## Tuesday Seminar of Analysis

Date, time & place Tuesday 16:00 - 17:30 126Room #126 (Graduate School of Math. Sci. Bldg.) ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi

Seminar information archive

### 2022/08/23

16:00-17:30   Room #126 (Graduate School of Math. Sci. Bldg.)
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 ]
https://forms.gle/V1wxbYhT4mkPF4gY9

### 2022/07/26

16:00-17:30   Room #126 (Graduate School of Math. Sci. Bldg.)
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 ]
https://forms.gle/ewZEy1jAXrAhWx1Q8

### 2022/06/28

16:00-17:30   Room #126 (Graduate School of Math. Sci. Bldg.)
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 ]
https://forms.gle/sBSeNH9AYFNypNBk9

### 2022/05/31

16:00-17:30   Room #126 (Graduate School of Math. Sci. Bldg.)
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 ]
https://forms.gle/wkCbqdmNuz9zr3vA8

### 2022/05/24

16:00-17:30   Online
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 ]
https://forms.gle/Cam3mpSSEKKVppZr9

### 2022/04/26

16:00-17:30   Room #126 (Graduate School of Math. Sci. Bldg.)
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 ]
https://forms.gle/mrXnjsgctSJJ1WSF6

### 2022/04/12

16:00-17:30   Online
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 ]
https://forms.gle/QbQKex12dbQrt2Lw6

### 2021/11/16

16:00-17:30   Online
KUBO Hideo (Hokkaido University)
TBA (Japanese)
[ Reference URL ]
https://forms.gle/6ZCp8hQxKA3vq3DB9

### 2021/10/19

16:00-17:30   Online
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 ]
https://forms.gle/hkfCd3fSW5A77mwv5

### 2021/07/13

16:00-17:30   Online
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 ]
https://forms.gle/gR4gfn8v59LEoqp38

### 2021/06/08

16:00-17:30   Online
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 ]
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)
[ 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 ]
https://forms.gle/wHpi7BSpppsiiguD6

### 2020/02/18

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

### 2020/01/14

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

### 2019/12/10

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
Tobias Barker (École Normale Supérieure)
Vorticity alignment vs vorticity creation at the boundary (English)
[ Abstract ]
The Navier-Stokes are used as a model for viscous incompressible fluids such as water. The question as to whether or not the equations in three dimensions form singularities is an open Millennium prize problem. In their celebrated paper in 1993, Constantin and Fefferman showed that (in the whole plane) if the vorticity is sufficiently well aligned in regions of high vorticity then the Navier-Stokes equations remain smooth. For the half-space it is commonly assumed that viscous fluids stick' to the boundary, which generates vorticity at the boundary. In such a setting, it is open as to whether Constantin and Fefferman's result remains to be true. In my talk I will present recent results in this direction. Joint work with Christophe Prange (CNRS, Université de Bordeaux)

### 2019/11/26

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
ASHIDA Sohei (Gakushuin University)
Accurate lower bounds for eigenvalues of electronic Hamiltonians (Japanese)
[ Abstract ]
Electronic Hamiltonians are differential operators depending on relative positions of nuclei as parameters. When we regard an eigenvalues of an electronic Hamiltonian as a function of relative positions of nuclei, minimum points correspond to shapes of molecules. Upper bounds for eigenvalues are obtained by variational methods. However, since the physical information as minimum points does not change when a reference point of energy changes, physical information can not be obtained by variational methods only. Combining lower and upper bounds physical information is obtained.
In this talk we discuss the Weinstein-Arnszajn intermediate problem methods for lower bounds of eigenvalues based on comparison of operators. A method for lower bounds of one-electronic Hamiltonians is also introduced. Some computations for two kinds of hydrogen molecule-ion are shown.

