## 解析学火曜セミナー

過去の記録 ～04/12｜次回の予定｜今後の予定 04/13～

開催情報 | 火曜日 16:00～17:30 数理科学研究科棟(駒場) 156号室 |
---|---|

担当者 | 石毛 和弘, 坂井 秀隆, 伊藤 健一 |

セミナーURL | https://www.ms.u-tokyo.ac.jp/seminar/analysis/ |

**過去の記録**

### 2016年01月05日(火)

16:50-18:20 数理科学研究科棟(駒場) 126号室

Stationary scattering theory on manifolds (English)

**Eric Skibsted 氏**(Aarhus University, Denmark)Stationary scattering theory on manifolds (English)

[ 講演概要 ]

We present a stationary scattering theory for the Schrödinger operator on Riemannian manifolds with the structure of ends each of which is equipped with an escape function (for example a convex distance function). This includes manifolds with ends modeled as cone-like subsets of the Euclidean space and/or the hyperbolic space. Our results include Rellich’s theorem, the limiting absorption principle, radiation condition bounds, the Sommerfeld uniqueness result, and we give complete characterization/asymptotics of the generalized eigenfunctions in a certain Besov space and show asymptotic completeness (with K. Ito).

We present a stationary scattering theory for the Schrödinger operator on Riemannian manifolds with the structure of ends each of which is equipped with an escape function (for example a convex distance function). This includes manifolds with ends modeled as cone-like subsets of the Euclidean space and/or the hyperbolic space. Our results include Rellich’s theorem, the limiting absorption principle, radiation condition bounds, the Sommerfeld uniqueness result, and we give complete characterization/asymptotics of the generalized eigenfunctions in a certain Besov space and show asymptotic completeness (with K. Ito).

### 2015年12月01日(火)

16:50-18:20 数理科学研究科棟(駒場) 126号室

On complex singularity analysis for some linear partial differential equations

**Stéphane Malek 氏**(Université de Lille, France)On complex singularity analysis for some linear partial differential equations

[ 講演概要 ]

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc outside some singular set S. The coefficients are written as linear combinations of powers of a solution X of some first order nonlinear partial differential equation following an idea :we have initiated in a previous joint work with C. Stenger. The solutions Y are shown to develop singularities along the singular set S with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series. (Joint work with A. Lastra and C. Stenger).

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc outside some singular set S. The coefficients are written as linear combinations of powers of a solution X of some first order nonlinear partial differential equation following an idea :we have initiated in a previous joint work with C. Stenger. The solutions Y are shown to develop singularities along the singular set S with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series. (Joint work with A. Lastra and C. Stenger).

### 2015年11月24日(火)

16:50-18:20 数理科学研究科棟(駒場) 126号室

A local analysis of the swirling flow to the axi-symmetric Navier-Stokes equations near a saddle point and no-slip flat boundary (English)

**許 本源 氏**(東大数理)A local analysis of the swirling flow to the axi-symmetric Navier-Stokes equations near a saddle point and no-slip flat boundary (English)

[ 講演概要 ]

As one of the violent flow, tornadoes occur in many place of the world. In order to reduce human losses and material damage caused by tornadoes, there are many research methods. One of the effective methods is numerical simulations. The swirling structure is significant both in mathematical analysis and the numerical simulations of tornado. In this joint work with H. Notsu and T. Yoneda we try to clarify the swirling structure. More precisely, we do numerical computations on axi-symmetric Navier-Stokes flows with no-slip flat boundary. We compare a hyperbolic flow with swirl and one without swirl and observe that the following phenomenons occur only in the swirl case: The distance between the point providing the maximum velocity magnitude $|v|$ and the $z$-axis is drastically changing around some time (which we call it turning point). An ``increasing velocity phenomenon'' occurs near the boundary and the maximum value of $|v|$ is obtained near the axis of symmetry and the boundary when time is close to the turning point.

