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過去の記録 ～04/30｜次回の予定｜今後の予定 05/01～
| 開催情報 | 木曜日 16:00～17:30 数理科学研究科棟(駒場) 002号室 |
|---|---|
| 担当者 | 石毛 和弘 |
過去の記録
2025年04月17日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
北野修平 氏 (東京大学大学院数理科学研究科)
On Calderón–Zygmund Estimates for Fully Nonlinear Equations (Japanese)
北野修平 氏 (東京大学大学院数理科学研究科)
On Calderón–Zygmund Estimates for Fully Nonlinear Equations (Japanese)
[ 講演概要 ]
The Calderón–Zygmund estimate provides a bound on the $L^p$ norms of second-order derivatives of solutions to elliptic equations. Caffarelli extended this result to fully nonlinear equations, requiring the exponent $p$ to be sufficiently large. In this work, we explore two generalizations of Caffarelli’s result: one concerning small values of $p$ and the other involving equations with $L^n$ drift terms.
The Calderón–Zygmund estimate provides a bound on the $L^p$ norms of second-order derivatives of solutions to elliptic equations. Caffarelli extended this result to fully nonlinear equations, requiring the exponent $p$ to be sufficiently large. In this work, we explore two generalizations of Caffarelli’s result: one concerning small values of $p$ and the other involving equations with $L^n$ drift terms.
2024年07月18日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
清水 良輔 氏 (早稲田大学)
Construction of a $p$-energy form and $p$-energy measures on the Sierpiński carpet (Japanese)
清水 良輔 氏 (早稲田大学)
Construction of a $p$-energy form and $p$-energy measures on the Sierpiński carpet (Japanese)
[ 講演概要 ]
In this talk, I will propose a new way of constructing the $(1,p)$-Sobolev space, $p$-energy functional and $p$-energy measures on the Sierpinski carpet for all $p \in (1,\infty)$. Our approach is mainly based on an idea in the work by Kusuoka--Zhou (1992), where the canonical regular Dirichlet forms (Brownian motions) on some self-similar sets are constructed as scaling limits of discrete $2$-energy forms. I will also explain some results related to the Ahlfors regular conformal dimension, which coincides with the critical value $p$ whether our $(1,p)$-Sobolev space is embedded in the set of continuous functions. This is based on joint work with Mathav Murugan (The University of British Columbia).
In this talk, I will propose a new way of constructing the $(1,p)$-Sobolev space, $p$-energy functional and $p$-energy measures on the Sierpinski carpet for all $p \in (1,\infty)$. Our approach is mainly based on an idea in the work by Kusuoka--Zhou (1992), where the canonical regular Dirichlet forms (Brownian motions) on some self-similar sets are constructed as scaling limits of discrete $2$-energy forms. I will also explain some results related to the Ahlfors regular conformal dimension, which coincides with the critical value $p$ whether our $(1,p)$-Sobolev space is embedded in the set of continuous functions. This is based on joint work with Mathav Murugan (The University of British Columbia).
2024年06月27日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
大泉 嶺 氏 (国立社会保障・人口問題研究所)
A Control Theory in Mathematical Demography (Japanese)
大泉 嶺 氏 (国立社会保障・人口問題研究所)
A Control Theory in Mathematical Demography (Japanese)
[ 講演概要 ]
Multistate Age-Structured Population Model is a fundamental mathematical model in mathematical demography that describes population structure and dynamics with state variables that are not uniform with age (e.g., body size, place of residence, genetic characteristics, etc.). The model's eigensystems have been used in various demographic analyses, providing essential indicators for discussing evolutionary theory. In this study, we derive a control equation (HJB equation) that maximizes the spectral radius from the eigensystem of the multistate age-structured population model and discuss the control process that generates an evolutionarily adaptive life history.
Multistate Age-Structured Population Model is a fundamental mathematical model in mathematical demography that describes population structure and dynamics with state variables that are not uniform with age (e.g., body size, place of residence, genetic characteristics, etc.). The model's eigensystems have been used in various demographic analyses, providing essential indicators for discussing evolutionary theory. In this study, we derive a control equation (HJB equation) that maximizes the spectral radius from the eigensystem of the multistate age-structured population model and discuss the control process that generates an evolutionarily adaptive life history.
2024年05月30日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
対面・オンラインハイブリッド開催
Tim Laux 氏 (University of Regensburg, Germany)
Energy convergence of the Allen-Cahn equation for mean convex mean curvature flow (English)
https://forms.gle/8KnFWfHFbkn9fAqaA
対面・オンラインハイブリッド開催
Tim Laux 氏 (University of Regensburg, Germany)
Energy convergence of the Allen-Cahn equation for mean convex mean curvature flow (English)
[ 講演概要 ]
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.
[ 参考URL ]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.
https://forms.gle/8KnFWfHFbkn9fAqaA
2024年05月23日(木)
16:00-17:30 数理科学研究科棟(駒場) 128号室
Adina Ciomaga 氏 (University Paris Cité (Laboratoire Jacques Louis Lions), France “O Mayer” Institute of the Romanian Academy, Iasi, Roumania)
Homogenization of nonlocal Hamilton Jacobi equations (English)
Adina Ciomaga 氏 (University Paris Cité (Laboratoire Jacques Louis Lions), France “O Mayer” Institute of the Romanian Academy, Iasi, Roumania)
Homogenization of nonlocal Hamilton Jacobi equations (English)
[ 講演概要 ]
I will present the framework of periodic homogenisation of nonlocal Hamilton-Jacobi equations, associated with Levy-Itô integro-differential operators. A typical equation is the fractional diffusion coupled with a transport term, where the diffusion is only weakly elliptical. Homogenization is established in two steps: (i) the resolution of a cellular problem - where Lipshitz regularity of the corrector plays a key role and (ii) the convergence of the oscillating solutions towards an averaged profile - where comparison principles are involved. I shall discuss recent results on the regularity of solutions and comparison principles for nonlocal equations, and the difficulties we face when compared with local PDEs. The talked is based on recent developments obtained in collaboration with D. Ghilli, O.Ley, E. Topp, T. Minh Le.
