## Tuesday Seminar of Analysis

Seminar information archive ～06/14｜Next seminar｜Future seminars 06/15～

Date, time & place | Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi |

**Seminar information archive**

### 2024/05/14

16:00-18:15 Room #128 (Graduate School of Math. Sci. Bldg.)

The Energy of Heavy Atoms: Density Functionals (English)

https://forms.gle/ZEyVso6wa9QpNfxH7

Well-posedness for Semilinear Heat Equations in Orlicz Spaces (English)

https://forms.gle/ZEyVso6wa9QpNfxH7

**Heinz Siedentop**(LMU University of Munich) 16:00-17:00The Energy of Heavy Atoms: Density Functionals (English)

[ Abstract ]

Since computing the energy of a system with $N$ particles requires solving a $4^N$ dimensional system of (pseudo-)differential equations in $3N$ independent variables, an analytic solution is practically impossible. Therefore density functionals, i.e., functionals that depend on the particle density (3 variables) only and yield the energy upon minimization, are of great interest.

This concept has been applied successfully in non-relativistic quantum mechanics. However, in relativistic quantum mechanics even the simple analogue of the Thomas-Fermi functional is not bounded from below for Coulomb potential. This problem was addressed eventually by Engel and Dreizler who derived a functional from QED. I will review some known mathematical properties of this functional and show that it yields basic features of physics, such as asymptotic correct energy, stability of matter, and boundedness of the excess charge.

[ Reference URL ]Since computing the energy of a system with $N$ particles requires solving a $4^N$ dimensional system of (pseudo-)differential equations in $3N$ independent variables, an analytic solution is practically impossible. Therefore density functionals, i.e., functionals that depend on the particle density (3 variables) only and yield the energy upon minimization, are of great interest.

This concept has been applied successfully in non-relativistic quantum mechanics. However, in relativistic quantum mechanics even the simple analogue of the Thomas-Fermi functional is not bounded from below for Coulomb potential. This problem was addressed eventually by Engel and Dreizler who derived a functional from QED. I will review some known mathematical properties of this functional and show that it yields basic features of physics, such as asymptotic correct energy, stability of matter, and boundedness of the excess charge.

https://forms.gle/ZEyVso6wa9QpNfxH7

**Robert Laister**(University of the West of England) 17:15-18:15Well-posedness for Semilinear Heat Equations in Orlicz Spaces (English)

[ Abstract ]

We consider the local well-posedness of semilinear heat equations in Orlicz spaces, the latter prescribed via a Young function $\Phi$. Many existence-uniqueness results exist in the literature for power-like or exponential-like nonlinearities $f$, where the natural setting is an Orlicz space of corresponding type; i.e. if $f$ is power-like then $\Phi$ is power-like (Lebesgue space), if $f$ is exponential-like then $\Phi$ is exponential-like. However, the general problem of prescribing a suitable $\Phi$ for a given, otherwise arbitrary $f$ is open. Our goal is to provide a suitable framework to resolve this problem and I will present some recent results in this direction. The key is a new (to the best of our knowledge) smoothing estimate for the heat semigroup between two arbitrary Orlicz spaces. Existence then follows familiar lines via monotonicity or contraction mapping arguments. Global solutions are also presented under additional assumptions. This work is part of a collaborative project with Prof Kazuhiro Ishige, Dr Yohei Fujishima and Dr Kotaro Hisa.

[ Reference URL ]We consider the local well-posedness of semilinear heat equations in Orlicz spaces, the latter prescribed via a Young function $\Phi$. Many existence-uniqueness results exist in the literature for power-like or exponential-like nonlinearities $f$, where the natural setting is an Orlicz space of corresponding type; i.e. if $f$ is power-like then $\Phi$ is power-like (Lebesgue space), if $f$ is exponential-like then $\Phi$ is exponential-like. However, the general problem of prescribing a suitable $\Phi$ for a given, otherwise arbitrary $f$ is open. Our goal is to provide a suitable framework to resolve this problem and I will present some recent results in this direction. The key is a new (to the best of our knowledge) smoothing estimate for the heat semigroup between two arbitrary Orlicz spaces. Existence then follows familiar lines via monotonicity or contraction mapping arguments. Global solutions are also presented under additional assumptions. This work is part of a collaborative project with Prof Kazuhiro Ishige, Dr Yohei Fujishima and Dr Kotaro Hisa.

https://forms.gle/ZEyVso6wa9QpNfxH7

### 2024/03/12

16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

On the generic regularity of min-max CMC hypersurfaces (English)

https://forms.gle/7mqzgLqhtBuAovKB8

**Kobe Marshall-Stevens**(University College London)On the generic regularity of min-max CMC hypersurfaces (English)

[ Abstract ]

Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.

