セミナーの趣旨

2013年4月，金沢大学の偏微分方程式研究者有志が集まり本セミナーを企画しました。各回の話題は，偏微分方程式の理論的な側面を中心に，セミナー幹事の関心に従い大らかに選択しています。参加者がセミナーを十分楽しみ，勉強し，新しい発見を得られるように，各回の最初の20分から30分程度，講演者の方にはその話題への導入となるような解説をお願いしています。ご関心がある方はどなたでもご自由にご参加ください。なお，基本的には月1回・金曜日に金沢大学で開催予定ですが，柔軟に対応して長く続けていくことを目標にしています。
どうぞよろしくお願いいたします。

2019年度

第84回 ※本セミナーは延期となりました 3月25日(水) 16:30～18:00 ※通常と曜日が異なります． Upanshu Sharma 氏（Freie Universität Berlin, Germany） Quantifying coarse-graining error for stochastic differential equations Coarse-graining is the procedure of approximating a complex and high-dimensional system by a simpler and lower dimensional one. Typically, such an approximation is achieved by using a coarse-graining map F, which projects the full state X of a system (representing for instance the position of particles in the system), onto a lower-dimensional state space (for instance a particular angle or centre of mass). Assuming that the state X evolves according to a stochastic differential equation (SDE), it is easy to identify the evolution of F(X) — however, this cannot be used in practice since the evolution still depends on the original state space. Legoll and Lelièvre (2010) addressed this by introducing a natural approximation, called the effective dynamics, and quantified the error of this approximation when X solves the overdamped Langevin equation. In this talk, I will present results which generalise the construction and the error estimates to the class of non-reversible SDEs. This is joint work with C. Hartmann and L. Neureither. コロキウム3 (自然科学5号館471)

第83回 3月9日(月) 16:30～18:00 ※通常と曜日が異なります． Michael Eden 氏 (University Bremen, Germany) Homogenization of a free boundary problem with prescribed normal velocity Phase transition processes (e.g, water/ice or different phases in steel) are typical examples of problems where the geometry is allowed to evolve and where microscopic effects (growing nucleation cells) are important to understand the macroscopic properties of the system.   In this talk, we present and analyze a Stefan type model describing such phase transition processes. Starting with a prescribed normal velocity of the interface separating the competing phases, a specific transformation of coordinates, the so-called Hanzawa transformation, is constructed. This is achieved by (i) solving a non-linear system of ODEs characterizing the motion of the interface and (ii) using the Implicit Function Theorem to arrive at the height function characterizing this motion.   Based on uniform estimates for the functions related to the transformation of coordinates, the strong two-scale convergence of these functions is shown.   Finally, these results are used to establish the corresponding effective model. コロキウム3 (自然科学5号館471)

第82回 1月15日(水) 16:30～17:30 ※通常と曜日・時間が異なります． 早川 知志 氏 (東京大学) Convergence analysis of approximation formulas for analytic functions via duality for potential energy minimization It is a basic procedure in numerical analysis to approximate a function with certain conditions on smoothness or asymptotic decay by its value at finite number of points. Such conditions are given by the weighted Hardy space, where the decay rate is controlled by a typical variable transformation such as the double exponential (DE) transformation. In the weighted Hardy space caused by transformations such as DE, [1] already showed near-optimality of the sinc formula. However, [2] recently proposed function approximation formulas in general weighted Hardy spaces which show in experiments much better accuracy than existing (sinc) methods. The construction of approximation formulas is based on discrete analogues of basic facts in potential theory. In this study, we give a proof of near-optimality for the approximation formula in [2]. At the same time, we also give an upper bound of the convergence rate of the approximation formula (when increasing the number of sampling points). These give theoretical guarantees to the procedure of [2] in designing approximate formulas based on discrete analogy of properties in the potential theory [3]. In particular, the evaluation of the latter convergence analysis obtained through duality with respect to an infinite dimensional convex quadratic programming. The presentation will focus on the latter evaluation. If we have time, we will mention that a theoretical guarantee is also given for numerical integration formulas obtained by a similar method. The presentation is based on our recent paper [4]. References: [1] Sugihara, M. (2003). Near optimality of the sinc approximation, Mathematics of Computation 72(242): 767–786. [2] Tanaka, K. & Sugihara, M. (2018). Design of accurate formulas for approximating functions in weighted hardy spaces by discrete energy minimization, IMA Journal of Numerical Analysis, published online (doi: 10.1093/imanum/dry056). [3] Levin, A. & Lubinsky, D. (2001). Green equilibrium measures and representations of an external field, Journal of Approximation Theory 113(2): 298 – 323. [4] Hayakawa, S. & Tanaka, K. (2019). Convergence analysis of approximation formulas for analytic functions via duality for potential energy minimization. arXiv:1906.03133. コロキウム3 (自然科学5号館471)

