## Applied Analysis

Seminar information archive ～08/07｜Next seminar｜Future seminars 08/08～

Date, time & place | Thursday 16:00 - 17:30 002Room #002 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2024/07/18

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Construction of a $p$-energy form and $p$-energy measures on the Sierpiński carpet (Japanese)

**Ryosuke SHIMIZU**(Waseda University)Construction of a $p$-energy form and $p$-energy measures on the Sierpiński carpet (Japanese)

[ Abstract ]

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).

### 2024/06/27

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

A Control Theory in Mathematical Demography (Japanese)

**Ryo OIZUMI**(National Institute of Population and Social Security Research)A Control Theory in Mathematical Demography (Japanese)

[ Abstract ]

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.

### 2024/05/30

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Energy convergence of the Allen-Cahn equation for mean convex mean curvature flow (English)

https://forms.gle/8KnFWfHFbkn9fAqaA

**Tim Laux**(University of Regensburg, Germany)Energy convergence of the Allen-Cahn equation for mean convex mean curvature flow (English)

[ Abstract ]

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.

[ Reference 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

### 2024/05/23

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Homogenization of nonlocal Hamilton Jacobi equations (English)

**Adina Ciomaga**(University Paris Cité (Laboratoire Jacques Louis Lions), France “O Mayer” Institute of the Romanian Academy, Iasi, Roumania)Homogenization of nonlocal Hamilton Jacobi equations (English)

[ Abstract ]

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.

### 2024/04/11

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Non-Markovian models of collective motion (English)

https://forms.gle/5cZ4WzqBjhsXrxgU6

**Jan Haskovec**(KAUST, Saudi Arabia)Non-Markovian models of collective motion (English)

[ Abstract ]

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.

[ Reference 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

### 2024/03/21

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Symmetry Results for Nonlinear PDEs (English)

**Mostafa Fazly**(University of Texas at San Antonio)Symmetry Results for Nonlinear PDEs (English)

[ Abstract ]

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.

### 2024/02/05

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Viscous Flow in Domains with Moving Boundaries - From Bounded to Unbounded Domains (English)

https://forms.gle/xKPKu1uw9PeHEEck9

**Reinhard Farwig**(Technische Universität Darmstadt)Viscous Flow in Domains with Moving Boundaries - From Bounded to Unbounded Domains (English)

[ Abstract ]

https://drive.google.com/file/d/1dJJU1ybE-n8yn3LZTReTeH2UFX9wXQv9/view?usp=drive_link

[ Reference URL ]https://drive.google.com/file/d/1dJJU1ybE-n8yn3LZTReTeH2UFX9wXQv9/view?usp=drive_link

https://forms.gle/xKPKu1uw9PeHEEck9

### 2024/01/30

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile. (English)

**Danielle Hilhorst**(CNRS / Université de Paris-Saclay)Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile. (English)

[ Abstract ]

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.

### 2023/11/30

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Einstein spacetimes: dispersion, localization, collapse, and bouncing (English)

https://forms.gle/HPsYinKweUW3AQGv9

**Philippe G. LeFloch**(Sorbonne University and CNRS)Einstein spacetimes: dispersion, localization, collapse, and bouncing (English)

[ Abstract ]

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

[ Reference 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

### 2023/09/14

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Bounds on the gradient of minimizers in variational denoising (English)

https://forms.gle/C39ZLdQNVHyVmJ4j8

**Michał Łasica**(The Polish Academy of Sciences)Bounds on the gradient of minimizers in variational denoising (English)

[ Abstract ]

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.

[ Reference 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

### 2023/09/07

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Uniform Convergence of Gradient Flows on a Stack of Banach Spaces (English)

https://forms.gle/T8yWr2gDTYzj8vkE7

**Samuel Mercer**(Delft University of Technology)Uniform Convergence of Gradient Flows on a Stack of Banach Spaces (English)

[ Abstract ]

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?

[ Reference 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

### 2023/06/22

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Convergence rate of periodic homogenization of forced mean curvature flow of graphs in the laminar setting (English)

https://forms.gle/BTuFtcmUVnvCLieX9

**Jiwoong Jang**(University of Wisconsin Madison)Convergence rate of periodic homogenization of forced mean curvature flow of graphs in the laminar setting (English)

[ Abstract ]

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.

[ Reference 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

### 2023/05/18

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On the wellposedness of generalized SQG equation in a half-plane (English)

https://forms.gle/Cezz3sicY7izDPfq8

**Junha Kim**(Korea Institute for Advanced Study)On the wellposedness of generalized SQG equation in a half-plane (English)

[ Abstract ]

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.

