Takeshi Fukao, Department of Mathematics, Kyoto University of Education

研究室: A棟4階 1A401
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非線形発展方程式セミナー@KUE

発展方程式

 無限次元関数空間上の常微分方程式を総称して発展方程式と呼ぶ。 特に時間変数についての常微分方程式を適切な関数空間上で考察することで、様々な現象を記述する偏微分方程式がこの枠組みで取り扱えることがよく知られている。 現代では時間微分を含む様な偏微分方程式を総称して発展方程式と呼ぶこともある。

セミナーの趣旨

 線形、非線形に関わらず発展方程式やその応用、さらに広くは時間発展偏微分方程式とその周辺に関連する話題について、研究者間の情報共有を目的としてセミナーを不定期で行っている。

    第8回非線形発展方程式セミナー@KUE

  • 日時: 2018年8月27日(月)15:00-
  • 場所: 京都教育大学 A棟4階 1A402教室(いつもと会場が違います)
  • 15:00-15:30
  • 講演者: Makoto Okumura(Graduate School of Osaka University)
  • 題目: A linear and structure-preserving scheme for a conservative Allen–Cahn equation with a time-dependent Lagrange multiplier
  • We propose a linear and structure-preserving scheme for a non-local conservative Allen– Cahn equation introduced by Rubinstein and Sternberg (1992) based on a combination of the discrete variational derivative method (DVDM) and a linearization technique. DVDM is a numerical method proposed by Furihata (2010). DVDM schemes inherit conservative or dissipative properties from the original PDEs in a discrete sense. By this approach, we obtain a nonlinear scheme in general. Then, we need some iterative solver to solve the system. This means that the computational cost is expensive. Therefore, we also use a linearization technique. The basic idea of our linearization technique is the decompositions of nonlinear terms by introducing extra time steps of numerical schemes. We expect that the proposed linear scheme is faster than the nonlinear one. In this talk, we show the stability, the existence and uniqueness of the solution for the proposed scheme, and the error estimate. We also show numerical experiments.

  • 15:40-16:30
  • 講演者: Hao Wu(School of Mathematical Sciences, Fudan University)
  • 題目: An introduction on the Energetic Variational Approach and its applications
  • In this talk, we will discuss the Energetic Variational Approach for modelling complex physical systems. This approach is based on a balance between the maximal dissipation principle and the least action principle. As a consequence, it naturally keeps the physical laws, such as the conservation of mass, energy dissipation and force balance. Then we will show its application to some well-known systems, for instance, the Cahn–Hilliard equation etc.

    プログラムはこちら

    第7回非線形発展方程式セミナー@KUE

  • 日時: 2018年5月28日(火)15:00-16:50
  • 場所: 京都教育大学 A棟4階 1A402教室
  • 15:00-15:50
  • 講演者: 赤川佳穂(金沢大学大学院自然科学研究科)
  • 題目: Rayleigh-Taylor不安定性を記述するある数理モデルの数値実験とその考察
  • 16:00-16:50
  • 講演者: 奥村真善美(大阪大学大学院情報科学研究科)
  • 題目: ある非局所項付き保存型Allen-Cahn方程式に対する構造保存スキームの誤差評価
  • プログラムはこちら

    第6回非線形発展方程式セミナー@KUE

  • 日時: 2017年8月24日(木)14:00-
  • 場所: 京都教育大学 A棟4階 1A413教室
  • 14:00-14:50
  • 講演者: Hao Wu(School of Mathematical Sciences, Fudan University)
  • 題目: Long-time behavior of nonlinear evolution equations: An introduction to the Lojasiewicz–Simon approach
  • In this talk we will introduce the Lojasiewicz–Simon approach, which is an efficient method to investigate the long-time behavior of nonlinear evolution equations with multiple steady states. We first explain the main idea by using a simple example, i.e., the Allen–Cahn equation. Then we discuss the possible extensions to evolution equations with different structure.

