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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