As one of the violent flow, tornadoes occur in many place of the world. In order to reduce human losses and material damage caused by tornadoes, there are many research methods. One of the effective methods is numerical simulations. The swirling structure is significant both in mathematical analysis and the numerical simulations of tornado. In this joint work with H. Notsu and T. Yoneda we try to clarify the swirling structure. More precisely, we do numerical computations on axi-symmetric Navier-Stokes flows with no-slip flat boundary. We compare a hyperbolic flow with swirl and one without swirl and observe that the following phenomenons occur only in the swirl case: The distance between the point providing the maximum velocity magnitude $|v|$ and the $z$-axis is drastically changing around some time (which we call it turning point). An ``increasing velocity phenomenon'' occurs near the boundary and the maximum value of $|v|$ is obtained near the axis of symmetry and the boundary when time is close to the turning point.

### 2015年10月20日(火)

16:50-18:20 数理科学研究科棟(駒場) 128号室

FMSPレクチャーズと共催／部屋が普段と異なりますのでご注意ください

Existence of an entropy solution in the sense of Young measures for a first order conservation law with a stochastic source term (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hilhorst151020.pdf

FMSPレクチャーズと共催／部屋が普段と異なりますのでご注意ください

**Danielle Hilhorst 氏**(CNRS / University of Paris-Sud)Existence of an entropy solution in the sense of Young measures for a first order conservation law with a stochastic source term (ENGLISH)

[ 講演概要 ]

We consider a finite volume scheme for a first order conservation law with a monotone flux function and a multiplicative source term involving a Q-Wiener process. We define a stochastic entropy solution in the sense of Young measures. We present some a priori estimates for the discrete solution including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities and show that the discrete solution converges along a subsequence to an entropy solution in the sense of Young measures.

This is joint work with T. Funaki, Y. Gao and H. Weber.

[ 参考URL ]We consider a finite volume scheme for a first order conservation law with a monotone flux function and a multiplicative source term involving a Q-Wiener process. We define a stochastic entropy solution in the sense of Young measures. We present some a priori estimates for the discrete solution including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities and show that the discrete solution converges along a subsequence to an entropy solution in the sense of Young measures.

This is joint work with T. Funaki, Y. Gao and H. Weber.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hilhorst151020.pdf

### 2015年10月13日(火)

16:50-18:20 数理科学研究科棟(駒場) 126号室

Nonlinear analysis with endlessly continuable functions (joint work with Shingo Kamimoto) (English)

**David Sauzin 氏**(CNRS, France)Nonlinear analysis with endlessly continuable functions (joint work with Shingo Kamimoto) (English)

[ 講演概要 ]

We give estimates for the convolution products of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power series.

We give estimates for the convolution products of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power series.

### 2015年09月29日(火)

16:50-18:20 数理科学研究科棟(駒場) 126号室

On the Phragmén-Lindelöf principle for holomorphic functions and factor classes of higher order complex forms in several complex variables

**Otto Liess 氏**(University of Bologna, Italy)On the Phragmén-Lindelöf principle for holomorphic functions and factor classes of higher order complex forms in several complex variables

[ 講演概要 ]

In this talk we will discuss maximum principles in unbounded domains in one or several complex variables. We will mainly be interested in these principles for plurisubharmonic (in the one-dimensional case, "subharmonic") or holomorphic functions, when the principles are of

Phragmen-Lindel{＼"o}f principle (henceforth called "PL") type. It will turn out that for 2 or more complex variables it will be useful to study our principles together with associated principles for factor classes of complex (0,q) forms with growth type conditions at infinity.

In this abstract we only say something concerning the case of functions. We consider then an open set U in C^n in one or several complex variables. We assume that we are given two real-valued continuous functions f and g on U. We say that PL holds for plurisubharmonic (respectively for holomorphic) functions, if the following implication is true for every plurisubharmonic function $ ＼rho $ (respectively for every $ ＼rho $ of form log |h| with h holomorphic) on U: if we know that $ ＼rho ＼leq f$ on the boundary of U and if $ (＼rho - f)$ is bounded on U, then it must follow that $ ＼rho ＼leq g$ on U. ($＼rho ＼leq f$ on the boundary has the following meaning: for ever z in the boundary of U and for every sequence of points y_j in U which tends to z, we have limsup (＼rho - f)(y_j) leq 0.) A trivial condition under which PL is true, is when there exists a plurisubharmonic function u on U such that

(*) -g(z) ＼leq u(z) ＼leq - f(z) for every z in U.

