Effect of hyperviscosity on the geometry of the vorticity

Abstract

Abstract.We consider the two-dimensional modified Navier-Stokes equations by adding a p-Laplacian. We provide upper bounds on the Hausdorff length of the level sets of the vorticity. We express them in terms of the Landau-Lifschitz estimates of the number of degrees of freedom in turbulent flow. The proof, based on very standard methods, relies on a straightforward relation between the size of the nodal set of the vorticity and the growth of hyperdissipation.Mathematics Subject Classification. 35K55, 35Q30, 76D05, 76F20, 76F70.