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PDE実解析研究会
過去の記録 ～04/18｜次回の予定｜今後の予定 04/19～
| 開催情報 | 火曜日 10:30～11:30 数理科学研究科棟(駒場) 056号室 |
|---|---|
| 担当者 | 儀我美一、石毛和弘、三竹大寿、米田剛 |
| セミナーURL | http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/ |
| 目的 | 首都圏の偏微分方程式、実解析の研究をさらに活発にするために本研究会を東大で開催いたします。 偏微分方程式研究者と実解析研究者の討論がより日常的になることを目指しています。 そのため、講演がその分野の概観をもわかるような形になるよう配慮いたします。 また講演者との1対1の討論がしやすいように講演は火曜の午前とし、午後に討論用の場所を用意いたします。 この研究会を通して皆様に気楽に東大を訪問していただければ幸いです。 北海道大学のHPには、第1回(2004年9月29日)～第38回(2008年10月15日)の情報が掲載されております。 |
過去の記録
2008年03月19日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Juergen Saal 氏 (Department of Mathematics and Statistics, University of Konstanz)
Maximal Regularity for Mixed Order Systems
Juergen Saal 氏 (Department of Mathematics and Statistics, University of Konstanz)
Maximal Regularity for Mixed Order Systems
[ 講演概要 ]
In classical boundary value problems the related symbols are homogeneous in space and time. This allows for the application of a standard compactness argument in order to obtain the important maximal regularity. However, quasilinear systems arising e.g. from free boundary problems are in general of mixed order. In other words the related symbols are of intricate structure and in particular highly inhomogeneous. Therefore, the standard compactness argument fails. The purpose of this talk is to introduce the Newton polygon method, which gives a systematic approach to such mixed order systems and to demonstrate its strength by applications to the Stefan problem and a free boundary problem for the Navier-Stokes equations.
In classical boundary value problems the related symbols are homogeneous in space and time. This allows for the application of a standard compactness argument in order to obtain the important maximal regularity. However, quasilinear systems arising e.g. from free boundary problems are in general of mixed order. In other words the related symbols are of intricate structure and in particular highly inhomogeneous. Therefore, the standard compactness argument fails. The purpose of this talk is to introduce the Newton polygon method, which gives a systematic approach to such mixed order systems and to demonstrate its strength by applications to the Stefan problem and a free boundary problem for the Navier-Stokes equations.
2008年01月30日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Oleg Yu. Emanouilov 氏 (Colorado State University)
Carleman estimates for parabolic equations, a Stokes system and the Navier-Stokes equations and applications to the control problem
Oleg Yu. Emanouilov 氏 (Colorado State University)
Carleman estimates for parabolic equations, a Stokes system and the Navier-Stokes equations and applications to the control problem
[ 講演概要 ]
We prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On basis of this estimate we obtain an improved Carleman estimate for the Stokes system and a system of parabolic equations with a parameter which can be viewed as an approximation of the Stokes system. We will discuss the applications to the control problem for these systems.
We prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On basis of this estimate we obtain an improved Carleman estimate for the Stokes system and a system of parabolic equations with a parameter which can be viewed as an approximation of the Stokes system. We will discuss the applications to the control problem for these systems.
2007年09月05日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Reinhard Farwig 氏 (Darmstadt University of Technology)
Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Reinhard Farwig 氏 (Darmstadt University of Technology)
Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion
[ 講演概要 ]
Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.
The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.
[ 参考URL ]Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.
The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2007年07月04日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Lars Diening 氏 (Universitat Freiburg)
The Lipschitz truncation method
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Lars Diening 氏 (Universitat Freiburg)
The Lipschitz truncation method
[ 講演概要 ]
We study the existence of weak solutions to the incompressible $p$-Navier Stokes equations. This system can be used to describe the flow of honey, ketchup, blood, suspensions, polymers, and glaciers. We are interested in small values of $p$, where the method of monotone operators fails. We establish weak solutions by means of the Lipschitz truncation technique, where Sobolev Functions are approximated by Lipschitz functions in a special way. We apply the technique also to electrorheological fluids, where the exponent $p$ depends on the electric field.
