京都大学NLPDEセミナー

本年（2024年）度のセミナーの記録

4 月 12 日: Timothy Candy 氏（University of Otago）

● 2024 年 4 月 12 日　(Fri)　16:00 ～ 17:00

講演者: Timothy Candy 氏（University of Otago）
講演題目: The non-relativistic limit for the cubic Dirac equation
講演要旨: The Dirac equation is the relativistic version of the Schrödinger equation, and hence unsurprisingly it plays a central role in relativistic (where the speed of light is finite) quantum mechanics. It is known that for certain nonlinear models, as the speed of light tends to infinity, the Dirac equation converges on finite time scales to the Schrödinger equation. Here I will explain how recent uniform (in the speed of light) estimates for small data solutions to the cubic Dirac equation can be used to prove that the non relativistic limit in fact holds on global time scales in dimensions d>1. In particular we have convergence of scattering states and wave operators from the Dirac equation to the corresponding Schrödinger equation. This is joint work with Sebastian Herr.

● 2024 年 4 月 19 日　(Fri)　16:00 ～ 17:00

講演者: 武内太貴氏（京都大学大学院理学研究科）
Taiki Takeuchi (Graduate School of Science, Kyoto University)
講演題目: On regularities of sign-changing solutions of the semilinear heat equation
講演要旨: 本講演では，全空間上の半線形熱方程式の初期値問題について扱う．方程式の非線形項は絶対値付きでべきの指数が整数でない場合を考察し，対応する方程式の解の正則性を詳細に調べる．特に，符号変化する特殊な関数を初期値に与えることで，解はある程度までの正則性を持つ一方で空間方向に関しては無限階微分可能ではないことが示される．本講演では，解の滑らかさを破綻させる為に必要なべき乗型非線形項の高階微分の Hölder 型評価を紹介し，解の高階微分の Hölder ノルムが発散することを具体的な評価によって示す．

● 2024 年 4 月 26 日　(Fri)　16:00 ～ 17:00

講演者: 青木基記氏（京都大学大学院理学研究科）
Motofumi Aoki (Graduate School of Science, Kyoto University)
講演題目: On the ill-posedness for the full system of compressible Navier--Stokes equations in three dimensions
講演要旨: 本講演では，理想気体の運動を表す温度付き圧縮性 Navier--Stokes 方程式の初期値問題について考察する．温度付き圧縮性 Navier--Stokes 方程式は質量保存則，運動量保存則，エネルギー保存則により構成される方程式である．近年，本方程式の初期値問題の適切性は尺度不変性から定まる関数空間で考察されてきた．実際，3 次元以上の空間において，可積分指数が次元より真に小さい場合は一意可解性が，可積分指数が次元より真に大きい場合は非適切性が知られている．本結果では，可積分指数が次元と一致するとき尺度不変となる Besov 空間で初期値連続依存性が成立しないことを示す．この結果は，岩渕司氏（東北大学）の共同研究に基づく．

● 2024 年 5 月 10 日　(Fri)　16:00 ～ 17:00

講演者: 大山広樹氏（京都大学大学院理学研究科）
Hiroki Ohyama (Graduate School of Science, Kyoto University)
講演題目: Fast rotation limit for the magnetohydrodynamics equations in a 3D layer
講演要旨: 本講演では，3 次元層状領域上の Coriolis 力付き非圧縮性磁気流体力学方程式の初期値問題に対して考察する．特に，スケール臨界な関数空間に属する初期値に対して，回転速度が十分大きい場合の同方程式の時間大域解の一意存在性を証明する．さらに，回転速度を無限大とする特異極限において，同方程式の時間大域解が 2 次元磁気流体力学方程式および 3 次元誘導方程式の連立系の解へ，時空間積分ノルムの意味で収束することを示す．本研究は，米田慧司氏（沼津高等専門学校）との共同研究に基づく．

● 2024 年 5 月 17 日　(Fri)　16:00 ～ 17:00

講演者: Michał Łasica 氏（Polish Academy of Sciences）
講演題目: A variational framework for singular limits of gradient flows
講演要旨: We consider a sequence of Hilbert spaces and convex, lower semicontinuous functionals defined on them. Any such functional generates a gradient flow on the underlying space. In the case of a fixed space, it is known that convergence of the sequence of gradient flows is guaranteed by convergence of the functionals in the sense of Mosco. However, in many interesting cases, such as discrete-to-continuum limits, thin domains, or boundary layer problems, the underlying space changes along the sequence.
We introduce a generalization of Mosco-convergence to such setting, based on a notion of "connecting operators". We prove a corresponding general weak convergence result for the flows, without assuming any coercivity of the sequence of functionals, and discuss some applications. Our main tool is the notion of "variational solutions" in the sense of Bögelein et al, which allows performing a "Gamma-convergence"-type argument in the evolutionary setting.
This is joint work with Y. Giga and P. Rybka.

