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。

2017年度

第60回
日時 8月24日(木) 16:30~18:00 ※通常と曜日が異なります
講演者 Riccardo Scala 氏
(Faculdade de Ciências da Universidade de Lisboa)
タイトル A variational approach to continuum dislocations
概要    In a series of papers in collaboration with Nicolas Van Goethem, we study the problem of minimizing an energy of a deformed elastic crystal in the presence of dislocations. These are material defects arising as loops or curves forming very complex networks, which prevents the deformation field to be a gradient, its curl being a Radon measure supported on the dislocations lines. As a consequence, in a neighbourhood of dislocations, the deformation field is allowed to be very large. For this reason our approach is based on a nonlinear framework, and the energy related to the deformation F is assumed to be polyconvex and to account for high order terms as well, depending on the first derivative of F. Our unknown are the deformation field F and the dislocation density too. In order to obtain existence of minimizers we study the graphs related to deformation and prove (under suitable hypotheses) that they are integral currents in the sense of Federer. This allows us to rely on the well-known theory of Cartesian currents and to adapt standard theorems of closeness for such graphs. This provides us the required compactness for minimizing sequence in order to get solutions of the minimum problems.
場所 コロキウム3 (自然科学5号館471)


第59回
日時 7月7日(金) 16:30~18:00
講演者 赤木 剛朗 氏 (東北大学)
タイトル Allen-Cahn equation with non-decreasing constraint
概要   In the context of Damage Mechanics, strongly irreversible (or unidirectional) gradient systems have been introduced in order to describe an irreversible character of damaging phenomena (e.g., crack propagation). In this talk, we particularly apply ourselves to a variant of the Allen-Cahn equation with non-decreasing constraint. The main purpose of this talk is to exhibit partial energy-dissipation structures hidden in the strongly irreversible gradient flow. Indeed, due to the strong irreversibility, full energy-dissipation structures as in the classical Allen-Cahn equation (without constraint) are not realized. On the other hand, (partial) dissipative phenomena of dynamics also emerge from the gradient flow structure as well as the parabolic nature of the equation. More precisely, we shall exhibit (partial) smoothing effect of solutions and (partial) energy-dissipation estimates, and then, we shall introduce a framework to construct a global attractor, which is different from a standard function space setting; indeed, no (compact) global attractor exists in any Lp spaces. If time permits, we shall also discuss the long-time behavior of each solution for the Cauchy-Dirichlet problem as well as Lyapunov stability for a certain class of equilibria. Finally, we shall further show some degenerate structure of entire solutions and equilibria due to the strong irreversibility.
  This talk is based on joint works with Messoud Efendiev (Helmholtz Zentrum München) and Christian Kuehn (TU München).
場所 コロキウム3 (自然科学5号館471)


第58回
日時 6月23日(金) 16:30~18:00
講演者 猪奥 倫左 氏 (愛媛大学)
タイトル 半線形熱方程式の可解性の分類
概要   特異性を伴う初期値を持つ半線形熱方程式について,ベキ乗型などの具体的な増大度を持つ非線形項に対しては F. B. Weisslerらの研究によって解の存在・非存在の分類がなされている.本講演では非線形項の増大度に具体的な仮定を置くことなく,一般の非線形項に対する時間局所解の存在・非存在性を分ける閾値を,初期値の可積分性を用いて明示的に決定する.分類のために中心的な役割を果たすスケール不変性,および一般化Cole-Hopf変換について解説し,具体的な非線形項に対する適用例を紹介する.本講演は静岡大学の藤嶋陽平氏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57回
日時 6月2日(金) 16:30~18:00
講演者 Matteo Negri 氏 (Pavia大学)
タイトル Gradient flows and quasi-static evolutions in phase-field fracture
概要   We will describe how gradient flows, in suitable norms, are a natural and flexible tool to generate quasi-static evolutions in brittle fracture. First, we will consider the classic case of ASTM-CT where a brittle sharp crack propagates along a straight line; we will see how a sequence of discrete incremental problems of gradient flow type can generate a quasi-static evolution of BV-type satisfying Griffith's criterion.
  Then, we will see how the same approach leads to quasi-static evolutions in the phase field setting. We will consider a couple of approaches, both based on time discretization. In the first we will employ the well known alternate minimization scheme. Recasting the algorithm as a gradient flow, with respect to a suitable family of intrinsic norms, we will characterize the time-continuous limit again in terms of a (parametrized) BV-evolution. Mechanically, we will see that this evolution is thermodynamically consistent (with respect to the irreversibility constraint) and that it satisfies a suitable phase-field Griffith's criterion. In the second, we will consider an L^2 gradient flow for the phase field variable, obtained again by time discretization and alternate minimization, and we will see how it converges to a quasi-static limit as the viscosity vanishes.
  Form the technical point of view it will be very useful, if not fundamental, to properly characterize both the gradient flow and the quasi-static BV-evolutions, in particular we will employ a parametrized setting and a suitable energy balance, combining in some sense Mielke's and De Giorgi's approaches. When possible, we will also provide the equivalent system of PDEs.
場所 コロキウム3 (自然科学5号館471)