第81回 12月20日(金) 16:30～18:00 張龍傑 氏 (東京大学) On the generalized Dirichlet problem for graph mean curvature flow with driving force We consider a generalized Dirichlet problem for graph forced mean curvature flow equations in the sense of viscosity solutions. As we know, viscosity solutions of Dirichlet problems may not satisfy the boundary condition pointwise in general. To investigate the behavior of viscosity solutions on the boundary, we study a relation of the generalized Dirichlet problem to the obstacle problem and we prove that if the viscosity solution loses the Dirichlet boundary condition, then the solution satisfies the singular Neumann boundary condition. We finally prove that the viscosity solution converges to the slowest traveling wave when the domain is a ball. コロキウム3 (自然科学5号館471)

第80回 8月19日(月) 16:30～18:00 ※通常と曜日が異なります． William Feldman 氏 (University of Chicago) Interfaces in inhomogeneous media: pinning, hysteresis, and facets I will discuss some models for the shape of liquid droplets on rough solid surfaces. These are elliptic free boundary problems with oscillatory coefficients. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets. コロキウム3 (自然科学5号館471)

第79回 6月7日(金) 16:30～18:00 Natali Hritonenko 氏 (Prairie View A&M University) Versatility, Connections, and Applications of Mathematical Models The talk emphasizes the importance of mathematical models in describing dynamics of various processes and solving a wide range of real-life problems as well as their versatility. It will be shown that similar models can be employed discusses several mathematical models and corresponding optimization problems applicable to environmental sciences, economics, biological sciences, and forestry. The models are presented in deterministic setting by means of differential or integral equations or their combination. Some of the models consider the age or size structure of assets or biological agents, others reflect climate change and natural disturbances and their impact on population development. Each problem reflects new developments in the related applied area. The investigation technique of the dynamic models and practical interpretation of obtained outcomes are provided.   Each model represents a new direction in the related applied area and is a result of collaborative efforts of mathematicians, economists, and environmentalists.   The development of science and technology demand new mathematical models and methods to address the emerging questions of critical societal importance such as profitable harvesting, optimal asset replacement, rational usage of natural recourses, sustainable development, environmental protection, and many other related problems. コロキウム3 (自然科学5号館471)

第78回 6月3日(月) 16:30～18:00 ※通常と曜日が異なります． 三竹 大寿 氏 (東京大学) 退化粘性ハミルトン・ヤコビ方程式とその解の漸近形について 力学系におけるAubry-Mather理論は，偏微分方程式論の粘性解理論を導入することで相互の理論がより明瞭なものとなった．この理論は，Kolmogorov-Arnold-Moser (KAM) 理論を背景に偏微分方程式論における弱解を利用した理論ということで，弱KAM 理論と呼ばれる．特に，一階の定常Hamilton-Jacobi (HJ) 方程式の粘性解の定性的性質の理解は大きく進展し，それはたとえば初期値問題の解の長時間挙動の解析に応用された．講演者は，最適確率制御問題に現れる退化粘性HJ方程式と呼ばれるクラスの方程式に適用できるよう，弱KAM理論の一般化に取り組んできた． 従来の弱KAM理論は決定論的な力学系しか扱えないため，新しい道具立てを必要とした．この点を偏微分方程式論から見直すことで，決定論及び確率論を統一できるための一つの枠組みを作ってきた．本講演では，退化粘性HJ方程式の初期値問題の粘性解の長時間後の漸近系に関して，最近得られた結果について紹介する． コロキウム3 (自然科学5号館471)