  • 15:10-16:00
  • 講演者: Hao Wu(School of Mathematical Sciences, Fudan University)
  • 題目: Analysis of the Cahn–Hilliard–Hele-Shaw system with singular potential
  • The Cahn–Hilliard–Hele-Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele-Shaw cell. In this talk, we will discuss the CHHS system with a physically relevant potential (i.e., of logarithmic type). We first prove the existence of global weak solutions with finite energy. Then in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any weak solution satisfies the so-called strict separation property. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we show that any global solution will converge to a single equilibrium as time goes to infinity. This is a joint work with A. Giorgini and M. Grasselli (Politecnico di Milano).

    プログラムはこちら

    第5回非線形発展方程式セミナー@KUE

  • 日時: 2017年5月26日(金)13:00-14:30
  • 場所: 京都教育大学 A棟4階 1A413教室
  • 講演者: 来間 俊介(東京理科大学大学院理学研究科)
  • 題目: Remarks on nonlinear diffusion equations on unbounded domains
  • プログラムはこちら

    第4回非線形発展方程式セミナー@KUE

  • 日時: 2017年5月23日(火)~24日(水)
  • 場所: 京都教育大学 A棟4階 1A413教室
  • プログラムはこちら

    第3回非線形発展方程式セミナー@KUE

  • 日時: 2017年1月20日(金)9:00-14:20
  • 場所: 京都教育大学 A棟4階 1A413教室
  • プログラムはこちら

    第2回非線形発展方程式セミナー@KUE

  • 日時: 2015年3月26日(木)16:20-17:20
  • 場所: 京都教育大学 A棟4階 1A413教室
  • 講演者: Pierluigi Colli (Department of Mathematics, The University of Pavia)
  • 題目: Optimal control of phase field equations with dynamic boundary conditions and possibly singular potentials
  • The talk reports on an optimal control problems for the Allen-Cahn or Cahn-Hiilliardequations with a nonlinear dynamic boundary condition involving the Laplace-Beltrami operator. The nonlinearities both in the bulk and on the boundary can be singular, i.e., they may range from the derivative of logarithmic potentials confined in [-1,1] to the subdifferentialof the indicator function of the interval [-1,1] up to a concave perturbation. Existence of optimal controls and first-order necessary optimality conditions are discussed. The presented results arise from collaborations with M. H. Farshbaf-Shaker, G. Gilardiand J. Sprekels.

    日本学術振興会 二国間交流事業 平成 26-27 年度 JSPS-CNR (日本-イタリア) 二国間共同研究 『発展方程式に対する新しい変分法の展開とその応用』によるセミナーです。詳細はこちら

    第1回非線形発展方程式セミナー@KUE

  • 日時: 2013年11月28日(木)16:20-17:20
  • 場所: 京都教育大学 A棟4階 1A413教室
  • 講演者: Pierluigi Colli (Department of Mathematics, The University of Pavia)
  • 題目: A phase field model for the Willmore flow with constraints applying to the evolution of membranes
  • Biological cell membranes define the border between the interior of the cell and its surrounding medium and can be roughly described as a bilayer in which several kinds of lipids are assembled and through which proteins diffuse. The size of the cell (a few microns) is typically much larger than the thickness of the membrane (a few nanometers) and a possible approach to model the geometric properties of the latter is to assume the membrane to be a two-dimensional embedded surface in the three-dimensional space with a shape at equilibrium being determined by the Canham-Helfrich elastic bending energy. In a simplified setting, this energy reduces to the Willmore functional which is a well-known object in differential geometry. Two natural geometric constraints come along with cell membranes: the inextensibility of the membrane fixes the total area while a volume constraint follows from its permeability properties.

    日本学術振興会 二国間交流事業 平成 24-25 年度 JSPS-CNR (日本-イタリア) 二国間共同研究 『発展方程式に対する新しい変分法とその応用』によるセミナーです。詳細はこちら

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