金沢解析セミナー

Kanazawa Analysis Seminar

2024

第125回

日時 Time
2025年3月19日（水）16:30〜18:00
場所 Place
自然科学5号館223室
NST Hall 5, room 223
講演者 Speaker
Gilbert Peralta (University of the Philippines Baguio)
タイトル Title
Optimal Control for the Stationary Navier-Stokes Equation Involving the Pressure
概要 Abstract
We consider optimal control problems for the two-dimensional stationary Navier-Stokes equation with observations involving the pressure. For point observations, it is known that such problems give rise to PDEs with measure data for the corresponding adjoint equation. In our analysis, we will examine the state equation locally near a regular (non-singular) solution. Roughly speaking, regular solutions are those that yield topological isomorphisms for the linearized operators and consequently lead to the uniqueness of solutions within a certain neighborhood. Local optimality systems for such control problems will be presented.

第124回

日時 Time
2025年3月12日（水）16:00〜18:15
場所 Place
自然科学５号館数学棟４階コロキウム
NST Hall 5, Math building, 4th floor, Colloqium room
Zoom registration: https://forms.gle/y876CNxkiyUdyoUy9

講演1: 16:00–17:00

講演者 Speaker
Yoshiho AKAGAWA (Gifu KOSEN(NAtional Institute of Technology))
タイトル Title
An evolution inclusion describing strain-hardening phenomenon
概要 Abstract
Strain-hardening is a phenomenon that results from elasto-plastic deformation. This phenomenon makes additional deformation difficult as plastic deformation progresses.In 1976, Duvaut and Lions introduced a prototype model of elasto-plasticity. The essential idea of this model is to characterise elasto-plastic deformation by imposing a constraint on the deviatoric part of the stress tensor. We describe strain-hardening by allowing the constraint to depend on an unknown function. The unknown dependencies of the constraint are the crucial points of this model. As one of them, the parallel shift of the constraint plays an important role. In the case when the parallel shift depends on time nonlocally or some unknowns, then the model represents a more realistic phenomenon. This system is related to Moreau’s sweeping process and is formulated as an evolution inclusion governed by a time-dependent subdifferential.

講演2: 17:15–18:15

講演者 Speaker
Megumi SANO (Hiroshima Univ.)
タイトル Title
Weighted Trudinger-Moser inequalities in the subcritical Sobolev spaces and their applications
概要 Abstract
We study boundedness, optimality and attainability of Trudinger-Moser type maximization problems in the subcritical radial Sobolev spaces. Our inequality converges to the original Trudinger-Moser inequality including the optimal exponent and the concentration limit in some sense. Finally, we consider applications of our inequality to elliptic and parabolic problems with exponential nonlinearity.This is a joint work with Masahiro Ikeda(Osaka Univ.) and Koichi Taniguchi(Shizuoka Univ.).

第123回

日時 Time
2025年1月6日（月）16:30〜18:00
場所 Place
自然科学５号館研究棟2階 209A室 + Zoom
Natural Science and Technology Hall 5, 2nd floor, Room 209A
Zoom registration: https://kanazawa-university.zoom.us/meeting/register/tZMtdeGprTItGdSVnSVyL2oVcqme0N4TEjkW
講演者 Speaker
Philip Schrader（Murdoch University）
タイトル Title
Curve shortening by Sobolev gradient flow
概要 Abstract
The classical curve shortening flow, or one-dimensional mean curvature flow, is the gradient descent of the length functional on curves when the gradient is taken with respect to a parametrisation invariant L^2 metric. In this talk I will discuss the gradient flows of length with respect to a family of parametrisation invariant Sobolev H^1 metrics with different degrees of homogeneity. The family of flows all turn out to be equivalent under time reparametrisations. Solutions preserve embeddedness, converge to points and appear to become round while converging, like the classical flow. But unlike the classical flow, we do not encounter singularities.