[ Reference URL ]Smooth constant mean curvature (CMC) hypersurfaces serve as effective tools to study the geometry and topology of Riemannian manifolds. In high dimensions however, one in general must account for their singular behaviour. I will discuss how such hypersurfaces are constructed via min-max techniques and some recent progress on their generic regularity, allowing for certain isolated singularities to be perturbed away.

https://forms.gle/7mqzgLqhtBuAovKB8

### 2023/11/14

16:15-17:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Resolvent expansions for magnetic Schrödinger operators (English)

https://forms.gle/qyEUeo4kVuPL1s289

**Arne Jensen**(Aalborg University)Resolvent expansions for magnetic Schrödinger operators (English)

[ Abstract ]

I will present some new results resolvent expansions around threshold zero for magnetic Schrödinger operators in dimension three. The magnetic field and the electric potential are assumed to decay sufficiently fast. Analogous results for Pauli operators will also be presented.

Joint work with H. Kovarik, Brescia, Italy.

[ Reference URL ]I will present some new results resolvent expansions around threshold zero for magnetic Schrödinger operators in dimension three. The magnetic field and the electric potential are assumed to decay sufficiently fast. Analogous results for Pauli operators will also be presented.

Joint work with H. Kovarik, Brescia, Italy.

https://forms.gle/qyEUeo4kVuPL1s289

### 2023/08/22

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

Towards a Levinson's Theorem for Discrete Magnetic operators on tubes under finite rank perturbations (English)

https://forms.gle/VBp4nSnYYKVpXFhB9

**Daniel Parra**(Universidad de Santiago de Chile)Towards a Levinson's Theorem for Discrete Magnetic operators on tubes under finite rank perturbations (English)

[ Abstract ]

In this talk we study a family of magnetic Hamiltonians on discrete tubes under a finite rank perturbation supported on its border. We go into detail for the case of rank $2$ and show how the eigenvalues can be related to the scattering matrix to exhibit an index theorem in the tradition of Levison’s theorem. We then turn to the general case, discuss the different spectral scenarios that can occur and explain the C*-algebraic framework that could allow us to treat this case. This is an ongoing work with S. Richard (Nagoya), V. Austen (Nagoya) and A. Rennie (Wollongong).

[ Reference URL ]In this talk we study a family of magnetic Hamiltonians on discrete tubes under a finite rank perturbation supported on its border. We go into detail for the case of rank $2$ and show how the eigenvalues can be related to the scattering matrix to exhibit an index theorem in the tradition of Levison’s theorem. We then turn to the general case, discuss the different spectral scenarios that can occur and explain the C*-algebraic framework that could allow us to treat this case. This is an ongoing work with S. Richard (Nagoya), V. Austen (Nagoya) and A. Rennie (Wollongong).

https://forms.gle/VBp4nSnYYKVpXFhB9

### 2023/07/11

16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Nodal solutions for a class of degenerate BVP’s (English)

https://forms.gle/S3VgMSWg9wUP69cY6

**Julian López-Gómez**(Complutense University of Madrid)Nodal solutions for a class of degenerate BVP’s (English)

[ Abstract ]

In this talk we characterize the existence of nodal solutions for a generalized class of one-dimensional diffusive logistic type equations, including

\[−u''=\lambda u−a(x)u^3,\quad x∈[0,L],\]

under the boundary conditions $u(0)=u(L)=0$, where $\lambda$ is regarded as a bifurcation parameter, and the non-negative weight function $a(x)$ vanishes on some subinterval

\[ [\alpha,\beta]\subset (0,L)\]

with $\alpha<\beta$.

At a later stage, the general case when $a(x)$ vanishes on finitely many subintervals with the same length is analyzed. Finally, we construct some examples with classical non-degenerate weights, with $a(x)>0$ for all $x∈[0,L]$, where the BVP has an arbitrarily large number of solutions with one node in $(0,L)$. These are the first examples of this nature constructed in the literature.

References:

P. Cubillos, J. López-Gómez and A. Tellini, Multiplicity of nodal solutions in classical non-degenerate logistic equations, El. Res. Archive 30 (2022), 898—928.

J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Disc. Cont. Dyn. Syst. B 22 (2017), 923—946.

J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP’s, Top. Meth. Nonl. Anal. 49 (2017), 359—376.

J. López-Gómez and P. H. Rabinowitz, The estructure of the set of 1-node solutions for a class of degenerate BVP’s, J. Differential Equations 268 (2020), 4691—4732.

P. H. Rabinowitz, A note on a anonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9 (1971), 536—548.

[ Reference URL ]In this talk we characterize the existence of nodal solutions for a generalized class of one-dimensional diffusive logistic type equations, including

\[−u''=\lambda u−a(x)u^3,\quad x∈[0,L],\]

under the boundary conditions $u(0)=u(L)=0$, where $\lambda$ is regarded as a bifurcation parameter, and the non-negative weight function $a(x)$ vanishes on some subinterval

\[ [\alpha,\beta]\subset (0,L)\]

with $\alpha<\beta$.

