Regularity conditions of 3D Navier–Stokes flow in terms of large spectral components. (April 2015) - Kim, Namkwon; Kwak, Minkyu; Yoo, Minha
Abstract: We develop Ladyzhenskaya–Prodi–Serrin type spectral regularity criteria for 3D incompressible Navier–Stokes equations in a torus. Concretely, for any N > 0, let w N be the sum of all spectral components of the velocity fields whose wave numbers | k i | > N for all i = 1, 2, 3 . Then, we show that for any N > 0, the finiteness of the Serrin type norm of w N implies the regularity of the flow. It implies that if the flow breaks down in a finite time, the energy of the velocity fields cascades down to the arbitrarily large spectral components of w N and corresponding energy spectrum, in some sense, roughly decays slower than κ − 2 .