第77回 4月24日(水) 16:00～18:10 ※通常と曜日が異なります※60分講演が2つ行われます 16:00～17:00 下條 昌彦 氏 (岡山理科大学) Center problem of reaction diffusion systems We study the initial boundary value problem for the two components reaction diffusion system having a nonlinear non-degenerate center. We prove that solutions near the center become spatially homogeneous and is subject to the ODE part asymptotically, provided that the diffusion coefficients of both components are the same. We also discuss blow-up of an parabolic system with quadratic nonlinearity having the origin as an uniform isochronous center. This is the joint work with Amy Poh Ai-Ling. 17:10～18:10 Jong-Shenq Guo氏 (Tamkang University) The spreading speed of a predator-prey system with nonlocal dispersal We are concerned with the propagation dynamics of a predator-prey system with nonlocal dispersal. We obtain a threshold phenomenon for the invasion of the predator into the habitat of the aborigine prey. It turns out that this threshold is the so-called spreading speed of the predator. コロキウム3 (自然科学5号館471)

第76回 4月19日(金) 16:30～18:00 Mark Peletier 氏 (Eindhoven University of Technology) Convergence of gradient flows: tilted is better It is a very common challenge to pass to the limit in a sequence of evolution equations $\dot x_\epsilon = f_\epsilon(x_\epsilon)$, and over the years I have proved many theorems of this kind. In particular, the theory of gradient flows provides various useful handles for proving such convergences.   In this talk I want to report on a slightly surprising observation: sometimes it is {better} not to study the original sequence $\dot x_\epsilon = f_\epsilon(x_\epsilon)$, but instead embed the sequence into a larger class of sequences, and study the convergence of this whole class instead. The resulting limiting gradient-flow structures may be different from those obtained by more classical methods, and in some cases they better represent the modelling aspects of the limits.   This is work together with Alexander Mielke and Alberto Montefusco. コロキウム3 (自然科学5号館471)

第75回 4月16日(火) 16:30～18:00 ※通常と曜日が異なります． Qing Liu 氏 (福岡大学) A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions In this talk, we give a deterministic discrete game-theoretic interpretation for a general class of fully nonlinear parabolic equations with nonlinear dynamic boundary conditions. It is well known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. Our game method can be used to drive general nonlinear dynamic boundary conditions. We also discuss the fast evolution asymptotics of the game values, which leads us to the associated elliptic problems.   This talk is based on joint work with Nao Hamamuki at Hokkaido University. コロキウム3 (自然科学5号館471)

第74回 4月2日(火) 16:30～18:00 ※通常と曜日が異なります． Jenn-Nan Wang 氏 (National Taiwan University) Quantitative unique continuation and applications to inverse problems Unique continuation property (UCP) is an important feature of solutions to partial differential equations. It reflects certain rigidity properties of solutions. UCP can be described both qualitatively and quantitatively. For example, in elliptic equations, quantitative uniqueness estimates give us the maximal vanishing order of any nontrivial solution at one point or the maximal decay rate at the infinity. Such maximal decay rate at the infinity is related to the Landis conjecture that has attracted a lot of attention recently. UCP is not only a theoretical property of the PDE solution but also has many interesting applications to inverse problems. In this talk, I would like to discuss its application to the estimate of the size of an embedded inclusion by measuring one pair of boundary data. コロキウム3 (自然科学5号館471)

セミナー幹事

• 今村 悠里
• Patrick van Meurs
• 大塚 浩史
• 小俣 正朗
• 木村 正人
• 中村 健一
• 野津 裕史
• 橋本 伊都子
• Norbert Pozar
• 和田出 秀光

お問い合わせ

• 中村 健一
• k-nakamura (at)
se.kanazawa-u.ac.jp