At a later stage, the general case when $a(x)$ vanishes on finitely many subintervals with the same length is analyzed. Finally, we construct some examples with classical non-degenerate weights, with $a(x)>0$ for all $x∈[0,L]$, where the BVP has an arbitrarily large number of solutions with one node in $(0,L)$. These are the first examples of this nature constructed in the literature.

References:

P. Cubillos, J. López-Gómez and A. Tellini, Multiplicity of nodal solutions in classical non-degenerate logistic equations, El. Res. Archive 30 (2022), 898—928.

J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Disc. Cont. Dyn. Syst. B 22 (2017), 923—946.

J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP’s, Top. Meth. Nonl. Anal. 49 (2017), 359—376.

J. López-Gómez and P. H. Rabinowitz, The estructure of the set of 1-node solutions for a class of degenerate BVP’s, J. Differential Equations 268 (2020), 4691—4732.

P. H. Rabinowitz, A note on a anonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9 (1971), 536—548.

https://forms.gle/S3VgMSWg9wUP69cY6

### 2023/06/06

17:00-18:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Stationary completeness; the many-body short-range case (English)

https://forms.gle/kWHDfb6J6kcjfSah8

**Erik Skibsted**(Aarhus University)Stationary completeness; the many-body short-range case (English)

[ Abstract ]

For a general class of many-body Schr\"odinger operators with short-range pair-potentials the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy. In fact this holds without imposing any a priori decay condition on channel eigenstates and even for models including long-range potentials of Derezi\'nski-Enss type. For short-range models we improve on the known \emph{weak continuity} statements in that we show that all non-threshold energies are \emph{stationary complete}, resolving in this case a recent conjecture. A consequence is that the above scattering quantities depend \emph{strongly continuously} on the energy parameter at all non-threshold energies (whence not only almost everywhere as previously demonstrated). Another consequence is that the scattering matrix is unitary at any such energy. Our procedure yields (as a side result) a new and purely stationary proof of asymptotic completeness for many-body short-range systems.

[ Reference URL ]For a general class of many-body Schr\"odinger operators with short-range pair-potentials the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy. In fact this holds without imposing any a priori decay condition on channel eigenstates and even for models including long-range potentials of Derezi\'nski-Enss type. For short-range models we improve on the known \emph{weak continuity} statements in that we show that all non-threshold energies are \emph{stationary complete}, resolving in this case a recent conjecture. A consequence is that the above scattering quantities depend \emph{strongly continuously} on the energy parameter at all non-threshold energies (whence not only almost everywhere as previously demonstrated). Another consequence is that the scattering matrix is unitary at any such energy. Our procedure yields (as a side result) a new and purely stationary proof of asymptotic completeness for many-body short-range systems.

https://forms.gle/kWHDfb6J6kcjfSah8

### 2023/03/14

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

Parameter reconstruction for degenerate parabolic equations (English)

https://forms.gle/nejpQS824vFKRbMQ6

**Piermarco Cannarsa**(University of Rome "Tor Vergata")Parameter reconstruction for degenerate parabolic equations (English)

[ Abstract ]

First, we study degenerate parabolic equations arising in climate dynamics, providing uniqueness and stability estimates for the determination of the insolation function. Then, we address several aspects of the reconstruction of the degenerate diffusion coefficient. Finally, we discuss systems of two equations including a vertical component into the model.

[ Reference URL ]First, we study degenerate parabolic equations arising in climate dynamics, providing uniqueness and stability estimates for the determination of the insolation function. Then, we address several aspects of the reconstruction of the degenerate diffusion coefficient. Finally, we discuss systems of two equations including a vertical component into the model.

https://forms.gle/nejpQS824vFKRbMQ6

### 2022/12/20

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

A commentary on J. Boman's recent two related results about the support of a distribution and its analyticity, and a special relationship between Radon transformations and ellipsoidal regions (Japanese)

https://forms.gle/BpciRTzKh9FPUV8D7

**KATAOKA Kiyoomi**(The University of Tokyo)A commentary on J. Boman's recent two related results about the support of a distribution and its analyticity, and a special relationship between Radon transformations and ellipsoidal regions (Japanese)

[ Abstract ]

Jan Boman's (Stockholm Univ.) recent two papers:

[1], Regularity of a distribution and of the boundary of its support, The Journal of Geometric Analysis vol.32, Article number: 300 (2022).

[2], A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case; J. Inverse Ill-Posed Probl. 2021; 29(3): 351–367.