[ Reference 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

### 2023/04/06

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Blowup solutions to the Keller-Segel system (English)

https://forms.gle/7ogZKyh1oXKkPbN56

**Van Tien Nguyen**(National Taiwan University)Blowup solutions to the Keller-Segel system (English)

[ Abstract ]

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 $d = 2,3,4$.

[ Reference 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

### 2023/02/22

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Long time decay of Fokker-Planck equations with confining drift (ENGLISH)

https://forms.gle/SCyZWtfC5bNGadxE8

**Alessio Porretta**(University of Rome Tor Vergata)Long time decay of Fokker-Planck equations with confining drift (ENGLISH)

[ Abstract ]

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.

[ Reference 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

### 2023/02/06

16:00-18:10 Room #126 (Graduate School of Math. Sci. Bldg.)

Solutions with moving singularities for nonlinear diffusion equations (ENGLISH)

Fast diffusion equation: uniqueness of solutions with a moving singularity (ENGLISH)

https://forms.gle/nKa4XATuuGPwZWbUA

**Marek Fila**(Comenius University) 16:00-17:00Solutions with moving singularities for nonlinear diffusion equations (ENGLISH)

[ Abstract ]

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.

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**(Comenius University) 17:10-18:10Fast diffusion equation: uniqueness of solutions with a moving singularity (ENGLISH)

[ Abstract ]

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.

[ Reference 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

### 2022/11/24

16:00-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)

Strong radiation condition and stationary scattering theory for 1-body Stark operators (Japanese)

[ Reference URL ]

https://forms.gle/admRaVnmPjFyp5op9

**Kyohei Itakura**(The University of Tokyo)Strong radiation condition and stationary scattering theory for 1-body Stark operators (Japanese)

[ Reference URL ]

https://forms.gle/admRaVnmPjFyp5op9

### 2022/06/30

16:00-17:00 Online

A brief introduction to a class of new phase field models (English)

https://forms.gle/esc7Y6KGASwbFro97

**Xingzhi Bian**(Shanghai University)A brief introduction to a class of new phase field models (English)

[ Abstract ]

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.

[ Reference 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

### 2022/04/21

16:00-17:30 Online

Effect of decay rates of initial data on the sign of solutions to Cauchy problems of some higher order parabolic equations (Japanese)

[ Reference URL ]

https://forms.gle/96bBNEAEHrsdXfH57

**( )**Effect of decay rates of initial data on the sign of solutions to Cauchy problems of some higher order parabolic equations (Japanese)

[ Reference URL ]

https://forms.gle/96bBNEAEHrsdXfH57

### 2021/12/16

16:00-17:00 Online

Existence of solutions for fractional semilinear parabolic equations in Besov-Morrey spaces (Japanese)

[ Reference URL ]

https://forms.gle/whpkgAwYvyQKQMzM8

**Zhanpeisov Erbol**( )Existence of solutions for fractional semilinear parabolic equations in Besov-Morrey spaces (Japanese)

[ Reference URL ]

https://forms.gle/whpkgAwYvyQKQMzM8

### 2021/12/02

### 2021/11/25

### 2021/10/28

16:00-17:00 Online

Quasiconformal and Sobolev mappings on metric measure

https://forms.gle/QATECqmwmWGvXoU56

**Xiaodan Zhou**(OIST)Quasiconformal and Sobolev mappings on metric measure

[ Abstract ]

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).

[ Reference 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

### 2021/10/14

### 2021/07/29

16:00-17:00 Online

Lotka-Volterra competition-diffusion system: the critical case

https://forms.gle/LHj5mVUdpQ3Jxkrd6

**Dongyuan Xiao**( )Lotka-Volterra competition-diffusion system: the critical case

[ Abstract ]

We consider the reaction-diffusion competition system u_t=u_{xx}+u(1-u-v), v_t=dv_{xx}+rv(1-v-u), which is the so-called critical case. The associated ODE system then admits infinitely many equilibria, which makes the analysis quite intricate. We first prove the non-existence of monotone traveling waves by applying the phase plane analysis. Next, we study the long time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the ''faster'' species excludes the ''slower'' species (with an identified ''spreading speed''), but also provide a sharp description of the profile of the solution, thus shedding light on a new ''bump phenomenon''.

[ Reference URL ]We consider the reaction-diffusion competition system u_t=u_{xx}+u(1-u-v), v_t=dv_{xx}+rv(1-v-u), which is the so-called critical case. The associated ODE system then admits infinitely many equilibria, which makes the analysis quite intricate. We first prove the non-existence of monotone traveling waves by applying the phase plane analysis. Next, we study the long time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the ''faster'' species excludes the ''slower'' species (with an identified ''spreading speed''), but also provide a sharp description of the profile of the solution, thus shedding light on a new ''bump phenomenon''.

https://forms.gle/LHj5mVUdpQ3Jxkrd6