## 2015年度

第37回 3月23日(水) 16:30～18:00 ※通常と曜日が異なります Matteo Negri 氏 (Pavia大学) Gradient flows and quasi-static evolutions in phase-field fracture 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 L2 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. From 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. コロキウム3 (自然科学5号館471)

第36回 3月7日(月) 16:30～18:00 ※通常と曜日が異なります Dohyun Kwon 氏 (National Institute for Mathematical Sciences, Republic of Korea/UCLA) Existence and Long-time Behavior of Normalized Mean Curvature Flow In this talk, we discuss about geometric stability properties and long-time behavior of a normalized mean curvature flow. We consider viscosity solutions of the level set equation for mean curvature flow introduced by Evans and Spruck. Based on a reflection comparison argument, we prove that the flow preserves the $\rho$-reflection property, which is a quantitative smalleness condition between the set and the nearest ball. Also, we apply variational method originated by Almgren, Taylor and Wang to investigate existence and the long-time behavior of solutions. We prove that viscosity and variational solutions yield the same solutions. As a consequence it is shown that the evolving set becomes smooth in finite time and uniformly converges to a ball in Hausdorff topology exponentially fast. This is joint work with Inwon Kim. コロキウム3 (自然科学5号館471)

第35回 2月18日(木)14:00～2月20日(土)11:40 拡大版として，研究集会を行います．プログラム等の詳細はこちらをご覧ください． 金沢大学サテライト・プラザ 3階集会室 (金沢市西町三番丁16番地 金沢市西町教育研修館内)

特別(院生向け講義) 2月16日(火) 10:00～12:15 ※通常と曜日・時間が異なります Mark Peletier 氏 (アイントホーフェン工科大学) Stochastic origins of gradient flows: relation with modelling In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently. In these talks I will discuss some of the modelling arguments that underlie the use of energies, entropies, and the Wasserstein gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other. * The participants of this lecture are recommended also to attend the Monday's talk (Feb. 15). * 本講義の受講者には，関連する内容の解析セミナー(2/15)への参加をお勧めします． コロキウム3 (自然科学5号館471)

第34回 2月15日(月) 16:30～18:00 ※通常と曜日が異なります Mark Peletier 氏 (アイントホーフェン工科大学) Stochastic origins of gradient flows: a general connection Although it has been known for some time that certain gradient-flow structures are related to the large deviations of a stochastic process, until recently we only understood this at the level of examples. In this lecture I will explain a general structure that gives rise to the following property: for every sequence of reversible stochastic processes with a large-deviations principle, the limiting equation is a generalized gradient flow that maps one-to-one to the large-deviations rate function. Therefore the large class of reversible stochastic processes generates a correspondingly large class of generalized gradient flows. This is a joint work with Michiel Renger and Alexander Mielke (both WIAS). コロキウム3 (自然科学5号館471)

第33回 11月20日(金) 16:30～18:00 澤田 宙広 氏 (岐阜大学) Thresholds between time-local well-posedness and ill-posedness for equations of fluid dynamics It was proved by Koch-Tataru that the Cauchy problem of the 3-dimensional incompressible Navier-Stokes equations (NS) is time-local well-posed in the critical space BMO-1 by iteration argument. Conversely, as shown by Bourgain-Pavlovic, (NS) is ill-posed in a broader space by norm-inflation argument. It is clarified where is the threshold between time-local well-posedness and ill-posedness of (NS) in terms of function spaces of initial data. It is also investigated that the similar threshold of the 3-d Euler equations (and other equations of fluid dynamics e.g. the barotropic compressible Navier-Stokes equations) due to the shear flow. コロキウム3 (自然科学5号館471)

第32回 11月6日(金) 16:30～18:00 田中 和永 氏 (早稲田大学) A new deformation argument for singular perturbation problems and applications 非線形シュレディンガー方程式における凝集解の存在問題等の特異摂動問題は非常に興味ある対象である. ここでは, 凝集解の存在を示すためのdeformation argument に関する新しい提案を行い, より広汎な非線形問題に対して凝集解の存在が期待できることを例をあげ示す. コロキウム3 (自然科学5号館471)

