## Applied Analysis

Seminar information archive ～09/27｜Next seminar｜Future seminars 09/28～

Date, time & place | Thursday 16:00 - 17:30 002Room #002 (Graduate School of Math. Sci. Bldg.) |
---|

**Seminar information archive**

### 2012/01/19

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Undercompressible shocks and moving phase boundaries

(ENGLISH)

**Philippe G. LeFloch**(Univ. Paris 6 / CNRS)Undercompressible shocks and moving phase boundaries

(ENGLISH)

[ Abstract ]

I will present a study of traveling wave solutions to third-order, diffusive-dispersive equations, which arise in the modeling of complex fluid flows and represent regularization-sensitive wave patterns, especially undercompressive shock waves and moving phase boundaries. The qualitative properties of these (possibly oscillatory) traveling waves are well-understood in terms of the so-called kinetic relation, and this has led to a new theory of (nonclassical) solutions to nonlinear hyperbolic systems. Relevant papers are available at the link: www.philippelefloch.org.

I will present a study of traveling wave solutions to third-order, diffusive-dispersive equations, which arise in the modeling of complex fluid flows and represent regularization-sensitive wave patterns, especially undercompressive shock waves and moving phase boundaries. The qualitative properties of these (possibly oscillatory) traveling waves are well-understood in terms of the so-called kinetic relation, and this has led to a new theory of (nonclassical) solutions to nonlinear hyperbolic systems. Relevant papers are available at the link: www.philippelefloch.org.

### 2011/11/10

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

Mathematical Modeling of Cellular Electrodiffusion and Osmosis (JAPANESE)

**Yoichiro Mori**(University of Minnesota )Mathematical Modeling of Cellular Electrodiffusion and Osmosis (JAPANESE)

### 2011/11/10

16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Schoenflies spheres in Sturm attractors (ENGLISH)

**Bernold Fiedler**(Free University of Berlin)Schoenflies spheres in Sturm attractors (ENGLISH)

[ Abstract ]

In gradient systems on compact manifolds the boundary of the unstable manifold of an equilibrium need not be homeomorphic to a sphere, or to any compact manifold.

For scalar parabolic equations in one space dimension, however, we can exlude complications like Reidemeister torsion and the Alexander horned sphere. Instead the boundary is a Schoenflies embedded sphere. This is due to Sturm nodal properties related to the Matano lap number.

In gradient systems on compact manifolds the boundary of the unstable manifold of an equilibrium need not be homeomorphic to a sphere, or to any compact manifold.

For scalar parabolic equations in one space dimension, however, we can exlude complications like Reidemeister torsion and the Alexander horned sphere. Instead the boundary is a Schoenflies embedded sphere. This is due to Sturm nodal properties related to the Matano lap number.

### 2011/06/30

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

On the macroscopic models for type-II superconductivity in 3D (JAPANESE)

**Yohei Kashima**(Graduate School of Mathematical Sciences, The University of Tokyo)On the macroscopic models for type-II superconductivity in 3D (JAPANESE)

### 2011/06/09

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Spectral representations and scattering for Schr\\"odinger operators on star graphs (JAPANESE)

**Kiyoshi Mochizuki**(Tokyo Metropolitan University, Emeritus Professor)Spectral representations and scattering for Schr\\"odinger operators on star graphs (JAPANESE)

[ Abstract ]

We consider Schr\\"odinger operators defined on star graphs with Kirchhoff boundary conditions. Under suitable decay conditions on the potential, we construct a complete set of eigenfunctions to obtain spectral representations of the operator. The results are applied to give a time dependent formulation of the scattering theory. Also we use the spectral representation to determine an integral equation of Marchenko which is fundamental to enter into the inverse scattering problems.

We consider Schr\\"odinger operators defined on star graphs with Kirchhoff boundary conditions. Under suitable decay conditions on the potential, we construct a complete set of eigenfunctions to obtain spectral representations of the operator. The results are applied to give a time dependent formulation of the scattering theory. Also we use the spectral representation to determine an integral equation of Marchenko which is fundamental to enter into the inverse scattering problems.

