## Seminar information archive

Seminar information archive ～10/14｜Today's seminar 10/15 | Future seminars 10/16～

### 2005/04/20

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Small modifications of quadrature domains around a cusp

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**酒井 良**(都立大学)Small modifications of quadrature domains around a cusp

[ Abstract ]

A flow which is produced by injection of fluid into the narrow gap between two parallel planes is called a Hele-Shaw flow. We regard the flow as an increasing family of plane domains and discuss the case that the initial domain has a cusp on the boundary. We give sufficient conditions for the cusp to be a laminar-flow point.

[ Reference URL ]A flow which is produced by injection of fluid into the narrow gap between two parallel planes is called a Hele-Shaw flow. We regard the flow as an increasing family of plane domains and discuss the case that the initial domain has a cusp on the boundary. We give sufficient conditions for the cusp to be a laminar-flow point.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/04/18

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

エネルギー有限な有理形関数の除外点の個数について

**厚地 淳**(慶大経済)エネルギー有限な有理形関数の除外点の個数について

### 2005/03/23

#### PDE Real Analysis Seminar

10:30-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Helmut Abels**(Max Planck Institute)Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

[ Abstract ]

We discuss an operator class that models elliptic differential boundary value problems as well as their solution operators and is closed under compositions. It was introduced by Boutet de Monvel in 1971 and provides a powerful tool to calculate with symbols associated to these operators. But the standard calculus and most of its further developments require that the symbols have smooth coefficient in the space and phase variable. We present some results which extend the calculus to symbols which have limited smoothness in the space variable. Such an extension is nescessary to apply the calculus to nonlinear partial differential boundary value problems and free boundary value problems.

[ Reference URL ]We discuss an operator class that models elliptic differential boundary value problems as well as their solution operators and is closed under compositions. It was introduced by Boutet de Monvel in 1971 and provides a powerful tool to calculate with symbols associated to these operators. But the standard calculus and most of its further developments require that the symbols have smooth coefficient in the space and phase variable. We present some results which extend the calculus to symbols which have limited smoothness in the space variable. Such an extension is nescessary to apply the calculus to nonlinear partial differential boundary value problems and free boundary value problems.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/03/02

#### PDE Real Analysis Seminar

10:30-11:30 Room #270 (Graduate School of Math. Sci. Bldg.)

The maximum principle in unbounded domains

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

Aubry set and applications

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Italo Capuzzo-Dolcetta**(Universita di Roma) 10:30-11:30The maximum principle in unbounded domains

[ Abstract ]

The issue of the talk is the validity of the Weak Maximum Principle for functions u satisfying a second-order partial differential inequality of the form

(*) F(x,u,Du,D^2u) ≧ 0

in a domain A of the n-dimensional euclidean space.

The main result presented in the lecture is that for bounded above upper semicontinuous functions verifying

(*) in the viscosity sense, the inequality u≦ 0 on the boundary ∂A is propagated in the interior of the domain itself, under suitable conditions on F and A.

These conditions include ellipticity of F, a general geometric condition on the (possibly) unbounded domain A and a joint requirement involving the spread of A and the decay of first order terms at infinity.

This result, contained in I.C.D, A.Leoni, A.Vitolo "The Alexandrov-Bakelman-Pucci weak Maximum Principle for fully nonlinear equations in unbounded domains", to appear in Comm.in PDE's, extends previous results due to X.Cabré and L.Caffarelli-X.Cabré.

In the second part of the talk we present different versions of Weak Maximum Principle, namely for solutions growing exponentially fast of (*) in narrow domains and for solutions of

(**) F(x,u,Du,D^2u) + c(x)u ≧ 0

(c changing sign) in domains of small measure.

[ Reference URL ]The issue of the talk is the validity of the Weak Maximum Principle for functions u satisfying a second-order partial differential inequality of the form

(*) F(x,u,Du,D^2u) ≧ 0

in a domain A of the n-dimensional euclidean space.

The main result presented in the lecture is that for bounded above upper semicontinuous functions verifying

(*) in the viscosity sense, the inequality u≦ 0 on the boundary ∂A is propagated in the interior of the domain itself, under suitable conditions on F and A.

These conditions include ellipticity of F, a general geometric condition on the (possibly) unbounded domain A and a joint requirement involving the spread of A and the decay of first order terms at infinity.

This result, contained in I.C.D, A.Leoni, A.Vitolo "The Alexandrov-Bakelman-Pucci weak Maximum Principle for fully nonlinear equations in unbounded domains", to appear in Comm.in PDE's, extends previous results due to X.Cabré and L.Caffarelli-X.Cabré.

In the second part of the talk we present different versions of Weak Maximum Principle, namely for solutions growing exponentially fast of (*) in narrow domains and for solutions of

(**) F(x,u,Du,D^2u) + c(x)u ≧ 0

(c changing sign) in domains of small measure.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Antonio Siconolfi**(Universita di Roma) 11:45-12:45Aubry set and applications

[ Abstract ]

For given Hamiltonian H(x, p) continuous and quasiconvex in the second argument, defined in Rn × Rn or on the cotangent bundle of a compact boundaryless manifold, we consider the equation

H= c

with c critical value, i.e. for which the equation admits locally Lipschitzcontinuous a.e. subsolutions, but not strict subsolutions. We show that there is a subset of the state variable space, called Aubry set and denoted by A, where the obstruction to the existence of such subsolutions is concentrated. We give a metric characterization of A, and we discuss its main properties.

