Upcoming

第129回

日時 Time: 2025年6月27日(金)
(27 June 2025 (Fri) talk1: 16:30～17:20, talk2: 17:30～18:20)
場所 Place: 金沢大学自然科学５号館数学棟４階コロキウム

【講演1】：16:30～17:20

講演者 Speaker: 清水扇丈先生(京都大学), Prof. Senjo Shimizu (Kyoto University)
タイトル Title: Local well-posedness of free boundary problems for the compressible Navier-Stokes equations in critical Besov spaces
概要 Abstract: We study local well-posedness of free boundary problems for the compressible Navier-Stokes equations in scaling critical homogeneous Besov spaces. For the density \dot B_{p,1}^{n/p}(R^n_+) and for the velocity \dot B_{p,1}^{-1+n/p}(R^n_+) we prove local well-posedness for the Lagrange transformed system with n-1< p< 2n-1 along Solonnikov’s formulation. A key ingredient in the proof is the end-point maximal L1-regularity for the associated linear initial-boundary value problem of the heat equation governing the velocity field. This is joint work with Takayoshi Ogawa (Waseda University).

【講演2】：17:30～18:20

講演者 Speaker

三沢正史先生(熊本大学), Prof. Masashi Misawa (Kumamoto University)

タイトル Title

p ソボレフ流のエネルギー体積集中と速い拡散型二重非線形放物型方程式の解の有限時間消滅
The energy-volume concentration for the p-Sobolev flow and the finite-time extinction for the fast diffusive doubly nonlinear parabolic equation

概要 Abstract

The p-Sobolev flow is the gradient flow associated with the Sobolev inequality and is described as a doubly nonlinear parabolic equation. In the case p=2 the p-Sobolev flow much related to the Yamabe flow. The asymptotic behavior at infinite-time of the p-Sobolev flow will be studied. We present the global existence for Cauchy-Dirichlet problem for the pSobolev flow, a boundedness, a positivity and a regularity of the solution. The local boundedness is the new ingredient obtained for the doubly nonlinear parabolic equation and the key for studying the energy-volume concentration phenomenon at infinite-time of the pSobolev flow. Our global existence of the p-Sobolev flow is based on the scaling transformation intrinsic to the doubly nonlinear parabolic equation and this our approach also eventually leads to an aplication to the finite-time extinction-behavior for the so-called fast and fast diffusive doubly nonlinear parabolic equation. This is based on a collaborative work with Tuomo Kuusi in University of Helsinki, Finland and Kenta Nakamura in Kumamoto University.

References:
T. Kuusi, M. Misawa, K. Nakamura: J. Geom. Anal. 30 (2020) 1918-1964; J.Differ. Equ. 279 (2021) 245-281;
M. Misawa, K. Nakamura: Adv. Calc. Var. (2021); J. Geom.Anal. 33: 33 (2023);
M. Misawa, K. Nakamura, Md Abu Hanif Sarkar: Nonlinear Differ. Eqn. Appl. 30 ; 43 (2023);
M. Misawa: Calc. Var. 62 (2023), no. 9, No. 265

第128回

日時 Time: 2025年6月6日（金）
(6 June 2025 (Fri)16:30〜17:30)
場所 Place: 自然科学５号館数学棟４階コロキウム
講演者 Speaker: 蚊戸宣幸先生　（金沢大学教育支援センター）
タイトル Title: サイズ構造をもつ個体数変動モデルと最適収穫問題における測度値最適解の存在
概要 Abstract:: 拡散を伴うサイズ構造を持つ個体数変動モデルについて，サイズゼロの供給と収穫を制御して利益を最大にする最適収穫問題を考える。サイズは植物や魚などの成長過程に重要な要素であり，この問題は農業や魚の養殖などで利益を最大にする問題に由来する。ここでは，サイズについてDirac測度含む測度値最適収穫率が存在することを示す。

第127回

日時 Time: 2025年5月22日（木）10:30〜11:30
May 22nd, 10:30-11:30 (JST)
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/dgeHysBjYkXcTrBh9
講演者 Speaker: JIANG Yu (School of Mathematics, Shanghai University of Finance and Economics)
タイトル Title: Recovery in vivo viscoelasticity from elastography measured data
概要 Abstract: This talk will briefly describe how to solve the inverse problem of recovering in vivo viscoelasticity from elastography (magnetic resonance elastography, ultrasound elastography)measurements. To solve it robustly, one need to have a proper partial differential equation model to describe the wave motion inside living body. And based on this PDE model and given interior measurements, theoretical and numerical inverse analyzes need to be performed. As the PDE model, we start with a dynamic viscoelastic model and several simplified models are given. For inversion analysis, we give several practical numerical inversion methods to identify viscoelasticity, such as regularized numerical differentiation method etc.

第126回

日時 Time: 2025年4月7日（月）16:30〜18:00
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: Björn Stinner (The University of Warwick)
タイトル Title: Finite element schemes and mesh smoothing for geometric evolution problems
概要 Abstract: Geometric evolutions can arise as simple models or fundamental building blocks in various applications with moving boundaries and time-dependent domains, such as grain boundaries in materials or deforming cell boundaries. Mesh-based methods require adaptation and smoothing, particularly in the case of strong deformations. We consider finite element schemes based on classical approaches for geometric evolution equations but augmented with the gradient of the Dirichlet energy or a variant of it, which is known to produce a tangential mesh movement beneficial for the mesh quality. We focus on the one-dimensional case, where convergence of semi-discrete schemes can be proved, and discuss two cases. For networks forming triple junctions, it is desirable to keep the impact any additional, mesh smoothing terms on the geometric evolution as small as possible, which can be achieved with a perturbation approach. Regarding the elastic flow of curves, the Dirichlet energy can serve as a replacement of the usual penalty in terms of the length functional in that, modulo rescaling, it yields the same minimisers in the long run.

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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。

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