### 2011/05/26

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Obstacle problem of Navier-Stokes equations in thermohydraulics (JAPANESE)

**Takeshi Fukao**(Kyoto University of Education)Obstacle problem of Navier-Stokes equations in thermohydraulics (JAPANESE)

[ Abstract ]

In this talk, we consider the well-posedness of a variational inequality for the Navier-Stokes equations in 2 or 3 space dimension with time dependent constraints. This problem is motivated by an initial-boundary value problem for a thermohydraulics model. The velocity field is constrained by a prescribed function,

depending on the space and time variables, so this is called the obstacle problem. The abstract theory of nonlinear evolution equations governed by subdifferentials of time dependent convex functionals is quite useful for showing their well-posedness. In their mathematical treatment one of the key is to specify the class of time-dependence of convex functionals. We shall discuss the existence and uniqueness questions for Navier-Stokes variational inequalities, in which a bounded constraint is imposed on the velocity field, in higher space dimensions. Especially, the uniqueness of a solution is due to the advantage of the prescribed constraint to the velocity fields.

In this talk, we consider the well-posedness of a variational inequality for the Navier-Stokes equations in 2 or 3 space dimension with time dependent constraints. This problem is motivated by an initial-boundary value problem for a thermohydraulics model. The velocity field is constrained by a prescribed function,

depending on the space and time variables, so this is called the obstacle problem. The abstract theory of nonlinear evolution equations governed by subdifferentials of time dependent convex functionals is quite useful for showing their well-posedness. In their mathematical treatment one of the key is to specify the class of time-dependence of convex functionals. We shall discuss the existence and uniqueness questions for Navier-Stokes variational inequalities, in which a bounded constraint is imposed on the velocity field, in higher space dimensions. Especially, the uniqueness of a solution is due to the advantage of the prescribed constraint to the velocity fields.

### 2011/05/19

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Wave front set defined by wave packet transform and its application (JAPANESE)

**Shingo Ito**(Tokyo University of Science)Wave front set defined by wave packet transform and its application (JAPANESE)

### 2011/04/14

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Homoclinic and heteroclinic orbits for a semilinear parabolic equation (ENGLISH)

**Marek FILA**(Comenius University (Slovakia))Homoclinic and heteroclinic orbits for a semilinear parabolic equation (ENGLISH)

[ Abstract ]

We study the existence of connecting orbits for the Fujita equation

u_t=\\Delta u+u^p

with a critical or supercritical exponent $p$. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.

We study the existence of connecting orbits for the Fujita equation

u_t=\\Delta u+u^p

with a critical or supercritical exponent $p$. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.

### 2011/02/24

16:00-18:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections (ENGLISH)

Some open problems in PDE control (ENGLISH)

**Arnaud Ducrot**(University of Bordeaux 2) 16:00-17:00Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections (ENGLISH)

[ Abstract ]

This work is devoted to the study of travelling wave solutions for some size structured model in population dynamics. The population under consideration is also spatially structured and has a nonlocal spatial reproduction. This phenomenon may model the invasion of plants within some empty landscape. Since the corresponding unspatially structured size structured models may induce oscillating dynamics due to Hopf bifurcations, the aim of this work is to prove the existence of point to sustained oscillating solution travelling waves for the spatially structured problem. From a biological viewpoint, such solutions represent the spatial invasion of some species with spatio-temporal patterns at the place where the population is established.

This work is devoted to the study of travelling wave solutions for some size structured model in population dynamics. The population under consideration is also spatially structured and has a nonlocal spatial reproduction. This phenomenon may model the invasion of plants within some empty landscape. Since the corresponding unspatially structured size structured models may induce oscillating dynamics due to Hopf bifurcations, the aim of this work is to prove the existence of point to sustained oscillating solution travelling waves for the spatially structured problem. From a biological viewpoint, such solutions represent the spatial invasion of some species with spatio-temporal patterns at the place where the population is established.

**Enrique Zuazua**(Basque Center for Applied Mathematics) 17:10-18:10Some open problems in PDE control (ENGLISH)

[ Abstract ]

The field of PDE control has experienced a great progress in the last decades, developing new theories and tools that have also influenced other disciplines as Inverse Problem and Optimal Design Theories and Numerical Analysis. PDE control arises in most applications ranging from classical problems in fluid mechanics or structural engineering to modern molecular design experiments.

