Dissipation Length Scale Estimates for Turbulent Flows: A Wiener Algebra Approach

Abstract

In this paper, a lower bound estimate on the uniform radius of spatial analyticity is established for solutions to the incompressible, forced Navier–Stokes system on an n-torus. This estimate matches previously known estimates provided that a certain bound on the initial data is satisfied. In particular, it is argued that for two-dimensional (2D) turbulent flows, the initial data is guaranteed to satisfy this hypothesized bound on a significant portion of the 2D global attractor, in which case, the estimate on the radius matches the best known one found in Kukavica (1998). A key feature in the approach taken here is the choice of the Wiener algebra as the phase space, i.e., the Banach algebra of functions with absolutely convergent Fourier series, whose structure is suitable for the use of the so-called Gevrey norms. We note that the method can also be applied with other phase spaces such as that of the functions with square-summable Fourier series, in which case the estimate on the radius matches that of Doering and Titi (1995). It can then similarly be shown that for three-dimensional (3D) turbulent flows, this estimate holds on a significant portion of the 3D weak attractor.