本年（2024年）度のセミナーの記録
- 4 月 12 日
- Timothy Candy 氏 （University of Otago）
- 4 月 19 日
- 武内 太貴 氏 （京都大学） Taiki Takeuchi (Kyoto University)
- 4 月 26 日
- 青木 基記 氏 （京都大学） Motofumi Aoki (Kyoto University)
- 5 月 10 日
- 大山 広樹 氏 （京都大学） Hiroki Ohyama (Kyoto University)
- 5 月 17 日
- Michał Łasica 氏 （Polish Academy of Sciences）
- 5 月 31 日
- 三浦 達哉 氏 （京都大学） Tatsuya Miura (Kyoto University)
- 6 月 10 日
- Tim Laux 氏 （University of Regensburg）
- 6 月 21 日
- 林 雅行 氏 （京都大学） Masayuki Hayashi (Kyoto University)
- 7 月 5 日
- Sanghyuk Lee 氏 （Seoul National University）
- 7 月 19 日
- Pierre Mackowiak 氏 （École Polytechnique）
- 7 月 26 日
- Yi C. Huang 氏 （Nanjing Normal University）
- 10 月 4 日
- 清水 一慶 氏 （京都大学） Ikkei Shimizu (Kyoto University)
● 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.
別の年度