From a mathematical viewpoint the problems arising in this field are extremely challenging since the existing theory of existence and uniqueness of solutions and the corresponding numerical schemes is insufficient when addressing realistic control problems. Indeed, an efficient controller requires of an in depth understanding of how solutions depend on the various parameters of the problem (shape of the domain, time of control, coefficients of the equation, location

of the controller, nonlinearity in the equation,...)

In this lecture we shall briefly discuss some important advances and some challenging open problems. All of them shear some features. In particular they are simple to state and very likely hard to solve. We shall discuss in particular:

1.- Semilinear wave equations and their control properties.

2.- Microlocal optimal design of wave processes

3.- Sharp observability estimates for heat processes.

4.- Robustness on the control of finite-dimensional systems.

5.- Unique continuation for discrete elliptic models

6.- Control of Kolmogorov equations and other hypoelliptic models.

The field of PDE control has experienced a great progress in the last decades, developing new theories and tools that have also influenced other disciplines as Inverse Problem and Optimal Design Theories and Numerical Analysis. PDE control arises in most applications ranging from classical problems in fluid mechanics or structural engineering to modern molecular design experiments.

From a mathematical viewpoint the problems arising in this field are extremely challenging since the existing theory of existence and uniqueness of solutions and the corresponding numerical schemes is insufficient when addressing realistic control problems. Indeed, an efficient controller requires of an in depth understanding of how solutions depend on the various parameters of the problem (shape of the domain, time of control, coefficients of the equation, location

of the controller, nonlinearity in the equation,...)

In this lecture we shall briefly discuss some important advances and some challenging open problems. All of them shear some features. In particular they are simple to state and very likely hard to solve. We shall discuss in particular:

1.- Semilinear wave equations and their control properties.

2.- Microlocal optimal design of wave processes

3.- Sharp observability estimates for heat processes.

4.- Robustness on the control of finite-dimensional systems.

5.- Unique continuation for discrete elliptic models

6.- Control of Kolmogorov equations and other hypoelliptic models.

### 2011/02/17

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Study of propagation phenomena in some reaction-diffusion systems (ENGLISH)

**Thomas Giletti**(University of Paul Cezanne (Marseilles))Study of propagation phenomena in some reaction-diffusion systems (ENGLISH)

[ Abstract ]

This talk deals with the existence and qualitative properties of traveling wave solutions of a nonlinear reaction-diffusion system with losses inside the domain, which has numerous applications in various fields ranging from chemical and biological contexts to combusion. Under some KPP type hypotheses, the existence of a continuum of admissible speeds for traveling waves can be shown, thus generalizing the single equation case. Lastly, by considering losses concentrated near the edge of the domain, those results can be compared with those of the boundary losses case.

This talk deals with the existence and qualitative properties of traveling wave solutions of a nonlinear reaction-diffusion system with losses inside the domain, which has numerous applications in various fields ranging from chemical and biological contexts to combusion. Under some KPP type hypotheses, the existence of a continuum of admissible speeds for traveling waves can be shown, thus generalizing the single equation case. Lastly, by considering losses concentrated near the edge of the domain, those results can be compared with those of the boundary losses case.

### 2011/02/10

15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Control and nonlinearity (ENGLISH)

**Jean-Michel Coron**(University of Paris 6)Control and nonlinearity (ENGLISH)

[ Abstract ]

We present methods to study the controllability and the stabilizability of nonlinear control systems. The emphasis is put on specific phenomena due to the nonlinearities. In particular we study cases where the nonlinearities are essential for the controllability or the stabilizability.

We illustrate these methods on control systems modeled by ordinary differential equations or partial differential equations (Euler and Navier-Stokes equations of incompressible fluids, shallow water equations, Korteweg de Vries equations).

We present methods to study the controllability and the stabilizability of nonlinear control systems. The emphasis is put on specific phenomena due to the nonlinearities. In particular we study cases where the nonlinearities are essential for the controllability or the stabilizability.

We illustrate these methods on control systems modeled by ordinary differential equations or partial differential equations (Euler and Navier-Stokes equations of incompressible fluids, shallow water equations, Korteweg de Vries equations).

