カテゴリー: 流体力学セミナー2018

  • 流体力学セミナー

    流 体 力 学 セ ミ ナ ー

    (京都大学応用数学セミナー(KUAMS)との共催)

    日時:  9月14日(金) 15:00 から 16:30

    場所:  京大 理学研究科 物理学教室(理5号館) 401号室

    講師: Uriel Frisch 氏
    Laboratoire Lagrange, Observatoire and Universite Cote d’Azur
    Nice, France

    講演題目:The mathematical and numerical construction of
    turbulent solutions for the 3D incompressible Euler
    equation and its perspectives

    講演要旨:

    Starting with Kolmogorov’s 1941 (K41) work, infinite Reynolds number
    flow is known to have velocity increments over a small distance r that
    vary roughly as the cubic root of r. Formally,  such  flow  is  expected  to
    satisfy  Euler’s  partial differential  equation,  but  the  flow  being
    not  spatially differentiable, the equation is satisfied only in
    a distributional sense. Since Leray’s 1934 work, such solutions are called
    weak. Actually  they  were  already  present  –very  briefly–  in
    Lagrange’s 1760/1761 work on non-smooth solutions of the wave equation.
    A  major  breakthrough  has  happened  recently: mathematicians  succeeded
    in  constructing  rigourously  weak solutions  of  the  Euler  equation
    whose  spatial  regularity  –measured by their Hölder continuity exponent–
    is arbitrarily close to the value predicted by K41 (Isett 2018), Buckmaster et
    al. 2017). Furthermore these solutions present the anomalous energy dissipation
    investigated by Onsager in 1949 (Ons49). We shall highlight some aspects of
    the derivation of these results which took about ten years and was started
    originally by Camillo de Lellis and Laszlo Szekelyhidi and continued with a
    number  of  collaborators.  On  the  mathematical  side  the derivation makes
    use of techniques developed by Nash (1954) for isometric embedding and by Gromov
    (1986, 2017) for convex  integration.  Fortunately,  many  features  of  the
    derivation  have  a  significant  fluid  mechanical  content.  In particular
    the successive introduction of finer and finer flow structures, called Mikados
    by Daneri and Szekelyhidi (2017) because  they  are  slender  and jetlike.
    The Mikados generate Reynolds stresses on larger scales; they can be chosen
    to cancel discrepancies between approximate and exact solutions of the Euler
    equation. A particular engaging aspect of the construction of weak solutions
    is its flexibility. The Mikados can be chosen not only to  reproduce  K41/Ons49
    selfsimilar  turbulence,  but  also  to synthesize  a  large  class  of
    turbulent  flows,  possessing,  for example,  small-scale  intermittency
    and  multifractal  scaling. This huge playground must of course be explored
    numerically for testing all manners of physical phenomena and theories, a
    process being started in a collaboration between Leipzig, Nice, Kyoto and Rome.

    (in collaboration with Laszlo Szekelyhidi,Department of Mathematics,
    University of Leipzig, Germany and Takeshi Matsumoto,Department of
    Physics, Kyoto University, Japan)

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    世話人:山田 道夫(京大数理研), 竹広 真一(京大数理研),
    藤 定義(京大理),松本 剛(京大理)
    連絡先:山田道夫 yamada_at_kurims.kyoto-u.ac.jp
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