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H 1 Determining the global dynamics of the two-dimensional Navier-Stokes equations by a scalar ODE

Auteurs : Titi, Edriss (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters - finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems to design finite-dimensional feedback control for stabilizing their solutions. Notably, it is observed that this very same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete measurements. In addition, I will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named determining form, which is governed by a globally Lipschitz vector field. Remarkably, as a result of this machinery I will eventually show that the global dynamics of the Navier-Stokes equations is being determining by only one parameter that is governed by an ODE. The Navier-Stokes equations are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative reaction-diffusion systems and geophysical models.

    Keywords : Navier-Stokes equation; turbulence; attractors; determining parameters

    Codes MSC :
    35Q30 - Stokes and Navier-Stokes equations
    35Q86 - PDEs in connection with geophysics

      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 29/11/2019
      Date de captation : 31/10/2019
      Collection : Research talks ; Analysis and its Applications ; Dynamical Systems and Ordinary Differential Equations ; Partial Differential Equations
      Format : MP4
      Durée : 00:33:15
      Domaine : Analysis and its Applications ; PDE ; Dynamical Systems & ODE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2019-10-31_Titi.mp4

    Informations sur la rencontre

    Nom de la rencontre : Evolution Equations: Applied and Abstract Perspectives / Equations d'évolution: perspectives appliquées et abstraites
    Organisateurs de la rencontre : Disser, Karoline ; Haller-Dintelmann, Robert ; Kyed, Mads ; Saal, Jürgen
    Dates : 28/10/2019 - 01/11/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/2071.html

    Citation Data

    DOI : 10.24350/CIRM.V.19575803
    Cite this video as: Titi, Edriss (2019). Determining the global dynamics of the two-dimensional Navier-Stokes equations by a scalar ODE. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19575803
    URI : http://dx.doi.org/10.24350/CIRM.V.19575803

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