[ 参考URL ]We study the existence of weak solutions to the incompressible $p$-Navier Stokes equations. This system can be used to describe the flow of honey, ketchup, blood, suspensions, polymers, and glaciers. We are interested in small values of $p$, where the method of monotone operators fails. We establish weak solutions by means of the Lipschitz truncation technique, where Sobolev Functions are approximated by Lipschitz functions in a special way. We apply the technique also to electrorheological fluids, where the exponent $p$ depends on the electric field.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2007年06月13日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Walter Strauss 氏 (Brown University)
Steady Water Waves with Vorticity
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Walter Strauss 氏 (Brown University)
Steady Water Waves with Vorticity
[ 講演概要 ]
Consider a classical 2D water wave under the influence of gravity with an arbitrary vorticity function. Assume such a wave is traveling at a constant speed over a flat bed. Then there exist many families of such waves of large amplitude. The proof is based on elliptic PDEs, bifurcation and degree theory. I will also exhibit some recent numerical computations. If the vorticity is sufficiently large, the first stagnation point occurs not at the crest (as with irrotational flows) but on the bed directly below the crest. For variable vorticity the first stagnation point can occur in the interior of the fluid.
[ 参考URL ]Consider a classical 2D water wave under the influence of gravity with an arbitrary vorticity function. Assume such a wave is traveling at a constant speed over a flat bed. Then there exist many families of such waves of large amplitude. The proof is based on elliptic PDEs, bifurcation and degree theory. I will also exhibit some recent numerical computations. If the vorticity is sufficiently large, the first stagnation point occurs not at the crest (as with irrotational flows) but on the bed directly below the crest. For variable vorticity the first stagnation point can occur in the interior of the fluid.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2007年03月22日(木)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Matteo Novaga 氏 (Hokkaido University / Universita di Pisa)
A semidiscrete scheme for the Perona Malik equation
http://coe.math.sci.hokudai.ac.jp/
Matteo Novaga 氏 (Hokkaido University / Universita di Pisa)
A semidiscrete scheme for the Perona Malik equation
[ 講演概要 ]
We discuss the convergence of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation. If the initial datum is 1-Lipschitz out of a finite number of jump points, we haracterize the problem satisfied by the limit solution. In the general case, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points.
[ 参考URL ]We discuss the convergence of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation. If the initial datum is 1-Lipschitz out of a finite number of jump points, we haracterize the problem satisfied by the limit solution. In the general case, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points.
http://coe.math.sci.hokudai.ac.jp/
2007年01月17日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Alex Mahalov 氏 (Department of Mathematics and Statistics, Department of Mechanical and Aerospace Engineering, Program in Environmental Fluid Dynamics, Arizona State University )
Fast Singular Oscillating Limits of Hydrodynamic PDEs: application to 3D Euler, Navier-Stokes and MHD equations
http://coe.math.sci.hokudai.ac.jp/
Alex Mahalov 氏 (Department of Mathematics and Statistics, Department of Mechanical and Aerospace Engineering, Program in Environmental Fluid Dynamics, Arizona State University )
Fast Singular Oscillating Limits of Hydrodynamic PDEs: application to 3D Euler, Navier-Stokes and MHD equations
[ 講演概要 ]
Methods of harmonic analysis and dispersive properties are applied
to 3d hydrodynamic equations to obtain long-time and/or global existence results to the Cauchy problem for special classes of 3d initial data. Smoothness assumptions for initial data are the same as in local existence theorems. Techniques for fast singular oscillating limits are used and large and/or infinite time regularity is obtained by bootstrapping from global regularity of the limit equations.
The latter gain regularity from 3d nonlinear cancellation of oscillations.
Applications include Euler, Navier-Stokes, Boussinesq and MHD equations, in infinite, periodic and bounded cylindrical domains.
