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6 - Miscellanies

Published online by Cambridge University Press:  05 October 2012

Sergei Kuksin
Affiliation:
Ecole Polytechnique, Palaiseau
Armen Shirikyan
Affiliation:
Université de Cergy-Pontoise
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Summary

This chapter is mostly devoted to the 3D Navier-Stokes equations with random perturbations. We begin with the problem in thin domains and state a result on the convergence of the unique stationary distribution to a unique measure which is invariant under the flow of the limiting 2D Navier-Stokes system. We next turn to the 3D problem in an arbitrary bounded domain or a torus. We describe two different approaches for constructing Markov processes whose trajectories are concentrated on weak solutions of the Navier-Stokes system and investigate the large-time asymptotics of their trajectories. Finally, we discuss some qualitative properties of solutions in the case of perturbations of low dimension. Almost all the results of this chapter are presented without proofs.

3D Navier-Stokes system in thin domains

In this section, we present a result that justifies the study of 2D Navier-Stokes equations in the context of hydrodynamical turbulence. Namely, we study the 3D Navier-Stokes system in a thin domain and prove that, roughly speaking, if the domain is sufficiently thin, then the problem in question has a unique stationary measure, which attracts exponentially all solutions in a large ball and converges to a limiting measure invariant under the 2D dynamics. Moreover, when the width of the domain shrinks to zero, the law of a 3D solution converges to that of a 2D solution uniformly in time. The accurate formulation of these results requires some preliminaries from the theory of Navier-Stokes equations in thin domains. They are discussed in the first subsection. We next turn to the large-time asymptotics of solutions and the limiting behaviour of stationary measures and solutions.

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Publisher: Cambridge University Press
Print publication year: 2012

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  • Miscellanies
  • Sergei Kuksin, Armen Shirikyan, Université de Cergy-Pontoise
  • Book: Mathematics of Two-Dimensional Turbulence
  • Online publication: 05 October 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139137119.007
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  • Miscellanies
  • Sergei Kuksin, Armen Shirikyan, Université de Cergy-Pontoise
  • Book: Mathematics of Two-Dimensional Turbulence
  • Online publication: 05 October 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139137119.007
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Miscellanies
  • Sergei Kuksin, Armen Shirikyan, Université de Cergy-Pontoise
  • Book: Mathematics of Two-Dimensional Turbulence
  • Online publication: 05 October 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9781139137119.007
Available formats
×