I will present the framework of periodic homogenisation of nonlocal Hamilton-Jacobi equations, associated with Levy-Itô integro-differential operators. A typical equation is the fractional diffusion coupled with a transport term, where the diffusion is only weakly elliptical. Homogenization is established in two steps: (i) the resolution of a cellular problem - where Lipshitz regularity of the corrector plays a key role and (ii) the convergence of the oscillating solutions towards an averaged profile - where comparison principles are involved. I shall discuss recent results on the regularity of solutions and comparison principles for nonlocal equations, and the difficulties we face when compared with local PDEs. The talked is based on recent developments obtained in collaboration with D. Ghilli, O.Ley, E. Topp, T. Minh Le.
2024年04月11日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Jan Haskovec 氏 (KAUST, Saudi Arabia)
Non-Markovian models of collective motion (English)
https://forms.gle/5cZ4WzqBjhsXrxgU6
対面・オンラインハイブリッド開催
Jan Haskovec 氏 (KAUST, Saudi Arabia)
Non-Markovian models of collective motion (English)
[ 講演概要 ]
I will give an overview of recent results for models of collective behavior governed by functional differential equations with non-Markovian structure. The talk will focus on models of interacting agents with applications in biology (flocking, swarming), social sciences (opinion formation) and engineering (swarm robotics), where latency (delay) plays a significant role. I will characterize two main sources of delay - inter-agent communications ("transmission delay") and information processing ("reaction delay") - and discuss their impacts on the group dynamics. I will give an overview of analytical methods for studying the asymptotic behavior of the models in question and their mean-field limits. In particular, I will show that the transmission vs. reaction delay leads to fundamentally different mathematical structures and requires appropriate choice of analytical tools. Finally, motivated by situations where finite speed of information propagation is significant, I will introduce an interesting class of problems where the delay depends nontrivially and nonlinearly on the state of the system, and discuss the available analytical results and open problems here.
[ 参考URL ]I will give an overview of recent results for models of collective behavior governed by functional differential equations with non-Markovian structure. The talk will focus on models of interacting agents with applications in biology (flocking, swarming), social sciences (opinion formation) and engineering (swarm robotics), where latency (delay) plays a significant role. I will characterize two main sources of delay - inter-agent communications ("transmission delay") and information processing ("reaction delay") - and discuss their impacts on the group dynamics. I will give an overview of analytical methods for studying the asymptotic behavior of the models in question and their mean-field limits. In particular, I will show that the transmission vs. reaction delay leads to fundamentally different mathematical structures and requires appropriate choice of analytical tools. Finally, motivated by situations where finite speed of information propagation is significant, I will introduce an interesting class of problems where the delay depends nontrivially and nonlinearly on the state of the system, and discuss the available analytical results and open problems here.
https://forms.gle/5cZ4WzqBjhsXrxgU6
2024年03月21日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
Mostafa Fazly 氏 (University of Texas at San Antonio)
Symmetry Results for Nonlinear PDEs (English)
Mostafa Fazly 氏 (University of Texas at San Antonio)
Symmetry Results for Nonlinear PDEs (English)
[ 講演概要 ]
The study of qualitative behavior of solutions of Partial Differential Equations (PDEs) started roughly in mid-18th century. Since then scientists and mathematicians from different fields have put in a great effort to expand the theory of nonlinear PDEs. PDEs can be divided into two kinds: (a) the linear ones, which are relatively easy to analyze and can often be solved completely, and (b) the nonlinear ones, which are much harder to analyze and can almost never be solved completely.
We begin this talk by an introduction on foundational ideas behind the De Giorgi’s conjecture (1978) for the Allen-Cahn equation that is inspired by the Bernstein’s problem (1910). This conjecture brings together three groups of mathematicians: (a) a group specializing in nonlinear partial differential equations, (b) a group in differential geometry, and more specially on minimal surfaces and constant mean curvature surfaces, and (c) a group in mathematical physics on phase transitions. We then present natural generalizations and counterparts of the problem. These generalizations lead us to introduce certain novel concepts, and we illustrate why these novel concepts seem to be the right concepts in the context and how they can be used to study particular systems and models arising in Sciences. We give a survey of recent results.
The study of qualitative behavior of solutions of Partial Differential Equations (PDEs) started roughly in mid-18th century. Since then scientists and mathematicians from different fields have put in a great effort to expand the theory of nonlinear PDEs. PDEs can be divided into two kinds: (a) the linear ones, which are relatively easy to analyze and can often be solved completely, and (b) the nonlinear ones, which are much harder to analyze and can almost never be solved completely.
