## 応用解析セミナー

開催情報 木曜日　16:00～17:30　数理科学研究科棟(駒場) 002号室 石毛 和弘

### 2011年02月24日(木)

16:00-18:10   数理科学研究科棟(駒場) 002号室
Arnaud Ducrot 氏 (University of Bordeaux 2) 16:00-17:00
Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections (ENGLISH)
[ 講演概要 ]
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 氏 (Basque Center for Applied Mathematics) 17:10-18:10
Some open problems in PDE control (ENGLISH)
[ 講演概要 ]
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年02月17日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Thomas Giletti 氏 (University of Paul Cezanne (Marseilles))
Study of propagation phenomena in some reaction-diffusion systems (ENGLISH)
[ 講演概要 ]
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年02月10日(木)

15:30-16:30   数理科学研究科棟(駒場) 002号室

Jean-Michel Coron 氏 (University of Paris 6)
Control and nonlinearity (ENGLISH)
[ 講演概要 ]
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年01月27日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Nitsan Ben-Gal 氏 (The Weizmann Institute of Science)
Attraction at infinity: Constructing non-compact global attractors in the slowly non-dissipative realm (ENGLISH)
[ 講演概要 ]
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年07月08日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Anna Vainchtein 氏 (University of Pittsburgh, Department of Mathematics)
Effect of nonlinearity on the steady motion of a twinning dislocation (ENGLISH)
[ 講演概要 ]
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年06月24日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
GCOE 共催

[ 講演概要 ]

### 2010年06月10日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Christian Klingenberg 氏 (Wuerzburg 大学 )
Hydrodynamic limit of microscopic particle systems to conservation laws to fluid models
[ 講演概要 ]
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年04月22日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Jens Starke 氏 (デンマーク工科大学)
Deterministic and stochastic modelling of catalytic surface processes (ENGLISH)
[ 講演概要 ]
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年04月15日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Alberto Tesei 氏 (University of Rome 1)
Long-time behaviour of solutions of a forward-backward parabolic equation
[ 講演概要 ]
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年02月18日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Bendong LOU 氏 (同済大学)
Homogenization limit of a parabolic equation with nonlinear boundary conditions
[ 講演概要 ]
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年01月28日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

[ 講演概要 ]

### 2010年01月21日(木)

16:00-17:30   数理科学研究科棟(駒場) 122号室
Danielle Hilhorst 氏 (パリ南大学 / CNRS)
A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation
[ 講演概要 ]
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年12月17日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Hatem Zaag 氏 (CNRS / パリ北大学)
A Liouville theorem for a semilinear heat equation with no gradient structure
[ 講演概要 ]
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年11月26日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

L^p 粘性解の弱ハルナック不等式の最近の進展
[ 講演概要 ]
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.

### 2009年11月05日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

A Mathematical Aspect of the One-Dimensional Keller and Rubinow Model for Liesegang Bands
[ 講演概要 ]
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年09月17日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Norayr MATEVOSYAN 氏 (ケンブリッジ大学・数理)
On a parabolic free boundary problem modelling price formation
[ 講演概要 ]
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年09月10日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Henrik SHAHGHOLIAN 氏 (王立工科大学・ストックホルム)
A two phase free boundary problem with applications in potential theory
[ 講演概要 ]
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.

### 2009年05月14日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

Particle trajectories around a running cylinder in Brinkman's porous-media flow
[ 講演概要 ]
Motion of fluid particles provides us with interesting problems of dynamical
systems. We consider here the movement of particles around a running cylinder.
Classically J. C. Maxwell (1870) considered the problem in irrotational flow of
inviscid fluid. He showed that the complete solution is given by the elliptic
functions and the trajectory forms one of the elastica curves. C. Darwin ('53)
considered a similar problem for a moving sphere. In this case, the solution
cannot be written in terms of elliptic functions but can be expressed by a
simple definite integral.
We consider a similar problem in Brinkman's porous-media flow which is proposed
by Brinkman ('49). Our numerical examinations reveals some new interesting
features of the particle trajectories which are not observed in the case of
irrotational flow. We will report them.