[ 参考URL ]Methods of harmonic analysis and dispersive properties are applied
to 3d hydrodynamic equations to obtain long-time and/or global existence results to the Cauchy problem for special classes of 3d initial data. Smoothness assumptions for initial data are the same as in local existence theorems. Techniques for fast singular oscillating limits are used and large and/or infinite time regularity is obtained by bootstrapping from global regularity of the limit equations.
The latter gain regularity from 3d nonlinear cancellation of oscillations.
Applications include Euler, Navier-Stokes, Boussinesq and MHD equations, in infinite, periodic and bounded cylindrical domains.
http://coe.math.sci.hokudai.ac.jp/
2006年11月01日(水)
10:30-11:30 数理科学研究科棟(駒場) 128号室
Tan Yongji 氏 (School of Mathematical Science, Fudan University )
A case study in petroleum industry: Mathematical modeling and numerical simulation in spontaneous potential well-logging
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Tan Yongji 氏 (School of Mathematical Science, Fudan University )
A case study in petroleum industry: Mathematical modeling and numerical simulation in spontaneous potential well-logging
[ 講演概要 ]
Spontaneous well-logging is an important technique in petroleum exploitation. The potential field is of strong discontinuity on the interface since the spontaneous potential differences. It causes difficulty in mathematical analysis and numerical computing.
New mathematical model and numerical method is designed to overcome the difficulty and good results is obtained.
[ 参考URL ]Spontaneous well-logging is an important technique in petroleum exploitation. The potential field is of strong discontinuity on the interface since the spontaneous potential differences. It causes difficulty in mathematical analysis and numerical computing.
New mathematical model and numerical method is designed to overcome the difficulty and good results is obtained.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2006年10月30日(月)
16:30-17:30 数理科学研究科棟(駒場) 128号室
Matti Lassas 氏 (Helsinki University of Technology, Institute of Mathematics)
Inverse Problems and Index Formulae for Dirac Operators
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Matti Lassas 氏 (Helsinki University of Technology, Institute of Mathematics)
Inverse Problems and Index Formulae for Dirac Operators
[ 講演概要 ]
We consider a selfadjoin Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M, g)$ with a nonempty boundary.
The operator $D_P$ is specified by a boundary condition $P(u|_{\\partial M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator.
We assume the existence of a chirality operator which decomposes $L2(M, V)$ into two orthogonal subspaces $X_+ \\oplus X_-$.
In the talk we consider the reconstruction of $(M, g)$, $V$, and $D_P$ from the boundary data on $\\partial M$.
The data used is either the Cauchy data, i.e. the restrictions to $\\partial M \\times R_+$ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e. the set of the eigenvalues and the boundary values of the eigenfunctions of $D_P$. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in $M\\times \\C4$, $M \\subset \\R3$. The presented results have been done in collaboration with Yaroslav Kurylev (Loughborough, UK).
[ 参考URL ]We consider a selfadjoin Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M, g)$ with a nonempty boundary.
The operator $D_P$ is specified by a boundary condition $P(u|_{\\partial M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator.
We assume the existence of a chirality operator which decomposes $L2(M, V)$ into two orthogonal subspaces $X_+ \\oplus X_-$.
In the talk we consider the reconstruction of $(M, g)$, $V$, and $D_P$ from the boundary data on $\\partial M$.
The data used is either the Cauchy data, i.e. the restrictions to $\\partial M \\times R_+$ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e. the set of the eigenvalues and the boundary values of the eigenfunctions of $D_P$. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in $M\\times \\C4$, $M \\subset \\R3$. The presented results have been done in collaboration with Yaroslav Kurylev (Loughborough, UK).
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2006年09月27日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Professor Vakhtang Kokilashvili 氏 (A. Razmadze Mathematical Institute, Georgian Academy of Science)
Integral operators in the weighted Lebesgue spaces with a variable exponent
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Professor Vakhtang Kokilashvili 氏 (A. Razmadze Mathematical Institute, Georgian Academy of Science)
Integral operators in the weighted Lebesgue spaces with a variable exponent
[ 講演概要 ]
We present a boundedness criteria of the maximal functions and the singular integral operators defined on Carleson curves in the weighted Lebesgue spaces with a variable exponent. There are also given the weighted estimates for the generalized singular integrals raised in the theory of generalized analytic functions of I.N.Vekua and the weighted Sobolev theorems for potentials on Carleson curves. The weight functions may be of power function type as well as oscillating type. The certain version of a Muckenhoupt-type condition for a variable exponent will be considered.