We begin this talk by an introduction on foundational ideas behind the De Giorgi’s conjecture (1978) for the Allen-Cahn equation that is inspired by the Bernstein’s problem (1910). This conjecture brings together three groups of mathematicians: (a) a group specializing in nonlinear partial differential equations, (b) a group in differential geometry, and more specially on minimal surfaces and constant mean curvature surfaces, and (c) a group in mathematical physics on phase transitions. We then present natural generalizations and counterparts of the problem. These generalizations lead us to introduce certain novel concepts, and we illustrate why these novel concepts seem to be the right concepts in the context and how they can be used to study particular systems and models arising in Sciences. We give a survey of recent results.
2024年02月05日(月)
16:00-17:30 数理科学研究科棟(駒場) 122号室
対面・オンラインハイブリッド開催（通常と開催曜日・会場が異なりますのでご注意ください）
Reinhard Farwig 氏 (Technische Universität Darmstadt)
Viscous Flow in Domains with Moving Boundaries - From Bounded to Unbounded Domains (English)
https://forms.gle/xKPKu1uw9PeHEEck9
対面・オンラインハイブリッド開催（通常と開催曜日・会場が異なりますのでご注意ください）
Reinhard Farwig 氏 (Technische Universität Darmstadt)
Viscous Flow in Domains with Moving Boundaries - From Bounded to Unbounded Domains (English)
[ 講演概要 ]
以下のPDFファイルをご参照ください：
https://drive.google.com/file/d/1dJJU1ybE-n8yn3LZTReTeH2UFX9wXQv9/view?usp=drive_link
[ 参考URL ]以下のPDFファイルをご参照ください：
https://drive.google.com/file/d/1dJJU1ybE-n8yn3LZTReTeH2UFX9wXQv9/view?usp=drive_link
https://forms.gle/xKPKu1uw9PeHEEck9
2024年01月30日(火)
16:30-17:30 数理科学研究科棟(駒場) 002号室
対面（通常と開催曜日・会場が異なりますのでご注意ください）
Danielle Hilhorst 氏 (CNRS / Université de Paris-Saclay)
Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile. (English)
対面（通常と開催曜日・会場が異なりますのでご注意ください）
Danielle Hilhorst 氏 (CNRS / Université de Paris-Saclay)
Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile. (English)
[ 講演概要 ]
We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary.
We construct a unique self-similar solution and show that for a large class of initial data, the solution of the time evolution problem converges to this self-similar solution as time tends to infinity. Similar results were already obtained by Bouguezzi, Hilhorst, Miyamoto, and Scheid in the case of Dirichlet data on the fixed boundary. However, they had to show that the space derivative of the solution uniformly converges to its limit. Here, our proof requires less regularity, which should make our arguments easier to adapt to different settings.
This is a joint work with Sabrina Roscani and Piotr Rybka.
We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary.
We construct a unique self-similar solution and show that for a large class of initial data, the solution of the time evolution problem converges to this self-similar solution as time tends to infinity. Similar results were already obtained by Bouguezzi, Hilhorst, Miyamoto, and Scheid in the case of Dirichlet data on the fixed boundary. However, they had to show that the space derivative of the solution uniformly converges to its limit. Here, our proof requires less regularity, which should make our arguments easier to adapt to different settings.
This is a joint work with Sabrina Roscani and Piotr Rybka.
2023年11月30日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Philippe G. LeFloch 氏 (Sorbonne University and CNRS)
Einstein spacetimes: dispersion, localization, collapse, and bouncing (English)
https://forms.gle/HPsYinKweUW3AQGv9
対面・オンラインハイブリッド開催
Philippe G. LeFloch 氏 (Sorbonne University and CNRS)
Einstein spacetimes: dispersion, localization, collapse, and bouncing (English)
[ 講演概要 ]
I will overview recent developments on Einstein's field equations of general relativity, especially the global evolution problem from initial data sets. A variety of phenomena may arise in this evolution: gravitational waves, dispersion, collapse, formation of singularities, and bouncing. While many problems remain widely open and very challenging, in the past decades major mathematical advances were made for several classes of spacetimes. I will review recent results on the (1) nonlinear stability of Minkowski spacetime, (2) localization problem at infinity, (3) collapse of spherically symmetric fields, and (4) scattering through quiescent singularity. This talk is based on joint work with Y. Ma (Xi'an), T.-C. Nguyen (Montpellier), F. Mena (Lisbon), B. Le Floch (Paris), and G. Veneziano (Geneva).
Blog: philippelefloch.org
[ 参考URL ]I will overview recent developments on Einstein's field equations of general relativity, especially the global evolution problem from initial data sets. A variety of phenomena may arise in this evolution: gravitational waves, dispersion, collapse, formation of singularities, and bouncing. While many problems remain widely open and very challenging, in the past decades major mathematical advances were made for several classes of spacetimes. I will review recent results on the (1) nonlinear stability of Minkowski spacetime, (2) localization problem at infinity, (3) collapse of spherically symmetric fields, and (4) scattering through quiescent singularity. This talk is based on joint work with Y. Ma (Xi'an), T.-C. Nguyen (Montpellier), F. Mena (Lisbon), B. Le Floch (Paris), and G. Veneziano (Geneva).
Blog: philippelefloch.org
https://forms.gle/HPsYinKweUW3AQGv9
2023年09月14日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Michał Łasica 氏 (The Polish Academy of Sciences)
Bounds on the gradient of minimizers in variational denoising (English)
https://forms.gle/C39ZLdQNVHyVmJ4j8
対面・オンラインハイブリッド開催
Michał Łasica 氏 (The Polish Academy of Sciences)
Bounds on the gradient of minimizers in variational denoising (English)
[ 講演概要 ]
We consider minimization problem for a class of convex integral functionals composed of two terms:
-- a regularizing term of linear growth in the gradient,
-- and a fidelity term penalizing the distance from a given function.