### 2009年04月30日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

ギーラー・マインハルト方程式に対するシャドウ系おける多重スポットの不安定性
[ 講演概要 ]

この系にはスポットパターンと呼ばれる定常解が存在することが知られており、この解は、その値が非常に大きい点(スポット)を持つこととその近傍の外側では急激に値が減少することにより特徴付けされる。

### 2009年02月05日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Jin CHENG (程 晋) 氏 (復旦大学)
Heat transfer in composite materials with Stenfen-Boltzmann conditions and related inverse problems
[ 講演概要 ]
In this talk, we will present our recent results on the mathematical model of the heat transfer in the composite materials. The related inverse problems are discussed. The numerical results show our methods are effective.

### 2009年01月29日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

Extension and Unification of Singular Perturbation Methods for ODE's Based on the Renormalization Gourp Method
[ 講演概要 ]
くりこみ群の方法は微分方程式に対する特異摂動法の一種であり,多重尺度法、平均化法、normal forms, 中心多様体縮約、位相縮約、WKB解析などの古くから知られる摂動法を統一的に扱うことができる.ここではくりこみ群の方法を数学的定式化を与え,結合振動子系などへのいくつかの応用も紹介したい.

### 2009年01月15日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

On a phase field model for mode III crack growth
[ 講演概要 ]
2次元弾性体の面外変形による亀裂の進展を記述する,ある
フェイズ・フィールド・モデルについて考える.モデルの

Griffithの破壊基準をもとに,Ambrosio-Tortorelliに
よるエネルギー正則化のアイデアを用いてなされる.

### 2008年11月20日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
(Vienna University of Technology(オーストリア))
Stochastic Particle Approximation for Measure Valued Solutions of the 2D Keller-Segel System
[ 講演概要 ]
We construct an approximation to the measure valued, global in time solutions to the Keller-Segel model in 2D, based on systems of stochastic interacting particles. The advantage of our approach is that it reproduces the well-known dichtomy in the qualitative behavior of the system and, moreover, captures the solution even after the possible blow-up events. We present a numerical method based on this approach and show some numerical results. Moreover, we make a first step toward the convergence analysis of our scheme by proving the convergence of the stochastic particle approximation for the Keller-Segel model with a regularized interaction potential. The proof is based on a BBGKY-like approach for the corresponding particle distribution function.

### 2008年11月13日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

Aronson-Benilan type estimate and the optimal Hoelder continuity of weak solutions for the 1D degenerate Keller-Segel systems
[ 講演概要 ]
We consider the Cauchy problem for the 1D Keller-Segel system of degenerate
type (KS)_m with $m>1$:
u_t= \\partial_x^2 u^m - \\partial_x (u^{q-2} \\partial_x v),
-\\partial_x^2 v + v - u=0.
We establish a uniform estimate from below of $\\partial_x^2 u^{m-1}$.
The corresponding estimate to the porous medium equation is well-known
as an Aronson-Benilan type.
As an application of our Aronson-Benilan type estimate,
we prove the optimal Hoelder continuity of the weak solution $u$ of (KS)_m.
In addition, we find that the positive region $D(t):=\\{x \\in \\R; u(x,t)>0\\}$
of $u$ is monotonically non-decreasing with respect to the time $t$.

### 2008年10月16日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Joseph F. Grotowski 氏 (University of Queensland)
Two-dimensional harmonic map heat flow versus four-dimensional Yang-Mills heat flow
[ 講演概要 ]
Harmonic map heat flow and Yang-Mills heat flow are the gradient flows associated to particular energy functionals. In the considered dimension, (i.e. dimension two for the harmonic map heat flow, dimension four for the Yang-Mills heat flow), the associated energy functional is (locally) conformally invariant, that is, the dimension is critical. This leads to a number of interesting phenomena when considering both the functionals and the associated flows. In this talk we discuss qualitative similarities and differences between the flows.