Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation

Abstract

We study Galerkin truncations of the two-dimensional Navier-Stokes equation under degenerate, large-scale, stochastic forcing. We identify the minimal set of modes that has to be forced in order for the system to be ergodic. Our results rely heavily on the structure of the nonlinearity.

Approximate controllability for Navier–Stokes equations in \rm3D cylinders under Lions boundary conditions by an explicit saturating set

“…In [4,Section 4] the authors presented an explicit saturating set for the 2D Navier-Stokes system. We would like to refer also to the works [7,9,22], where the notion of saturating set was used to derive ergodicity for the Navier-Stokes system under degenerate stochastic forcing (compare the sequence of subsets Z n in [9,Section 4] with the sequence of subsets K n in [3,Section 8]).…”

Approximate Controllability for Navier–Stokes Equations in 3D Rectangles Under Lions Boundary Conditions

“…Further we know that ẑ = 0 solves system (10) with η b = 0, for time t ∈ I T . Therefore, from [28, Remark 1.9], we can conclude that there exists a unique solution for system (10), for time…”

“…In [4, section 4] the authors present an explicit saturating set for the 2D Navier-Stokes system. We would like to refer also to the works [7,9,22], where the notion of saturating set was used to derive ergodicity for the Navier-Stokes system under degenerate stochastic forcing (compare the sequence of subsets Z n in [9, section 4] with the sequence of subsets K n in [3, section 8]).…”

“…The divergence free condition follows from the way the vectors w j(k),k are chosen in (7). It is also clear that u • n vanishes at the boundary ∂C.…”

Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing

“…Nowadays ergodicity research on infinite-dimensional systems driven by degenerate stochastic forcing has attracted considerable attention ( [1,7,8,9,10,11,14,15,18,20]), not only because this poses many interesting mathematical challenges, but also provides rigorous justification for the explicit or implicit statistical measurement assumptions invoked in a physical environment. It is exciting that recently there have been remarkable breakthroughs (c.f.…”