To ensure that such functionals attain their minima, one needs to extend their domain to the BV space. In particular minimizers may exhibit jump discontinuities. I will discuss estimates on the gradient of minimizers in terms of the data, focusing on singular part of the gradient measure.
The talk is based on joint works with P. Rybka, Z. Grochulska and A. Chambolle.
[ 参考URL ]We consider minimization problem for a class of convex integral functionals composed of two terms:
-- a regularizing term of linear growth in the gradient,
-- and a fidelity term penalizing the distance from a given function.
To ensure that such functionals attain their minima, one needs to extend their domain to the BV space. In particular minimizers may exhibit jump discontinuities. I will discuss estimates on the gradient of minimizers in terms of the data, focusing on singular part of the gradient measure.
The talk is based on joint works with P. Rybka, Z. Grochulska and A. Chambolle.
https://forms.gle/C39ZLdQNVHyVmJ4j8
2023年09月07日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Samuel Mercer 氏 (Delft University of Technology)
Uniform Convergence of Gradient Flows on a Stack of Banach Spaces (English)
https://forms.gle/T8yWr2gDTYzj8vkE7
対面・オンラインハイブリッド開催
Samuel Mercer 氏 (Delft University of Technology)
Uniform Convergence of Gradient Flows on a Stack of Banach Spaces (English)
[ 講演概要 ]
Within this talk I will recall the classical result: Given a sequence of convex functionals on a Hilbert space, Gamma-convergence of this sequence implies uniform convergence on finite time-intervals for their gradient flows. I will then discuss a generalisation for this result. In particular our functionals are defined on a sequence of distinct Banach spaces that can be stacked together inside of a unifying space. We will study a kind of gradient flow for our functionals inside their respective Banach space and ask the following question. What structure is necessary within our unifying space to attain uniform convergence of gradient flows?
[ 参考URL ]Within this talk I will recall the classical result: Given a sequence of convex functionals on a Hilbert space, Gamma-convergence of this sequence implies uniform convergence on finite time-intervals for their gradient flows. I will then discuss a generalisation for this result. In particular our functionals are defined on a sequence of distinct Banach spaces that can be stacked together inside of a unifying space. We will study a kind of gradient flow for our functionals inside their respective Banach space and ask the following question. What structure is necessary within our unifying space to attain uniform convergence of gradient flows?
https://forms.gle/T8yWr2gDTYzj8vkE7
2023年06月22日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Jiwoong Jang 氏 (University of Wisconsin Madison)
Convergence rate of periodic homogenization of forced mean curvature flow of graphs in the laminar setting (English)
https://forms.gle/BTuFtcmUVnvCLieX9
対面・オンラインハイブリッド開催
Jiwoong Jang 氏 (University of Wisconsin Madison)
Convergence rate of periodic homogenization of forced mean curvature flow of graphs in the laminar setting (English)
[ 講演概要 ]
Mean curvature flow with a forcing term models the motion of a phase boundary through media with defects and heterogeneities. When the environment shows periodic patterns with small oscillations, an averaged behavior is exhibited as we zoom out, and providing mathematical treatment for the behavior has received a great attention recently. In this talk, we discuss the periodic homogenization of forced mean curvature flows, and we give a quantitative analysis for the flow starting from an entire graph in a laminated environment.
[ 参考URL ]Mean curvature flow with a forcing term models the motion of a phase boundary through media with defects and heterogeneities. When the environment shows periodic patterns with small oscillations, an averaged behavior is exhibited as we zoom out, and providing mathematical treatment for the behavior has received a great attention recently. In this talk, we discuss the periodic homogenization of forced mean curvature flows, and we give a quantitative analysis for the flow starting from an entire graph in a laminated environment.
https://forms.gle/BTuFtcmUVnvCLieX9
2023年05月18日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Junha Kim 氏 (Korea Institute for Advanced Study)
On the wellposedness of generalized SQG equation in a half-plane (English)
https://forms.gle/Cezz3sicY7izDPfq8
対面・オンラインハイブリッド開催
Junha Kim 氏 (Korea Institute for Advanced Study)
On the wellposedness of generalized SQG equation in a half-plane (English)
[ 講演概要 ]
In this talk, we investigate classical solutions to the $\alpha$-SQG in a half-plane, which reduces to the 2D Euler equations and SQG equation for $\alpha=0$ and $\alpha=1$, respectively. When $\alpha \in (0,1/2]$, we establish that $\alpha$-SQG is well-posed in appropriate anisotropic Lipschitz spaces. Moreover, we prove that every solution with smooth initial data that is compactly supported and not vanishing on the boundary has infinite $C^{\beta}$-norm instantaneously where $\beta > 1-\alpha$. In the case of $\alpha \in (1/2,1]$, we show the nonexistence of solutions in $C^{\alpha}$. This is a joint work with In-Jee Jeong and Yao Yao.
