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H 1 Ill-posedness for Leray solutions of the ipodissipative Navier-Stokes equations

Auteurs : De Lellis, Camillo (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : In a joint work with Maria Colombo and Luigi De Rosa we consider the Cauchy problem for the ipodissipative Navier-Stokes equations, where the classical Laplacian $-\Delta$ is substited by a fractional Laplacian $(-\Delta)^\alpha$. Although a classical Hopf approach via a Galerkin approximation shows that there is enough compactness to construct global weak solutions satisfying the energy inequality à la Leray, we show that such solutions are not unique when $\alpha$ is small enough and the initial data are not regular. Our proof is a simple adapation of the methods introduced by Laszlo Székelyhidi and myself for the Euler equations. The methods apply for $\alpha < \frac{1}{2}$, but in order to show that they produce Leray solutions some more care is needed and in particular we must take smaller exponents.

    Keywords : Navier-Stokes equation; fractional dissipation; Leray solutions; non-uniqueness

    Codes MSC :
    35D30 - Weak solutions of PDE
    35Q31 - Euler equations
    35A01 - Existence problems: global existence, local existence, non-existence

    Informations sur la rencontre

    Nom du congrès : Harmonic analysis and geometric measure theory / Analyse harmonique et théorie géométrique de la mesure
    Organisteurs Congrès : Bernicot, Frédéric ; Durand-Cartagena, Estibalitz ; Lemenant, Antoine ; Pajot, Hervé ; Rigot, Séverine
    Dates : 02/10/2017 - 06/10/2017
    Année de la rencontre : 2017
    URL Congrès : http://conferences.cirm-math.fr/1685.html

    Citation Data

    DOI : 10.24350/CIRM.V.19226503
    Cite this video as: De Lellis, Camillo (2017). Ill-posedness for Leray solutions of the ipodissipative Navier-Stokes equations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19226503
    URI : http://dx.doi.org/10.24350/CIRM.V.19226503


    Voir aussi

    Bibliographie

    1. Colombo, M., De Lellis, C., & De Rosa, L. (2017). Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations. - https://arxiv.org/abs/1708.05666

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