Gevrey class regularity and exponential decay property for Navier-Stokes-\(\alpha\) equations
The Navier-Stokes-\(\alpha\) equations were introduced as a closed model of turbulence [see e.g. \textit{C. Foias}, \textit{D. D. Holm} and \textit{E. S. Titi}, Physica D 152--153, 505--519 (2001; Zbl 1037.76022)]. Here the authors study these equations with periodic boundary conditions, and prove analyticity in time for a class of solutions belonging to a Gevrey class. The authors also establish exponential decay of the spatial Fourier spectrum for the analytic solutions, and find the lower bounds for the decay rate. The proof is based on representing the solution as Fourier series, and on introducing some quantities with a clear physical sense (kinetic energy, the instantaneous global rate of dissipation, Kolmogorov length).
scientific article; zbMATH DE number 5134775