[ 参考URL ]In this talk, we investigate classical solutions to the $\alpha$-SQG in a half-plane, which reduces to the 2D Euler equations and SQG equation for $\alpha=0$ and $\alpha=1$, respectively. When $\alpha \in (0,1/2]$, we establish that $\alpha$-SQG is well-posed in appropriate anisotropic Lipschitz spaces. Moreover, we prove that every solution with smooth initial data that is compactly supported and not vanishing on the boundary has infinite $C^{\beta}$-norm instantaneously where $\beta > 1-\alpha$. In the case of $\alpha \in (1/2,1]$, we show the nonexistence of solutions in $C^{\alpha}$. This is a joint work with In-Jee Jeong and Yao Yao.
https://forms.gle/Cezz3sicY7izDPfq8
2023年04月06日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催
Van Tien Nguyen 氏 (National Taiwan University)
Blowup solutions to the Keller-Segel system (English)
https://forms.gle/7ogZKyh1oXKkPbN56
対面・オンラインハイブリッド開催
Van Tien Nguyen 氏 (National Taiwan University)
Blowup solutions to the Keller-Segel system (English)
[ 講演概要 ]
I will present constructive examples of finite-time blowup solutions to the Keller-Segel system in $\mathbb{R}^d$. For $d = 2$ ($L^1$-critical), there are finite time blowup solutions that are of Type II with finite mass. Blowup rates are completely quantized according to a discrete spectrum of a linearized operator around the rescaled stationary solution in the self-similar setting. There is a stable blowup mechanism which is expected to be generic among others. For $d \geq 3$ ($L^1$-supercritical), we construct finite time blowup solutions that are completely unrelated to the self-similar scale, in particular, they are of Type II with finite mass. Interestingly, the radial blowup profile is linked to the traveling-wave of the 1D viscous Burgers equation. Our constructed solution actually has the form of collapsing-ring which consists of an imploding, smoothed-out shock wave moving towards the origin to form a Dirac mass at the singularity. I will also discuss other blowup patterns that possibly occur in the cases $d = 2,3,4$.
[ 参考URL ]I will present constructive examples of finite-time blowup solutions to the Keller-Segel system in $\mathbb{R}^d$. For $d = 2$ ($L^1$-critical), there are finite time blowup solutions that are of Type II with finite mass. Blowup rates are completely quantized according to a discrete spectrum of a linearized operator around the rescaled stationary solution in the self-similar setting. There is a stable blowup mechanism which is expected to be generic among others. For $d \geq 3$ ($L^1$-supercritical), we construct finite time blowup solutions that are completely unrelated to the self-similar scale, in particular, they are of Type II with finite mass. Interestingly, the radial blowup profile is linked to the traveling-wave of the 1D viscous Burgers equation. Our constructed solution actually has the form of collapsing-ring which consists of an imploding, smoothed-out shock wave moving towards the origin to form a Dirac mass at the singularity. I will also discuss other blowup patterns that possibly occur in the cases $d = 2,3,4$.
https://forms.gle/7ogZKyh1oXKkPbN56
2023年02月22日(水)
16:00-17:30 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催（通常と開催曜日が異なりますのでご注意下さい）
Alessio Porretta 氏 (University of Rome Tor Vergata)
Long time decay of Fokker-Planck equations with confining drift (ENGLISH)
https://forms.gle/SCyZWtfC5bNGadxE8
対面・オンラインハイブリッド開催（通常と開催曜日が異なりますのでご注意下さい）
Alessio Porretta 氏 (University of Rome Tor Vergata)
Long time decay of Fokker-Planck equations with confining drift (ENGLISH)
[ 講演概要 ]
The convergence to equilibrium of Fokker-Planck equations with confining drift is a classical issue, starting with the basic model of the Ornstein-Uhlenbeck process. I will discuss a new approach to obtain estimates on the time decay rate, which applies to both local and nonlocal diffusions. This is based on duality arguments and oscillation estimates for transport-diffusion equations, which are reminiscent of coupling methods used in probabilistic approaches.
[ 参考URL ]The convergence to equilibrium of Fokker-Planck equations with confining drift is a classical issue, starting with the basic model of the Ornstein-Uhlenbeck process. I will discuss a new approach to obtain estimates on the time decay rate, which applies to both local and nonlocal diffusions. This is based on duality arguments and oscillation estimates for transport-diffusion equations, which are reminiscent of coupling methods used in probabilistic approaches.
https://forms.gle/SCyZWtfC5bNGadxE8
2023年02月06日(月)
16:00-18:10 数理科学研究科棟(駒場) 126号室
対面・オンラインハイブリッド開催（通常と開催曜日が異なりますのでご注意下さい）
Marek Fila 氏 (Comenius University) 16:00-17:00
Solutions with moving singularities for nonlinear diffusion equations (ENGLISH)
Fast diffusion equation: uniqueness of solutions with a moving singularity (ENGLISH)
https://forms.gle/nKa4XATuuGPwZWbUA
対面・オンラインハイブリッド開催（通常と開催曜日が異なりますのでご注意下さい）
Marek Fila 氏 (Comenius University) 16:00-17:00
Solutions with moving singularities for nonlinear diffusion equations (ENGLISH)
[ 講演概要 ]
We give a survey of results on solutions with singularities moving along a prescribed curve for equations of fast diffusion or porous medium type. These results were obtained in collaboration with J.R. King, P. Mackova, J. Takahashi and E. Yanagida.
Petra Mackova 氏 (Comenius University) 17:10-18:10We give a survey of results on solutions with singularities moving along a prescribed curve for equations of fast diffusion or porous medium type. These results were obtained in collaboration with J.R. King, P. Mackova, J. Takahashi and E. Yanagida.
