We provide a method to select flows of solutions the Cauchy problem for linear and nonlinear Fokker–Planck–Kolmogorov equations (FPK equations) measures on Euclidean space. In case, our improves similar results previous work author. Our consideration flow selections equations, including particularly interesting case Nemytskii-type coefficients, seems be new. also characterize (restricted) well-...

Several relations from the Kolmogorov theory of fully-developed threedimensional turbulence are rigorously established for finite-time averages over LerayHopf weak solutions of the Navier-Stokes equations. The Navier-Stokes equations are considered with periodic or no-slip boundary conditions and an external forcing term. The main parameter is the Grashof number associated with the forcing term...

The discussion on the relation between the Kolmogorov system, considered as low-order approximation of Navier–Stokes equations, and the well-known Lorenz equations is still not completely understood. In this Letter, referring to the mathematical theory of motion on Lie groups, a particular class of Kolmogorov systems, largely studied in low-dimensional models of geophysical fluid dynamics, is e...

We consider a signal process X taking values in a complete, separable metric space E. X is assumed to be a Markov process charachterized via the martingale problem for an operator A. In the context of the finitely additive white noise theory of filtering, we show that the optimal filter Γt(y) is the unique solution of the analogue of the Zakai equation for every y, not necessarily continuous. T...

We obtain Calderón-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous nonlinear coefficients. We then derive the well-posedness of the martingale problem associated with related degenerate operators, and therefore uniqueness in law for the corresponding stochastic differential equations. Some density estimates are established as well.