In fact, if such a function exists, then we can apply the classical maximal principle for unbounded domains to the function $ ＼rho'= ＼rho+u$ to obtain at first $ ＼rho' ＼leq 0$ and then $ ＼rho ＼leq - u ＼leq g$. It is one of the main goals of the talk to explain how far (*) is from being also a necessary condition for PL. Some examples are intended to justify our approach and applications will be given to problems in convex analysis.

In this talk we will discuss maximum principles in unbounded domains in one or several complex variables. We will mainly be interested in these principles for plurisubharmonic (in the one-dimensional case, "subharmonic") or holomorphic functions, when the principles are of

Phragmen-Lindel{＼"o}f principle (henceforth called "PL") type. It will turn out that for 2 or more complex variables it will be useful to study our principles together with associated principles for factor classes of complex (0,q) forms with growth type conditions at infinity.

In this abstract we only say something concerning the case of functions. We consider then an open set U in C^n in one or several complex variables. We assume that we are given two real-valued continuous functions f and g on U. We say that PL holds for plurisubharmonic (respectively for holomorphic) functions, if the following implication is true for every plurisubharmonic function $ ＼rho $ (respectively for every $ ＼rho $ of form log |h| with h holomorphic) on U: if we know that $ ＼rho ＼leq f$ on the boundary of U and if $ (＼rho - f)$ is bounded on U, then it must follow that $ ＼rho ＼leq g$ on U. ($＼rho ＼leq f$ on the boundary has the following meaning: for ever z in the boundary of U and for every sequence of points y_j in U which tends to z, we have limsup (＼rho - f)(y_j) leq 0.) A trivial condition under which PL is true, is when there exists a plurisubharmonic function u on U such that

(*) -g(z) ＼leq u(z) ＼leq - f(z) for every z in U.

In fact, if such a function exists, then we can apply the classical maximal principle for unbounded domains to the function $ ＼rho'= ＼rho+u$ to obtain at first $ ＼rho' ＼leq 0$ and then $ ＼rho ＼leq - u ＼leq g$. It is one of the main goals of the talk to explain how far (*) is from being also a necessary condition for PL. Some examples are intended to justify our approach and applications will be given to problems in convex analysis.

### 2015年09月08日(火)

16:50-18:20 数理科学研究科棟(駒場) 126号室

長距離型斥力ポテンシャルを持つシュレディンガー方程式の時間大域的ストリッカーツ評価 (Japanese)

**水谷治哉 氏**(大阪大学・理学研究科)長距離型斥力ポテンシャルを持つシュレディンガー方程式の時間大域的ストリッカーツ評価 (Japanese)

[ 講演概要 ]

We will discuss a resent result on global-in-time Strichartz estimates for Schr\"odinger equations with slowly decreasing repulsive potentials. If the potential is of very short-range type, there is a simple method due to Rodnianski-Schlag or Burq et al, which seems to be difficult to apply for the present case. The proof instead follows a similar line as in speaker’s resent joint work with J.-M. Bouclet. In particular, we employ both Morawetz type estimates and the methods of micro local analysis such as the Isozaki-Kitada parametrix, even in the low frequency regime.

We will discuss a resent result on global-in-time Strichartz estimates for Schr\"odinger equations with slowly decreasing repulsive potentials. If the potential is of very short-range type, there is a simple method due to Rodnianski-Schlag or Burq et al, which seems to be difficult to apply for the present case. The proof instead follows a similar line as in speaker’s resent joint work with J.-M. Bouclet. In particular, we employ both Morawetz type estimates and the methods of micro local analysis such as the Isozaki-Kitada parametrix, even in the low frequency regime.

