2024/09/26 by Zdzisław Brzeźniak, Brzeźniak, Zdzisław, Matteo Ferrari +1
Mathematics · Engineering · Economics, Econometrics and Finance · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2409.17697
We prove the existence and some moment estimates for an invariant measure μ for the two-dimensional (2D) deterministic Euler equations on the unbounded domain \mathbb R2 and with highly regular initial data. The result is achieved by first showing the existence of Markov stationary processes which solve the hyperviscous 2D Navier-Stokes equations with kinematic viscosity ν>0 and an additive stochastic noise scaling as √ ν. We then study the inviscid limit and prove that, as ν tends to 0, these processes converge, in an appropriate trajectory space, to a pathwise stationary solution to the Euler equations. Its law is the sought invariant measure μ.