第56回
日時 5月12日(金) 16:30~18:00
講演者 三竹 大寿 氏 (広島大学)
タイトル 平均曲率流の擬似3Dモデルの導出とその解析
概要   Allen-Cahn方程式の特異極限として平均曲率流が得られることはよく知られている.本講演では,一方向のみ差分化したAllen-Cahn方程式に対して,その特異極限を考えることで形式的に多層的界面方程式を導出する.この得られた多層的界面方程式を等高線法を利用して解析する.ここで得られる等高面方程式は,不連続な係数を含む準単調的退化放物型方程式系となる.本講演では,まずこの導出方法について説明をして,導出された方程式に対して,粘性解を利用して解の存在,一意性に関する議論をする.この方程式系を3次元上で考え,その粘性解の等高面を平均曲率流の擬似3Dモデルとみなす.このような取り組みの動機の一つは,3次元(以上)の平均曲率流の挙動を理解したいことにある.その端緒として,ダンベル型の初期界面に対して,この擬似3Dモデルの(粘性解の範疇での)特殊解を考えて,その特異性に関して解析する.
 なお,本講演は明治大学の二宮広和氏,轟賢太氏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55回
日時 4月26日(水) 16:30~18:00 ※通常と曜日が異なります
講演者 Pierluigi Cesana 氏 (九州大学)
タイトル Modeling and Analysis of microstructure in smart materials: the case of Nematic Elastomers
概要    Microstructure and pattern formation occurs in a large family of materials undergoing solid-to-solid phase-transformations whose technological applications span from biophysics to metallurgy. Mathematically, microstructure is described by using nonlinear elasticity models and it is associated to deep questions of the multi-dimensional calculus of variations (some of which still unsolved) such as lower semicontinuity and quasiconvexity. In this talk I will present a case study arising from Nematic Liquid Crystal Elastomers (NLCEs). This is a class of optically active biopolymers and gels with applications in aeronautics in which a complex interplay of material and structural non-linearities is observed. By adopting an energy-minimization approach I will describe the effective energy density of large 3D samples of NLCEs and provide a detailed characterization of their fine-scale features. By constructing all the energetically optimal and kinematically compatible microstructures, prove that the minima of the relaxed functional exhibit an effective biaxial (isotropic) texture. This modeling result implies that, at a sufficiently macroscopic scale, the response of the material is soft even if the order of the system is assumed to be fixed at the microscopic scale. Moving on from order-strain investigation in the bulk, I will consider the geometrically constrained problem for thin membranes of NLCEs. In this regime, membranes can display fine-scale features both due to wrinkling that one expects in thin elastic membranes and oscillations of the local optical axis that one expects in NLCEs. Existence of solutions to Boundary Value Problems turns out to be an extremely delicate and challenging problem in the calculus of variations due to the intricate coupling of the optical microstructure with the high curvature and high stress regions in the deformed membrane. Gamma-convergence and Relaxation theory shed some light on the analysis of the dimension reduction problem: I show existence of a regime where one has shear strain but no shear stress and all the fine-scale features are in-plane with no wrinkling. This may act as a mechanism preventing formation of wrinkles in active membranes under complex boundary conditions.
場所 コロキウム3 (自然科学5号館471)


第54回
日時 4月14日(金) 16:30~18:00
講演者 Giulio Giusteri 氏 (Okinawa Institute of Science and Technology Graduate University)
タイトル Mathematical modeling and characterization of non-Newtonian viscous fluids
概要   A general representation of the stress tensor in an incompressible fluid in terms of material functions and local kinematical parameters of the flow will be presented. It provides a comprehensive theoretical framework for consistently organizing rheological measurements and establishing complete models for fluids with instantaneous response, generalizing the classical theory of viscometric flows to any flow geometry. The most promising directions for experiments and simulations aimed at exploring the properties of non-Newtonian fluids will be highlighted. A direct application to the interpretation of computational data for hard-sphere suspensions and their constitutive modeling will be discussed. Finally, some preliminary results concerning the nonlinear PDEs associated with general fluid models will be presented, together with a first extension of the framework to viscoelastic fluids.
場所 コロキウム3 (自然科学5号館471)


第53回(拡大版)
日時 4月8日(土) 13:30~17:30 ※通常と曜日・時間・場所が異なります
講演者 13:30--14:20 Yihong Du 氏 (University of New England)
  Logarithmic shifting in spreading governed by the Fisher-KPP porous medium equation

14:30--15:20 田中 吉太郎 氏 (北海道大学)
  Reaction-diffusion approximation to nonlocal evolution equations

15:40--16:30 生駒 典久 氏 (金沢大学)
  Existence of nontrivial solutions for equations with fractional operator

16:40--17:30 俣野 博 氏 (東京大学)
  Front propagation in predator-prey models

アブストラクトはこちら
場所 金沢大学サテライト・プラザ 2階講義室
(金沢市西町三番丁16番地 金沢市西町教育研修館内)


セミナー幹事

  • 生駒 典久
  • 大塚 浩史
  • 小俣 正朗
  • 木村 正人
  • 中村 健一
  • 野津 裕史
  • 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

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