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。
2018年度
| 第73回 | |
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| 日時 | 1月4日(金) 16:30～18:00 |
| 講演者 | 榊原 航也 氏 (京都大学理学研究科・理化学研究所数理創造プログラム) |
| タイトル | Numerical analysis of constrained total variation flow |
| 概要 | Constrained total variation flow, which means that the value of the function is constrained to a prescribed manifold, appears in several fields of application such as color image denoising, continuum model for the time evolution of grain boundaries in a crystal, and denoising diffusion tensor MRI. In applications as mentioned earlier, it is natural to consider spatially discretized constrained total variation flow. Therefore, in this talk, we consider a numerical scheme for spatially discretized constrained total variation flow. Among several numerical methods, the minimizing movement scheme is a popular one, which requires to solve a non-smooth Riemannian optimization problem in each step and it is not easy to address it. To overcome this difficulty, we modify the energy functional by localizing it to the tangent space, which yields a convex optimization problem. We then prove that the Rothe interpolation of numerical solution converges to the original flow. Finally, we show some results of numerical experiments. This talk is based on joint work with Prof. Yoshikazu Giga (The University of Tokyo), Mr. Kazutoshi Taguchi (The University of Tokyo), and Dr. Masaaki Uesaka (Hokkaido University). |
| 場所 | コロキウム3 (自然科学5号館471) |
| 第72回 | |
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| 日時 | 11月30日(金) 16:30～18:00 |
| 講演者 | Yueyuan Gao 氏 (東北大学) |
| タイトル | Finite volume methods for deterministic and stochastic partial differential equations |
| 概要 | In this talk, we start by presenting numerical simulations for the first order Burgers equation on the one-dimensional torus forced by a stochastic source term. The source term corresponds to the Q-Brownian motion. It turns out that the empirical mean introduce a smoothing effect to the shock and it converges to the space-average of the deterministic initial condition for large t. In the theoretical study, we prove the existence and uniqueness of the weak entropy solution of a first-order stochastic conservation law involving a Q-Brownian motion. The existence is proved as the discrete solution obtained by a finite volume method convergences along a subsequence in the sense of Young measure; and the uniqueness is deduced as a corollary of the Kato's inequality. The theoretical part is joint work with Tadahisa Funaki and Danielle Hilhorst. In the third part we present numerical simulations for density driven flows in porous media. It amounts to solving a nonlinear convection-diffusion parabolic equation for the concentration coupled with an elliptic equation for the pressure. We apply a semi-implicit time scheme together with a generalized finite volume method SUSHI, for which the orthogonality condition and volume matching condition of the mesh are not necessary. It is joint work with Huy Cuong Vu Do and Danielle Hilhorst. |
| 場所 | コロキウム3 (自然科学5号館471) |
| 第71回 | |
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| 日時 | 11月12日(月) 16:30～18:00 ※通常と曜日が異なります |
| 講演者 | Cassio M. Oishi 氏（サンパウロ州立大学） |
| タイトル | Numerical methods for dealing with non-Newtonian materials: moving interface flows and confined benchmark problems |
| 概要 | In this talk we will discuss some recent trends for constructing numerical methods to solve fluid flows considering non-Newtonian materials, e.g., viscoelastic and yield-stress fluids. Firstly, we will describe numerical techniques for treating the incompressible Navier-Stokes equations combined with the fluid interface models. Numerical simulations for moving interface problems will be presented in order to describe two different interface representation frameworks. Moreover, we will use computational results for analysing peculiar behaviours encountered in Non-Newtonian flows. The influence of rheological parameters will also be studied in complex fluid flows. In the final part of this talk, we will explore an alternative methodology to represent the viscoelastic constitutive equations which is more appropriated in flows with singularities, as for example, the contraction flow and the stick-slip problem. |
