## Applied Analysis

Seminar information archive ～06/03｜Next seminar｜Future seminars 06/04～

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

### 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

### 2021/06/17

### 2021/04/22

16:30-18:00 Online

Relaxation of Optimal Transport problem on finite state space via Bregman divergence (Japanese)

[ Reference URL ]

https://forms.gle/yg9XZDVdxYG6qMos8

**( )**Relaxation of Optimal Transport problem on finite state space via Bregman divergence (Japanese)

[ Reference URL ]

https://forms.gle/yg9XZDVdxYG6qMos8

### 2021/04/15

### 2020/11/05

16:00-17:30 Room #オンライン開催 (Graduate School of Math. Sci. Bldg.)

Hölder gradient estimates on L^p-viscosity solutions of fully nonlinear parabolic equations with VMO coefficients (Japanese)

https://docs.google.com/forms/d/e/1FAIpQLSf4Rmd6B0m9_t_-xdy2hT1ZC1Ziz2qEc3yLRCQNZBilAOB1Ag/viewform?usp=sf_link

**( )**Hölder gradient estimates on L^p-viscosity solutions of fully nonlinear parabolic equations with VMO coefficients (Japanese)

[ Abstract ]

We discuss fully nonlinear second-order uniformly parabolic equations, including parabolic Isaacs equations. Isaacs equations arise in the theory of stochastic differential games. In 2014, N.V. Krylov proved the existence of L^p-viscosity solutions of boundary value problems for equations with VMO (vanishing mean oscillation) “coefficients” when p>n+2. Furthermore, the solutions were in the parabolic Hölder space C^{1,α} for 0<α<1. Our purpose is to show C^{1,α} estimates on L^p-viscosity solutions of fully nonlinear parabolic equations under the same conditions as in Krylov’s result.

[ Reference URL ]We discuss fully nonlinear second-order uniformly parabolic equations, including parabolic Isaacs equations. Isaacs equations arise in the theory of stochastic differential games. In 2014, N.V. Krylov proved the existence of L^p-viscosity solutions of boundary value problems for equations with VMO (vanishing mean oscillation) “coefficients” when p>n+2. Furthermore, the solutions were in the parabolic Hölder space C^{1,α} for 0<α<1. Our purpose is to show C^{1,α} estimates on L^p-viscosity solutions of fully nonlinear parabolic equations under the same conditions as in Krylov’s result.

https://docs.google.com/forms/d/e/1FAIpQLSf4Rmd6B0m9_t_-xdy2hT1ZC1Ziz2qEc3yLRCQNZBilAOB1Ag/viewform?usp=sf_link

### 2020/10/08

16:00-17:30 Room #オンライン開催 (Graduate School of Math. Sci. Bldg.)

(Japanese)

[ Reference URL ]

https://docs.google.com/forms/d/e/1FAIpQLSd7MT077191TeM4aQzeo2hK9Bqn6HQudr3pjLRdmEqND2heqQ/viewform?usp=sf_link

**( )**(Japanese)

[ Reference URL ]

https://docs.google.com/forms/d/e/1FAIpQLSd7MT077191TeM4aQzeo2hK9Bqn6HQudr3pjLRdmEqND2heqQ/viewform?usp=sf_link

### 2019/12/19

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

### 2019/10/31

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

Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)

**Marius Ghergu**(University College Dublin)Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)

[ Abstract ]

We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).

We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).

### 2019/10/24

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

### 2019/06/20

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

### 2019/04/25

16:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

The porous medium equation on noncompact Riemannian manifolds with initial datum a measure

(English)

On sharp large deviations for the bridge of a general diffusion

(English)

**Matteo Muratori**(Polytechnic University of Milan) 16:00-17:00The porous medium equation on noncompact Riemannian manifolds with initial datum a measure

(English)

[ Abstract ]

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.

**Maurizia Rossi**(University of Pisa) 17:00-18:00On sharp large deviations for the bridge of a general diffusion

(English)

[ Abstract ]

In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.

In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.

### 2018/11/15

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)

**Nakao Hayashi**(Osaka University)Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)

[ Abstract ]

We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

### 2018/10/11

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

(Japanese)

**Takahito Kashiwabara**(University of Tokyo)(Japanese)