● 2024 年 5 月 31 日　(Fri)　16:00 ～ 17:00

講演者: 三浦達哉氏（京都大学大学院理学研究科）
Tatsuya Miura (Graduate School of Science, Kyoto University)
講演題目: 距離関数の特異点集合のデルタ凸構造
Delta-convex structure of the singular set of distance functions
講演要旨: This talk is about the structure of the singular set of the distance function from an arbitrary closed subset of the standard Euclidean space, or more generally of a complete Finsler manifold. In terms of PDE, the distance function can be viewed as a viscosity solution to the classical eikonal equation or its generalization. Our main result, obtained jointly with Minoru Tanaka (Tokai University), shows that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. A finer structure theorem is given in two dimensions. Those results are new even in the standard Euclidean space and optimal in view of regularity.

● 2024 年 6 月 10 日　(Mon)　13:30 ～ 14:30

講演者: Tim Laux 氏（University of Regensburg）
講演題目: Energy convergence of the Allen-Cahn equation for mean convex mean curvature flow
講演要旨: In this talk, I'll present a work in progress in which I positively answer a question of Ilmanen (JDG 1993) on the strong convergence of the Allen-Cahn equation to mean curvature flow when starting from well-prepared initial data around a mean convex surface. As a corollary, the conditional convergence result with Simon (CPAM 2018) becomes unconditional in the mean convex case.

● 2024 年 6 月 21 日　(Fri)　16:00 ～ 17:00

講演者: 林雅行氏（京都大学大学院人間・環境学研究科）
Masayuki Hayashi (Kyoto University)
講演題目: Global H²-solutions for the generalized derivative NLS on the torus
講演要旨: We prove global existence of H² solutions to the Cauchy problem for the generalized derivative nonlinear Schrödinger equation on the 1-d torus. This answers an open problem posed by Ambrose and Simpson (2015). The key in the proof is to extract the terms that cause the problem in energy estimates and construct modified energies so as to cancel them out by effectively using integration by parts and the equation. This talk is based on a joint work with Tohru Ozawa and Nicola Visciglia.

● 2024 年 7 月 5 日　(Fri)　16:00 ～ 17:00

講演者: Sanghyuk Lee 氏（Seoul National University）
講演題目: Bounds on the strong spherical maximal functions
講演要旨: This talk concerns Lp bounds on the strong spherical maximal functions that are multi-parametric maximal functions defined by averages over ellipsoids. We obtain Lp bounds on those maximal operators for a certain range of nontrivial p. No such maximal bounds have been known until recently. In particular, our results extend Stein’s celebrated spherical maximal bounds to multi-parametric versions.

● 2024 年 7 月 19 日　(Fri)　16:00 ～ 17:00

講演者: Pierre Mackowiak 氏（CMAP, École Polytechnique, Institut Polytechnique de Paris）
講演題目: Standing waves for the Anderson-Gross-Pitaevskii equation in dimension 1 and 2
講演要旨: The Gross-Pitaevskii equation is a non-linear Schrödinger equation with confining potential that appears in the study of Bose-Einstein condensate. Adding a random potential to this equation is a way to model spatial inhomogeneities during the condensation process. With this in mind, the Anderson-Gross-Pitaevskii, that is the Gross-Pitaevskii with a spatial white noise potential, can be seen as a toy model for spatial inhomogeneities with small correlation length.
In this talk, I will briefly present the outline of the construction of the Anderson-Hermite operator and some of its spectral properties. Then, I will sum up some of the known results on standing waves of the Anderson-Gross-Pitaevskii equation in dimension 1: existence, regularity, localization, stability, bifurcation. I will finish by presenting what is known for 2d standing waves and what still needs to be investigated. This talk is based on an ongoing work.

