記録 

名古屋微分方程式セミナー

日時:月曜 16:30 〜(1時間半〜2時間位)
場所:多元数理科学棟 109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


2023 年度 秋学期

10月2日(月)
講師:井波 虎太郎 氏 (名古屋大学 多元数理科学研究科 D1)
題目:Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients

本講演では, 摩擦項をもつ Klein-Gordon 型方程式のエネルギーの時間減衰評価と摩擦係数の幾何学的条件との同値性について議論する. この時間減衰評価は, ラプラシアンに対するあるレゾルベント評価によって, 特徴づけられることが知られている. そして, このレゾルベント評価は, Paneah-Logvinenko-Sereda 型不確定性原理を用いることで証明される. 本講演では不確定性原理が, どのようにして, エネルギーの時間減衰評価に結びつくかを紹介する. 本講演の内容は, 中央大学の鈴木聡一郎氏との共同研究に基づく.


10月16日(月)
講師:柘植 直樹 氏 (岐阜大学 教育学研究科)
題目:Decay of solutions of isentropic gas dynamics for large data

In this talk, we are concerned with the Cauchy problem for isentropic gas dynamics. Through the contribution of many researchers such as Lax, P. D., Glimm, J., DiPerna, R. J., and Liu, T. P., the decay of solutions was established. They treated with initial data with the small total variation. On the other hand, the decay for large initial data has been open for half a century. Our goal is to provide a new method to deduce a decay of solutions with a large oscillation.


10月30日(月)
講師:生駒 真 氏 (名古屋大学 多元数理科学研究科 PD)
題目:Optimal constants of smoothing estimates for the 3D Dirac equation

In this talk, we find the optimal constants of smoothing estimates for the free 3D Dirac equation. The smoothing estimates (Kato-type smoothing estimates) which express the smoothing effect of dispersive equations such as Schrödinger equation have been studied for a long time, and now many facts are known for more general partial differential equations not limited to dispersive equations. On the other hand, the optimal constants of smoothing estimates for the Dirac equation is unknown in 3D or more. We will solve this problem in the free 3D case.


[集中講義 ]  11月13日〜11月17日
講師:肥田野 久二男 氏 (三重大学 教育学部)
題目:零条件と空間3次元半線形波動方程式系の小さな時間大域解の存在

波動方程式は弦, 膜, 弾性体の振動の現象の単純化モデルの役割を果たします. 本授業では, 小さくなめらかな初期値に対する非線形波動方程式のコーシー問題を扱います. 時間大域解の存在のための2次の非線形項の形状に関する十分条件の研究の展開について解説します. そのための証明方法の工夫と進展にとくに重点を置きます. 波動方程式の場合を手本にして, 受講生が将来, 自分自身のテーマを深める上で役立つような知見が得られます.

1. 零条件と空間 3 次元半線形波動方程式系の初期値問題に対する時間大域解の存在. Klainerman の証明の振り返り.
2. Alinhac の ghost weight method による新証明.
3. $u^3$ のような高次の項のもとでの安定性の問題. Li-Yu の不等式の応用として.
4. 零条件を満たさないような空間 3 次元半線形波動方程式系に対する初期値問題の時間大域解の存在.
5. $u^3$ のような高次の項のもとでの4の問題の解の安定性.


11月27日(月)
講師:大石 健太 氏 (早稲田大学 理工学術院)
題目:On the global well-posedness and decay of a free boundary problem of the Navier-Stokes equation in the two-dimensional half space

We establish the global well-posedness and some decay properties for a free boundary problem of the incompressible Navier-Stokes equations in the two-dimensional half space. Since the solution of the free boundary problem decays as fast as the heat semigroup, it decays slowly for low dimensions and this makes it difficult to estimate the nonlinear terms on the boundary. We overcome this difficulty by obtaining some decay from the derivative arising from the trace estimate in the half space.


12月11日(月)
講師:村松 亮 氏 (東北大学 理学研究科)
題目:Estimates on modulation spaces for solutions to Schrödinger equations with magnetic fields

In this talk, we study the solutions of the Schrödinger equation in a magnetic field by initial data in the modulation space. For the Schrödinger equation of free particles or with scalar potentials, estimates and the existence of the solutions on the modulation space have been shown. However, for the case with magnetic fields, it have not been obtained such results yet due to the first-order derivative term of the solutions. In this presentation, we talk about the estimates on modulation spaces for the solutions to Schrödinger equations with spatially decaying magnetic fields and spatially constant magnetic fields, along with its proof method. Additionally, we present results for the case where the magnetic field can be considered “short-range.” This talk is based on the joint work with Prof. Keiichi Kato (Tokyo University of Science).


