流体数学セミナー

第 15 回　流体数学セミナー

  • 日　  時:  2024 年 12月5日(木) 15時30分 ～ 16時30分
  • 場所・教室:　お茶の水女子大学理学部1号館629教室
    (通常と教室が異なりますのでご注意ください)
  • 講 演 者:　Prof. Thomas Eiter　 (Freie Universitat Berlin)
  • 講演題目:　Existence of time-periodic flow past a rotating body by uniform
    resolvent estimates
  • 講演要旨:  We consider the time-periodic viscous flow around a rotating rigid  body. Since the linearization of this problem is not well-posed in a setting of classical Sobolev spaces, we introduce a framework of homogeneous Sobolev spaces where the corresponding resolvent problems are uniquely solvable. In the case of a pure rotation, one can further derive uniform resolvent estimates, which lead to the existence of solutions to the time-periodic problem. However, in the case of a rotating and translating body, the uniformity of the resolvent estimates requires additional restrictions, and the existence of time-periodic solutions merely follows if the two present oscillating processes are compatible, that is, if the rotational velocity of the body and the angular velocity  of the time-periodic forcing are rational multiples of each other. A counterexample suggests that this restriction is even necessary for existence of time-periodic solutions in the proposed functional framework.
  • 日　  時:  2024 年 12月6日(金) 14時15分 ～ 16時30分
  • 場所・教室:　お茶の水女子大学理学部1号館633教室
  • 講 演 者:　Prof. Manuel V. Gnann (Delft University of Technology)
  • 講演題目:　Classical solutions to the thin-film equation with general mobility in the perfect-wetting regime
  • 講演要旨:  We prove well-posedness, partial regularity, and stability of the thin-film equation h_t + (m(h) h_{zzz})_z = 0 with general mobility m(h) = h^n and mobility exponent n∈ (1,3/2)∪(3/2,3) in the regime of perfect wetting (zero contact angle). After a suitable coordinate transformation to fix the free boundary (the contact line where liquid, air, and solid coalesce), the thin-film equation is rewritten as an abstract Cauchy problem and we obtain maximal L^p_t-regularity for the linearized evolution. Partial regularity close to the free boundary is obtained by studying the elliptic regularity of the spatial part of the linearization. This yields solutions that are non-smooth in the distance to the free boundary, in line with previous findings for source-type self-similar solutions. In a scaling-wise quasi-minimal norm for the initial data, we obtain a well-posedness and asymptotic stability result for perturbations of traveling waves.The novelty of this work lies in the usage of L^p-estimates in time, where 1 <p<∞, while the existing literature mostly deals with p = 2 at least for nonlinear mobilities. This turns out to be essential to obtain for the first time a well-posedness result in the perfect-wetting regime for all physical nonlinear slip conditions except for a strongly degenerate case at n = 3/2 and the well-understood Greenspan-slip case n = 1. Furthermore, compared to [J. Differential Equations, 257(1):15-81, 2014] by Giacomelli, the speaker, Kn\”upfer, and Otto, where a PDE approach yields L^2_t-estimates, well-posedness, and stability for 1.8384 ~ 3(15-√21)/17<n< 3(7+√5)/11 ~ 2.5189, our functional-analytic approach is significantly shorter while at the same time giving a more general result.This talk is based on joint work with Anouk C. Wisse (TU Delft): \href{https://doi.org/10.48550/arXiv.2310.20400}{arXiv:2310.20400}.
  • 講 演 者:　Prof. Piotr Bogusław Mucha (University of Warsaw)
  • 講演題目:　The Compressible Euler System with Nonlocal Pressure
  • 講演要旨: In this talk, I will present a modified version of the compressible barotropic Euler system with friction, where a nonlocal, “fuzzy” pressure term replaces the traditional pressure. This nonlocal pressure is parameterized by ϵ>0ϵ>0, with the system formally converging to the classical pressure model as ϵϵ approaches zero. The main objective is to demonstrate that this modified system reliably approximates the classical compressible Euler system. Our findings are parameter-independent, allowing us to rigorously establish the convergence of solutions to those of the classical Euler system. An additional result is a rigorous derivation of the mass equation converging to various forms of the porous media equation as friction tends to infinity. This analysis is carried out in the whole space, which requires the use of an appropriate L1L1-in-time framework.The talk is based on joint work with Raphael Danchin from Paris.

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流体数学セミナー　世話人

    • 齋藤平和(電気通信大学)
    • 村田美帆(静岡大学)
    • 渡邊圭市(公立諏訪東京理科大学)
    • 久保隆徹(お茶の水女子大学)