They are applied to a projection problem in a Banach space, to the study of the largetime behaviour of subsolutions to a timedependent HamiltonJacobi equation, and to construct a Lyapunov function for a perturbed dynamics, under suitable stability assumptions.

[ Reference URL ]For given Hamiltonian H(x, p) continuous and quasiconvex in the second argument, defined in Rn × Rn or on the cotangent bundle of a compact boundaryless manifold, we consider the equation

H= c

with c critical value, i.e. for which the equation admits locally Lipschitzcontinuous a.e. subsolutions, but not strict subsolutions. We show that there is a subset of the state variable space, called Aubry set and denoted by A, where the obstruction to the existence of such subsolutions is concentrated. We give a metric characterization of A, and we discuss its main properties.

They are applied to a projection problem in a Banach space, to the study of the largetime behaviour of subsolutions to a timedependent HamiltonJacobi equation, and to construct a Lyapunov function for a perturbed dynamics, under suitable stability assumptions.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/01/26

#### PDE Real Analysis Seminar

10:30-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

L^p-Theory of the Navier-Stokes flow past rotating or moving obstacles

**Matthias Hieber**(ダルムシュタット工科大学)L^p-Theory of the Navier-Stokes flow past rotating or moving obstacles

[ Abstract ]

In this talk we consider the equation of Navier-Stokes in the exterior of a rotating or moving domain. Using techniques from the analysis of Ornstein-Uhlenbeck operators it is shown that, after rewriting the problem on a fixed domain $\\Omega$, the solution of the linearized equation is governed by a $C_0$-semigroup on $L^p_\\sigma(\\Omega)$ for $1

In this talk we consider the equation of Navier-Stokes in the exterior of a rotating or moving domain. Using techniques from the analysis of Ornstein-Uhlenbeck operators it is shown that, after rewriting the problem on a fixed domain $\\Omega$, the solution of the linearized equation is governed by a $C_0$-semigroup on $L^p_\\sigma(\\Omega)$ for $1

[ Reference URL ]

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/01/24

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Hausdorff measure and exceptional sets in Dirichlet space theory on local fields

**金子 宏**(東京理科大)Hausdorff measure and exceptional sets in Dirichlet space theory on local fields

### 2005/01/17

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

複素函数論の情報ネットワーク特性評価への応用

**中川健治**(長岡技術科学大学)複素函数論の情報ネットワーク特性評価への応用

[ Abstract ]

情報ネットワークの特性評価を目的として,離散型 および連続型確率変数 X の裾確率 P(X > x) の指数的 減少について調べる。特に X が連続型の場合,X の確率 分布関数 F(x) = P(X ≧ x)のLaplace-Stieltjes変換を φ(s) とし,φ(s) の収束座標を σ とする。 -∞ < σ < 0 を仮定する。φ(s) の収束軸上の 特異点が高々有限個の極のみならば P(X > x) が指数的に 減少することを示す。その解析のために Ikehara による Tauber 型定理を拡張して適用する。

情報ネットワークの特性評価を目的として,離散型 および連続型確率変数 X の裾確率 P(X > x) の指数的 減少について調べる。特に X が連続型の場合,X の確率 分布関数 F(x) = P(X ≧ x)のLaplace-Stieltjes変換を φ(s) とし,φ(s) の収束座標を σ とする。 -∞ < σ < 0 を仮定する。φ(s) の収束軸上の 特異点が高々有限個の極のみならば P(X > x) が指数的に 減少することを示す。その解析のために Ikehara による Tauber 型定理を拡張して適用する。

### 2004/12/17

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

A simple proof of a theorem by Uhlenbeck and Yau

**D. Popovici**(Warwick)A simple proof of a theorem by Uhlenbeck and Yau

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Holomorphic curves into algebraic varieties

**Min Ru**(Houston)Holomorphic curves into algebraic varieties

### 2004/12/15

#### PDE Real Analysis Seminar

10:30-12:45 Room #122 (Graduate School of Math. Sci. Bldg.)

Hamilton-Jacobi-Bellman equations for optimal control of stochastic Navier-Stokes equations.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

A GEOMETRICAL APPROACH TO FRONT PROPAGATION PROBLEMS IN BOUNDED DOMAINS WITH NEUMANN-TYPE BOUNDARY AND APPLICATIONS

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Andrzej Swiech**(ジョージア工科大学) 10:30-11:30Hamilton-Jacobi-Bellman equations for optimal control of stochastic Navier-Stokes equations.