[研究集会 ]  1月8日 (月・祝日)
「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: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), 三浦英之 (東京工業大学)


1月22日(月)
講師:Reinhard Farwig 氏 (Technische Universität Darmstadt, Germany)
題目:Viscous fluid flow in domains with moving boundaries

We consider the Navier-Stokes system modelling the flow of a viscous incompressible fluid in a domain with moving boundary $\partial\Omega(t)$ and Dirichlet boundary conditions. Fixing a reference domain $\Omega_0$, we reduce the problem via a coordinate transform $x=\phi(t,\xi): \Omega_0\to\Omega(t)$ to a modified non-autonomous Navier-Stokes system \[ \partial_t u(t)+A(t)u(t)=P(t)F-P(t)u\cdot\nabla^{\phi(t)}u,\quad u(0)=u_0\quad \text{in}\ \Omega_0. \] Here $A(t)$ is a $t$-dependent modified Stokes operator on $L^q_\sigma(\Omega_0)$, $P(t)$ a modified Helmholtz projection with range $L^q_\sigma(\Omega_0)$, and $\nabla^{\phi(t)}$ denotes a $\phi(t)$-dependent gradient.

To solve the initial-boundary value problem or find time-periodic solutions the construction of the fundamental operator $\{U(t,s)\}_{t>s}$ of the linear non-autonomous system \[ \partial_t u(t)+A(t)u(t)=0,\ t>s,\quad u(s)=u_0 \] poses new problems in unbounded domains since then the operators $A(t)$ are not boundedly invertible. Another important property are $t$-independent estimates of $A(t)$, e.g. Sobolev embeddings for fractional Stokes operators $A(t)^\theta$ with $t$-independent bounds. The adjoint operators $A(t)^*$ will be analyzed similarly. The final aim is to get global-in-time estimates of $\{U(t,s)\}$ and to establish sufficiently fast decay rates.

The focus of the talk is put on the half space $\mathbb R^n_+$ with compact perturbations. The results are based on joint papers with K. Tsuda (Kyushu Sangyo University, Fukuoka).   [PDF file]


[学位審査セミナー ]  2月16日(金) 15:30〜17:00
講師:東條 理 氏 (名古屋大学 多元数理科学研究科)
題目:Small energy scattering for radial solutions to the generalized Zakharov system
   (一般化ザハロフ方程式に対する球対称解の小エネルギー散乱)
会場:多-309


[研究集会 ]  3月14日 (木) 〜 15日 (金)
「第15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:PDF file


2023 年度 春学期

4月10日(月)
講師:菱田 俊明 氏 (名古屋大学 多元数理科学研究科)
題目:Stability of time-dependent motions for fluid-rigid ball interaction

We aim at stability of time-dependent motions, such as time-periodic ones, of a rigid body in a viscous fluid filling the exterior to it in 3D. The fluid motion obeys the incompressible Navier-Stokes system, whereas the motion of the body is governed by the balance for linear and angular momentum. Both motions are affected by each other at the boundary. Assuming that the rigid body is a ball, we adopt a monolithic approach to deduce $L^q$-$L^r$ decay estimates of solutions to a non-autonomous linearized system by adaptation from the method developed by the speaker on the similar estimates, however, for the exterior problem without interaction with a moving obstacle. We then apply those estimates to the full nonlinear initial-value problem to find temporal decay properties of the disturbance. Although the shape of the body is not allowed to be arbitrary, the present contribution is the first attempt at analysis of the large time behavior of solutions around nontrivial basic states, that can be time-dependent, for fluid-structure interaction problem and provides us with a stability theorem which is indeed new even for steady motions with wake structure or self-propelling condition.


4月27日(月)
講師:至田 直人 氏 (名古屋大学 多元数理科学研究科)
題目:Boundedness of bilinear pseudo-differential operators of $S_{0,0}$-type on Sobolev and Besov spaces

We consider the boundedness of bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $BS^m_{0,0}$. The boundedness of these operators on Lebesgue spaces has been established by Miyachi-Tomita (2013), Kato-Miyachi-Tomita (2022), and so on. In this talk, the boundedness of those operators in the settings of Sobolev spaces and Besov spaces will be discussed. We also mention that, in contrast to the linear case, some restrictions on the exponents of function spaces are necessary to prove the boundedness and discuss the necessity of them.


