consistent with Kraichnan’s theory of two-dimensional turbulence [35]. The notion of determining modes was introduced by Foias and Prodi in the seminal work [25] where it was shown that high modes of a solution to the 2D NSE are controlled by low modes asymptotically as time goes to infinity. Then the number of such determining modes N was estimated by Foias, Manley, Temam, and Treve [24] and later improved by Jones and Titi [32, 33]. We refer the readers to [13, 14, 19, 20, 21, 23, 26, 27, 28, 29, 37, 38] and references therein for more background and related results. In the remarkable work of Jones and Titi [33], the number of determining modes N was estimated as