第122回

日時 Time
2024年12月18日(水) 16:00-18:10
December 18th 2024 (Wed) 16:00-18:10JST
場所 Place
自然科学5号館2階209A室
NST Hall 5, main building, 2th floor, room 209A

講演1: 16:00–17:00

講演者 Speaker
Jiří Minarčík（Czech Technical University in Prague）
タイトル Title
Trajectory Surfaces of Space Curve Flows
概要 Abstract
This talk will examine the dynamics of geometric flows in three-dimensional Euclidean space, focusing on new methods for generating surfaces with specific properties, such as constant mean curvature, through the motion of space curves. We will discuss challenges inherent to higher codimension flows, including their analytical and topological properties, and conclude with an exploration of potential applications in computational geometry.

講演2: 17:10–18:10

講演者 Speaker
Maciej Buze（Heriot-Watt University in Edinburgh）
タイトル Title
Barycenters in unbalanced optimal transport
概要 Abstract
The theory of optimal transportation, dating back to Gaspard Monge’s work in 1781, continues to develop at pace as one of the fundamental mathematical theories with an ever- growing list of diverse applications in fields such as economics, computer vision, image processing and machine learning. A central challenge in many applications concerns finding a representative, or barycentric (probability) distribution, which provides some average description of a given set of distributions. The basic optimal transport approach to this problem is to find the barycenter by minimizing the sum of weighted two-marginal optimal transport costs between the barycenter and each input distributions. In a seminal contribution by Agueh and Carlier [1], it was subsequently shown that an equivalent and computationally favourable approach is to instead solve a single least-cost multi-marginal optimal transport problem. If the input distributions do not all have equal mass, an unbalanced barycenter can be found via a recourse to the emerging theory of unbalanced optimal transportation. This, however, can be done in a number of ways, depending on how one penalises mass deviations, what cost function is employed and whether one wishes to consider the conic formulation — see the detailed discussion in [2]. In this talk, I will introduce the above ideas in an accessible manner, followed by presenting several results on how to recover the celebrated least-cost multi-marginal formulation of Agueh and Carlier in the unbalanced setting [3]. [1] M. Agueh and G. Carlier. Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis 43.2 (2011), pp. 904–924. [2] M. Liero, A. Mielke, and G. Savaré. Optimal entropy-transport problems and a new Hellinger– Kantorovich distance between positive measures. Inventiones mathematicae 211.3 (2018), pp. 969–
[3] M. Buze. Constrained Hellinger-Kantorovich barycenters: least-cost soft and conic multi-marginal formulations. arXiv e-prints, 2402.11268, 2024 (to appear in SIAM Journal on Mathematical Analysis).

第121回

日時 Time
2024年12月13日(金) 16:30-18:00
Dec 13, 2024 (Fri) 16:30-18:00JST
場所 Place
自然科学５号館数学棟４階コロキウム + Zoom
Zoom registration: https://forms.gle/JVb1F1TnYAvyqurUA
講演者 Speaker
渡辺 樹 氏（大分大学）
タイトル Title
Markov chain approximation in nonlocal diffusion models
概要 Abstract
マルコフ過程で構成される拡散現象を表す粒子系 (ゼロレンジプロセス)の連続極限について考察する. 特に空間分割数を無限大にすると同時に粒子が細分化されるという流体化極限に着目する. これはKurtz(1970) によって常微分方程式の近似手法として初めて導入され, その後Arnold and Theodosopulu(1980)やKotelenez(1986), Blount(1991)らによって局所拡散を表す粒子系の連続極限が2階の反応拡散方程式として記述されることが明らかにされている. 本講演では, ネットワーク上での人の流動といった非局所拡散現象に着目し, 再生核理論を用いることにより連続極限が積分型拡散方程式で与えられる(大数の法則)ことを示す. さらにスケールの違いによって極限として確率微分方程式が現れる(中心極限定理)ことを紹介する. 時間が許せば, 連立系に対応する交差拡散を伴う2種粒子系に関する結果についても紹介する.

第120回

日時 Time
2024年11月18日(月) 16:30-18:00
November 18, 2024 (Mon) 16:30-18:00JST
場所 Place
自然科学５号館数学棟４階コロキウム + Zoom
Zoom link: https://us06web.zoom.us/j/81724187213?pwd=f9boPu1dGaKiwDwIZDTk4dNT8qbSaF.1
講演者 Speaker
利根川 吉廣 氏（東京科学大学）
タイトル Title
Brakke flow with a forcing term
概要 Abstract
Given a compact smooth hypersurface and a non-smooth time-dependent vector field, one can prove some existence of evolving hypersurface whose velocity is equal to the mean curvature and the given vector field. I discuss the existence and regularity of such hypersurface under a certain general regularity assumption on the vector field which is subcritical in the sense of parabolic scaling. I also present a recent existence result which may be considered a critical case.