We also expect to treat two-weight problems for the classical integral operators in the variable Lebesgue spaces and to give some applications of the obtained results to the summability problems of Fourier series in two-weighted setting.
[ 参考URL ]We present a boundedness criteria of the maximal functions and the singular integral operators defined on Carleson curves in the weighted Lebesgue spaces with a variable exponent. There are also given the weighted estimates for the generalized singular integrals raised in the theory of generalized analytic functions of I.N.Vekua and the weighted Sobolev theorems for potentials on Carleson curves. The weight functions may be of power function type as well as oscillating type. The certain version of a Muckenhoupt-type condition for a variable exponent will be considered.
We also expect to treat two-weight problems for the classical integral operators in the variable Lebesgue spaces and to give some applications of the obtained results to the summability problems of Fourier series in two-weighted setting.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2006年07月12日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Piotr Rybka 氏 (Warsaw University)
Analysis of a crystal growth model
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Piotr Rybka 氏 (Warsaw University)
Analysis of a crystal growth model
[ 講演概要 ]
We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.
We are mostly concerned with the shape-persitency problem of the
evolution. The problem is, the Gibbs-Thomson relation is in fact a
driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.
It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow ``too big''.
[ 参考URL ]We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.
We are mostly concerned with the shape-persitency problem of the
evolution. The problem is, the Gibbs-Thomson relation is in fact a
driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.
It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow ``too big''.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2006年06月28日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Jian Zhai 氏 (Zhejiang University)
Uniqueness of Constant Anisotropic Mean Curvature Immersion of Sphere $S^2$ In $\\Bbb E^3$
Jian Zhai 氏 (Zhejiang University)
Uniqueness of Constant Anisotropic Mean Curvature Immersion of Sphere $S^2$ In $\\Bbb E^3$
[ 講演概要 ]
We prove that the constant anisotropic mean curvature immersion of sphere $S^2$ in $\\Bbb E^3$ is unique, provided that the energy density function $\\gamma$ satisfies some reasonable assumptions.
We prove that the constant anisotropic mean curvature immersion of sphere $S^2$ in $\\Bbb E^3$ is unique, provided that the energy density function $\\gamma$ satisfies some reasonable assumptions.
2006年06月14日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Qing-Ming Cheng 氏 (Saga University)
Bounds on eigenvalues of Dirichlet aplacian
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Qing-Ming Cheng 氏 (Saga University)
Bounds on eigenvalues of Dirichlet aplacian
[ 講演概要 ]
In this talk, I shall consider the eigenvalue problem of the Dirichlet Laplacian. I shall mention the Weyl asymptotic formula,
Polya conjecture and its partial solution. Furthermore, I shall talk about Bochner-Kac problem.
For universal inequalities for eigenvalues, I shall consider
conjectures of Payne, Polya and Weinberger and their development. In the final, I shall talk the universal bounds for eigenvalues as main part of my talk, which is my recent joint work with rofessor Yang.
[ 参考URL ]In this talk, I shall consider the eigenvalue problem of the Dirichlet Laplacian. I shall mention the Weyl asymptotic formula,
Polya conjecture and its partial solution. Furthermore, I shall talk about Bochner-Kac problem.