### 2015年07月21日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

Born-Oppenheimer approximation for an atom in constant magnetic fields (Japanese)

**蘆田 聡平 氏**(京都大学理学研究科)Born-Oppenheimer approximation for an atom in constant magnetic fields (Japanese)

[ 講演概要 ]

We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. Martinez and Sordoni also dealt with such a case but their reduced Hamiltonian includes the vector potential terms. Using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.

We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. Martinez and Sordoni also dealt with such a case but their reduced Hamiltonian includes the vector potential terms. Using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.

### 2015年07月14日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

Small-time Asymptotics for Subelliptic Heat Kernels (English)

**Li Yutian 氏**(Department of Mathematics, Hong Kong Baptist University)Small-time Asymptotics for Subelliptic Heat Kernels (English)

[ 講演概要 ]

Subelliptic operators are the natural generalizations of the Laplace- Beltrami operators, and they play important roles in geometry, several complex variables, probability and physics. As in the classical spectral theory for the elliptic operators, some geometrical properties of the induced subRiemannian geometry can be extracted from the analysis of the heat kernels for subelliptic operators. In this talk we shall review the recent progress in the heat kernel asymptotics for subelliptic operators. We concentrate on the small-time asymptotics of the heat kernel on the diagonal, or equivalently, the asymptotics for the trace. Our interest is to find the exact form of the leading term, and this will lead to a Weyl’s asymptotic formula for the subelliptic operators. This is a joint work with Professor Der-Chen Chang.

Subelliptic operators are the natural generalizations of the Laplace- Beltrami operators, and they play important roles in geometry, several complex variables, probability and physics. As in the classical spectral theory for the elliptic operators, some geometrical properties of the induced subRiemannian geometry can be extracted from the analysis of the heat kernels for subelliptic operators. In this talk we shall review the recent progress in the heat kernel asymptotics for subelliptic operators. We concentrate on the small-time asymptotics of the heat kernel on the diagonal, or equivalently, the asymptotics for the trace. Our interest is to find the exact form of the leading term, and this will lead to a Weyl’s asymptotic formula for the subelliptic operators. This is a joint work with Professor Der-Chen Chang.

### 2015年05月12日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

Brakkeの平均曲率流に対する制約条件付きAllen-Cahn方程式の収束について (Japanese)

**高棹 圭介 氏**(東京大学大学院数理科学研究科)Brakkeの平均曲率流に対する制約条件付きAllen-Cahn方程式の収束について (Japanese)

[ 講演概要 ]

In this talk we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.

In this talk we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.

### 2015年04月21日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

今年度から部屋が１２６に変わりました．

留数カレントと定数係数線形遅延微分方程式系の一般論について (Japanese)

今年度から部屋が１２６に変わりました．

**松原 宰栄 氏**(東京大学大学院数理科学研究科)留数カレントと定数係数線形遅延微分方程式系の一般論について (Japanese)

[ 講演概要 ]

We introduce the ring of differential operators with constant coefficients and commensurate time lags (we use the terminology D$\Delta$ operators from now) initially defined by H. Gl\"using-L\"ur\ss en for ordinary $D\Delta$ operators and observe that various function modules enjoy good cohomological properties over this ring. %After revising the notion of the residue current in the spirit of M. Andersson and E. Wulcan, we introduce the multidimensional version of the ring D$\Delta$ operators.

Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $D\Delta$ equations.

We introduce the ring of differential operators with constant coefficients and commensurate time lags (we use the terminology D$\Delta$ operators from now) initially defined by H. Gl\"using-L\"ur\ss en for ordinary $D\Delta$ operators and observe that various function modules enjoy good cohomological properties over this ring. %After revising the notion of the residue current in the spirit of M. Andersson and E. Wulcan, we introduce the multidimensional version of the ring D$\Delta$ operators.

Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $D\Delta$ equations.

