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H 1 Adding small viscosity to hyperbolic (stochastic) conservation laws

Auteurs : Boritchev, Alexandre (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : The mechanism responsible for blow-up is well-understood for many hyperbolic conservation laws. Indeed, for a whole class of problems including the Burgers equation and many aggregation-diffusion equations such as the 1D parabolic-elliptic Keller-Segel system, the time and nature of the blow-up can be estimated by ODE arguments. It is, however, a much more delicate question to understand the small-scale behaviour of the viscous layers appearing in (classic or fractional) parabolic regularisations of these conservation laws.
    Here we give sharp estimates for Sobolev norms and for a class of small-scale quantities such as increments and energy spectrum (which are relevant for the theory of turbulence), for solutions of these conservation laws. Moreover, many of our results can be generalised for perturbations of the viscous conservation laws by random additive noise, and some of them admit a simpler formulation in this case. To our best knowledge, these are the only sharp results of this type for small-scale behaviour of solutions of nonlinear PDEs.
    The work on the aggregation-diffusion equations is an ongoing collaboration with Piotr Biler and Grzegorz Karch (Wroclaw) and Philippe Laurençot (Toulouse).

    Keywords : Burgers equation; stochastic PDEs; scalar conservation laws; turbulence

    Codes MSC :
    35Q53 - KdV-like (Korteweg-de Vries) equations
    35R60 - PDEs with randomness, stochastic PDE
    60H15 - Stochastic partial differential equations
    37L40 - Invariant measures

      Informations sur la Vidéo

      Réalisateur : Recanzone, Luca
      Langue : Anglais
      Date de publication : 04/11/2019
      Date de captation : 15/10/2019
      Collection : Research talks ; Partial Differential Equations
      Format : MP4
      Durée : 00:57:36
      Domaine : PDE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2019-10-15_Boritchev.mp4

    Informations sur la rencontre

    Nom de la rencontre : PDE/Probability Interactions: Particle Systems, Hyperbolic Conservation Laws / Interactions EDP/Probabilités : systèmes de particules, lois de conservation hyperboliques
    Organisateurs de la rencontre : Caputo, Pietro ; Fathi, Max ; Guillin, Arnaud ; Reygner, Julien
    Dates : 14/10/2019 - 18/10/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/2083.html

    Citation Data

    DOI : 10.24350/CIRM.V.19571703
    Cite this video as: Boritchev, Alexandre (2019). Adding small viscosity to hyperbolic (stochastic) conservation laws. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19571703
    URI : http://dx.doi.org/10.24350/CIRM.V.19571703


    Voir aussi

    Bibliographie

    1. BILER, Piotr, BORITCHEV, Alexandre, KARCH, Grzegorz, LAURENCOT Philippe. Concentration in an aggregation model when diffusion vanishes. Currently drafted. -

    2. BORITCHEV, Alexandre. Sharp estimates for turbulence in white-forced generalised Burgers equation. Geometric and Functional Analysis, 2013, vol. 23, no 6, p. 1730-1771. - https://doi.org/10.1007/s00039-013-0245-4

    3. BORITCHEV, Alexandre. Decaying turbulence in the generalised Burgers equation. Archive for Rational Mechanics and Analysis, 2014, vol. 214, no 1, p. 331-357. - https://doi.org/10.1007/s00205-014-0766-5

    4. BORITCHEV, Alexandre. Multidimensional potential Burgers turbulence. Communications in Mathematical Physics, 2016, vol. 342, no 2, p. 441-489. - https://doi.org/10.1007/s00220-015-2521-7

    5. BORITCHEV, Alexandre, KUKSIN, Sergei. Introduction to SPDEs, stochastic Burgers equation and burgulence. Currently drafted. -

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