### 2019/11/19

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
Wenjia Jing (Tsinghua University)
Quantitative homogenization for the Dirichlet problem of Stokes system in periodic perforated domain - a unified approach (English)
[ Abstract ]
We present a new unified approach for the quantitative homogenization of the Stokes system in periodically perforated domains, that is domains outside a periodic array of holes, with Dirichlet data at the boundary of the holes. The method is based on the (rescaled) cell-problem and is adaptive to the ratio between the typical distance and the typical side length of the holes; in particular, for the critical ratio identified by Cioranescu-Murat, we recover the “strange term from nowhere”termed by them, which, in the context of Stokes system, corresponds to the Brinkman’s law. An advantage of the method is that it can be systematically quantified using the periodic layer potential technique. We will also report some new correctors to the homogenization problem using this approach. The talk is based on joint work with Yong Lu and Christophe Prange.

### 2019/11/12

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
KAMIMOTO Shingo (Hiroshima University)
Mould expansion and resurgent structure (Japanese)

### 2019/11/05

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
Ngô Quốc Anh (Vietnam National University, Hanoi / the University of Tokyo)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
[ Abstract ]
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity $\Delta^m u = \pm u^\alpha$ in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.

### 2019/06/11

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
Antonio De Rosa (Courant Institute of Mathematical Sciences)
Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)
[ Abstract ]
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)

### 2019/04/09

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
Fabio Punzo (Politecnico di Milano)
The Poisson equation on Riemannian manifolds (English)
[ Abstract ]
The talk is concerned with the existence of solutions to the Poisson equation on complete non-compact Riemannian manifolds. In particular, the interplay between the Ricci curvature and the behaviour at infinity of the source function will be discussed. This is a joint work with G. Catino and D.D. Monticelli.

### 2019/03/05

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
Nicholas Edelen (Massachusetts Institute of Technology)
The structure of minimal surfaces near polyhedral cones (English)
[ Abstract ]
We prove a regularity theorem for minimal varifolds which resemble a cone $C_0$ over an equiangular geodesic net. For varifold classes admitting a `no-hole'' condition on the singular set, we additionally establish regularity near the cone $C_0 \times R^m$. Our result implies the following generalization of Taylor's structure theorem for soap bubbles: given an $n$-dimensional soap bubble $M$ in $R^{n+1}$, then away from an $(n-3)$-dimensional set, $M$ is locally $C^{1,\alpha}$ equivalent to $R^n$, a union of three half-$n$-planes meeting at $120$ degrees, or an $(n-2)$-line of tetrahedral junctions. This is joint work with Maria Colombo and Luca Spolaor.

### 2019/01/22

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
KATO Keiichi (Tokyo University of Science)
Construction of solutions to Schrodinger equations with sub-quadratic potential via wave packet transform (Japanese)
[ Abstract ]
In this talk, we consider linear Schrodinger equations with sub-quadratic potentials, which can be transformed by the wave packet transform with time dependent wave packet to a PDE of first order with inhomogeneous terms including unknown function and second derivatives of the potential. If the second derivatives of the potentials are bounded, the homogenous term of the first oder equation gives a construction of solutions to Schrodinger equations with sub-quadratic potentials by the similar way as in D. Fujiwara's work for Feynman path integral. We will show numerical computations by using our construction, if we have enough time.

### 2018/12/25

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
MASAKI Satoshi (Osaka University)
Modified scattering for nonlinear dispersive equations with critical non-polynomial nonlinearities (Japanese)
[ Abstract ]
In this talk, I will introduce resent progress on modified scattering for Schrodinger equation and Klein-Gordon equation with a non-polynomial nonlinearity. We use Fourier series expansion technique to find the resonant part of the nonlinearity which produces phase correction factor.

### 2018/11/06

16:50-18:20   Room #128 (Graduate School of Math. Sci. Bldg.)
SHIBATA Tetsutaro (Hiroshima University)
Global behavior of bifurcation curves and related topics (日本語)
[ Abstract ]
In this talk, we consider the asymptotic behavior of bifurcation curves for ODE with oscillatory nonlinear term. First, we study the global and local behavior of oscillatory bifurcation curves. We also consider the bifurcation problems with nonlinear diffusion.