For universal inequalities for eigenvalues, I shall consider
conjectures of Payne, Polya and Weinberger and their development. In the final, I shall talk the universal bounds for eigenvalues as main part of my talk, which is my recent joint work with rofessor Yang.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2006年06月07日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
高坂 良史 氏 (室蘭工業大学)
On phase boundary motion by surface diffusion with triple junction
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/022.html
高坂 良史 氏 (室蘭工業大学)
On phase boundary motion by surface diffusion with triple junction
[ 講演概要 ]
The phase boundary motion by a geometrical evolution law in a bounded domain is studied in this talk. We consider the surface diffusion flow equation, which has the gradient flow structure with respect to $H^{-1}$-inner product and the area-preserving property. This equation was derived by Mullins to model the motion of interfaces in the case that the motion of interfaces is governed purely by mass diffusion within the interfaces. We study the three-phase problem with triple junction in a bounded domain and analyze the stability of the stationary solutions for this problem.
[ 参考URL ]The phase boundary motion by a geometrical evolution law in a bounded domain is studied in this talk. We consider the surface diffusion flow equation, which has the gradient flow structure with respect to $H^{-1}$-inner product and the area-preserving property. This equation was derived by Mullins to model the motion of interfaces in the case that the motion of interfaces is governed purely by mass diffusion within the interfaces. We study the three-phase problem with triple junction in a bounded domain and analyze the stability of the stationary solutions for this problem.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/022.html
2006年01月18日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Juergen Saal 氏 (TU Darmstadt)
Analyticity of the interface of the classical two-phase Stefan problem
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Juergen Saal 氏 (TU Darmstadt)
Analyticity of the interface of the classical two-phase Stefan problem
[ 講演概要 ]
The Stefan problem is a model for phase transitions in liquid-solid systems, as e.g. ice surrounded by water, and accounts for heat diffusion and exchange of latent heat in a homogeneous medium.
The strong formulation of this model corresponds to a free boundary problem involving a parabolic diffusion equation for each phase and a transmission condition prescribed at the interface separating the phases.
We prove that under mild regularity assumptions on the initial data the two-phase classical Stefan problem admits a unique solution that is analytic in space and time.
The result is based on $L_p$ maximal regularity for a linearized problem, which is proved first, and the implicit function theorem.
[ 参考URL ]The Stefan problem is a model for phase transitions in liquid-solid systems, as e.g. ice surrounded by water, and accounts for heat diffusion and exchange of latent heat in a homogeneous medium.
The strong formulation of this model corresponds to a free boundary problem involving a parabolic diffusion equation for each phase and a transmission condition prescribed at the interface separating the phases.
We prove that under mild regularity assumptions on the initial data the two-phase classical Stefan problem admits a unique solution that is analytic in space and time.
The result is based on $L_p$ maximal regularity for a linearized problem, which is proved first, and the implicit function theorem.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2006年01月11日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
伊東 一文 氏 (North Carolina State University) 10:30-11:30
On Fluid Mechanics Formulation of Monge-Kantorovich Mass Transfer Problem
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Oleg Yu. Imanuvilov 氏 (Colorado State University) 11:45-12:45
Local and Global Exact Controllability of Evolution Equations
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
伊東 一文 氏 (North Carolina State University) 10:30-11:30
On Fluid Mechanics Formulation of Monge-Kantorovich Mass Transfer Problem
[ 講演概要 ]
The Monge-Kantorovich mass transfer problem is equivalently formulated as an optimal control problem for the mass transport equation. The equivalency of the two problems is establish using the Lax-Hopf formula and the optimal control theory arguments. Also, it is shown that the optimal solution to the equivalent control problem is given in a gradient form in terms of the potential solution to the Monge-Kantorovich problem. It turns out
that the control formulation is a dual formulation of the Kantrovich distance problem via the Hamilton-Jacobi equations.
[ 参考URL ]The Monge-Kantorovich mass transfer problem is equivalently formulated as an optimal control problem for the mass transport equation. The equivalency of the two problems is establish using the Lax-Hopf formula and the optimal control theory arguments. Also, it is shown that the optimal solution to the equivalent control problem is given in a gradient form in terms of the potential solution to the Monge-Kantorovich problem. It turns out
that the control formulation is a dual formulation of the Kantrovich distance problem via the Hamilton-Jacobi equations.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Oleg Yu. Imanuvilov 氏 (Colorado State University) 11:45-12:45
Local and Global Exact Controllability of Evolution Equations
[ 講演概要 ]
We discuss rcent global and local controlability results for the Navier-Stokes system and Bousinesq system. The control is acting on the part of the boundary or locally distributed over subdomain.