Fast diffusion equation: uniqueness of solutions with a moving singularity (ENGLISH)
[ 講演概要 ]
This talk focuses on open questions in the area of the uniqueness of distributional solutions of the fast diffusion equation with a given source term. The existence of different sets of such solutions is known from previous research, and the natural next issue is to examine their uniqueness. Assuming that the source term is a measure, the existence of different classes of solutions is known, however, their uniqueness is an open problem. The existence of a class of asymptotically radially symmetric solutions with a singularity that moves along a prescribed curve was proved by M. Fila, J. Takahashi, and E. Yanagida. More recently, it has been established by M. Fila, P. M., J. Takahashi, and E. Yanagida that these solutions solve the corresponding problem with a moving Dirac source term. In this talk, we discuss the uniqueness of these solutions. This is a joint work with M. Fila.
[ 参考URL ]This talk focuses on open questions in the area of the uniqueness of distributional solutions of the fast diffusion equation with a given source term. The existence of different sets of such solutions is known from previous research, and the natural next issue is to examine their uniqueness. Assuming that the source term is a measure, the existence of different classes of solutions is known, however, their uniqueness is an open problem. The existence of a class of asymptotically radially symmetric solutions with a singularity that moves along a prescribed curve was proved by M. Fila, J. Takahashi, and E. Yanagida. More recently, it has been established by M. Fila, P. M., J. Takahashi, and E. Yanagida that these solutions solve the corresponding problem with a moving Dirac source term. In this talk, we discuss the uniqueness of these solutions. This is a joint work with M. Fila.
https://forms.gle/nKa4XATuuGPwZWbUA
2022年11月24日(木)
16:00-17:30 数理科学研究科棟(駒場) 370号室
対面・オンラインハイブリッド開催
板倉 恭平 氏 (東京大学 大学院数理科学研究科)
シュタルク・シュレディンガー作用素に対する放射条件評価と定常散乱理論 (Japanese)
https://forms.gle/admRaVnmPjFyp5op9
対面・オンラインハイブリッド開催
板倉 恭平 氏 (東京大学 大学院数理科学研究科)
シュタルク・シュレディンガー作用素に対する放射条件評価と定常散乱理論 (Japanese)
[ 講演概要 ]
本講演では1体粒子系のシュタルク・シュレディンガー作用素に対し，古典力学から類推される最良な重み付き放射条件評価の導出を行い，これを土台として定常波動作用素の存在性と完全性を調べる．さらに関連する話題として，定常散乱行列のユニタリ性，一般化フーリエ変換の構成，および最小増大度をもつ一般化固有関数に対する定常散乱行列と近似外向・内向波を用いた空間遠方での漸近挙動の特徴づけについても考察する．本研究では，対応する古典力学を適切に反映させたエスケープ関数と，それに付随するアグモン-ヘルマンダー空間の使用が肝要となる．本講演の内容は足立匡義氏(京都大学)，伊藤健一氏(東京大学)，Skibsted Erik氏(オーフス大学)との共同研究に基づく．
[ 参考URL ]本講演では1体粒子系のシュタルク・シュレディンガー作用素に対し，古典力学から類推される最良な重み付き放射条件評価の導出を行い，これを土台として定常波動作用素の存在性と完全性を調べる．さらに関連する話題として，定常散乱行列のユニタリ性，一般化フーリエ変換の構成，および最小増大度をもつ一般化固有関数に対する定常散乱行列と近似外向・内向波を用いた空間遠方での漸近挙動の特徴づけについても考察する．本研究では，対応する古典力学を適切に反映させたエスケープ関数と，それに付随するアグモン-ヘルマンダー空間の使用が肝要となる．本講演の内容は足立匡義氏(京都大学)，伊藤健一氏(東京大学)，Skibsted Erik氏(オーフス大学)との共同研究に基づく．
https://forms.gle/admRaVnmPjFyp5op9
2022年06月30日(木)
16:00-17:00 オンライン開催
Xingzhi Bian 氏 (Shanghai University)
A brief introduction to a class of new phase field models (English)
https://forms.gle/esc7Y6KGASwbFro97
Xingzhi Bian 氏 (Shanghai University)
A brief introduction to a class of new phase field models (English)
[ 講演概要 ]
Existence of weak solutions for a type of new phase field models, which are the system consisting of a degenerate parabolic equation of order parameter coupled to a linear elasticity sub-system. The models are applied to describe the phase transitions in elastically deformable solids.
[ 参考URL ]Existence of weak solutions for a type of new phase field models, which are the system consisting of a degenerate parabolic equation of order parameter coupled to a linear elasticity sub-system. The models are applied to describe the phase transitions in elastically deformable solids.
https://forms.gle/esc7Y6KGASwbFro97
2022年04月21日(木)
16:00-17:30 オンライン開催
参加を希望される方は［参考URL］をご覧ください。
三宅庸仁 氏 (東大数理)
Effect of decay rates of initial data on the sign of solutions to Cauchy problems of some higher order parabolic equations (Japanese)
https://forms.gle/96bBNEAEHrsdXfH57
参加を希望される方は［参考URL］をご覧ください。
三宅庸仁 氏 (東大数理)
Effect of decay rates of initial data on the sign of solutions to Cauchy problems of some higher order parabolic equations (Japanese)
[ 講演概要 ]
本講演では高階放物型方程式に対する初期値問題の解の正値性について考察する. 二階放物型問題では, 「非負である任意の初期値に対する解は時空大域的に正値となる」という正値性保存則が広く成立することが知られている. 一方で高階放物型問題においては, 最も単純な重調和熱方程式に対する初期値問題においてさえ, 正値性保存則は一般に成立しない. 同問題の正値性に関する一般次元における結果としては, 時間終局的かつ空間局所的な正値性が成り立つような初期値のクラスが幾つか構成されているのみである.本講演では, 多重調和熱方程式に対する初期値問題の解が時空間大域的に正値関数となるための初期値に対する十分条件を与える. また, 初期値の空間遠方での減衰速度に応じて解が時間終局的かつ空間大域的に正値となるか否か別れることを示す. さらに, 冪乗型非線形項をもつ半線形多重調和熱方程式の初期値問題に対して同様の性質を有する解を構成する. なお, 本講演の内容の一部は, Hans-Christoph Grunau 氏 (University of Magdeburg) と岡部真也氏 (東北大学) との共同研究に基づく.