### 2014年12月16日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

Global Strichartz estimates for Schr¥”odinger equations on

asymptotically conic manifolds (Japanese)

**水谷 治哉 氏**(大阪大学・理学研究科)Global Strichartz estimates for Schr¥”odinger equations on

asymptotically conic manifolds (Japanese)

### 2014年12月02日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

New isoperimetric inequalities with densities arising in reaction-diffusion problems (English)

**Xavier Cabre 氏**(ICREA and UPC, Barcelona)New isoperimetric inequalities with densities arising in reaction-diffusion problems (English)

[ 講演概要 ]

In joint works with X. Ros-Oton and J. Serra, the study of the

regularity of stable solutions to reaction-diffusion problems

has led us to certain Sobolev and isoperimetric inequalities

with weights. We will present our results in these new

isoperimetric inequalities with the best constant, that we

establish via the ABP method. More precisely, we obtain

a new family of sharp isoperimetric inequalities with weights

(or densities) in open convex cones of R^n. Our results apply

to all nonnegative homogeneous weights satisfying a concavity

condition in the cone. Surprisingly, even that our weights are

not radially symmetric, Euclidean balls centered at the origin

(intersected with the cone) minimize the weighted isoperimetric

quotient. As a particular case of our results, we provide with

new proofs of classical results such as the Wulff inequality and

the isoperimetric inequality in convex cones of Lions and Pacella.

Furthermore, we also study the anisotropic isoperimetric problem

for the same class of weights and we prove that the Wulff shape

always minimizes the anisotropic weighted perimeter under the

weighted volume constraint.

In joint works with X. Ros-Oton and J. Serra, the study of the

regularity of stable solutions to reaction-diffusion problems

has led us to certain Sobolev and isoperimetric inequalities

with weights. We will present our results in these new

isoperimetric inequalities with the best constant, that we

establish via the ABP method. More precisely, we obtain

a new family of sharp isoperimetric inequalities with weights

(or densities) in open convex cones of R^n. Our results apply

to all nonnegative homogeneous weights satisfying a concavity

condition in the cone. Surprisingly, even that our weights are

not radially symmetric, Euclidean balls centered at the origin

(intersected with the cone) minimize the weighted isoperimetric

quotient. As a particular case of our results, we provide with

new proofs of classical results such as the Wulff inequality and

the isoperimetric inequality in convex cones of Lions and Pacella.

Furthermore, we also study the anisotropic isoperimetric problem

for the same class of weights and we prove that the Wulff shape

always minimizes the anisotropic weighted perimeter under the

weighted volume constraint.

### 2014年11月25日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

Stationary scattering theory on manifold with ends (JAPANESE)

**伊藤 健一 氏**(神戸大学理学研究科)Stationary scattering theory on manifold with ends (JAPANESE)

### 2014年09月09日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

FMSPレクチャーズとの共催です

Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

FMSPレクチャーズとの共催です

**Hatem Zaag 氏**(CNRS / University of Paris Nord)Energy methods and blow-up rate for semilinear wave equations in the superconformal case (ENGLISH)

[ 講演概要 ]

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

In a series of papers with Mohamed Ali Hamza (University of Tunis-el Manar), we consider the semilinear wave equations with power nonlinearity.

In the subconformal and the conformal case, we consider perturbations with lower order terms and modify the Lyapunov functional Antonini and Merle designed for the unperturbed case. We also find a blow-up criterion for the equation. As a consequence, we bound the Lyapunov functional. Thanks to interpolations in Sobolev spaces and a Gagliardo-Nirenberg inequality, we bound the solution in the self-similar variable, which gives a sharp bound on the blow-up rate.

Surprisingly, our approach works in the superconformal case (still Sobolev subcritical), leading to a new bound on the blow-up rate, which improves the bound of Killip, Stoval and Visan.

### 2014年06月10日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

On estimates for the Stokes flow in a space of bounded functions (JAPANESE)

**阿部 健 氏**(名古屋大学)On estimates for the Stokes flow in a space of bounded functions (JAPANESE)

[ 講演概要 ]

The Stokes equations are well understood on $L^p$ space for various kinds of domains such as bounded or exterior domains, and fundamental to study the nonlinear Navier-Stokes equations. The situation is different for the case $p=\\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. In this talk, we show some a priori estimate for a composition operator of the Stokes semigroup and the Helmholtz projection on a space of bounded functions.