In [1] he proved "Let $f(x_1,…,x_n,y)$ be a non-zero distribution with support in a $C^1$ surface $N=\{y=F(x)\}$. If $f(x,y)$ is depending real analytically on x-variables, then $F(x)$ is analytic". As an application, he reinforced the main result of [2]. These results are obtained essentially by means of matrix algebra and a number theoretic method.

[ Reference URL ]Jan Boman's (Stockholm Univ.) recent two papers:

[1], Regularity of a distribution and of the boundary of its support, The Journal of Geometric Analysis vol.32, Article number: 300 (2022).

[2], A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case; J. Inverse Ill-Posed Probl. 2021; 29(3): 351–367.

In [1] he proved "Let $f(x_1,…,x_n,y)$ be a non-zero distribution with support in a $C^1$ surface $N=\{y=F(x)\}$. If $f(x,y)$ is depending real analytically on x-variables, then $F(x)$ is analytic". As an application, he reinforced the main result of [2]. These results are obtained essentially by means of matrix algebra and a number theoretic method.

https://forms.gle/BpciRTzKh9FPUV8D7

### 2022/12/13

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

Continuum limit problem of discrete Schrödinger operators on square lattices (Japanese)

https://forms.gle/CRha8hydEuXzh71S7

**TADANO Yukihide**(Tokyo University of Science)Continuum limit problem of discrete Schrödinger operators on square lattices (Japanese)

[ Abstract ]

We consider discrete Schrödinger operators on the square lattice with its mesh size very small. The aim of this talk is to introduce the rigorous setting of continuum limit problems in the view point of operator theory and then to give its proof for the above operators, the one of which is defined on the vertices and the other of which is defined on the edges. This talk is based on joint works with Shu Nakamura (Gakushuin University) and Pavel Exner (Czech Academy of Science, Czech Technical University).

[ Reference URL ]We consider discrete Schrödinger operators on the square lattice with its mesh size very small. The aim of this talk is to introduce the rigorous setting of continuum limit problems in the view point of operator theory and then to give its proof for the above operators, the one of which is defined on the vertices and the other of which is defined on the edges. This talk is based on joint works with Shu Nakamura (Gakushuin University) and Pavel Exner (Czech Academy of Science, Czech Technical University).

https://forms.gle/CRha8hydEuXzh71S7

### 2022/11/29

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

Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions (Japanese)

https://forms.gle/93YQ9C6DGYt5Vjuf7

**TAKIMOTO Kazuhiro**(Hiroshima University)Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions (Japanese)

[ Abstract ]

In the early twentieth century, Bernstein proved that a minimal surface which can be expressed as the graph of a function defined in $\mathbb{R}^2$ must be a plane. For Monge-Ampère equation, it is known that a convex solution to $\det D^2 u=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Such kind of theorems, which we call Bernstein type theorems in this talk, have been extensively studied for various PDEs. For the parabolic $k$-Hessian equation, Bernstein type theorem has been proved by Nakamori and Takimoto (2015, 2016) under the convexity and some growth assumptions on the solution. In this talk, we shall obtain Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions.

[ Reference URL ]In the early twentieth century, Bernstein proved that a minimal surface which can be expressed as the graph of a function defined in $\mathbb{R}^2$ must be a plane. For Monge-Ampère equation, it is known that a convex solution to $\det D^2 u=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Such kind of theorems, which we call Bernstein type theorems in this talk, have been extensively studied for various PDEs. For the parabolic $k$-Hessian equation, Bernstein type theorem has been proved by Nakamori and Takimoto (2015, 2016) under the convexity and some growth assumptions on the solution. In this talk, we shall obtain Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions.

https://forms.gle/93YQ9C6DGYt5Vjuf7

### 2022/10/04

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

The Cahn-Hilliard equation with forward-backward dynamic boundary condition via vanishing viscosity (Japanese)

[ Reference URL ]

https://forms.gle/nPfEgKUX2tfUrg5LA

**FUKAO Takeshi**(Kyoto University of Education)The Cahn-Hilliard equation with forward-backward dynamic boundary condition via vanishing viscosity (Japanese)

[ Reference URL ]

https://forms.gle/nPfEgKUX2tfUrg5LA

### 2022/08/23

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

Quantitative homogenization for monotone, uniformly elliptic systems with random coefficients (English)

https://forms.gle/V1wxbYhT4mkPF4gY9

**Stefan Neukamm**(Dresden University/RIMS)Quantitative homogenization for monotone, uniformly elliptic systems with random coefficients (English)

[ Abstract ]

Motivated by homogenization of nonlinearly elastic composite materials, we study homogenization rates for elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. Under the assumption of a fast decay of correlations on scales larger than the microscale $\varepsilon$, we establish estimates of optimal order for the approximation of the homogenized operator by the method of representative volumes. Moreover, we discuss applications to nonlinear elasticity random laminates.