● 2024 年 7 月 26 日　(Fri)　16:00 ～ 17:00

講演者: Yi C. Huang 氏（Nanjing Normal University）
講演題目: Calderón-Zygmund Decompositions in Tent Spaces
講演要旨: We prove some Calderón-Zygmund type decompositions for functions in the tent spaces introduced by Coifman, Meyer and Stein. These decompositions are useful in developing further the operator theory on tent spaces (OTTS) of Auscher, Kriegler, Monniaux and Portal. (It should be pointed out that the OTTS is motivated by the solvability of elliptic boundary value problems and the Maximal Regularity of evolution equations. This motivation will be explained at the beginning of the talk.) As an application of these decompositions to the study of quadratic functionals on tent spaces, we give a unified proof for tent space generalizations of C. Fefferman’s endpoint weak-type estimates for grand square functions and of C. Fefferman and Stein’s endpoint weak-type estimates for box maximal functions.

● 2024 年 10 月 4 日　(Fri)　16:00 ～ 17:00

講演者: 清水一慶氏（京都大学大学院理学研究科）
Ikkei Shimizu (Graduate School of Science, Kyoto University)
講演題目: Time decay estimate for localized perturbation around helical state for 2D Landau-Lifshitz-Gilbert equation
講演要旨: We consider time-decay estimate for solutions of 2D Landau-Lifshitz-Gilbert (LLG) equation around the helical state. This state is a stationary solution which is periodic in one spatial direction, while constant in the other direction. In the linearized analysis, we apply frequency decomposition in terms of the Bloch wave numbers, obtaining time-decay estimate for low-frequency part and energy inequality for high-frequency part.

● 2024 年 10 月 11 日　(Fri)　16:00 ～ 17:00

講演者: 岡本潤氏（京都大学高等研究院ヒト生物学高等研究拠点）
Jun Okamoto (ASHBi, Kyoto University)
講演題目: On a singular limit of the Kobayashi--Warren--Carter energy and its gradient flow
講演要旨: 本講演では，多結晶物質の結晶粒界のダイナミクスを記述するモデルである，Kobayashi--Warren--Carter(KWC) エネルギーの特異極限問題について考察する． KWC エネルギーに含まれる重み付き全変動エネルギーの効果により，KWC エネルギーの最小元は不連続点以外では定数であるが，不連続点の低次元領域で別の値を持つ関数に収束する．したがって低次元の測度 0 の集合を無視してしまうような Lebesgue 空間などの位相で特異極限を考察することは不十分である．我々はスライス-グラフ収束というより細かい位相を導入することにより，KWC エネルギーの精密な特異極限の特徴づけに成功した．また空間一次元における KWC エネルギーの勾配流の特異極限が，分数階微分を含む系として特徴づけられた結果も報告する．本講演の内容は，儀我美一氏（東京大学），久保絢斗氏（北海道大学），黒田紘敏氏（北海道大学），榊原航也氏（金沢大学），上坂正晃氏（DataLabs 株式会社）との共同研究に基づく．

● 2024 年 10 月 18 日　(Fri)　16:00 ～ 17:00

講演者: 清水良輔氏（早稲田大学）
Ryosuke Shimizu (Waseda University)
講演題目: Self-similar $p$-energy forms and $p$-energy measures on the Sierpinski carpet
講演要旨: 1980 年代後半から急速に発展した「フラクタル上の解析学/確率論」では Sierpinski gasket や Sierpinski carpet を始めとした自己相似フラクタル上での熱拡散（Brown 運動）の研究が中心であったが，多くの評価が確率論的解釈に依存していることが障害となり，単純な $L^p$-拡張，すなわち $(1,p)$-Sobolev 空間と対応する $p$-エネルギーの定式化，すらままならない状況であった．本講演では Sierpinski carpet のグラフ近似列上の離散エネルギーの（部分列）スケール極限としての $(1,p)$-Sobolev 空間と自己相似性を有する $p$-エネルギーの構成，可分反射性や正則性（連続関数の中で稠密）などといった Sobolev 空間の基本的性質に関する結果，及び Ahlfors 正則等角次元と呼ばれる幾何学的量との関連を述べる．本研究は Mathav Murugan 氏 (University of British Columbia) との共同研究に基づく．

● 2024 年 11 月 1 日　(Fri)　16:00 ～ 17:00

講演者: Fabian Rupp 氏（University of Vienna）
講演題目: On the Dynamics of Elastic Cell Membranes
講演要旨: The Canham-Helfrich model characterizes the equilibrium shapes of lipid bilayers, such as red blood cells, as constrained critical points of a curvature-dependent bending energy. In this talk, we present a time-dependent approach via a non-local geometric gradient flow. For initial data below an explicit threshold, we show global existence and convergence. Our proof is based on an adapted Li-Yau-type multiplicity inequality, a classification of blow-ups, and an appropriate version of the Lojasiewicz-Simon gradient inequality.

