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

By E. Weinan, Jonathan C. Mattingly. 2001

141 citations recorded by the source corpus.

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. © 2001 John Wiley & Sons, Inc.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/f8c5922fa282fbfa7b7d6ec683cb030c24cb2088

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In Computational Mechanics

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Citations in the corpus

Exponential Convergence for the Stochastically Forced Navier-Stokes Equations and Other Partially Dissipative Dynamics (2002, background): “In [EM01], it was shown that arbitrary finite dimensional projections of the Navier Stokes equations were ergodic under quite general assumptions.”
On ergodicity of some Markov processes (2008, background): “A different approach to prove the uniqueness of an invariant measure for a stochastic evolution equation is based on the strong Feller property of the transition semigroup (see [4], [24], [7], and [12]) or in a more refined version on the asymptotic strong Feller property (see [13]).”
Statistical mechanics of two-dimensional and geophysical flows (2011, background): “In the case α = 0, many interesting mathematical results have been recently obtained for the stochastic Navier–Stokes equations (91): the existence of an invariant measure, its properties in the Euler limit ν → 0, the validity of the law of large numbers, central limit theorems, ergodicity (see [115”
Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity (2008, background): “For the proof we need the following lemma, whose proof in turn is inspired by [6].”
Navier–Stokes Equations: Controllability by Means of Low Modes Forcing (2005, methodology): “We would like to mention paper [8] by W.”
An Introduction to 3D Stochastic Fluid Dynamics (2008, background): “Weinan E, J.C. Mattingly, Ergodicity for the Navier–Stokes equation with degenerate random forcing: finite-dimensional approximation, Comm. Pure Appl.”