[ 参考URL ]本講演では高階放物型方程式に対する初期値問題の解の正値性について考察する. 二階放物型問題では, 「非負である任意の初期値に対する解は時空大域的に正値となる」という正値性保存則が広く成立することが知られている. 一方で高階放物型問題においては, 最も単純な重調和熱方程式に対する初期値問題においてさえ, 正値性保存則は一般に成立しない. 同問題の正値性に関する一般次元における結果としては, 時間終局的かつ空間局所的な正値性が成り立つような初期値のクラスが幾つか構成されているのみである.本講演では, 多重調和熱方程式に対する初期値問題の解が時空間大域的に正値関数となるための初期値に対する十分条件を与える. また, 初期値の空間遠方での減衰速度に応じて解が時間終局的かつ空間大域的に正値となるか否か別れることを示す. さらに, 冪乗型非線形項をもつ半線形多重調和熱方程式の初期値問題に対して同様の性質を有する解を構成する. なお, 本講演の内容の一部は, Hans-Christoph Grunau 氏 (University of Magdeburg) と岡部真也氏 (東北大学) との共同研究に基づく.
https://forms.gle/96bBNEAEHrsdXfH57
2021年12月16日(木)
16:00-17:00 オンライン開催
Zhanpeisov Erbol 氏 (東大数理)
Existence of solutions for fractional semilinear parabolic equations in Besov-Morrey spaces (Japanese)
https://forms.gle/whpkgAwYvyQKQMzM8
Zhanpeisov Erbol 氏 (東大数理)
Existence of solutions for fractional semilinear parabolic equations in Besov-Morrey spaces (Japanese)
[ 講演概要 ]
分数冪拡散を伴う半線型放物型方程式の局所解の存在について考える。これらの方程式については、時間局所解の存在と初期値に許容される特異性の関係が知られているが、本講演では局所ベゾフモレイ空間で解を構成する事でデルタ関数の微分を含むような初期値に対して解を構成する。講演の前半では既存の研究や局所ベゾフモレイ空間の性質について触れ、後半では基本解の減衰評価と不動点定理を用いた証明について紹介する。
[ 参考URL ]分数冪拡散を伴う半線型放物型方程式の局所解の存在について考える。これらの方程式については、時間局所解の存在と初期値に許容される特異性の関係が知られているが、本講演では局所ベゾフモレイ空間で解を構成する事でデルタ関数の微分を含むような初期値に対して解を構成する。講演の前半では既存の研究や局所ベゾフモレイ空間の性質について触れ、後半では基本解の減衰評価と不動点定理を用いた証明について紹介する。
https://forms.gle/whpkgAwYvyQKQMzM8
2021年12月02日(木)
16:00-17:00 オンライン開催
寺井健悟 氏 (東大数理)
平均場ゲームに現れる1階の非線形偏微分方程式系の割引消去問題
https://forms.gle/6cKyu9meCxSv72N19
寺井健悟 氏 (東大数理)
平均場ゲームに現れる1階の非線形偏微分方程式系の割引消去問題
[ 講演概要 ]
平均場ゲーム理論から導出される1階のハミルトン・ヤコビ・ベルマン方程式と連続方程式の連立系を扱い，割引率をゼロに近づけたときの解の漸近挙動を考察する．この漸近問題の特徴は極限方程式が多重解を持つことであり，部分列に依らず解が収束するか否かは非自明である．本講演では，弱解のコンパクト性および収束の意味での安定性を示し, 任意の収束部分列の極限が満たすべき条件を与える． そしてこれを用いて部分列に依らず解が収束する具体例を紹介する．本講演は三竹大寿氏（東京大学）との共同研究に基づく．
[ 参考URL ]平均場ゲーム理論から導出される1階のハミルトン・ヤコビ・ベルマン方程式と連続方程式の連立系を扱い，割引率をゼロに近づけたときの解の漸近挙動を考察する．この漸近問題の特徴は極限方程式が多重解を持つことであり，部分列に依らず解が収束するか否かは非自明である．本講演では，弱解のコンパクト性および収束の意味での安定性を示し, 任意の収束部分列の極限が満たすべき条件を与える． そしてこれを用いて部分列に依らず解が収束する具体例を紹介する．本講演は三竹大寿氏（東京大学）との共同研究に基づく．
https://forms.gle/6cKyu9meCxSv72N19
2021年11月25日(木)
16:00-17:30 オンライン開催
清水雄貴 氏 (東大数理)
Euler方程式のカレント値弱解とその応用 (日本語)
https://forms.gle/xBAgncTERzYfauJE6
清水雄貴 氏 (東大数理)
Euler方程式のカレント値弱解とその応用 (日本語)
[ 講演概要 ]
二次元非圧縮Euler方程式に対し，初期渦度がデルタ関数の線形結合で与えられる際の形式的な解は点渦系と呼ばれ，局在化した渦構造を持つ流体運動を記述する簡易モデルとして応用上重要である．しかしながら，点渦系はEuler流から派生して得られるモデルである以上，点渦系が数学的に適切な意味でEuler流となることを保障する必要がある．本講演ではEuler方程式に対し，カレント値弱解を定式化することで，点渦系がEuler方程式のカレント値弱解として正当化されることを紹介する．
[ 参考URL ]二次元非圧縮Euler方程式に対し，初期渦度がデルタ関数の線形結合で与えられる際の形式的な解は点渦系と呼ばれ，局在化した渦構造を持つ流体運動を記述する簡易モデルとして応用上重要である．しかしながら，点渦系はEuler流から派生して得られるモデルである以上，点渦系が数学的に適切な意味でEuler流となることを保障する必要がある．本講演ではEuler方程式に対し，カレント値弱解を定式化することで，点渦系がEuler方程式のカレント値弱解として正当化されることを紹介する．
https://forms.gle/xBAgncTERzYfauJE6
2021年10月28日(木)
16:00-17:00 オンライン開催
Xiaodan Zhou 氏 (OIST)
Quasiconformal and Sobolev mappings on metric measure
https://forms.gle/QATECqmwmWGvXoU56
Xiaodan Zhou 氏 (OIST)
Quasiconformal and Sobolev mappings on metric measure
[ 講演概要 ]