● 2024 年 11 月 8 日　(Fri)　16:00 ～ 17:00

講演者: 牛島健夫氏（東京理科大学創域理工学部数理科学科）
Takeo Ushijima (Department of mathematics, Faculty of science and technology, Tokyo university of science)
講演題目: 曲線短縮問題に現れるある準線形放物型方程式の解の爆発について
On blow-up solutions of a quasilinear parabolic equation arising in a curve-shortening problem
講演要旨: ある種の曲線短縮運動を記述する準線形放物型方程式 $u_t = u^p (u_{xx}+u)$ は，いわゆるタイプ II の特異性を持つ爆発解を有することが知られている．しかしながら，$p>2$ の場合には，その爆発解の詳細な挙動についてはあまり多くのことは知られていない．我々は，この方程式を周期境界条件下で考察し，$2<p\leq 3$ の場合のタイプ II の爆発解の爆発レートの上からの評価を得た．本講演では，この結果について報告する．
この講演は，穴田浩一氏（早稲田大学高等学院），石渡哲哉氏（芝浦工業大学）との共同研究に基づくものである．
The quasilinear parabolic equation $u_t = u^p (u_{xx}+u)$ describing a certain type of curve-shortening motion is known to have a blow-up solution with so-called type II singularity. However, only a little is known about the detailed behavior of its blow-up solution, especially in the case $p>2$. We consider this equation under periodic boundary conditions with a class of initial data and obtain an evaluation from above of the blow-up rate of the type II solution in the case of $2<p\leq 3$. In this talk, we report on these results.
This talk is based on joint research with Koichi Anada (Waseda University Senior High School) and Tetsuya Ishiwata (Shibaura Institute of Technology).

● 2024 年 11 月 15 日　(Fri)　16:00 ～ 17:00

講演者: Zihua Guo 氏（Monash University）
講演題目: On the well-posedness of the compressible Navier-Stokes equations
講演要旨: We consider the Cauchy problem to the barotropic compressible Navier-Stokes equations. We obtain optimal local well-posedness in the sense of Hadamard in the critical Besov space. The main new result is the continuity of the solution maps from B to C([0,T]:B), which was not proved in previous works. To prove our results, we combine the method of frequency envelope developed by Tao but in the transport-parabolic setting and the Lagrangian approach for the compressible Navier-Stokes equations developed by Danchin. As a by-product, the Lagrangian transform used by Danchin is a continuous bijection and hence bridges the Euler methods and Lagrangian approach.

● 2024 年 11 月 29 日　(Fri)　16:00 ～ 17:00

講演者: 高橋仁氏（東京科学大学情報理工学院）
Jin Takahashi (School of Computing, Institute of Science Tokyo)
講演題目: Critical norm blow-up for supercritical semilinear heat equation
講演要旨: We consider the scaling critical Lebesgue norm of finite-time blow-up solutions to the energy supercritical semilinear heat equation. Our result shows that the critical norm also blows up at the blow-up time. In addition, we also give estimates on the blow-up rate of the critical norm. This talk is based on joint works with Tobias Barker (University of Bath) and Hideyuki Miura (Institute of Science Tokyo).

● 2024 年 12 月 6 日　(Fri)　16:00 ～ 17:00

講演者: 比佐幸太郎氏（東京大学大学院数理科学研究科）
Kotaro Hisa (Graduate School of Mathematical Sciences, The University of Tokyo)
講演題目: Initial traces of solutions to a semilinear heat equation under the Dirichlet boundary condition
講演要旨: In this talk, we study qualitative properties of initial traces of nonnegative solutions to a semilinear heat equation in a smooth domain under the Dirichlet boundary condition. Furthermore, for the corresponding Cauchy--Dirichlet problem, we obtain sharp necessary conditions and sufficient conditions on the existence of nonnegative solutions and identify optimal singularities of solvable nonnegative initial data. This talk is based on the joint work with Prof. K. Ishige (The University of Tokyo).

