| Affiliation: | (1) Department of Mathematics, Texas A & M University, College Station, TX 77843, USA;(2) Department of Mathematics, Indiana University, Bloomington, IN 47405, USA |
| Abstract: | Relations central to Kraichnan’s theory of fully developed two-dimensional turbulence are rigorously established for finite time averages. In particular, we prove that if the ratio of the averages of palinstrophy to enstrophy is large, then a large inertial range displaying an enstrophy cascade exists. Moreover, if this ratio is comparable (up to a logarithm) to the dissipation wave number (a necessary condition for turbulence), then the power law for the energy spectrum, until now derived only heuristically, is rigorously shown to provide (up to a logarithm) an upper bound for the energy spectrum. Finally we show that, deep in the dissipation range, the palinstrophy contributed by eddies smaller than a specified length decays exponentially in the corresponding wave number. The averaging times needed for these relations are bounded in terms of the generalized Grashof number, independent of the solution for which the time averages are taken. Solutions are not assumed to be on the global attractor, merely within the absorbing ball, an easily verified condition. This work was partially supported by NSF grant number DMS-0139874. This work, though completed after O.P.M. passed away, was initiated by his insistence that the physical content of our paper [FJMR] be made independent of the esoteric Hahn-Banach extension of the classical concept of a limit. An erratum to this article is available at . |