[ Reference URL ]Motivated by homogenization of nonlinearly elastic composite materials, we study homogenization rates for elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. Under the assumption of a fast decay of correlations on scales larger than the microscale $\varepsilon$, we establish estimates of optimal order for the approximation of the homogenized operator by the method of representative volumes. Moreover, we discuss applications to nonlinear elasticity random laminates.

https://forms.gle/V1wxbYhT4mkPF4gY9

### 2022/07/26

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

Periodic homogenization of non-symmetric jump-type processes with drifts (Japanese)

https://forms.gle/ewZEy1jAXrAhWx1Q8

**KUMAGAI Takashi**(Waseda University)Periodic homogenization of non-symmetric jump-type processes with drifts (Japanese)

[ Abstract ]

Homogenization problem is one of the classical problems in analysis and probability which is very actively studied recently. In this talk, we consider homogenization problem for non-symmetric Lévy-type processes with drifts in periodic media. Under a proper scaling, we show the scaled processes converge weakly to Lévy processes on ${\mathds R}^d$. In particular, we completely characterize the limiting processes when the coefficient function of the drift part is bounded continuous, and the decay rate of the jumping measure is comparable to $r^{-1-\alpha}$ for $r>1$ in the spherical coordinate with $\alpha \in (0,\infty)$. Different scaling limits appear depending on the values of $\alpha$.

This talk is based on joint work with Xin Chen, Zhen-Qing Chen and Jian Wang (Ann. Probab. 2021).

[ Reference URL ]Homogenization problem is one of the classical problems in analysis and probability which is very actively studied recently. In this talk, we consider homogenization problem for non-symmetric Lévy-type processes with drifts in periodic media. Under a proper scaling, we show the scaled processes converge weakly to Lévy processes on ${\mathds R}^d$. In particular, we completely characterize the limiting processes when the coefficient function of the drift part is bounded continuous, and the decay rate of the jumping measure is comparable to $r^{-1-\alpha}$ for $r>1$ in the spherical coordinate with $\alpha \in (0,\infty)$. Different scaling limits appear depending on the values of $\alpha$.

This talk is based on joint work with Xin Chen, Zhen-Qing Chen and Jian Wang (Ann. Probab. 2021).

https://forms.gle/ewZEy1jAXrAhWx1Q8

### 2022/06/28

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

Mourre inequality for non-local Schödinger operators (Japanese)

https://forms.gle/sBSeNH9AYFNypNBk9

**ISHIDA Atsuhide**(Tokyo University of Science)Mourre inequality for non-local Schödinger operators (Japanese)

[ Abstract ]

We consider the Mourre inequality for the following self-adjoint operator $H=\Psi(-\Delta/2)+V$ acting on $L^2(\mathbb{R}^d)$, where $\Psi: [0,\infty)\rightarrow\mathbb{R}$ is an increasing function, $\Delta$ is Laplacian and $V: \mathbb{R}^d\rightarrow\mathbb{R}$ is an interaction potential. Mourre inequality immediately yields the discreteness and finite multiplicity of the eigenvalues. Moreover, Mourre inequality has the application to the absence of the singular continuous spectrum by combining the limiting absorption principle and, in addition, Mourre inequality is also used for proof of the minimal velocity estimate that plays an important role in the scattering theory. In this talk, we report that Mourre inequality holds under the general $\Psi$ and $V$ by choosing the conjugate operator $A=(p\cdot x+x\cdot p)/2$ with $p=-\sqrt{-1}\nabla$, and that the discreteness and finite multiplicity of the eigenvalues hold. This talk is a joint work with J. Lőrinczi (Hungarian Academy of Sciences) and I. Sasaki (Shinshu University).

[ Reference URL ]We consider the Mourre inequality for the following self-adjoint operator $H=\Psi(-\Delta/2)+V$ acting on $L^2(\mathbb{R}^d)$, where $\Psi: [0,\infty)\rightarrow\mathbb{R}$ is an increasing function, $\Delta$ is Laplacian and $V: \mathbb{R}^d\rightarrow\mathbb{R}$ is an interaction potential. Mourre inequality immediately yields the discreteness and finite multiplicity of the eigenvalues. Moreover, Mourre inequality has the application to the absence of the singular continuous spectrum by combining the limiting absorption principle and, in addition, Mourre inequality is also used for proof of the minimal velocity estimate that plays an important role in the scattering theory. In this talk, we report that Mourre inequality holds under the general $\Psi$ and $V$ by choosing the conjugate operator $A=(p\cdot x+x\cdot p)/2$ with $p=-\sqrt{-1}\nabla$, and that the discreteness and finite multiplicity of the eigenvalues hold. This talk is a joint work with J. Lőrinczi (Hungarian Academy of Sciences) and I. Sasaki (Shinshu University).

