Browsing by Subject "PDES"

Filter results by typing the first few letters
Now showing 1 - 2 of 2
  • Thumbnail Image
    ItemOpen Access
    Ergodic properties of highly degenerate 2D stochastic Navier-Stokes equations
    (Comptes Rendus Mathématique. Académie des Sciences. Paris, 2004) Hairer, Martin; Mattingly, Jonathan C
    This Note presents the results from "Ergodicity of the degenerate stochastic 2D Navier-Stokes equation"; by M. Hairer and J.C. Mattingly. We study the Navier-Stokes equation on the two-dimensional torus when forced by a finite dimensional Gaussian white noise and give conditions under which the system is ergodic. In particular, our results hold for specific choices of four-dimensional Gaussian white noise. © 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.
  • Thumbnail Image
    ItemOpen Access
    Spectral gaps in wasserstein distances and the 2d stochastic navier-stokes equations
    (Annals of Probability, 2008-11-01) Hairer, Martin; Mattingly, Jonathan C
    We develop a general method to prove the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł p-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1-Wasserstein distance. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared toward total variation convergence, often fail to hold. In the first part of this paper, we consider semigroups that have uniform behavior which one can view as the analog of Doeblin's condition. We then proceed to study situations where the behavior is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier-Stokes equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show that the stochastic Navier-Stokes equations' invariant measures depend continuously on the viscosity and the structure of the forcing. © Institute of Mathematical Statistics, 2008.