| 場所 | コロキウム3 (自然科学5号館471) |
| 第70回（拡大版） | |
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| 日時 | 11月9日(金) 15:00～18:00 ※通常と開始時刻が異なります |
| 講演者 | 15:00--15:10 Opening 15:10--15:50 Hirofumi Notsu (Kanazawa University) Development of an accurate Lagrange-Galerkin scheme with adaptive mesh refinement 15:50--16:30 Martin Lind (Karlstad University) BV-like descriptions of Sobolev spaces 16:40--17:20 Hiroshi Ohtsuka (Kanazawa University) On the impulse response for solutions of two-dimensional Liouville type equations 17:20--18:00 Adrian Muntean (Karlstad University) Towards understanding nonlinear transport through heterogeneous thin layers |
| 概要 | Notsu: An accurate Lagrange-Galerkin scheme with adaptive mesh refinement for convection problems is presented. We prove an error estimate of second-order in time and first-order in space in the framework of L2-theory and a mass-conservation property. Two and three dimensional numerical results are shown to see the accuracy and property of the scheme. This is a joint work with Mr. Kouta Futai (Kanazawa University). Lind: We discuss a variational characterization of certain Sobolev spaces [1] which provides an extension of a classical theorem of F. Riesz. Some applications to the study of fine properties of functions [1], and the mapping properties of the Hardy-Littlewood maximal operator [2] will be outlined. This is joint work with Sorina Barza (Karlstad University, Sweden). [1] S. Barza and M. Lind, "A new variational characterization of Sobolev spaces", J. Geom. Anal. , vol 25 (2015) [2] E. Carneiro and J. Madrid, "Derivative bounds for fractional maximal functions", Trans. Amer. Math. Soc., vol 369 (2017) Ohtsuka: Motivated by the experimental facts observed in confined non-neutral plasma, we are interested in the impulse response for solutions of two-dimensional Liouville type equations, that is, the asymptotic behavior for solutions of Liouville type equations with one singular source as the singularity vanishes. In this talk, we observe some basic facts on the exact solution for singular Liouville type equation in R2 established by Prajapat and Tarantello (2001) and try to generalize some of them. Muntean: We study the diffusion of particles through a thin heterogenous membrane under a one--directional nonlinear drift. Using mean-field equations derived from a Monte Carlo lattice dynamics for the problem at hand, we study the possibility to upscale the system and to compute the effective transport coefficients accounting for the presence of the membrane. For a special scaling regime, we perform a simultaneous homogenization asymptotics together with dimension reduction, allowing us to replace completely the heterogenous membrane by an homogeneous obstacle line provided with effective transmission conditions. The heterogeneities we account for in this context are assumed to be arranged periodically, but the same working techniques can cover as well the locally periodic case. This is joint work with Ida de Bonis (Benevento Telematica Univ., Italy), Emilio Cirillo (La Sapienza Univ. Rome, Italy) and Omar Richardson (Karlstad University, Sweden). |
| 場所 | コロキウム3 (自然科学5号館471) |
| 第69回 | |
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| 日時 | 10月5日(金) 16:30～18:00 |
| 講演者 | 加須栄 篤 氏 (金沢大学) |
| タイトル | 非線形抵抗無限ネットワークのコンパクト化とディリクレ境界値問題 |
| 概要 | 非線形抵抗無限ネットワーク上のポテンシャル論を紹介する．エネルギー有限な関数からなるバナッハ空間を導入し，それと結びついたコンパクト化と理想境界を考える．与えられた境界値を持つ調和関数(局所的にエネルギー最小性を持つ関数)を求めるディリクレ境界値問題に対して，ペロンによる方法を適用する．解の境界挙動についてはわかりやすい場合に限定してすこし触れる．p-ネットワークは1980年代から研究されてきたが，これを含むネットワークを対象とする．具体的例を示すとともに，擬等長不変量を扱う幾何との関連について説明する． |
| 場所 | コロキウム3 (自然科学5号館471) |
| 第68回 | |
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| 日時 | 9月3日(月) 16:00～18:00 ※通常と曜日・時間が異なります ※60分講演が2つ行われます |
| 講演者 | Luca Rondi 氏 (Università di Trieste) |
| タイトル | Multiscale decompositions in imaging |