[ Abstract ]

We consider a parameterized family of continuous functions, which containsas its members Bourbai's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

[ Reference URL ]We consider a parameterized family of continuous functions, which containsas its members Bourbai's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Francesca Da Lio**(Dipartimento di Matematica P. e A.Universit di Padova researcher) 11:45-12:45A GEOMETRICAL APPROACH TO FRONT PROPAGATION PROBLEMS IN BOUNDED DOMAINS WITH NEUMANN-TYPE BOUNDARY AND APPLICATIONS

[ Abstract ]

We talk about a new definition of weak solution for the global-in-time motion of a front in bounded domains with normal velocity depending not only on its curvature but also on the measure of the set it encloses and with a contact angle boundary condition. We apply this definition to study the asymptotic behaviour of the solutions of some local and nonlocal reaction-diffusion equations.

[ Reference URL ]We talk about a new definition of weak solution for the global-in-time motion of a front in bounded domains with normal velocity depending not only on its curvature but also on the measure of the set it encloses and with a contact angle boundary condition. We apply this definition to study the asymptotic behaviour of the solutions of some local and nonlocal reaction-diffusion equations.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2004/12/13

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

正則曲線、小林双曲性とアーベル多様体の川又特徴付け

**野口潤次郎**(東大数理)正則曲線、小林双曲性とアーベル多様体の川又特徴付け

### 2004/12/01

#### PDE Real Analysis Seminar

10:30-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A remark on continuous, nowhere differentiable functions

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**岡本 久**(京都大学)A remark on continuous, nowhere differentiable functions

[ Abstract ]

We consider a parameterized family of continuous functions, which containsas its members Bourbai's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

[ Reference URL ]We consider a parameterized family of continuous functions, which containsas its members Bourbai's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2004/10/20

#### PDE Real Analysis Seminar

10:30-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Some recent results on the Navier-Stokes equations

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Hermann Sohr**(University of Paderborn)Some recent results on the Navier-Stokes equations

[ Abstract ]

The aim of this talk is to explain some new results in particular on local regularity properties of Hopf type weak solutions to the Navier-Stokes equations for arbitrary domains. Further we explain a new existence result for nonhomogeneous data and a result for global regular solutions with "slightly" modified forces.

[ Reference URL ]The aim of this talk is to explain some new results in particular on local regularity properties of Hopf type weak solutions to the Navier-Stokes equations for arbitrary domains. Further we explain a new existence result for nonhomogeneous data and a result for global regular solutions with "slightly" modified forces.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2004/10/13

#### PDE Real Analysis Seminar

10:30-11:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Existence, uniqueness, and continuous dependence of entropy solutions to hyperbolic systems

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Philippe G. LeFloch**(University of Paris 6)Existence, uniqueness, and continuous dependence of entropy solutions to hyperbolic systems

[ Abstract ]

I will review the well-posedness theory of nonlinear hyperbolic systems, in conservative or in non-conservative form, by focusing attention on the existence and properties of entropy solutions with sufficiently small total variation.

New results and perspectives on the following issues will be discussed: Glimm's existence theorem,

Bressan-LeFloch's uniqueness theorem,and the L1 continuous dependence property (established by Bressan, LeFloch, Liu, and Yang).

[ Reference URL ]I will review the well-posedness theory of nonlinear hyperbolic systems, in conservative or in non-conservative form, by focusing attention on the existence and properties of entropy solutions with sufficiently small total variation.

New results and perspectives on the following issues will be discussed: Glimm's existence theorem,

Bressan-LeFloch's uniqueness theorem,and the L1 continuous dependence property (established by Bressan, LeFloch, Liu, and Yang).

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2004/09/29

#### PDE Real Analysis Seminar

10:30-11:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Global Regularity of the 3D Navier-Stokes with Uniformly Large Initial Vorticity

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Alex Mahalov**(Arizona State University)Global Regularity of the 3D Navier-Stokes with Uniformly Large Initial Vorticity

[ Abstract ]

We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for fully three-dimensional initial data characterized by uniformly large vorticity with periodic boundary conditions and in bounded cylindrical domains; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold.

The global existence is proven using techniques of fast singular oscillating limits and the Littlewood-Paley dyadic decomposition. The approach is based on the computation of singular limits of rapidly oscillating operators and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, we obtain fully 3D limit resonant Navier-Stokes equations. Using Lemmas on restricted convolutions, we establish the global regularity of the latter without any restriction on the size of 3D initial data.

With strong convergence theorems, we bootstrap this into the global regularity of the 3D Navier-Stokes Equations with uniformly large initial vorticity.

[ Reference URL ]We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for fully three-dimensional initial data characterized by uniformly large vorticity with periodic boundary conditions and in bounded cylindrical domains; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold.

The global existence is proven using techniques of fast singular oscillating limits and the Littlewood-Paley dyadic decomposition. The approach is based on the computation of singular limits of rapidly oscillating operators and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, we obtain fully 3D limit resonant Navier-Stokes equations. Using Lemmas on restricted convolutions, we establish the global regularity of the latter without any restriction on the size of 3D initial data.

With strong convergence theorems, we bootstrap this into the global regularity of the 3D Navier-Stokes Equations with uniformly large initial vorticity.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2004/07/05

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

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