第119回

日時 Time
2024年11月5日(火) 16:00-18:10
November 5th 2024 (Tue) 16:00-18:10JST
場所 Place
自然科学５号館数学棟４階コロキウム
NST Hall 5, Math building, 4th floor, Colloqium room

講演1: 16:00–17:00

講演者 Speaker
Scott McCue (Queensland University of Technology)
タイトル Title
Continuing Burgers’ equation into the complex plane
概要 Abstract
Many of you will know the viscous Burgers’ equation, a very well-studied parabolic pde which sets up a competition between nonlinear advection that tends to steepen the solution profile and linear diffusion that tends to smooth it out. This pde is a simple toy model for the Navier-Stokes equation in 1D. A simpler version is the inviscid Burgers’ equation, which is a first-order nonlinear pde that can be solved exactly by an undergraduate using the method of characteristics. For this version of Burgers’ equation, there is no diffusion and so nonlinear advection drives the solution to continue to steepen until the derivative blows up somewhere in finite time. We shall revisit these models, but instead of restricting ourselves to the real line, we shall continue the solution out to the complex plane. In this way, we observe directly the culprit for the finite-time blow-up in the inviscid Burgers’ equation, which is a branch point that moves towards the real axis and touches it at the blow-up time. In the viscous Burgers case, the singularity structure in the complex plane is much more complicated, but with some investigative tools we can track the motion of complex-plane singularities and show why blow-up does not occur.

講演2: 17:10–18:10

講演者 Speaker
Liam Morrow (University of Oxford)
タイトル Title
Recent developments in pattern formation in a Hele-Shaw cell
概要 Abstract
The Hele-Shaw cell is a simple experimental device that has been used extensively to study the development of patterns in a wide variety of physical phenomena. In this talk, I will present a state of the art numerical scheme for studying a generalised model of fluid flow in a Hele-Shaw cell. The scheme is based on the level set method, which is a popular numerical framework for solving moving boundary problems. By making simple adjustments, I will show how this scheme can be used to study a wide variety of problems that have previously only been studied by making significant simplifications to the model. Further, I will demonstrate how using this scheme, we have been able to provide significant new advancements into our understanding of how patterns develop and, in turn, can be controlled.

第118回

日時 Time
2024年8月9日(金) 16:00-17:00(講演1), 17:10-18:10 (講演2)
August 9st 2024 (Fri) 16:00-17:00(Talk 1), 17:10-18:10 (Talk 2）JST
場所 Place
Hybrid

講演1　

講演者 Speaker
山田哲也氏 (福井工業高等専門学校)
タイトル Title
2次元放物型attraction-repulsion Keller-Segel方程式に対する初期値問題の解の有界性
概要 Abstract
3本の放物型方程式で記述された2次元誘引反発型Keller-Segel方程式の初期値問題を考察する。この初期値問題において、H.Y. Jin-Z. Liu (2015)や山田(2022)は誘引効果を表す係数が反発効果を表す係数よりも小さいかまたは等しいとき, 非負解は有界であることを証明した。本講演では、誘引効果を表す係数が反発効果を表す係数より大きい場合について考察し、誘引および反発効果を表す係数に依存して決まる定数より初期値の積分量が小さいならば、非負解が有界であることを示す。なお、本講演の内容は永井敏隆氏(広島大名誉教授)と関行宏氏(東京都立大)との共同研究に基づく。