| 概要 | We extend the hierarchical decomposition of an image as a sum of constituents of different scales, introduced by Tadmor, Nezzar and Vese in 2004, to a general setting. We develop a theory for multiscale decompositions which, besides settling an open problem on the decomposition of Tadmor, Nezzar and Vese for arbitrary L2 functions, is applicable to a wide range of other imaging problems or strictly related ones, such as nonlinear inverse problems. A significant application of our theory is to image registration, in the framework of the so-called Large Deformation Diffeomorphic Metric Mapping (LDDMM). Image registration is an important problem in medical imaging, where one seeks to align images obtained at different times or with different instrumentation by transforming one to the other through an optimal diffeomorphism. We construct analogous hierarchical expansions for such diffeomorphisms, with the sum replaced by composition of maps. This is a joint work with Klas Modin and Adrian Nachman. |
| 講演者 | Maria Giovanna Mora 氏 (Università degli Studi di Pavia) |
| タイトル | A corrected Sadowsky functional for inextensible elastic ribbons |
| 概要 | In 1930 Sadowsky gave a constructive proof for the existence of a developable Möbius band and posed the problem of determining the equilibrium configuration of a Möbius strip made of an unstretchable material. He tackled this latter problem variationally and he deduced the bending energy for a strip whose width is much smaller than the length. This energy, now known as the Sadowsky energy, depends on the curvature and torsion of the centerline of the band and it is singular at points with zero curvature. In this talk we will re-examine the derivation of the Sadowsky energy using Gamma-convergence. The energy deduced in this way generalizes and corrects the classical Sadowsky functional. |
| 場所 | コロキウム3 (自然科学5号館471) |
| 第67回 | |
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| 日時 | 8月3日(金) 16:30～18:00 |
| 講演者 | 橋本 伊都子 氏 (関西大学) |
| タイトル | 高次元空間上におけるバーガーズ方程式の球対称解の漸近挙動について |
| 概要 | 高次元空間上におけるバーガーズ方程式の球対称解について初期値境界値問題を考察する. 最近の研究を通し，1次元空間上と高次元空間上におけるバーガーズ方程式の解の漸近形に相違があることが明らかになってきた．特に1次元バーガーズ方程式の初期値境界値問題の漸近形としては現れない様々な形状の定常波が，高次元空間上におけるバーガーズ方程式の球対称問題において現れることが明らかになってきた. 講演では，1次元バーガーズ方程式には見られない漸近形の紹介と共に，球対称解の漸近挙動を決定する種々の境界条件の相関を論じていく. さらに空間3次元空間の漸近形の考察を通し，衝撃波の形状も1次元とは異なることを紹介する． 本研究は，松村昭孝氏（大阪大学名誉教授）との共同研究に基づく． |
| 場所 | コロキウム3 (自然科学5号館471) |
| 第66回 | |
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| 日時 | 7月20日(金) 16:30～18:00 |
| 講演者 | Marco Morandotti 氏 (ミュンヘン工科大学) |
| タイトル | Spatially inhomogeneous evolutionary games |
| 概要 | We study an interaction model of a large population of players based on an evolutionary game, which describes the dynamical process of how the distribution of strategies changes in time according to their individual success. Differently from spatially homogeneous dynamical games, we assume that the population of players is distributed over a state space and that they are each endowed with probability distributions of pure strategies, which they draw at random to evolve their states. Simultaneously, the mixed strategies evolve according to a replicator dynamics, modeling the success of pure strategies according to a payoff functional. We establish existence, uniqueness, and stability of Lagrangian and Eulerian solutions of this dynamical game by using methods of ODE and optimal transport on Banach spaces. |
| 場所 | コロキウム3 (自然科学5号館471) |
| 第65回 | |
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| 日時 | 6月22日(金) 16:30～18:00 |
| 講演者 | Quentin Griette 氏 (明治大学) |
| タイトル | Studying the spread of evolving diseases: traveling waves and pulsating fronts |
| 概要 | I will talk about a system of two coupled reaction-diffusion equations modeling the spread of evolving diseases. In this scenario, a pathogen propagates within a population of susceptible hosts while a fast mutation process allows its phenotype to change in the same time scale as the invasion process. I will consider a special case where only two phenotypes exists, leading to a system of two coupled KPP-type equations. I will first talk about the case of a homogeneous space, where the reaction coefficients do not depend on the space variable, and present a construction of traveling waves that allow us to characterize the propagation. Then, I will investigate the case of a periodically heterogeneous space, and show how we constructed pulsating fronts in this situation. In both cases, there is competition between the two pathogens, which we treated as a non-local term; in particular, we are not in a situation where a comparison principle is available, which is a challenging mathematical problem. |
| 場所 | コロキウム3 (自然科学5号館471) |