### 2011/01/27

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Attraction at infinity: Constructing non-compact global attractors in the slowly non-dissipative realm (ENGLISH)

**Nitsan Ben-Gal**(The Weizmann Institute of Science)Attraction at infinity: Constructing non-compact global attractors in the slowly non-dissipative realm (ENGLISH)

[ Abstract ]

One of the primary tools for understanding the much-studied realm of reaction-diffusion equations is the global attractor, which provides us with a qualitative understanding of the governing behaviors of solutions to the equation in question. Nevertheless, the classic global attractor for such systems is defined to be compact, and thus attractor theory has previously excluded such analysis from being applied to non-dissipative reaction-diffusion equations.

In this talk I will present recent results in which I developed a non-compact analogue to the classical global attractor, and will discuss the methods derived in order to obtain a full decomposition of the non-compact global attractor for a slowly non-dissipative reaction-diffusion equation. In particular, attention will be paid to the nodal property techniques and reduction methods which form a critical underpinning of asymptotics research in both dissipative and non-dissipative evolutionary equations. I will discuss the concepts of the ‘completed inertial manifold’ and ‘non-compact global attractor’, and show how these in particular allow us to produce equivalent results for a class of slowly non-dissipative equations as have been achieved for dissipative equations. Additionally, I will address the behavior of solutions to slowly non-dissipative equations approaching and at infinity, the realm which presents both the challenges and rewards of removing the necessity of dissipativity.

One of the primary tools for understanding the much-studied realm of reaction-diffusion equations is the global attractor, which provides us with a qualitative understanding of the governing behaviors of solutions to the equation in question. Nevertheless, the classic global attractor for such systems is defined to be compact, and thus attractor theory has previously excluded such analysis from being applied to non-dissipative reaction-diffusion equations.

In this talk I will present recent results in which I developed a non-compact analogue to the classical global attractor, and will discuss the methods derived in order to obtain a full decomposition of the non-compact global attractor for a slowly non-dissipative reaction-diffusion equation. In particular, attention will be paid to the nodal property techniques and reduction methods which form a critical underpinning of asymptotics research in both dissipative and non-dissipative evolutionary equations. I will discuss the concepts of the ‘completed inertial manifold’ and ‘non-compact global attractor’, and show how these in particular allow us to produce equivalent results for a class of slowly non-dissipative equations as have been achieved for dissipative equations. Additionally, I will address the behavior of solutions to slowly non-dissipative equations approaching and at infinity, the realm which presents both the challenges and rewards of removing the necessity of dissipativity.

### 2010/07/08

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Effect of nonlinearity on the steady motion of a twinning dislocation (ENGLISH)

**Anna Vainchtein**(University of Pittsburgh, Department of Mathematics)Effect of nonlinearity on the steady motion of a twinning dislocation (ENGLISH)

[ Abstract ]

We consider the steady motion of a twinning dislocation in a Frenkel-Kontorova lattice with a double-well substrate potential that has a non-degenerate spinodal region. Semi-analytical traveling wave solutions are constructed for the piecewise quadratic potential, and their stability and further effects of nonlinearity are investigated numerically. We show that the width of the spinodal region and the nonlinearity of the potential have a significant effect on the dislocation kinetics, resulting in stable steady motion in some low-velocity intervals and lower propagation stress. We also conjecture that a stable steady propagation must correspond to an increasing portion of the kinetic relation between the applied stress and dislocation velocity.

We consider the steady motion of a twinning dislocation in a Frenkel-Kontorova lattice with a double-well substrate potential that has a non-degenerate spinodal region. Semi-analytical traveling wave solutions are constructed for the piecewise quadratic potential, and their stability and further effects of nonlinearity are investigated numerically. We show that the width of the spinodal region and the nonlinearity of the potential have a significant effect on the dislocation kinetics, resulting in stable steady motion in some low-velocity intervals and lower propagation stress. We also conjecture that a stable steady propagation must correspond to an increasing portion of the kinetic relation between the applied stress and dislocation velocity.

### 2010/06/24

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Reaction-diffusion approximation to nonlinear diffusion problems (JAPANESE)

**Hideki Murakawa**(University of Toyama)Reaction-diffusion approximation to nonlinear diffusion problems (JAPANESE)

### 2010/06/10

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Hydrodynamic limit of microscopic particle systems to conservation laws to fluid models

**Christian Klingenberg**(Wuerzburg 大学 )Hydrodynamic limit of microscopic particle systems to conservation laws to fluid models

[ Abstract ]

In this talk we discuss the hydrodynamic limit of a microscopic description of a fluid to its macroscopic PDE description.