第31回 10月30日(金) 16:30～18:00 Thomas Geert de Jong (アイントホーフェン工科大学) Modelling fungal hyphae growth: a quest for travelling waves in an extension of the thin viscous sheet equations If you would take a leaf of the nearest plant or tree and would put it under a microscope then there is a good chance that all the cells have a spherical-like shape. This makes sense because the pressure inside the cell is high compared to the outside. An exception to this type of growth are hyphae cells. They exhibit an extreme lengthwise growth while maintaining a high pressurized environment inside the cell. Hence their cell-shape is governed by a completely different mechanism then the spherical-like cells. My work concerns constructing a mathematical model which explains why this type of growth can occur in hyphae. Many models have already been constructed which describe the growth of hyphae. However, these models do not take into account the dynamics inside the cell wall and consequently have to prescribe explicitly how the wall grows as new material arrives. Our new model overcomes this by modelling the cell wall as a thin viscous sheet where ageing of the cell wall results in hardening of the wall by making the wall more viscous over time. Mathematically it turns out that the growth of hyphae is best described by travelling waves. Existence of the desired travelling wave then corresponds to finding a specific global solution in a highly non-linear 5-dimensional first order ODE. It turns out that this ODE behaves singular close to the boundaries of the desired solution. At the boundaries our local existence and uniqueness proofs yield solutions with the desired boundary behavior. Furthermore, our numerical work suggests that globally the desired solutions exist. コロキウム3 (自然科学5号館471)

第30回 10月23日(金) 16:30～18:00 隠居 良行 氏 (九州大学) On Chorin's method for stationary solutions of the Oberbeck-Boussinesq equation To find a stationary solution of the incompressible Navier-Stokes equation, A. Chorin proposed an artificial compressible system which is obtained by adding the time derivative of the pressure $¥epsilon ¥partial_t p$ to the continuity equation for the incompressible fluid, where $¥epsilon>0$ is a small parameter. If the solution of the artificial compressible system converges to a stationary solution, then the stationary solution is also a stationary solution of the incompressible Navier-Stokes equation. By using this method, Chorin numerically obtained stationary cellular convection solutions of the Oberbeck-Boussinesq equation in a domain between two parallel plates. In this talk I will consider a mathematical justification of Chorin's method. It will be shown that if a stationary solution of the Oberbeck-Boussinesq equation is asymptotically stable and the velocity field of the stationary solution satisfies some smallness condition, then it is also asymptotically stable as a stationary solution of the artificial compressible system for sufficiently small $¥epsilon$. This result is applicable for stable cellular convection solution near the onset of convection of the Benard problem. Problem is formulated as a kind of singular perturbation problem. The point of the proof is to control the spectrum of the "compressbile part" of the linearized operator. This talk is based on a joint work with Takaaki Nishida (Kyoto university). コロキウム3 (自然科学5号館471)

第29回 10月16日(金) 16:45～17:45 ※通常と時間が異なります 鈴木 厚 氏 (元 パリ第六大学リオンス研究所研究員) A dissection solver with kernel detection for unsymmetric matrices in FreeFem++ A direct solver for unsymmetric sparse matrices from finite element problem is developed. The solver has capability to detect the kernel dimension of the matrix when the matrix has an LDU-factorization by a symmetric partial pivoting. The kernel detection algorithm is based on measurement of residual with orthogonal projection onto supposed image spaces and it was developed for symmetric matrices (cf. DOI:10.1002/nme.4729). With small modification by replacing the orthogonal projection onto image space of the matrix by one onto the image space of the transposed matrix, the algorithm works well even in case the kernels of the matrix and of the transposed matrix are not same. The direct solver, "dissection", is coded by C++ and it uses POSIX threads library to manage parallel tasks on shared memory computers. Since most of arithmetic operations are performed by BLAS library, it has high efficiency on modern CPUs with cache memory.　FreeFem++ is a software to solve several kinds of finite element problems, which has been developed in LJLL, Paris 6. "dissection" is called as a sparse direct solver of FreeFem++ by means of a dynamic loading module. The kernel detection capability simplifies FreeFem++ script, e.g. incompressible fluid problem can be solved without a penalization which eliminates ambiguity of the constant pressure. コロキウム3 (自然科学5号館471)

第28回 10月2日(金) 16:30～18:00 Patrick J.P. van Meurs 氏 (金沢大学数物科学系 JSPS外国人特別研究員) Discrete-to-continuum limits of interacting dislocations I will present the discrete-to-continuum limit passage of a non-locally interacting particle system where the unknowns are the positions of the particles on the one-dimensional half line. This particle system arises as a model for understanding plasticity of crystals (the particles represent dislocation walls). Interestingly, five different continuum limits are found by using different scalings. I will show how to obtain these limits by proving Gamma-convergence of the energy describing the particle system in the many-particle limit. コロキウム3 (自然科学5号館471)