講演2　

講演者 Speaker
和久井洋司氏 (福井大学 学術研究院工学系部門)
タイトル Title
誘引反発型移流拡散方程式の定数定常解の安定性
概要 Abstract
誘引反発型の移流項を持つ移流拡散方程式のn次元ユークリッド空間における定数定常解の安定性を考察する。誘引型移流項を持つ移流拡散方程式の定数定常解は無数に存在し、誘引効果の強さを表す係数に応じて安定となる定数の範囲が制限されることが知られている。一方で反発型移流項を持つ移流拡散方程式では、反発効果を表す係数の大きさに依存せずに、正値な定数定常解は安定となる。誘引反発混合型の移流項を持つ場合は、誘引効果と反発効果の影響が、それぞれの効果の強さを表す係数の関係によって決定されることがいくつかの先行研究によって示唆されている。本講演では、それらの係数の関係に着目し、安定な定数定常解となる定数の範囲とその閾値を述べる。本研究は、山田哲也氏(福井高専)との共同研究に基づく。 ## 第116回
日時 Time
2024年6月21日(金) 16:30-18:00
June 21st 2024 (Fri) 16:30-18:00JST
場所 Place
自然科学５号館数学棟４階コロキウム
NST Hall 5, Math building, 4th floor, Colloqium room
講演者 Speaker
Professor Zhouping Xin (Chinese University of Hong Kong)
タイトル Title
On some free boundary value problems arising from subsonic-sonic jet flows and rigidity
概要 Abstract
In this talk, I will discuss some results on steady compressible potential jet flows from a finite converging nozzle, which are free boundary problems for a nonlinear degenerate elliptic equation. An important feature is that such problems do not have a variational structure. Formulation of the problems and the existence (and non-existence) of solutions will be discussed. Both finite jets and infinite jets can be obtained by a PDE approach and regularity and properties of the solutions. In particular, a general result on the rigidity of the location of sonic degeneracy will be established. This talk is based on joint works with Chunpeng Wang.

第117回

日時 Time
2024年7月25日(木) 16:30-18:00
July 25th 2024 (Thu) 16:30-18:00JST
場所 Place
自然科学５号館数学棟４階コロキウム
NST Hall 5, Math building, 4th floor, Colloqium room
講演者 Speaker
Eric Kim (UCLA)
タイトル Title
Long Term Behavior of Area-Preserving Anisotropic Curvature Flow in the Plane
概要 Abstract
I will present joint work with Dohyun Kwon, in which we establish the exponential convergence of the area-preserving anisotropic flat flow in the plane to a disjoint union of Wulff shapes of equal area, extending the results of Julin-Morini-Ponsiglione-Spadaro in the isotropic setting.

第115回

日時 Time
2024年4月4日(木) 16:30-18:00
April 4th 2024 (Thu) 16:30-18:00JST
場所 Place
Online seminar
Zoom registration: https://us06web.zoom.us/meeting/register/tZ0qc-qprj8oG9eN-7TH6etjYMkOz9NCM0Pn
講演者 Speaker
Koondanibha Mitra (Eindhoven University of Technology, the Netherlands)
Work in collaboration with:
S. Sonner, Radboud University, Nijmegen, the Netherlands
J.M. Hughes, University of British Columbia, Canada
H.J. Eberl, University of Guelph, Canada
J.D. Dockery, Montana State University, USA
R.K.H. Smeets, University of Amsterdam, the Netherlands
I.S. Pop, Hasselt University, Belgium
タイトル Title
Analysis of coupled degenerate and singular systems in biofilm modelling: Well-posedness, special solutions, and numerical strategies
概要 Abstract
Coupled nonlinear degenerate systems arise in various applications of societal relevance, such as biofilm growth, wildfires, and reactive transport in porous media. One of the equations in the system, describing the evolution of a biomass, exhibits degenerate and singular diffusion behaviour. The other equations are either of advection-reaction-diffusion type or ordinary differential equations modelling nutrient dispersion. In our analysis, we first propose a backward Euler time-discretization of the problem where the reactive terms coupling the equations are estimated semi-implicitly, and thus, the equations can be solved sequentially. Properties such as well-posedness, boundedness, and positivity of the time-discrete solutions are proven, and the existence of the time-continuous solutions is shown by passing the time-step size to zero. Global- in-time well-posedness is established for Dirichlet and mixed boundary conditions, whereas only local well-posedness can be shown for homogeneous Neumann boundary conditions. Using a suitable barrier function and comparison theorems we formulate sufficient conditions for finite-time blow-up or uniform boundedness of solutions. Assuming additional structural assumptions we also prove the uniqueness of solutions. The time-discretization method serves as an efficient way to solve such problems and numerical experiments are provided that support this. Then, we show the existence of traveling wave (TW) solutions for a special case of PDE-ODE coupled models arising in the growth of cellulolytic biofilms. TW solutions for such systems have previously been observed numerically as well as in experiments. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase-plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. We present numerical results that exhibit the existence and stability of the TWs, along with corroborating predictions on parametric dependence. References
[1] K. Mitra, J.M. Hughes, S. Sonner, H.J. Eberl, & J.D. Dockery (2023). Travelling Waves in a PDE– ODE Coupled Model of Cellulolytic Biofilms with Nonlinear Diffusion. Journal of Dynamics and Differential Equations, 1-35.
[2] K. Mitra, & S. Sonner (2023). Well-posedness and properties of nonlinear coupled evolution problems modelling biofilm growth. arXiv:2304.00175.