In the first part we consider flow through porous media, i.e. the macroscopic description is a scalar conservation law. Here the new feature is that we allow sudden changes in porosity and thereby the flux may have discontinuities in space. Microscopically this is described through an interacting particle system having only one conserved quantity, namely the total mass. Macroscopically this gives rise to a scalar conservation laws with space dependent flux functions

u_t + f(u, x)_x = 0 .

We are able to derive the PDE together with an entropy condition as a hydrodynamic limit from a microscopic interacting particle system.

In the second part we consider a Hamiltonian system with boundary conditions. Microscopically this is described through a system of coupled oscillators. Macroscopically this will lead to a system of conservation laws, namely the p-system. The proof of the hydrodynamic limit is restricted to smooth solutions. The new feature is that we can derive this with boundary conditions.

In this talk we discuss the hydrodynamic limit of a microscopic description of a fluid to its macroscopic PDE description.

In the first part we consider flow through porous media, i.e. the macroscopic description is a scalar conservation law. Here the new feature is that we allow sudden changes in porosity and thereby the flux may have discontinuities in space. Microscopically this is described through an interacting particle system having only one conserved quantity, namely the total mass. Macroscopically this gives rise to a scalar conservation laws with space dependent flux functions

u_t + f(u, x)_x = 0 .

We are able to derive the PDE together with an entropy condition as a hydrodynamic limit from a microscopic interacting particle system.

In the second part we consider a Hamiltonian system with boundary conditions. Microscopically this is described through a system of coupled oscillators. Macroscopically this will lead to a system of conservation laws, namely the p-system. The proof of the hydrodynamic limit is restricted to smooth solutions. The new feature is that we can derive this with boundary conditions.

### 2010/04/22

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Deterministic and stochastic modelling of catalytic surface processes (ENGLISH)

**Jens Starke**(Technical University of Denmark)Deterministic and stochastic modelling of catalytic surface processes (ENGLISH)

[ Abstract ]

Three levels of modelling, the microscopic, the mesoscopic and the macroscopic level are discussed for the CO oxidation on low-index platinum single crystal surfaces. The introduced models on the microscopic and mesoscopic level are stochastic while the model on the macroscopic level is deterministic. The macroscopic description can be derived rigorously for low pressure conditions as limit of the stochastic many particle model for large particle numbers. This is in correspondence with the successful description of experiments under low pressure conditions by deterministic reaction-diffusion equations while for intermediate pressures phenomena of stochastic origin can be observed in experiments. The introduced models include a new approach for the platinum phase transition which allows for a unification of existing models for Pt(100) and Pt(110).

The rich nonlinear dynamical behaviour of the macroscopic reaction kinetics is investigated and shows good agreement with low pressure experiments. Furthermore, for intermediate pressures, noise-induced pattern formation, so-called raindrop patterns which are not captured by earlier models, can be reproduced and are shown in simulations.

This is joint work with M. Eiswirth, H. Rotermund, G. Ertl,

Frith Haber Institut, Berlin, K. Oelschlaeger, University of

Heidelberg and C. Reichert, INSA, Lyon.

Three levels of modelling, the microscopic, the mesoscopic and the macroscopic level are discussed for the CO oxidation on low-index platinum single crystal surfaces. The introduced models on the microscopic and mesoscopic level are stochastic while the model on the macroscopic level is deterministic. The macroscopic description can be derived rigorously for low pressure conditions as limit of the stochastic many particle model for large particle numbers. This is in correspondence with the successful description of experiments under low pressure conditions by deterministic reaction-diffusion equations while for intermediate pressures phenomena of stochastic origin can be observed in experiments. The introduced models include a new approach for the platinum phase transition which allows for a unification of existing models for Pt(100) and Pt(110).

The rich nonlinear dynamical behaviour of the macroscopic reaction kinetics is investigated and shows good agreement with low pressure experiments. Furthermore, for intermediate pressures, noise-induced pattern formation, so-called raindrop patterns which are not captured by earlier models, can be reproduced and are shown in simulations.

This is joint work with M. Eiswirth, H. Rotermund, G. Ertl,

Frith Haber Institut, Berlin, K. Oelschlaeger, University of

Heidelberg and C. Reichert, INSA, Lyon.