https://forms.gle/sBSeNH9AYFNypNBk9

### 2022/05/31

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

Convergence of Sobolev gradient trajectories to elastica (Japanese)

https://forms.gle/wkCbqdmNuz9zr3vA8

**OKABE Shinya**(Tohoku University)Convergence of Sobolev gradient trajectories to elastica (Japanese)

[ Abstract ]

In this talk we consider a higher order Sobolev gradient flow for the modified elastic energy defined on closed space curves. The $L^2$-gradient flow for the modified elastic energy has been well studied, and standard results are solvability of the flow for smooth initial curve and subconvergence of solutions to elastica. Moreover, stronger convergence results, so called full limit convergence, are generally up to reparametrisation and sometimes translation. In this talk, we consider $H^2$-gradient flow for the modified elastic energy and prove (i) the solvability of the flow for initial curve in the energy class, (ii) full limit convergence to elastica by way of a Lojasiewicz—Simon gradient inequality. This talk is based on a joint work with Philip Schrader (Murdoch University).

[ Reference URL ]In this talk we consider a higher order Sobolev gradient flow for the modified elastic energy defined on closed space curves. The $L^2$-gradient flow for the modified elastic energy has been well studied, and standard results are solvability of the flow for smooth initial curve and subconvergence of solutions to elastica. Moreover, stronger convergence results, so called full limit convergence, are generally up to reparametrisation and sometimes translation. In this talk, we consider $H^2$-gradient flow for the modified elastic energy and prove (i) the solvability of the flow for initial curve in the energy class, (ii) full limit convergence to elastica by way of a Lojasiewicz—Simon gradient inequality. This talk is based on a joint work with Philip Schrader (Murdoch University).

https://forms.gle/wkCbqdmNuz9zr3vA8

### 2022/05/24

16:00-17:30 Online

Stability analysis for the surface diffusion flow on double bubbles using the Lojasiewicz-Simon (English)

https://forms.gle/Cam3mpSSEKKVppZr9

**Michael Goesswein**(The University of Tokyo/University of Regensburg)Stability analysis for the surface diffusion flow on double bubbles using the Lojasiewicz-Simon (English)

[ Abstract ]

Many strategies for stability analysis use precise knowledge of the set of equilibria. For example, Escher, Mayer, and Simonett used center manifold analysis to study the surface diffusion flow on closed manifolds. Especially in higher dimensional situations with boundaries, this can cause problems as the set of equilibria will have a lot of degrees of freedom. In such situations approaches with a Lojasiewicz-Simon inequality gives an elegant way to avoid this problem. In this talk, we will both explain the general tools and ideas for this strategy and use them to prove the stability of standard double bubbles with respect to the surface diffusion flow. The talk is based on joint work with H. Garcke.

[ Reference URL ]Many strategies for stability analysis use precise knowledge of the set of equilibria. For example, Escher, Mayer, and Simonett used center manifold analysis to study the surface diffusion flow on closed manifolds. Especially in higher dimensional situations with boundaries, this can cause problems as the set of equilibria will have a lot of degrees of freedom. In such situations approaches with a Lojasiewicz-Simon inequality gives an elegant way to avoid this problem. In this talk, we will both explain the general tools and ideas for this strategy and use them to prove the stability of standard double bubbles with respect to the surface diffusion flow. The talk is based on joint work with H. Garcke.

https://forms.gle/Cam3mpSSEKKVppZr9

### 2022/04/26

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

Existence of a bounded forward self-similar solution to a minimal Keller-Segel model (Japanese)

https://forms.gle/mrXnjsgctSJJ1WSF6

**WAKUI Hiroshi**(Tokyo University of Science)Existence of a bounded forward self-similar solution to a minimal Keller-Segel model (Japanese)

[ Abstract ]

In this talk, we consider existence of a bounded forward self-similar solution to the initial value problem of a minimal Keller-Segel model. It is well known that the mass conservation law plays an important role to classify its large time behavior of solutions to Keller-Segel models. On the other hand, we could not expect existence of self-similar solutions to our problem with the mass conservation law except for the two dimensional case due to the scaling invariance of our problem. We will show existence of a forward self-similar solution to our problem. The key idea to guarantee boundedness of its self-similar solution is to choose a concrete upper barrier function using the hypergeometric function.

[ Reference URL ]In this talk, we consider existence of a bounded forward self-similar solution to the initial value problem of a minimal Keller-Segel model. It is well known that the mass conservation law plays an important role to classify its large time behavior of solutions to Keller-Segel models. On the other hand, we could not expect existence of self-similar solutions to our problem with the mass conservation law except for the two dimensional case due to the scaling invariance of our problem. We will show existence of a forward self-similar solution to our problem. The key idea to guarantee boundedness of its self-similar solution is to choose a concrete upper barrier function using the hypergeometric function.

https://forms.gle/mrXnjsgctSJJ1WSF6

### 2022/04/12

16:00-17:30 Online

Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications (English)

https://forms.gle/QbQKex12dbQrt2Lw6

**Amru Hussein**(Technische Universität Kaiserslautern)Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications (English)

[ Abstract ]

Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix}A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part though with possibly large relative bound. For such operators, the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time-dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.

