日時：月曜 16:30 〜（1時間半〜2時間位）
場所：多元数理科学棟 509号室
組織委員：杉本充 菱田俊明 加藤淳 寺澤祐高
世話人：至田直人
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2024 年度
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4月15日（月）
講師：田中 智之 氏 (同志社大学 理工学研究科)
題目：Improved bilinear Strichartz estimates and generalized KdV type equations
We consider the Cauchy problem for generalized KdV type equations on the torus. In order to show the unconditional well-posedness, we introduce improved bilinear Strichartz estimates which are used to recover the derivative loss for resonant nonlinear interactions. Their proofs are based on counting estimates on a certain set. Since we work on the torus, we have an unfavorable term when we use a counting estimate. We overcome this difficulty by a kind of scaling argument, which is reminiscent of the uncertainty principle.
4月22日（月）
講師：加藤 睦也 氏 (岐阜大学 工学部)
題目：Boundedness of some bilinear wave operators
フーリエ積分作用素の有界性は, 例えば, Seeger--Sogge--Stein (1991) によって示されているが, 最近, Grafakos--Peloso (2010) や Rodriguez-Lopez--Rule--Staubach (2014) はその双線形版を考え, ある双線形フーリエ積分作用素に関する有界性を得ている. 本講演では, その双線形作用素の典型例である波動作用素に由来する双線形フーリエ乗子作用素について考え, 彼らの結果を改良できることを紹介したい. また, もし時間が許せば, そこでの結果は Rodriguez-Lopez らのような双線形フーリエ積分作用素への結果へと拡張できることも述べたい. 本講演は宮地晶彦先生（東京女子大学）と冨田直人先生（大阪大学）との共同研究に基づく．
5月13日（月）
講師：谷口 晃一 氏 (静岡大学 工学部)
題目：Reservoir computing with the Kuramoto model and its approximation ability
Reservoir computing is a type of recurrent machine learning with dynamical systems for computations on time series data. We study the reservoir computing with the Kuramoto model, which is the most typical mathematical model for synchronization phenomena. We provide an explicit expression of the Kuramoto reservoir and discuss its approximation ability based on the bifurcation theory of Kuramoto model. This talk is based on the joint work with H. Chiba (Tohoku Univ.) and T. Sumi (Tohoku Univ.).
[集中講義 ] 5月20日〜5月24日
講師：前川 泰則 氏 (京都大学 理学研究科)
題目：プラントル境界層展開の数学解析
流体力学における基礎方程式である非圧縮性ナヴィエ・ストークス方程式を粘着境界条件下で考察する. 流体の粘性が非常に小さい場合における固体壁近傍での解の漸近挙動を調べることは, 理論的にも応用上も重要である. この授業では, その基礎となるプラントル境界層展開に対する数学理論の概要を学ぶことを目的とする.
1. プラントル境界層展開の基礎理論.
2. シアー型境界層周りにおける線形化問題.
3. 凸シアー型境界層周りでのプラントル境界層展開.
6月3日（月）
講師：寺澤 祐高 氏 (名古屋大学 多元数理科学研究科)
題目：Liouville-type theorems for the Taylor-Couette-Poiseuille flow of the stationary Navier-Stokes equations
We study the stationary Navier--Stokes equations in the region between two rotating concentric cylinders.
We first prove that, under the small Reynolds number, if the fluid is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily the Taylor-Couette-Poiseuille flow.
If, in addition, the associated pressure is bounded or periodic in the $z$-axis, then it coincides with the well-known Taylor-Couette flow.
We also give a certain upper bound of the Reynolds number and the $L^\infty$-norm of the velocity under which the fluid is indeed, necessarily axisymmetric.
As the result, it is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the Taylor-Couette-Poiseuille flow with the exact form of the pressure.
This talk is based on a joint work with Professor Hideo Kozono (Waseda University / Tohoku University) and Professor Yuta Wakasugi (Hiroshima University).
6月17日（月）
講師：青山 和寛 氏 (名古屋大学 多元数理科学研究科 D3)
題目：Asymptotic representation of time-periodic Navier-Stokes flows in 3D with asymptotically homogeneous forcing
We study the asymptotic behavior of solutions of the periodic Navier-Stokes equations in the whole space and in exterior domains. We consider the particular case when the forcing term involves a vector field of divergence form with homogeneous potential of degree (-2). The goal is to find out the leading term of the flow at infinity, which is give by a stationary Navier-Stokes flows being homogeneous of degree (-1).
7月8日（月）
講師：三浦英之 氏 (東京工業大学 理学院)
題目：Critical norm blow-up for the energy supercritical nonlinear heat equation
We consider the critical norm blow-up problem for the nonlinear heat equation with power type nonlinearity $|u|^{p-1}u$ in $\mathbb{R}^n$. In the energy supercritical range $p>(n+2)/(n−2)$, we show that if the maximal existence time $T$ is finite, the scaling critical $L^q$ norm of the solution becomes infinite at $t=T$. This is a joint work with Jin Takahashi (Tokyo institute of technology).