### 2010/04/15

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Long-time behaviour of solutions of a forward-backward parabolic equation

**Alberto Tesei**(University of Rome 1)Long-time behaviour of solutions of a forward-backward parabolic equation

[ Abstract ]

We discuss some recent results concerning the asymptotic behaviour of entropy measure-valued solutions for a class of ill-posed forward-backward parabolic equations, which arise in the theory of phase transitions.

We discuss some recent results concerning the asymptotic behaviour of entropy measure-valued solutions for a class of ill-posed forward-backward parabolic equations, which arise in the theory of phase transitions.

### 2010/02/18

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Homogenization limit of a parabolic equation with nonlinear boundary conditions

**Bendong LOU**(同済大学)Homogenization limit of a parabolic equation with nonlinear boundary conditions

[ Abstract ]

We consider a quasilinear parabolic equation with the following nonlinear Neumann boundary condition:

"the slope of the solution on the boundary is a function $g$ of the value of the solution". Here $g$ takes values near its supremum with the frequency of $\\epsilon$. We show that the homogenization limit of the solution, as $\\epsilon$ tends to 0, is the solution satisfying the linear Neumann boundary condition: "the slope of the solution on the boundary is the supremum of $g$".

We consider a quasilinear parabolic equation with the following nonlinear Neumann boundary condition:

"the slope of the solution on the boundary is a function $g$ of the value of the solution". Here $g$ takes values near its supremum with the frequency of $\\epsilon$. We show that the homogenization limit of the solution, as $\\epsilon$ tends to 0, is the solution satisfying the linear Neumann boundary condition: "the slope of the solution on the boundary is the supremum of $g$".

### 2010/01/28

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

相転移を伴う非圧縮性2相流の線形化問題について

**清水扇丈**(静岡大学理学部)相転移を伴う非圧縮性2相流の線形化問題について

[ Abstract ]

氷が常圧で0度以上になると水になるなどの相転移を伴う非圧縮性2相流に対し,質量保存則, 運動量保存則, エネルギー保存則を界面を含む系全体に適用し, 線形化した方程式系について考察する. 本講演では, 線形化方程式系のL_p-L_q 最大正則性定理について述べる.

密度が異なる場合は, 法線方向の高さ関数は表面張力つき2相Stokes問題の高さ関数と同じ正則性をもち, 系は流速が支配するのに対し,密度が等しい場合は, Gibbs-Thomson補正された表面張力つき2相Stefan問題の高さ関数と同じ正則性をもち, 系は温度が支配する.

氷が常圧で0度以上になると水になるなどの相転移を伴う非圧縮性2相流に対し,質量保存則, 運動量保存則, エネルギー保存則を界面を含む系全体に適用し, 線形化した方程式系について考察する. 本講演では, 線形化方程式系のL_p-L_q 最大正則性定理について述べる.

密度が異なる場合は, 法線方向の高さ関数は表面張力つき2相Stokes問題の高さ関数と同じ正則性をもち, 系は流速が支配するのに対し,密度が等しい場合は, Gibbs-Thomson補正された表面張力つき2相Stefan問題の高さ関数と同じ正則性をもち, 系は温度が支配する.

### 2010/01/21

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation

**Danielle Hilhorst**(パリ南大学 / CNRS)A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation

[ Abstract ]

We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.

We propose a finite volume method on general meshes for degenerate parabolic convection-reaction-diffusion equations. Such equations arise for instance in the modeling of contaminant transport in groundwater. After giving a convergence proof, we present the results of numerical tests.

### 2009/12/17

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A Liouville theorem for a semilinear heat equation with no gradient structure

**Hatem Zaag**(CNRS / パリ北大学)A Liouville theorem for a semilinear heat equation with no gradient structure

[ Abstract ]

We prove a Liouville Theorem for entire solutions of a vector

valued semilinear heat equation with no gradient structure. Classical tools such as the maximum principle or energy techniques break down and have to be replaced by a new approach. These tools involve a very good understanding of the dynamical system formulation of the equation in the selfsimilar setting. Using the Liouville Theorem, we derive uniform estimates for blow-up solutions of the same equation.

