## Applied Analysis

Seminar information archive ～07/21｜Next seminar｜Future seminars 07/22～

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

### 2015/04/23

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

The importance of being just late (ENGLISH)

**Bernold Fiedler**(Free University of Berlin)The importance of being just late (ENGLISH)

[ Abstract ]

Delays are a ubiquitous nuisance in control. Delays increase finite-dimensional phase spaces to become infinite-dimensional. But, are delays all that bad?

Following an idea of Pyragas, we attempt noninvasive and model-independent stabilization of unstable p-periodic phenomena $u(t)$ by a friendly delay $r$ . Our feedback only evaluates differences $u(t-r)-u(t)$. When the time delay $r$ is chosen to be an integer multiple $np$ of the minimal period $p$, the difference and the feedback vanish alike: the control strategy becomes noninvasive on the target periodic orbit.

We survey promise and limitations of this idea, including applications and an example of delay control of delay equations.

The results are joint work with P. Hoevel, W. Just, I. Schneider, E. Schoell, H.-J. Wuensche, S. Yanchuk, and others. See also

http://dynamics.mi.fu-berlin.de/

Delays are a ubiquitous nuisance in control. Delays increase finite-dimensional phase spaces to become infinite-dimensional. But, are delays all that bad?

Following an idea of Pyragas, we attempt noninvasive and model-independent stabilization of unstable p-periodic phenomena $u(t)$ by a friendly delay $r$ . Our feedback only evaluates differences $u(t-r)-u(t)$. When the time delay $r$ is chosen to be an integer multiple $np$ of the minimal period $p$, the difference and the feedback vanish alike: the control strategy becomes noninvasive on the target periodic orbit.

We survey promise and limitations of this idea, including applications and an example of delay control of delay equations.

The results are joint work with P. Hoevel, W. Just, I. Schneider, E. Schoell, H.-J. Wuensche, S. Yanchuk, and others. See also

http://dynamics.mi.fu-berlin.de/

### 2015/01/22

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

On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data

(ENGLISH)

**Arnaud Ducrot**(University of Bordeaux)On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data

(ENGLISH)

[ Abstract ]

In this talk we discuss the asymptotic behaviour of a multi-dimensional Fisher-KPP equation posed in an asymptotically homogeneous medium and supplemented together with a compactly supported initial datum. We derive precise estimates for the location of the front before proving the convergence of the solutions towards travelling front. In particular we show that the location of the front drastically depends on the rate at which the medium become homogeneous at infinity. Fast rate of convergence only changes the location by some constant while lower rate of convergence induces further logarithmic delay.

In this talk we discuss the asymptotic behaviour of a multi-dimensional Fisher-KPP equation posed in an asymptotically homogeneous medium and supplemented together with a compactly supported initial datum. We derive precise estimates for the location of the front before proving the convergence of the solutions towards travelling front. In particular we show that the location of the front drastically depends on the rate at which the medium become homogeneous at infinity. Fast rate of convergence only changes the location by some constant while lower rate of convergence induces further logarithmic delay.

### 2014/07/24

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

Decomposition of the Mobius energy (JAPANESE)

**Nagasawa Takeyuki**(Saitama University)Decomposition of the Mobius energy (JAPANESE)

### 2014/07/03

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

Parabolic power concavity and parabolic boundary value problems (JAPANESE)

**Kazuhiro Ishige**(Tohoku University)Parabolic power concavity and parabolic boundary value problems (JAPANESE)

### 2014/01/23

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

Inside dynamics of pushed and pulled fronts (ENGLISH)

**Thomas Giletti**(Univ. of Lorraine at Nancy)Inside dynamics of pushed and pulled fronts (ENGLISH)

[ Abstract ]

Mathematical analysis of reaction-diffusion equations is a powerful tool in the understanding of dynamics of many real-life propagation phenomena. A feature of particular interest is the fact that dynamics and their underlying mechanisms vary greatly, depending on the choice of the nonlinearity in the reaction term. In this talk, we will discuss the pushed/pulled front terminology, based upon the role of each component of the front inside the whole propagating structure.

Mathematical analysis of reaction-diffusion equations is a powerful tool in the understanding of dynamics of many real-life propagation phenomena. A feature of particular interest is the fact that dynamics and their underlying mechanisms vary greatly, depending on the choice of the nonlinearity in the reaction term. In this talk, we will discuss the pushed/pulled front terminology, based upon the role of each component of the front inside the whole propagating structure.

### 2013/12/12

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

A theoretical study on the spontaneous radiation of atmospheric gravity waves using the renormalization group method (JAPANESE)

**Yuki Yasuda**(University of Tokyo (Department of Earth and Planetary Science))A theoretical study on the spontaneous radiation of atmospheric gravity waves using the renormalization group method (JAPANESE)

### 2013/11/14

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

Singular limit of a damped wave equation with a bistable nonlinearity (ENGLISH)

**Danielle Hilhorst**(Université de Paris-Sud / CNRS)Singular limit of a damped wave equation with a bistable nonlinearity (ENGLISH)

[ Abstract ]

We study the singular limit of a damped wave equation with

a bistable nonlinearity. In order to understand interfacial

phenomena, we derive estimates for the generation and the motion

of interfaces. We prove that steep interfaces are generated in

a short time and that their motion is governed by mean curvature

flow under the assumption that the damping is sufficiently strong.

To this purpose, we prove a comparison principle for the damped

wave equation and construct suitable sub- and super-solutions.

This is joint work with Mitsunori Nata.

We study the singular limit of a damped wave equation with

a bistable nonlinearity. In order to understand interfacial

phenomena, we derive estimates for the generation and the motion

of interfaces. We prove that steep interfaces are generated in

a short time and that their motion is governed by mean curvature

flow under the assumption that the damping is sufficiently strong.

To this purpose, we prove a comparison principle for the damped

wave equation and construct suitable sub- and super-solutions.

This is joint work with Mitsunori Nata.

### 2013/06/06

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

The Geometry of Critical Points of Green functions On Tori (ENGLISH)

**Chang-Shou Lin**(National Taiwan University)The Geometry of Critical Points of Green functions On Tori (ENGLISH)

[ Abstract ]

The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.

We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.

We can generalize our results to a sum of two Green functions.

The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.

We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.

We can generalize our results to a sum of two Green functions.

### 2012/09/20

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

Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes

(ENGLISH)

**Bernold Fiedler**(Free University of Berlin)Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes

(ENGLISH)

[ Abstract ]

Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutation. These permutation relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.

We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.

This is joint work with Juliette Hell, Brian Smith, Carlos Rocha, Pablo Castaneda, and Matthias Wolfrum.

Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutation. These permutation relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.

We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.

This is joint work with Juliette Hell, Brian Smith, Carlos Rocha, Pablo Castaneda, and Matthias Wolfrum.

### 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.