Level Sets of the Vorticity and the Stream Function for the 2D Periodic Navier–Stokes Equations with Potential Forces

By a result of Foias and Saut, the quotient of the enstrophy and the energy of a solution of the Navier–Stokes equations with potential forces converges to the square root of an eigenvalueΛkof the Stokes operator. We study the Hausdorff length ( 1) of the level sets of the vorticity and the stream function of solutions. We provide upper bounds for these quantities, and for largetwe express them in terms of the corresponding eigenvalueΛk.