We prove a Liouville Theorem for entire solutions of a vector

valued semilinear heat equation with no gradient structure. Classical tools such as the maximum principle or energy techniques break down and have to be replaced by a new approach. These tools involve a very good understanding of the dynamical system formulation of the equation in the selfsimilar setting. Using the Liouville Theorem, we derive uniform estimates for blow-up solutions of the same equation.

### 2009/11/26

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

L^p 粘性解の弱ハルナック不等式の最近の進展

**小池 茂昭**(埼玉大学・理学部数学科)L^p 粘性解の弱ハルナック不等式の最近の進展

[ Abstract ]

Caffarelli による粘性解の regularity 研究 (1989 年) を基に, 1996 年に Caffarelli- Crandall-Kocan-Swiech によって L^p 粘性解の概念が導入された. L^p 粘性解とは, 通 常の粘性解理論では扱えなかった, 非有界非斉次項を持つ (非発散型) 偏微分方程 式にも適用可能な弱解である.

しかしながら, 係数に関しては有界係数しか研究されていなかった. その後, Swiech との共同研究により, 係数が非有界だが適当なべき乗可積分性を仮定して Aleksandrov-Bakelman-Pucci 型の最大値原理を導くことが可能になった.

本講演では, 非有界係数・非斉事項を持った, 完全非線形 2 階一様楕円型方程式 の L^p 粘性解の弱ハルナック不等式に関する最近のSwiech との共同研究の結果を紹 介する.

Caffarelli による粘性解の regularity 研究 (1989 年) を基に, 1996 年に Caffarelli- Crandall-Kocan-Swiech によって L^p 粘性解の概念が導入された. L^p 粘性解とは, 通 常の粘性解理論では扱えなかった, 非有界非斉次項を持つ (非発散型) 偏微分方程 式にも適用可能な弱解である.

しかしながら, 係数に関しては有界係数しか研究されていなかった. その後, Swiech との共同研究により, 係数が非有界だが適当なべき乗可積分性を仮定して Aleksandrov-Bakelman-Pucci 型の最大値原理を導くことが可能になった.

本講演では, 非有界係数・非斉事項を持った, 完全非線形 2 階一様楕円型方程式 の L^p 粘性解の弱ハルナック不等式に関する最近のSwiech との共同研究の結果を紹 介する.

### 2009/11/05

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A Mathematical Aspect of the One-Dimensional Keller and Rubinow Model for Liesegang Bands

**大西 勇**(広島大学大学院理学研究科)A Mathematical Aspect of the One-Dimensional Keller and Rubinow Model for Liesegang Bands

[ Abstract ]

In 1896, colloid-chemist R.E. Liesegang [4] observed strikingly

regular patterns in precipitation-reaction processes, which are referred to as Liesegang bands or rings, according to their shape. In this talk I introduce an attempt to understand from a mathematical viewpoint the experiments in which regularized structures with spatially distinct bands of precipitated material are exhibited, with clearly visible scaling properties. This study is a result [1] of a collaboration with Professors D. Hilhorst, R. van der Hout, and M. Mimura.

References:

[1] Hilhorst, D., van der Hout, R., Mimura, M., and Ohnishi, I.: A Mathematical Study of the One-Dimensional Keller and Rubinow Model for Liesegang Bands. J. Stat Phys 135: 107-132 (2009)

[2] Kai, S., Muller, S.C.: Spatial and temporal macroscopic structures in chemical reaction system: precipitation patterns and interfacial motion. Sci. Form 1, 8-38 (1985)

[3] Keller, J.B., Rubinow, S.I.: Recurrent precipitation and Liesegang rings. J. Chem. Phys. 74, 5000-5007 (1981)

[4] Liesegang, R.E.: Chemische Fernwirkung. Photo. Archiv 800, 305-309 (1896)

[5] Mimura, M., Ohnishi, I., Ueyama, D.: A mathematical aspect of Liesegang phenomena in two space dimensions. Res. Rep. Res. Inst. Math. Sci. 1499, 185-201 (2006)

[6] Ohnishi, I.,Mimura, M.: A mathematical aspect of Liesegang phenomena. In: Proceedings of Equadiff-11, pp. 343-352 (2005).

In 1896, colloid-chemist R.E. Liesegang [4] observed strikingly

regular patterns in precipitation-reaction processes, which are referred to as Liesegang bands or rings, according to their shape. In this talk I introduce an attempt to understand from a mathematical viewpoint the experiments in which regularized structures with spatially distinct bands of precipitated material are exhibited, with clearly visible scaling properties. This study is a result [1] of a collaboration with Professors D. Hilhorst, R. van der Hout, and M. Mimura.