5月8日(月)
講師:石田 あかり 氏 (名古屋大学 多元数理科学研究科 D1)
題目:A depth-dependent stability estimate in an iterative method for solving a Cauchy problem for the Laplace equation

In this talk, we consider the Cauchy problem for the Laplace operator. We construct approximate solutions by using the iterative method proposed by Bastay, Kozlov and Turesson. In the iterative method, we solve the corresponding boundary value problems repeatedly. Then, we show that we construct them more stably when we choose the smaller domain where we consider the boundary value problems. Furthermore, since the iterative method is applicable to the case where we know only approximations to the exact data with error, we also deal with this case.


5月22日(月)
講師:山崎 陽平 氏 (九州大学 数理学研究院)
題目:Center stable manifold for ground states of nonlinear Schrödinger equations with internal modes

We consider the behavior of solutions around unstable ground states of nonlinear Schrödinger equations in the 3D whole space. In the case of the cubic nonlinear Schrödinger equation, Schlag constructed center stable manifolds around the unstable ground states and showed the asymptotic behavior of solutions on the center stable manifolds. In the cubic case, the linearized operator around the ground states has no internal modes which are non-zero eigenvalues between the negative essential spectrum and the positive essential spectrum. In this talk, we prove the asymptotic behavior of solutions on center stable manifolds around unstable ground states in the case of the existence of internal modes. This is a joint work with Masaya Maeda in Chiba University.


6月5日(月)
講師:東條 理 氏 (名古屋大学 多元数理科学研究科 D3)
題目:Uniform estimate of the radial global solutions to the generalized Klein-Gordon-Zakharov system

In this talk, we consider the Cauchy problem for the generalized Klein-Gordon-Zakharov system in the energy space with radial initial data. The system, which has two large parameters representing the plasma frequency and the sound speed, is expected to converge to the Schrödinger equation with the Hartree type nonlinearity in the simultaneous high-frequency and subsonic limit.
The purpose of this talk is to show the existence of the small energy global solutions which have the uniform bound with respect to the large parameters in the appropriate norm. Our method is based on the normal form reduction and the radial improved Strichartz estimates, which is used by Guo-Nakanishi-Wang (2014) for the Klein-Gordon-Zakharov system. To derive the uniform estimate, more delicate frequency decomposition is required.


6月19日(月)
講師:橋詰 雅斗 氏 (広島大学 先進理工系科学研究科)
題目:Asymptotic properties of critical points for Trudinger-Moser functional involving scale parameter

In this talk, we consider asymptotic behavior of critical points for the Trudinger-Moser functional. We derive several properties of the critical points both for large scale parameter and for small scale parameter. For large scale parameter, we prove that shape of maximizers depends on the exponent in the Trudinger-Moser functional. If the exponent is close to the critical exponent, then sequence of maximizers concentrates at one point, while if the exponent is small, then the limit of sequence of maximizers vanishes. For small scale parameter, we show that any positive critical point is close to a constant solution of a nonlocal elliptic equation. Besides these results, we obtain properties of maximizers for the Trudinger-Moser inequality in the whole space.


7月10日(月)10:00~12:00,  会場:多-409 号室
講師:Helmut Abels 氏 (University Regenburg)
題目:Sharp Interface Limits of Diffuse Interface Models

We consider the dynamics of two components, which are partly miscible on a small length scale proportional to some parameter $\varepsilon>0$. This is described by an order parameter as e.g. a concentration difference, which is close to two distinct values (e.g. $1$ and $-1$) in most of the domain, and varies smoothly, but with a large gradient, between these two values in a small interfacial region of thickness proportional to $\varepsilon$. Such models are called diffuse interface models and have various applications. We are interested in rigorous results on convergence to classical sharp interface models as $\varepsilon$ tends to zero, where the components fill two disjoint regions is separated by a lower dimensional interface. We will start with the case of the scalar Allen-Cahn equation, discuss the method of formally matched asymptotics and show how to use this method in a rigorous proof of convergence. Afterwards we will consider results on a coupled Navier-Stokes/Allen-Cahn system, which describes a two-phase flow of macroscopically immiscible viscous Newtonian fluids. In this case the results depend essentially on the choice of a mobility coefficients in Allen-Cahn equation.


[集中講義 ]  7月10日〜7月14日
講師:倉坪 茂彦 氏 (弘前大学 名誉教授)
題目:多重フーリエ級数と格子点問題

多次元(2次元以上)のフーリエ級数(多重フーリエ級数)の点毎収束問題を考えるとき, 1次元のそれとは様相のことなることに注目する. その解析のために解析的整数論の一分野である格子点問題が重要な手がかりになることを解説する.

1. フーリエ級数の収束問題について
2. 格子点問題について
3. 2つの問題の接点について