In this talk, we will discuss higher-order regularity of the solutions to both the Navier-Stokes equations (NSE) and supercritical Quasi-geostrophic equations (QG). In the case of the NSE, analytic Gevrey regularity is established, which consequently provides a lower bound estimate on the maximal radius of spatial analyticity. In the context of turbulence, one can couch this quantity in terms of the dissipation length scale of Kolmogorov (in 3D) or Kriachnan (in 2D). In fact, under certain turbulent assumptions, we recover the best-to-date estimates of these length scales by Doering-Titi (3D) and Kukavica (2D).
In the case of the supercritical QG equations, only subanalytic Gevrey regularity is established due to the low order of dissipation. The presence of low dissipation presents its own mathematical difficulties, which, as we will discuss, can be overcome by using some elementary harmonic analysis.
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