References:

[1] Hilhorst, D., van der Hout, R., Mimura, M., and Ohnishi, I.: A Mathematical Study of the One-Dimensional Keller and Rubinow Model for Liesegang Bands. J. Stat Phys 135: 107-132 (2009)

[2] Kai, S., Muller, S.C.: Spatial and temporal macroscopic structures in chemical reaction system: precipitation patterns and interfacial motion. Sci. Form 1, 8-38 (1985)

[3] Keller, J.B., Rubinow, S.I.: Recurrent precipitation and Liesegang rings. J. Chem. Phys. 74, 5000-5007 (1981)

[4] Liesegang, R.E.: Chemische Fernwirkung. Photo. Archiv 800, 305-309 (1896)

[5] Mimura, M., Ohnishi, I., Ueyama, D.: A mathematical aspect of Liesegang phenomena in two space dimensions. Res. Rep. Res. Inst. Math. Sci. 1499, 185-201 (2006)

[6] Ohnishi, I.,Mimura, M.: A mathematical aspect of Liesegang phenomena. In: Proceedings of Equadiff-11, pp. 343-352 (2005).

### 2009/09/17

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

On a parabolic free boundary problem modelling price formation

**Norayr MATEVOSYAN**(ケンブリッジ大学・数理)On a parabolic free boundary problem modelling price formation

[ Abstract ]

We will discuss existence and uniqueness of solutions for a one dimensional parabolic evolution equation with a free boundary. This problem was introduced by J.-M. Lasry and P.-L. Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in time-extension of the local solution which is intimately connected to the regularity of the free boundary.

We also present numerical results.

We will discuss existence and uniqueness of solutions for a one dimensional parabolic evolution equation with a free boundary. This problem was introduced by J.-M. Lasry and P.-L. Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in time-extension of the local solution which is intimately connected to the regularity of the free boundary.

We also present numerical results.

### 2009/09/10

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

A two phase free boundary problem with applications in potential theory

**Henrik SHAHGHOLIAN**(王立工科大学・ストックホルム)A two phase free boundary problem with applications in potential theory

[ Abstract ]

In this talk I will present some recent directions, still to be developed, in potential theory, that are connected to a two-phase free boundary problems. The potential theoretic topic that I will discuss is the so called Quadrature Domains.

The most simple free boundary/potential problem that we can present is the following. Given constants $a_\\pm, \\lambda_\\pm >0$ and two points $x^\\pm$ in ${\\bf R}^n$. Find a function $u$ such that

$$\\Delta u = \\left( \\lambda_+ \\chi_{\\{u>0 \\}} - a_+\\delta_{x^+}\\right) - \\left( \\lambda_- \\chi_{\\{u<0 \\}} - a_-\\delta_{x^-}\\right),$$

where $\\delta$ is the Dirac mass.

In general this problem is solvable for two Dirac masses. The requirement, somehow implicit in the above equation, is that the support of the measures (in this case the Dirac masses) is to be in included in the positivity and the negativity set (respectively).

In general this problem does not have a solution, and there some strong restrictions on the measures, in order to have some partial results.

In this talk I will present some recent directions, still to be developed, in potential theory, that are connected to a two-phase free boundary problems. The potential theoretic topic that I will discuss is the so called Quadrature Domains.

The most simple free boundary/potential problem that we can present is the following. Given constants $a_\\pm, \\lambda_\\pm >0$ and two points $x^\\pm$ in ${\\bf R}^n$. Find a function $u$ such that

$$\\Delta u = \\left( \\lambda_+ \\chi_{\\{u>0 \\}} - a_+\\delta_{x^+}\\right) - \\left( \\lambda_- \\chi_{\\{u<0 \\}} - a_-\\delta_{x^-}\\right),$$

where $\\delta$ is the Dirac mass.

In general this problem is solvable for two Dirac masses. The requirement, somehow implicit in the above equation, is that the support of the measures (in this case the Dirac masses) is to be in included in the positivity and the negativity set (respectively).

In general this problem does not have a solution, and there some strong restrictions on the measures, in order to have some partial results.