The Stokes equations are well understood on $L^p$ space for various kinds of domains such as bounded or exterior domains, and fundamental to study the nonlinear Navier-Stokes equations. The situation is different for the case $p=\\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. In this talk, we show some a priori estimate for a composition operator of the Stokes semigroup and the Helmholtz projection on a space of bounded functions.

### 2014年05月27日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

楕円型方程式の正則性定理と領域の滑らかさ (JAPANESE)

**宮崎 洋一 氏**(日本大学歯学部)楕円型方程式の正則性定理と領域の滑らかさ (JAPANESE)

[ 講演概要 ]

We consider the Dirichlet boundary problem for a strongly elliptic operator of order $2m$ with non-smooth coefficients, and prove the regularity theorem for $L_p$-based Sobolev spaces when the domain has a boundary of limited smoothness. Compared to the known results, we can weaken the smoothness assumption on the boundary by $m-1$.

We consider the Dirichlet boundary problem for a strongly elliptic operator of order $2m$ with non-smooth coefficients, and prove the regularity theorem for $L_p$-based Sobolev spaces when the domain has a boundary of limited smoothness. Compared to the known results, we can weaken the smoothness assumption on the boundary by $m-1$.

### 2014年05月13日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

Ultra-differentiable classes and intersection theorems (JAPANESE)

**岡田 靖則 氏**(千葉大学大学院理学研究科)Ultra-differentiable classes and intersection theorems (JAPANESE)

[ 講演概要 ]

There are two ways to define notions of

ultra-differentiability: one in terms of estimates on derivatives, and

the other in terms of growth properties of Fourier transforms of

suitably localized functions.

In this talk, we study the relation between BMT-classes and

inhomogeneous Gevrey classes, which are examples of such two kinds of

notions of ultra-differentiability.

We also mention intersection theorems on these classes.

This talk is based on a joint work with Otto Liess (Bologna University).

There are two ways to define notions of

ultra-differentiability: one in terms of estimates on derivatives, and

the other in terms of growth properties of Fourier transforms of

suitably localized functions.

In this talk, we study the relation between BMT-classes and

inhomogeneous Gevrey classes, which are examples of such two kinds of

notions of ultra-differentiability.

We also mention intersection theorems on these classes.

This talk is based on a joint work with Otto Liess (Bologna University).

### 2014年04月22日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

拡散性を有しない誘因因子に対する走化性方程式の小さな有界な解 (JAPANESE)

**筒井 容平 氏**(東大 数理)拡散性を有しない誘因因子に対する走化性方程式の小さな有界な解 (JAPANESE)

[ 講演概要 ]

We consider a chemotaxis system with a logarithmic

sensitivity and a non-diffusive chemical substance. For some chemotactic

sensitivity constants, Ahn and Kang proved the existence of bounded

global solutions to the system. An entropy functional was used in their

argument to control the cell density by the density of the chemical

substance. Our purpose is to show the existence of bounded global

solutions for all the chemotactic sensitivity constants. Assuming the

smallness on the initial data in some sense, we can get uniform

estimates for time. These estimates are used to extend local solutions.

This talk is partially based on joint work with Yoshie Sugiyama (Kyusyu

Univ.) and Juan J. L. Vel\\'azquez (Univ. of Bonn).

We consider a chemotaxis system with a logarithmic

sensitivity and a non-diffusive chemical substance. For some chemotactic

sensitivity constants, Ahn and Kang proved the existence of bounded

global solutions to the system. An entropy functional was used in their

argument to control the cell density by the density of the chemical

substance. Our purpose is to show the existence of bounded global

solutions for all the chemotactic sensitivity constants. Assuming the

smallness on the initial data in some sense, we can get uniform

estimates for time. These estimates are used to extend local solutions.