This talk is based on a joint work with Antonio Agresti, see https://arxiv.org/abs/2108.01962

[ Reference URL ]Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix}A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part though with possibly large relative bound. For such operators, the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time-dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.

This talk is based on a joint work with Antonio Agresti, see https://arxiv.org/abs/2108.01962

https://forms.gle/QbQKex12dbQrt2Lw6

### 2021/11/16

16:00-17:30 Online

TBA (Japanese)

[ Reference URL ]

https://forms.gle/6ZCp8hQxKA3vq3DB9

**KUBO Hideo**(Hokkaido University)TBA (Japanese)

[ Reference URL ]

https://forms.gle/6ZCp8hQxKA3vq3DB9

### 2021/10/19

16:00-17:30 Online

Global structure of steady-states for a cross-diffusion limit in the Shigesada-Kawasaki-Teramoto model (Japanese)

https://forms.gle/hkfCd3fSW5A77mwv5

**KUTO Kousuke**(Waseda University)Global structure of steady-states for a cross-diffusion limit in the Shigesada-Kawasaki-Teramoto model (Japanese)

[ Abstract ]

In 1979, Shigesada, Kawasaki and Teramoto proposed a Lotka-Volterra competition model with cross-diffusion terms in order to realize the segregation phenomena of two competing species. This talk concerns the asymptotic behavior of steady-states to the Shigesada-Kawasaki-Teramoto model in the full cross-diffusion limit where both coefficients of cross-diffusion terms tend to infinity at the same rate. In the former half of this talk, we derive a uniform estimate of all steady-states independent of the cross-diffusion terms. In the latter half, we show the global structure of steady-states of a shadow system in the full cross-diffusion limit.

[ Reference URL ]In 1979, Shigesada, Kawasaki and Teramoto proposed a Lotka-Volterra competition model with cross-diffusion terms in order to realize the segregation phenomena of two competing species. This talk concerns the asymptotic behavior of steady-states to the Shigesada-Kawasaki-Teramoto model in the full cross-diffusion limit where both coefficients of cross-diffusion terms tend to infinity at the same rate. In the former half of this talk, we derive a uniform estimate of all steady-states independent of the cross-diffusion terms. In the latter half, we show the global structure of steady-states of a shadow system in the full cross-diffusion limit.

https://forms.gle/hkfCd3fSW5A77mwv5

### 2021/07/13

16:00-17:30 Online

Li-Yau type inequality for curves and applications (Japanese)

https://forms.gle/gR4gfn8v59LEoqp38

**MIURA Tatsuya**(Tokyo Institute of Technology)Li-Yau type inequality for curves and applications (Japanese)

[ Abstract ]

A classical result of Li and Yau asserts an optimal relation between the bending energy and multiplicity of a closed surface in Euclidean space. Here we establish an analogue for curves in a completely general form, and observe new phenomena due to low dimensionality. We also discuss its applications to elastic flows, networks, and knots.

[ Reference URL ]A classical result of Li and Yau asserts an optimal relation between the bending energy and multiplicity of a closed surface in Euclidean space. Here we establish an analogue for curves in a completely general form, and observe new phenomena due to low dimensionality. We also discuss its applications to elastic flows, networks, and knots.

https://forms.gle/gR4gfn8v59LEoqp38

### 2021/06/08

16:00-17:30 Online

Local well-posedness for the Landau-Lifshitz equation with helicity term (Japanese)

https://forms.gle/nc85Mw9Jd6NgJzT98

**SHIMIZU Ikkei**(Osaka University)Local well-posedness for the Landau-Lifshitz equation with helicity term (Japanese)

[ Abstract ]

We consider the initial value problem for the Landau-Lifshitz equation with helicity term (chiral interaction term), which arises from the Dzyaloshinskii-Moriya interaction. We show that it is locally well-posed in Sobolev spaces $H^s$ when $s>2$. The key idea is to reduce the problem to a system of semi-linear Schr\"odinger equations, called modified Schr\"odinger map equation. The problem here is that the helicity term appears as quadratic derivative nonlinearities, which is known to be difficult to treat as perturbation of the free evolution. To overcome that, we consider them as magnetic terms, then apply the energy method by introducing the differential operator associated with magnetic potentials.