第114回

日時 Time
2024年1月22日(月) 17:00-18:00
January 22nd 2024 (Mon) 17:00-18:00JST
場所 Place
自然科学５号館研究棟2階 209A室 + Zoom
Natural Science and Technology Hall 5, 2nd floor, Room 209A
Zoom registration: https://kanazawa-university.zoom.us/meeting/register/tZYld-qsrjsrHtX3xOL9QF_F0npy8qb55O9O
講演者 Speaker
Jyrki Jauhiainen (University of Helsinki)
タイトル Title
Mumford-Shah functional in electrical impedance tomography
概要 Abstract
In electrical impedance tomography, we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This problem, known as the inverse conductivity problem or Calderon problem, is notoriously ill-posed, especially in real applications with only partial boundary data available. We discuss various approaches to modeling the inverse conductivity problem, the characteristics of these models, and the known regularity results. We then focus on the variational approach by considering the minimization of an energy regularized by the Mumford-Shah functional. As this minimization poses a significant challenge, we take the classical Ambrosio-Tortorelli approximation approach, showing its approximation properties for the inverse conductivity problem. Finally, we take a look at a few numerical examples.

第113回

日時 Time
2024年1月17日（水）16:30〜18:00
場所 Place
自然科学５号館数学棟４階コロキウム＋Zoom によるオンライン配信 登録リンク：https://kanazawa-university.zoom.us/meeting/register/tZYrcOmqqjsqGdfKalOqCoGol6OobOzSNMP8
講演者 Speaker
西口 純矢 氏（東北大学）
タイトル Title
On a theory of “mild solutions” and its application to the linearized stability for delay differential equations
概要 Abstract
A retarded functional differential equation (RFDE) is a differential equation describing the dependence of the time-derivative \dot{x}(t) of an unknown function x on the history segment x_t. The RFDEs gives a mathematical model of various delay differential equations (DDEs), and it determines an important class of infinite-dimensional dynamical systems as the time evolution of the history segment x_t in a space of continuous functions. In this talk, we develop a theory of “mild solutions” to autonomous linear RFDEs in order to resolve difficulties about discontinuous history functions in the theory of RFDEs. Furthermore, we apply it to an extension of the Poincaré–Lyapunov theorem to RFDEs. 参考文献 (References)：
[1] J. Nishiguchi, Mild solutions, variation of constants formula, and linearized stability for delay differential equations, Electron. J. Qual. Theory Differ. Equ. 2023, No.~32, 1–77. https://doi.org/10.14232/ejqtde.2023.1.32.
[2] J. Nishiguchi, On regularity of mild solutions for linear delay differential equations, submitted.

Information

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。 どうぞよろしくお願いいたします。

セミナー幹事 Organizers

Patrick van Meurs・大塚 浩史・小俣 正朗・蚊戸 宣幸・木村 正人・榊原 航也・Thomas Geert De Jong・野津 裕史・Norbert Pozar・Julius Fergy Tiongson Rabago

リンク Links

アクセス Access・数学コース Math course・計算数理プログラム Applied Math program

お問い合わせ Contact

Norbert Pozar ・npozar (at) se.kanazawa-u.ac.jp

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