This talk is partially based on joint work with Yoshie Sugiyama (Kyusyu

Univ.) and Juan J. L. Vel\\'azquez (Univ. of Bonn).

### 2014年01月28日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

This seminar is cosponsored by the GCOE program.

Asymptotic behaviour of a non-local diffusive logistic equation (ENGLISH)

This seminar is cosponsored by the GCOE program.

**Arnaud Ducrot 氏**(ボルドー大学)Asymptotic behaviour of a non-local diffusive logistic equation (ENGLISH)

[ 講演概要 ]

In this talk we investigate the long time behaviour of a logistic type equation modelling the motion of cells. The equation we consider takes into account birth and death process using a simple logistic effect as well as a non-local motion of cells using non-local Darcy’s law with regular kernel. Using the periodic framework we first investigate the well-posedness of the problem before deriving some information about its long time behaviour. The lack of asymptotic compactness of the system is overcome by making use of Young measure theory. This allows us to conclude that the semiflow converges for the Young measure topology.

In this talk we investigate the long time behaviour of a logistic type equation modelling the motion of cells. The equation we consider takes into account birth and death process using a simple logistic effect as well as a non-local motion of cells using non-local Darcy’s law with regular kernel. Using the periodic framework we first investigate the well-posedness of the problem before deriving some information about its long time behaviour. The lack of asymptotic compactness of the system is overcome by making use of Young measure theory. This allows us to conclude that the semiflow converges for the Young measure topology.

### 2014年01月21日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

符号付き距離関数との比較に基づく改良された等高面法 (JAPANESE)

**浜向 直 氏**(東大数理)符号付き距離関数との比較に基づく改良された等高面法 (JAPANESE)

[ 講演概要 ]

等高面法では、各時刻における曲面をある補助関数のゼロ等高面として表示し、

その補助関数に対する偏微分方程式を解くことで曲面の動きを求める。 しかし

時間が経つにつれて解の傾きが小さくなることがあり、 計算機では等高面を取

り出すことが困難となる。 そこで本講演では、方程式を適当に修正すること

で、 ゼロ等高面付近で傾きが小さくならない解が得られることを、 符号付き距

離関数との比較を通して示す。

等高面法では、各時刻における曲面をある補助関数のゼロ等高面として表示し、

その補助関数に対する偏微分方程式を解くことで曲面の動きを求める。 しかし

時間が経つにつれて解の傾きが小さくなることがあり、 計算機では等高面を取

り出すことが困難となる。 そこで本講演では、方程式を適当に修正すること

で、 ゼロ等高面付近で傾きが小さくならない解が得られることを、 符号付き距

離関数との比較を通して示す。

### 2014年01月14日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

Threshold properties for one-dimensional discrete Schr\\"odinger operators (JAPANESE)

**伊藤 健一 氏**(筑波大)Threshold properties for one-dimensional discrete Schr\\"odinger operators (JAPANESE)

[ 講演概要 ]

We study the relation between the generalized eigenspace and the asymptotic expansion of the resolvent around the threshold $0$ for the one-dimensional discrete Schr\\"odinger operator on $\\mathbb Z$. We decompose the generalized eigenspace into the subspaces corresponding to the eigenstates and the resonance states only by their asymptotics at infinity, and classify the coefficient operators of the singlar part of resolvent expansion completely in terms of these eigenspaces. Here the generalized eigenspace we consider is largest possible. For an explicit computation of the resolvent expansion we apply the expansion scheme of Jensen-Nenciu (2001). This talk is based on the recent joint work with Arne Jensen (Aalborg University).

We study the relation between the generalized eigenspace and the asymptotic expansion of the resolvent around the threshold $0$ for the one-dimensional discrete Schr\\"odinger operator on $\\mathbb Z$. We decompose the generalized eigenspace into the subspaces corresponding to the eigenstates and the resonance states only by their asymptotics at infinity, and classify the coefficient operators of the singlar part of resolvent expansion completely in terms of these eigenspaces. Here the generalized eigenspace we consider is largest possible. For an explicit computation of the resolvent expansion we apply the expansion scheme of Jensen-Nenciu (2001). This talk is based on the recent joint work with Arne Jensen (Aalborg University).

