Stabilizing the long-time behavior of the forced Navier–Stokes and damped Euler systems by large mean flow

Abstract

The paper studies the issue of stability of solutions to the forced Navier–Stokes and damped Euler systems in periodic boxes. It is shown that for large, but fixed, Grashoff (Reynolds) number the turbulent behavior of all Leray–Hopf weak solutions of the three-dimensional Navier–Stokes equations, in periodic box, is suppressed, when viewed in the right frame of reference, by large enough average flow of the initial data; a phenomenon that is similar in spirit to the Landau damping. Specifically, we consider an initial data which have large enough spatial average, then by means of the Galilean transformation, and thanks to the periodic boundary conditions, the large time independent forcing term changes into a highly oscillatory force; which then allows us to employ some averaging principles to establish our result. Moreover, we also show that under the action of fast oscillatory-in-time external forces all two-dimensional regular solutions of the Navier–Stokes and the damped Euler equations converge to a unique time-periodic solution.