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
|
2024 年度・秋学期
|
10月7日（月）
講師：原 宇信 氏 (東北大学 理学研究科 PD)
題目：Uniformly Elliptic Equations on Domains with Capacity Density Conditions: Existence of Continuous Solutions and Homogenization Results
We discuss Dirichlet problems of uniformly elliptic Poisson-type equations on domains with capacity density conditions (CDC domains). We give a brief summary of known facts of CDC domains, including Hardy's inequality, and review our previous work of existence of solutions. Additionally, we apply the result to homogenization problems and provide a convergence rate of $L^{\infty}$ norms in the periodic setting.
[談話会 ] 10月9日 15:00〜16:00
講師：Neal Bez 氏 (名古屋大学 多元数理科学研究科 / 埼玉大学 理工学研究科)
題目：幾何学的 Brascamp-Lieb 不等式
幾何学的 Brascamp-Lieb 不等式は1980 年代後半に Keith Ball 氏によって証明され, 彼はこれを応用して凸幾何学における問題に大きな進展を得た. 幾何学的 Brascamp-Lieb 不等式は, Brascamp-Lieb 不等式の一般理論において重要な役割を果たすことが明らかになっている. 本講演では, このようなテーマを広い観点から説明し, 最後に幾何学的 Brascamp-Lieb データの「稠密性」に関する最近の研究を紹介する.
10月21日（月）
講師：千頭 昇 氏 (名古屋工業大学)
題目：Asymptotically self-similar global solutions for Hardy-Hénon parabolic equations
We construct global asymptotically self-similar solutions to the Hardy-Hénon parabolic equation for a large class of initial data belonging to weighted Lorentz spaces. The solution may be asymptotic to a self-similar solution of the linear heat equation or to a self-similar solution to the Hardy-Hénon equation depending on the decay of the initial data at infinity. The asymptotic results are new for the Hénon case. Furthermore, for complex-valued initial data, a more intricate asymptotic behaviors is shown; if either one of the real part or the imaginary part of the initial data has a spatial decay faster than the critical decay, then the solution exhibits a combined Nonlinear-"Modified Linear" asymptotic behavior, which is new even for the Fujita case. This talk is based on a joint work with M. Ikeda (RIKEN), K. Taniguchi (Shizuoka U.) and S. Tayachi (Université de Tunis El Manar).
11月11日（月）
講師：津原 駿 氏 (神奈川大学)
題目：On the well-posedness of the Schrödinger equation with a nonlinear boundary condition in the half space
We consider the initial boundary value problem for the nonlinear Schrödinger equation in the half plane with a nonlinear boundary condition. The one-dimensional problem was considered by Hayashi--Ogawa--Sato. In this talk, we show the problem is locally well-posed in the higher dimensional half space via the anisotropic Strichartz estimate for a boundary term. This talk is based on a joint work with Prof. Takayoshi Ogawa (Waseda University) and Prof. Takuya Sato (Kumamoto University).
12月9日（月）
講師：山本 立規 氏 (早稲田大学 理工学術院 D3)
題目：The flux problem for the steady-state Navier-Stokes equations with nonhomogeneous slip boundary conditions
We consider the nonhomogeneous boundary value problem for the steady Navier-Stokes equations under the slip boundary conditions in a two-dimensional bounded domain with multiple boundary components. By the incompressibility condition of the fluid, the total flux of the given boundary datum through the boundary must be zero. We prove that this problem has a solution if the friction coefficient is sufficiently large compared with the kinematic viscosity constant and the curvature of the boundary. No additional assumption (other than the necessary requirement of zero total flux through the boundary) is imposed on the boundary datum. This talk is based on the joint work with Prof. Giovanni P. Galdi (University of Pittsburgh).
[研究集会 ] 1月6日 (月)〜7 (火)
「若手による流体力学の基礎方程式研究集会」
会場：多-109 号室
プログラム：PDF file
世話人：三浦英之 (東京科学大学), 加藤淳 (名古屋大学), 鈴木政尋 (名古屋工業大学), 寺澤祐高 (名古屋大学)
1月20日（月）
講師：Neal Bez 氏 (名古屋大学 多元数理科学研究科 / 埼玉大学 理工学研究科)
題目：On the ubiquity of geometric Brascamp-Lieb data
The Brascamp-Lieb inequality unifies several important inequalities such as Holder's inequality, Young's convolution inequality and the Loomis-Whitney inequality. A particularly important special case is the geometric Brascamp-Lieb inequality. This inequality goes back to pioneering work of Keith Ball in the 1980s in convex geometry, and it turns out to play a fundamental role in the general theory of the Brascamp-Lieb inequality. For instance, it was shown by Bennett, Carbery, Christ and Tao that a Brascamp-Lieb inequality which possesses maximizers is equivalent to a geometric Brascamp-Lieb inequality. Relying heavily on work of Garg, Gurvits, Oliveira and Wigderson, here we present another sense in which the class of geometric Brascamp-Lieb data may be considered large. This addresses a question of Bennett and Tao in their recent work on the adjoint Brascamp-Lieb inequality. The talk will focus on the proofs, but I will not assume any prior knowledge of the Brascamp-Lieb inequality. Joint work with Anthony Gauvan (Saitama University) and Hiroshi Tsuji (Saitama University).
1月27日（月）
講師：藤井 幹大 氏 (名古屋市立大学 理学研究科)
題目：Ill-posedness of the two-dimensional stationary Navier-Stokes equations on the whole plane
本講演では 2 次元全平面上の Navier-Stokes 方程式の定常問題を考察する. 空間次元が 3 以上である高次元の場合にはスケール臨界空間における適切性が詳しく調べられている一方で, 2 次元の場合は未解決問題として残されていた. 本講演ではこの問題に対する否定的な解答の一つを与える. 具体的には高次元の場合に適切である臨界 Besov 空間の枠組みにて, 2 次元では適切性が破綻する結果が得られたことを述べる. 講演の前半では主結果と問題の困難点を乗り越えるアイデアを紹介し, 後半に鍵となる命題の証明を説明することを予定している.
[研究集会 ] 3月10日 (月) 〜 12日 (水)
「第16回 名古屋微分方程式研究集会」 (Web サイト)
会場：多-509 号室
プログラム：TBA
|
2024 年度・春学期
|
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).