京都大学応用数学セミナー(KUAMS)

京都大学理学研究科数学教室では,数学の諸分野への応用研究や応用につながる可能性のある数学,また将来において数学の応用が期待される研究分野などから幅広く講師を国内外からお招きし,月一度のペースでセミナーを開いています.応用数学の特性を活かした幅広いテーマで刺激に満ちた講演を多数行っていますので,専門家の方のみならず興味をもたれる関係分野の方の参加を歓迎いたします.

 

今後のセミナー予定

 

第55回:2018年9月14日(金)15:00−16:30 (更新しました!)

Uriel Frisch(Laboratoire Lagrange, Observatoire and Universite Cote d'Azur, Nice)
「The mathematical and numerical construction of turbulent solutions for the 3D incompressible Euler equation and its perspectives」

概要: 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)
備考:本セミナーは京都大学の流体力学セミナーとの共催で行われます.場所は京都大学理学研究科物理学教室(理5号館)401号室です.

 

第56回:2018年10月23日(火)16:30−18:00

青木 高明(香川大学 教育学部)
「実地形空間における都市・道路網のパターン形成」

概要: 都市と道路網は社会インフラの根幹であり,その発生原理の探求を目指して数理的研究が行われてきた.先行研究では簡単化のため,周期境界かつ一様平面条件を仮定していたが,現実の地形は高低があり,河川や海岸線がある.本課題では仮説として,外部条件としての地形要因 + 都市と道路網の共発展を記述する単純な時間発展方程式で,現実の都市・道路網の概形が再現できると提案する.これを検証するため,約30m格子の詳細な地形データを活用し,都市・道路形成をパターン形成過程として定式化し,その結果を現実の人口地理分布・道路網と比較する.

 

第57回:2018年12月11日(火)16:30−18:00

望月 敦史(京都大学 ウイルス・再生医科学研究所)
「TBA」

 

第58回:2019年1月29日(火)16:30−18:00

伊藤 祥司(大阪電気通信大学 工学部)
「TBA」