[ 参考URL ]We discuss rcent global and local controlability results for the Navier-Stokes system and Bousinesq system. The control is acting on the part of the boundary or locally distributed over subdomain.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年11月09日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
藤田安啓 氏 (富山大学)
Asymptotic solutions and Aubry sets for Hamilton-Jacobi equations
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
藤田安啓 氏 (富山大学)
Asymptotic solutions and Aubry sets for Hamilton-Jacobi equations
[ 講演概要 ]
In this talk, we consider the asymptotic behavior of the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation $u_t + \\alpha x\\cdot Du + H(Du) =f(x)$ in ${\\rm I}\\!{\\rm R}^N \\times (0,\\infty)$, where $\\alpha$ is a positive constant and $H$ is a convex function on ${\\rm I} \\!{\\rm R}^N$. We show that, under some assumptions, $u(x,t) - ct - v(x)$ converges to $0$ locally uniformly in ${\\rm I}\\!{\\rm R}^N$ as $t \\to \\infty$, where $c$ is a constant and $v$ is a viscosity solution of the Hamilton-Jacobi equation $c + \\alpha x\\cdot Dv + H(Dv) = f(x)$ in ${\\rm I}\\!{\\rm R}^N$. A set in ${\\rm I}\\!{\\rm R}^N$, which is called the {\\it Aubry set}, gives a concrete representation of the viscosity solution $v$. We also discuss convergence rates of this asymptotic behavior. This is a joint work with Professors H. Ishii and P. Loreti.
[ 参考URL ]In this talk, we consider the asymptotic behavior of the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation $u_t + \\alpha x\\cdot Du + H(Du) =f(x)$ in ${\\rm I}\\!{\\rm R}^N \\times (0,\\infty)$, where $\\alpha$ is a positive constant and $H$ is a convex function on ${\\rm I} \\!{\\rm R}^N$. We show that, under some assumptions, $u(x,t) - ct - v(x)$ converges to $0$ locally uniformly in ${\\rm I}\\!{\\rm R}^N$ as $t \\to \\infty$, where $c$ is a constant and $v$ is a viscosity solution of the Hamilton-Jacobi equation $c + \\alpha x\\cdot Dv + H(Dv) = f(x)$ in ${\\rm I}\\!{\\rm R}^N$. A set in ${\\rm I}\\!{\\rm R}^N$, which is called the {\\it Aubry set}, gives a concrete representation of the viscosity solution $v$. We also discuss convergence rates of this asymptotic behavior. This is a joint work with Professors H. Ishii and P. Loreti.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年10月26日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
利根川吉廣 氏 (北海道大学)
On Mullins-Sekerka as singular limit of Cahn-Hilliard, some mathematical progress and open problems
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
利根川吉廣 氏 (北海道大学)
On Mullins-Sekerka as singular limit of Cahn-Hilliard, some mathematical progress and open problems
[ 講演概要 ]
The Cahn-Hilliard equation and its variants have been widely used in materials science community to model coarse graining phenomena in mesoscopic scale. The equation has a parameter corresponding the order of thickness of phase boundaries. When the parameter is close to zero, the phase boundary and the chemical potential field are known to evolve by the so-called Mullins-Sekerka problem. The rigorous justification for the latter statement is known only for short-time so far. I describe some recent progress as well as some difficulties on the long-time case, relateing my recent works and those by M. Roeger and R. Schaetzle.
