We prove that the small scale structures of the stochastically forced Navier-Stokes equations approach those of the naturally associated Ornstein-Uhlenbeck process as the scales get smaller. Precisely, we prove that the rescaled k-th spatial Fourier mode converges weakly on path space to an associated Ornstein-Uhlenbeck process as |k| --> infty .
In addition, we prove that the Navier-Stokes equations and the naturally associated Ornstein-Uhlenbeck process induce equivalent transition densities if the viscosity is replaced with sufficient hyperviscosity. This gives a simple proof of unique ergodicity for the hyperviscous Navier-Stokes system. We show how different strengthened hyperviscosity produce varying levels of equivalence.
We study the high Reynolds number limit of a viscous fluid in the presence of a rough boundary. We consider the two-dimensional incompressible Navier-Stokes equations with Navier slip boundary condition, in a domain whose boundaries exhibit fast oscillations in the form $x_2 = \varepsilon^{1+α} η(x_1/\varepsilon)$, $α> 0$. Under suitable conditions on the oscillating parameter $\varepsilon$ and the viscosity $ν$, we show that solutions of the Navier-Stokes system converge to solutions of the Euler system in the vanishing limit of both $ν$ and $\varepsilon$. The main issue is that the curvature…
In this paper we focus on the Cauchy problem for the incompressible Navier-Stokes equation with a rough external force. If the given rough external force is small, we prove the local-in-time existence of this system for any initial data belonging to the critical Besov space $\dot{B}_{p,p}^{-1+\frac{3}{p}}$, where $3<p<\infty$. Moreover, We show the long-time behavior of the priori global solutins constructed by us. Also, we give three kinds of uniqueness results of the forced Navier-Stokes equations.
Concentration of Stationary Measures onto Large Scales in the Fast-Advection Limit of the Stochastically Forced Two-Dimensional Navier-Stokes Equations
We study the asymptotic behavior of the forced linear Euler and nonlinear Navier-Stokes equations close to Couette flow in a periodic channel. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $H^{s}, s>-1$. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier-Stokes equations. Hence, these results provide an indication that nonli…
The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in R2. This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains bounded noise. The algorithm designs a time-dependent gain such that the estimation error converges to zero exponentially. The sufficient condition for the existence of the gain are formulated in the form of algebraic Riccati equations. To demonstrate th…
The NS equation is considered (in 2 & 3 dimensions) with a fixed forcing on large scale; the stationary states form a family of probability distributions on the fluid velocity fields depending on a parameter R (Reynolds number). It is proposed that other equations could lead to -- exactly -- the same distributions via a mechanism closely analogous to the coincidence of the canonical and microcanonical averages of local observables in the statistical mechanics thermodynamic limit (proposed, here, to correspond to the limit in which the UV cut-off N, regularizing the equations, is removed to inf…