[ Reference URL ]We consider the initial value problem for the Landau-Lifshitz equation with helicity term (chiral interaction term), which arises from the Dzyaloshinskii-Moriya interaction. We show that it is locally well-posed in Sobolev spaces $H^s$ when $s>2$. The key idea is to reduce the problem to a system of semi-linear Schr\"odinger equations, called modified Schr\"odinger map equation. The problem here is that the helicity term appears as quadratic derivative nonlinearities, which is known to be difficult to treat as perturbation of the free evolution. To overcome that, we consider them as magnetic terms, then apply the energy method by introducing the differential operator associated with magnetic potentials.

https://forms.gle/nc85Mw9Jd6NgJzT98

### 2021/05/25

16:00-17:30 Online

Asymptotic limit of fast rotation for the incompressible Navier-Stokes equations in a 3D layer (Japanese)

https://forms.gle/wHpi7BSpppsiiguD6

**TAKADA Ryo**(Kyushu University)Asymptotic limit of fast rotation for the incompressible Navier-Stokes equations in a 3D layer (Japanese)

[ Abstract ]

In this talk, we consider the initial value problem for the Navier-Stokes equation with the Coriolis force in a three-dimensional infinite layer. We prove the unique existence of global solutions for initial data in the scaling invariant space when the speed of rotation is sufficiently high. Furthermore, we consider the asymptotic limit of the fast rotation, and show that the global solution converges to that of 2D incompressible Navier-Stokes equations in some global in time space-time norms. This talk is based on the joint work with Hiroki Ohyama (Kyushu University).

[ Reference URL ]In this talk, we consider the initial value problem for the Navier-Stokes equation with the Coriolis force in a three-dimensional infinite layer. We prove the unique existence of global solutions for initial data in the scaling invariant space when the speed of rotation is sufficiently high. Furthermore, we consider the asymptotic limit of the fast rotation, and show that the global solution converges to that of 2D incompressible Navier-Stokes equations in some global in time space-time norms. This talk is based on the joint work with Hiroki Ohyama (Kyushu University).

https://forms.gle/wHpi7BSpppsiiguD6

### 2020/02/18

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Long time behavior of mean field games systems (English)

**Alessio Porretta**(Tor Vergata university of Rome)Long time behavior of mean field games systems (English)

[ Abstract ]

I will review several aspects related to the long time ergodic behavior of mean field game systems: the turnpike property, the exponential rate of convergence, the role of monotonicity of the couplings, the convergence of u up to translations, the limit of the vanishing discounted problem, the long time behavior of the master equation. All those aspects have independent interest and are correlated at the same time.

I will review several aspects related to the long time ergodic behavior of mean field game systems: the turnpike property, the exponential rate of convergence, the role of monotonicity of the couplings, the convergence of u up to translations, the limit of the vanishing discounted problem, the long time behavior of the master equation. All those aspects have independent interest and are correlated at the same time.

### 2020/01/14

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Scattering near a two-cluster threshold (English)

**Erik Skibsted**(Aarhus University)Scattering near a two-cluster threshold (English)

[ Abstract ]

For a one-body Schr\"odinger operator with an attractive slowly decaying potential the scattering matrix is well-defined at the energy zero, and the structure of its singularities is well-studied. The usual (non-relativistic) model for the Hydrogen atom is a particular example of such Schr\"odinger operator.

Less is known on scattering at a two-cluster threshold of an $N$-body Schr\"odinger operator for which the effective interaction between the two bound clusters is attractive Coulombic. An example of interest is scattering at a two-cluster threshold of a neutral atom/molecule. We present results of an ongoing joint work with X.P. Wang on the subject, including a version of the Sommerfeld uniqueness result and its applications.

We shall also present general results on spectral theory at a two-cluster threshold (not requiring the effective interaction to be attractive Coulombic). This includes a general structure theorem on the bound and resonance states at the threshold as well as a resolvent expansion in weighted spaces above the threshold (under more restrictive conditions). Applications to scattering theory will be indicated.

For a one-body Schr\"odinger operator with an attractive slowly decaying potential the scattering matrix is well-defined at the energy zero, and the structure of its singularities is well-studied. The usual (non-relativistic) model for the Hydrogen atom is a particular example of such Schr\"odinger operator.

Less is known on scattering at a two-cluster threshold of an $N$-body Schr\"odinger operator for which the effective interaction between the two bound clusters is attractive Coulombic. An example of interest is scattering at a two-cluster threshold of a neutral atom/molecule. We present results of an ongoing joint work with X.P. Wang on the subject, including a version of the Sommerfeld uniqueness result and its applications.

We shall also present general results on spectral theory at a two-cluster threshold (not requiring the effective interaction to be attractive Coulombic). This includes a general structure theorem on the bound and resonance states at the threshold as well as a resolvent expansion in weighted spaces above the threshold (under more restrictive conditions). Applications to scattering theory will be indicated.