[ 参考URL ]The Cahn-Hilliard equation and its variants have been widely used in materials science community to model coarse graining phenomena in mesoscopic scale. The equation has a parameter corresponding the order of thickness of phase boundaries. When the parameter is close to zero, the phase boundary and the chemical potential field are known to evolve by the so-called Mullins-Sekerka problem. The rigorous justification for the latter statement is known only for short-time so far. I describe some recent progress as well as some difficulties on the long-time case, relateing my recent works and those by M. Roeger and R. Schaetzle.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年09月28日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Matthias Geissert 氏 (ダルムシュタット工科大学)
The Navier-Stokes flow in the exterior of a rotating obstacle
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Matthias Geissert 氏 (ダルムシュタット工科大学)
The Navier-Stokes flow in the exterior of a rotating obstacle
[ 講演概要 ]
We show the existence of solutions of the Navier-Stokes flow in the exterior of a rotating obstacle. In the first step we transform the Navier-Stokes equations to a problem in a time independent domain. In this talk we present two different change of coordinates to do this. Finally, we discuss the advantages of both approaches and show the local existence and uniqueness of mild and strong $L^p$ solutions.
[ 参考URL ]We show the existence of solutions of the Navier-Stokes flow in the exterior of a rotating obstacle. In the first step we transform the Navier-Stokes equations to a problem in a time independent domain. In this talk we present two different change of coordinates to do this. Finally, we discuss the advantages of both approaches and show the local existence and uniqueness of mild and strong $L^p$ solutions.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年07月20日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
相川弘明 氏 (島根大学)
Equivalence between the boundary Harnack principle and the Carleson estimate
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
相川弘明 氏 (島根大学)
Equivalence between the boundary Harnack principle and the Carleson estimate
[ 講演概要 ]
Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this talk is to show that the boundary Harnack principle and the Carleson estimate are equivalent for arbitrary domains.
[ 参考URL ]Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this talk is to show that the boundary Harnack principle and the Carleson estimate are equivalent for arbitrary domains.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年07月13日(水)
10:30-11:30 数理科学研究科棟(駒場) 122号室
Yonggeun Cho 氏 (北海道大学)
On classical solutions of the compressible Navier-Stokes equation with nonnegative density
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Yonggeun Cho 氏 (北海道大学)
On classical solutions of the compressible Navier-Stokes equation with nonnegative density
[ 講演概要 ]
In this talk, we discuss a recent progress on the regularity of solution of compressible Navier-Stokes equations with nonnegative density. The nonnegativity of density cauases a problem in using the usual parabolicity of momentum equations and hence in general makes it hard to gain a regularity of solution. To overcome the difficulty, we develop a natural compatibility condition. Then observing a smoothing effect for positive time, we obtain classical solutions of the compressible Navier-Stokes equations.
[ 参考URL ]In this talk, we discuss a recent progress on the regularity of solution of compressible Navier-Stokes equations with nonnegative density. The nonnegativity of density cauases a problem in using the usual parabolicity of momentum equations and hence in general makes it hard to gain a regularity of solution. To overcome the difficulty, we develop a natural compatibility condition. Then observing a smoothing effect for positive time, we obtain classical solutions of the compressible Navier-Stokes equations.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年06月15日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
中井英一 氏 (大阪教育大学)
Singular and fractional integral operators on function spaces related to Morrey spaces
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
中井英一 氏 (大阪教育大学)
Singular and fractional integral operators on function spaces related to Morrey spaces
[ 講演概要 ]
It is known that the Hardy-Littlewood maximal operator, singular integral operators and fractional integral operators are bounded on L^p spaces and Morrey spaces. We extend the boundedness to generalized Morrey spaces, Orlicz-Morrey spaces, etc.
[ 参考URL ]It is known that the Hardy-Littlewood maximal operator, singular integral operators and fractional integral operators are bounded on L^p spaces and Morrey spaces. We extend the boundedness to generalized Morrey spaces, Orlicz-Morrey spaces, etc.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年06月08日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
宮地晶彦 氏 (東京女子大学)
Weighted Hardy spaces on an interval and Jacobi series
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
宮地晶彦 氏 (東京女子大学)
Weighted Hardy spaces on an interval and Jacobi series
[ 講演概要 ]
For the classical Hardy class consisting of functions holomorphic in the unit disc, the Burkholder-Gundy-Silverstein theorem gives a characterization of the class in terms of certain maximal functions. We give a variant of this theorem related to weighted Hardy spaces on the interval(0,$\\pi$) and generalized holomorphic functions efined through ultraspherical (Gegenbauer) polynomials.