臨時 9月28日(月) 16:30～18:00 ※通常と曜日が異なります 藤森 祥一 氏 (岡山大学) 周期的極小曲面の退化極限 本講演では, 3次元ユークリッド空間の周期的極小曲面の極限に関する考察をする. 特に, ある種の重要な2重周期的極小曲面が, Meeks族と呼ばれる3重周期的極小曲面の退化極限として得られることを示す. 本講演は名城大学の江尻典雄氏, 佐賀大学の庄田敏宏氏との共同研究の成果に基づく. コロキウム3 (自然科学5号館471)

第27回 8月27日(木)9:30～8月28日(金)11:50 拡大版として，研究集会を行います．プログラム等の詳細はこちらをご覧ください． 金沢大学サテライト・プラザ 3階集会室 (金沢市西町三番丁16番地 金沢市西町教育研修館内)

第26回 7月27日(月) 16:30～18:00 ※通常と曜日が異なります Pavel Strachota氏， Dejan Kirda 氏 (チェコ工科大学) Mathematical Models and Numerical Simulations in Complex Industrial Applications Our MMG research group at the Czech Technical University deals with numerical solution of mathematical models describing complex processes in industrial applications as well as in nature. In the first part of the contribution, we briefly introduce some of the models and demonstrate the results of the performed simulations. Next, we focus on the problems in validation and verification of numerical models as well as the usual procedures during confrontation of the obtained results with observations and measurements. In the second part, we will describe the details and mathematical background of one such particular model - the quasi-1D CFD model of combustion in a furnace of a circulating fluidized bed boiler. コロキウム3 (自然科学5号館471)

第25回 7月10日(金) 16:30～18:00 三沢 正史 氏 (熊本大学) p調和写像熱流の正則性について コンパクトリーマン多様体に値をとる調和写像およびそのL2勾配流である調和写像熱流の先見的評価は, 局所的に解が正則になるための正則性条件を与えることでほとんど最良に証明されている. 定常p調和写像はp調和作用素を含む非線形退化特異楕円型とはいえ, p=2と同様に最良の正則性条件が証明できる. これら評価で重要なことは, 方程式系の変分(勾配流)構造と同次性である. しかし, p調和写像熱流については, 退化特異方物型に加え, 解のオーダーについて非同次性があり, 局所評価をする上で本質的な障害となる. p調和写像熱流に対して, 時空および解のサイズに関する2つのスケーリング変数をうまく選択して, スケールエネルギーの単調型評価を構成し, 正則性条件を与える. その最良性についても議論する. コロキウム3 (自然科学5号館471)

臨時 7月7日(火) 16:30～17:30 ※通常と曜日・時間が異なります Michal Benes 氏 (チェコ工科大学) Constrained mean curvature flow and the rain droplets In the contribution we discuss the formulation and numerical solution of the planar mean curvature flow with the area constraint. The motion law is treated by means of the phase-field method with non-local terms. This problem is originally mentioned in [1] and analyzed partially in [2]. We identify basic properties of this motion, generalize it by including anisotropy in relative geometry and discuss the quantitative numerical results, their mutual relation, and present qualitative behavior of the solution. [1] J. Rubinstein and P. Sternberg: Nonlocal reaction-diffusion equation and nucleation, IMA J.Appl. Math. (1992) 48, 249--264. [2] L. Bronsard and B. Stoth: Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM Journal on Mathematical Analysis (1997) 28 No. 4, 769--807. コロキウム3 (自然科学5号館471)

第24回 6月12日(金) 16:30～18:00 石毛 和弘 氏 (東北大学) Parabolic Minkowski convolutions of solutions to parabolic boundary value problems We introduce a new kind of convolution, which is a sort of parabolic version of the classical supremal convolution of convex analysis. This operation allows us to compare solutions of different parabolic problems in different domains. As examples of applications of our main result, we study the parabolic concavity of solutions to parabolic boundary value problems, analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term. コロキウム3 (自然科学5号館471)

第23回 5月22日(金) 16:30～18:00 Natali Hritonenko 氏 (Prairie View A&M University) Optimal control of differential and integral equations with applications The talk 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. コロキウム3 (自然科学5号館471)

第22回 4月21日(火) 16:30～18:00 ※通常と曜日が異なります Andrea Malchiodi 氏 (SISSA) Embedded Willmore tori in three-manifolds with small area constraint While there are lots of contributions on Willmore surfaces in the three-dimensional Euclidean space, the literature on curved manifolds is still relatively limited. One of the main aspects of the Willmore problem is the loss of compactness under conformal transformations. We construct embedded Willmore tori in manifolds with a small area constraint by analysing how the Willmore energy under the action of the Möbius group is affected by the curvature of the ambient manifold. The loss of compactness is then taken care of using minimisation arguments or Morse theory. コロキウム3 (自然科学5号館471)

## 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
• 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