Ergodicity of Stochastic Navier-Stokes Equations 01/01/2025

Mathematics

The onset of turbulence is often related to the randomness of background movement, for instance, structural vibrations, magnetic fields, and other environmental disturbances. One way to model this is to consider randomly forced Navier-Stokes equations. The stochastic forcing that is added to the deterministic Navier-Stokes equation models the influence of the random environment on the fluid in fully developed turbulence. Small noise prevalent in nature is magnified by the instabilities in the flow and therefore, it becomes more useful to consider the velocity in turbulent flow to be a stochastic process. In this context, we considered the Navier-Stokes equations subject random noise which is the sum of Gaussian noise for the continuous process, and Levy noise to model the jump part of the process. We studied the rigorous mathematical justification of the following ergodic principle for the 3D stochastic Navier-Stokes equations which lies at the foundation of the statistical approach to the theory of fluid dynamics: there exists an equilibrium measure over the phase space consisting of velocity fields such that, for every regular observable defined over the phase space, and for every initial velocity field, the time average of the observable tends to the mean value of the observable with respect to the equilibrium measure as time goes to infinity.

Since we are considering the 3D stochastic Navier-Stokes equations with Levy noise, we first construct a Markov family of martingale solutions for this equation. It is then constructively used to obtain the existence of a unique invariant measure, which is ergodic and strongly mixing. Indeed, ergodicity and strongly mixing properties of the invariant measure are obtained from the strong Feller property and irreducibility of the transition semigroup.