● 2024 年 12 月 13 日　(Fri)　16:00 ～ 17:00

講演者: 久藤衡介氏（早稲田大学理工学術院基幹理工学研究科）
Kousuke Kuto (School of Fundamental Science and Engineering, Waseda University)
講演題目: SKT モデルの定常解の大域分岐構造と安定性
講演要旨: 競争種の棲み分けの数理モデルとして交差拡散 (cross-diffusion) を伴うロトカ・ボルテラ系が重定・川崎・寺本 (1979) によって提唱され SKT モデルとよばれている． SKT モデルは拡散の相互作用のプロトタイプとして様々な観点から研究が続けられており，棲み分けを示唆するパターンを再現する行程として交差拡散係数を無限大にしたときの解の漸近挙動が重要視されている．本講演においては，両種の交差拡散係数を無限大にしたときの定常解の漸近挙動をノイマン，ディレクレのそれぞれの境界条件の下で考える．前半部では，定常解の交差拡散係数に依らないアプリオリ評価，交差拡散極限系の解集合に関する結果を紹介する．後半部では，極限系の解集合の摂動によって構成される交差拡散係数が大きいケースの分岐図と各分岐枝のモース指数に関する結果を紹介する．結果の一部は講演者がかつて研究指導にあたった井上順平氏（キオクシア），佐藤誉氏（セガ）との共同研究に基づく．

● 2024 年 12 月 20 日　(Fri)　16:00 ～ 17:00

講演者: 江藤徳宏氏（東京大学大学院数理科学研究科）
Tokuhiro Eto (Graduate School of Mathematical Sciences, The University of Tokyo)
講演題目: Mean Curvature Flow with Contact Angles: Convergence of the Level Set Formulation to Viscosity Solutions
講演要旨: In this talk, we introduce a capillary Chambolle-type scheme for mean curvature flow (MCF) with prescribed contact angle conditions in smooth bounded domains. This advanced scheme incorporates a capillary functional in place of total variation and demonstrates consistency with the Almgren-Taylor-Wang energy minimizing framework. We establish the well-posedness of the scheme and provide numerical examples based on the split Bregman method for planar motions. Furthermore, we prove that the approximate solutions generated by the scheme converge to the level-set formulation of MCF with prescribed contact angles. Specifically, under convexity of the domain and suitable control of the derivatives of the prescribed angle, the auxiliary function associated with the scheme is shown to uniformly converge to the unique viscosity solution of the level-set equation with an oblique derivative boundary condition. This talk is based on joint works with Yoshikazu Giga.

● 2025 年 1 月 10 日　(Fri)　16:00 ～ 17:00

講演者: 尾上文彦氏（ミュンヘン工科大学）
Fumihiko Onoue (Technische Universität München)
講演題目: Variational models involving nonlocal perimeter and prescribed nonlocal mean curvature problems
講演要旨: It is well-known in physics that the shape of liquid drops subject to the action of a potential minimizes (under volume constraint) their free energy at equilibrium. The classical free energy consists of a surface tension and a potential energy induced by an external force field. In this talk, we consider a nonlocal extension of this model in which a "nonlocal" surface tension of liquid drops appears. We prove that for small mass, minimizers of the nonlocal energy are convex if a potential is coercive. Moreover, we see some connection between the nonlocal model and prescribed "nonlocal" mean curvature problems and show several results on this geometric problem. This talk is based on a joint work with K. Bessas (Università di Pavia) and M. Novaga (Università di Pisa).

● 2025 年 1 月 31 日　(Fri)　13:30 ～ 14:30

講演者: Van Tien Nguyen 氏（National Taiwan University）
講演題目: Multiple collapsing solutions to the 2D Keller-Segel system
講演要旨: So far, except in some integrable PDEs whose explicit formulas for multiple-soliton solutions are known, the question of the collision of solitons (soliton-soliton interaction) is much less understood. Most previous studies have focused on global-in-time solutions. The talk presents a rigorous framework to construct such blowup solutions involving simultaneously the non-radial collision and concentration of several solitons in the 2D Keller-Segel system. It's a brand new mechanism of singularity formation formed by a collision of two sub-collapses, resulting in a finite-time blowup solution with a 16-pi mass concentration at a single point.; The talk is drawn from joint works with C. Collot, T. Ghoul and N. Masmoudi from the references:
https://arxiv.org/abs/2409.05363
https://doi.org/10.1002/cpa.21988
https://link.springer.com/article/10.1007/s40818-022-00118-5

別の年度

〒606-8502 京都市左京区北白川追分町
京都大学大学院理学研究科数学教室（理学研究科 3 号館）

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京都大学 NLPDE セミナー

本年（2024年）度のセミナーの記録