### 2013年12月17日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

Dispersion and observability for completely integrable Schrödinger flows (ENGLISH)

**Fabricio Macia 氏**(Universidad Politécnica de Madrid)Dispersion and observability for completely integrable Schrödinger flows (ENGLISH)

[ 講演概要 ]

I will present some results on weak dispersion and unique continuation (observability) for linear Schrödinger

equations that are obtained as the quantization of a completely integrable Hamiltonian system.

The model case corresponds to the linear Schrödinger equation (with a potential) on the flat torus.

Our results are obtained through a detailed analysis of semiclassical measures corresponding to

sequences of solutions, which is performed using a two-microlocal approach.

This is a joint work with Nalini Anantharaman and Clotilde Fermanian-Kammerer.

I will present some results on weak dispersion and unique continuation (observability) for linear Schrödinger

equations that are obtained as the quantization of a completely integrable Hamiltonian system.

The model case corresponds to the linear Schrödinger equation (with a potential) on the flat torus.

Our results are obtained through a detailed analysis of semiclassical measures corresponding to

sequences of solutions, which is performed using a two-microlocal approach.

This is a joint work with Nalini Anantharaman and Clotilde Fermanian-Kammerer.

### 2013年12月10日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

Quantitative unique continuation principle, local behavior of solutions, and bounds on the density of states for Schr\\"odinger operators (ENGLISH)

**Abel Klein 氏**(UC Irvine)Quantitative unique continuation principle, local behavior of solutions, and bounds on the density of states for Schr\\"odinger operators (ENGLISH)

[ 講演概要 ]

We establish bounds on the density of states measure for Schr\\"odinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a ``density of states outer-measure'' that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-H\\"older continuity for this density of states outer-measure in one, two, and three dimensions for Schr\\"odinger operators, and in any dimension for discrete Schr\\"odinger operators. Our proofs use a quantitative unique continuation principle and the local behavior of approximate solutions of the stationary Schr\\"odinger equation.

(Joint work with Jean Bourgain.)

References: Jean Bourgain and Abel Klein: Bounds on the density of states for Schr\\"odinger operators. Invent. Math. 194, 41-72 (2013).

We establish bounds on the density of states measure for Schr\\"odinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a ``density of states outer-measure'' that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-H\\"older continuity for this density of states outer-measure in one, two, and three dimensions for Schr\\"odinger operators, and in any dimension for discrete Schr\\"odinger operators. Our proofs use a quantitative unique continuation principle and the local behavior of approximate solutions of the stationary Schr\\"odinger equation.

(Joint work with Jean Bourgain.)

References: Jean Bourgain and Abel Klein: Bounds on the density of states for Schr\\"odinger operators. Invent. Math. 194, 41-72 (2013).

### 2013年11月26日(火)

16:30-18:00 数理科学研究科棟(駒場) 128号室

Global Strichartz estimates for Schr\\"odinger equations with long range metrics (JAPANESE)

**水谷 治哉 氏**(学習院大学)Global Strichartz estimates for Schr\\"odinger equations with long range metrics (JAPANESE)

[ 講演概要 ]

We consider Schr\\"odinger equations on the asymptotically Euclidean space

with the long-range condition on the metric.

We show that if the high energy resolvent has at most polynomial growth with respect to the energy,

then global-in-time Strichartz estimates, outside a large compact set, hold.

Under the non-trapping condition we also discuss global-in-space Strichartz estimates.

This talk is based on a joint work with J.-M. Bouclet (Toulouse University).

We consider Schr\\"odinger equations on the asymptotically Euclidean space

with the long-range condition on the metric.

We show that if the high energy resolvent has at most polynomial growth with respect to the energy,

then global-in-time Strichartz estimates, outside a large compact set, hold.

Under the non-trapping condition we also discuss global-in-space Strichartz estimates.

This talk is based on a joint work with J.-M. Bouclet (Toulouse University).