[ 参考URL ]For the classical Hardy class consisting of functions holomorphic in the unit disc, the Burkholder-Gundy-Silverstein theorem gives a characterization of the class in terms of certain maximal functions. We give a variant of this theorem related to weighted Hardy spaces on the interval(0,$\\pi$) and generalized holomorphic functions efined through ultraspherical (Gegenbauer) polynomials.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年06月01日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Jong-Shenq-Guo 氏 (国立台湾師範大学)
Annihilation of wave fronts of a reaction-diffusion equation
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Jong-Shenq-Guo 氏 (国立台湾師範大学)
Annihilation of wave fronts of a reaction-diffusion equation
[ 講演概要 ]
We shall present some recent results on the existence and uniqueness of 2-front entire solutions of a reaction-diffusion equation. These entire solutions behave as two opposite wave fronts approaching each other from both sides of the x-axis and then annihilating in a finite time.
[ 参考URL ]We shall present some recent results on the existence and uniqueness of 2-front entire solutions of a reaction-diffusion equation. These entire solutions behave as two opposite wave fronts approaching each other from both sides of the x-axis and then annihilating in a finite time.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
2005年05月25日(水)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Vincenzo Vespri 氏 (Dipartimento di Matematica Ulisse Dini Viale Morgagni) 10:30-11:30
Some regularity results for Stefan equation
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Paolo Marcellini 氏 (Università degli Studi di Firenze) 11:45-12:45
Nonlinear elliptic systems with general growth
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Vincenzo Vespri 氏 (Dipartimento di Matematica Ulisse Dini Viale Morgagni) 10:30-11:30
Some regularity results for Stefan equation
[ 講演概要 ]
We consider the eqation $\\beta (u)_t = A(u)$ where $A$ is an elliptic operator and $\\beta$ is a maximal graph. Under suitable hypothesis on $\\beta$ and $A$ we prove the continuity of local solutions extendind some techniques introduced in the 80's.
[ 参考URL ]We consider the eqation $\\beta (u)_t = A(u)$ where $A$ is an elliptic operator and $\\beta$ is a maximal graph. Under suitable hypothesis on $\\beta$ and $A$ we prove the continuity of local solutions extendind some techniques introduced in the 80's.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Paolo Marcellini 氏 (Università degli Studi di Firenze) 11:45-12:45
Nonlinear elliptic systems with general growth
[ 講演概要 ]
We prove \\textit{local Lipschitz-continuity} and, as a consequence, $C^{k}$%\\textit{\\ and }$C^{\\infty }$\\textit{\\ regularity} of \\textit{weak} solutions $u$ for a class of \\textit{nonlinear elliptic differential systems} of the form $\\sum_{i=1}^{n}\\frac{\\partial }{\\partial x_{i}}a_{i}^{\\alpha}(Du)=0,\\;\\alpha =1,2\\dots m$. The \\textit{growth conditions} on the dependence of functions $a_{i}^{\\alpha }(\\cdot )$ on the gradient $Du$ are so mild to allow us to embrace growths between the \\textit{linear} and the \\textit{exponential} cases, and they are more general than those known in the literature.
[ 参考URL ]We prove \\textit{local Lipschitz-continuity} and, as a consequence, $C^{k}$%\\textit{\\ and }$C^{\\infty }$\\textit{\\ regularity} of \\textit{weak} solutions $u$ for a class of \\textit{nonlinear elliptic differential systems} of the form $\\sum_{i=1}^{n}\\frac{\\partial }{\\partial x_{i}}a_{i}^{\\alpha}(Du)=0,\\;\\alpha =1,2\\dots m$. The \\textit{growth conditions} on the dependence of functions $a_{i}^{\\alpha }(\\cdot )$ on the gradient $Du$ are so mild to allow us to embrace growths between the \\textit{linear} and the \\textit{exponential} cases, and they are more general than those known in the literature.
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