The study of quasiconformal mappings has been an important and active topic since its introduction in the 1930s and the theory has been widely applied to different fields including differential geometry, harmonic analysis, PDEs, etc. In the Euclidean space, it is a fundamental result that three definitions (metric, geometric and analytic) of quasiconformality are equivalent. The theory of quasiconformal mappings has been extended to metric measure spaces by Heinonen and Koskela in the 1990s and their work laid the foundation of analysis on metric spaces. In general, the equivalence of the three characterizations will no longer hold without appropriate assumptions on the spaces and mappings. It is a question of general interest to find minimal assumptions on the metric spaces and on the mapping to guarantee the metric definition implies the analytic characterization or geometric characterization. In this talk, we will give an brief review of the above mentioned classical theory and present some recent results we achieved in obtaining the analytic property, in particular, the Sobolev regularity of a metric quasiconformal mapping with relaxed spaces and mapping conditions. Unexpectedly, we can apply this to prove results that are new even in the classical Euclidean setting. This is joint work with Panu Lahti (Chinese Academy of Sciences).
[ 参考URL ]The study of quasiconformal mappings has been an important and active topic since its introduction in the 1930s and the theory has been widely applied to different fields including differential geometry, harmonic analysis, PDEs, etc. In the Euclidean space, it is a fundamental result that three definitions (metric, geometric and analytic) of quasiconformality are equivalent. The theory of quasiconformal mappings has been extended to metric measure spaces by Heinonen and Koskela in the 1990s and their work laid the foundation of analysis on metric spaces. In general, the equivalence of the three characterizations will no longer hold without appropriate assumptions on the spaces and mappings. It is a question of general interest to find minimal assumptions on the metric spaces and on the mapping to guarantee the metric definition implies the analytic characterization or geometric characterization. In this talk, we will give an brief review of the above mentioned classical theory and present some recent results we achieved in obtaining the analytic property, in particular, the Sobolev regularity of a metric quasiconformal mapping with relaxed spaces and mapping conditions. Unexpectedly, we can apply this to prove results that are new even in the classical Euclidean setting. This is joint work with Panu Lahti (Chinese Academy of Sciences).
https://forms.gle/QATECqmwmWGvXoU56
2021年10月14日(木)
16:00-17:00 オンライン開催
立石 優二郎 氏 (東大数理)
逆二乗冪ポテンシャル項を持つ Schrödinger 熱半群に対する最適時間減衰評価 (Japanese)
https://forms.gle/s4zMhkwpih3FrdhE7
立石 優二郎 氏 (東大数理)
逆二乗冪ポテンシャル項を持つ Schrödinger 熱半群に対する最適時間減衰評価 (Japanese)
[ 講演概要 ]
本講演では, 逆二乗冪ポテンシャル項を持つ楕円型作用素に対して, その熱半群及び導関数に対して作用素ノルムの時間減衰評価を考える. 楕円型作用素の正値調和関数の可積分性は熱半群の時間減衰率と密接な関係があり, 本研究では, 球面調和関数を利用した初期データのフーリエ級数展開によって, ポテンシャル項付き熱方程式の空間球対称解及び付随する正値調和関数の解析に帰着させる方法をとる. 結果として, 熱半群及びその導関数の Lorentz 空間上の作用素ノルムについて, 最適な時間減衰評価を導出した. 本講演は石毛和弘氏 (東京大学) との共同研究に基づく.
[ 参考URL ]本講演では, 逆二乗冪ポテンシャル項を持つ楕円型作用素に対して, その熱半群及び導関数に対して作用素ノルムの時間減衰評価を考える. 楕円型作用素の正値調和関数の可積分性は熱半群の時間減衰率と密接な関係があり, 本研究では, 球面調和関数を利用した初期データのフーリエ級数展開によって, ポテンシャル項付き熱方程式の空間球対称解及び付随する正値調和関数の解析に帰着させる方法をとる. 結果として, 熱半群及びその導関数の Lorentz 空間上の作用素ノルムについて, 最適な時間減衰評価を導出した. 本講演は石毛和弘氏 (東京大学) との共同研究に基づく.
https://forms.gle/s4zMhkwpih3FrdhE7
