## 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

### 2017年12月21日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

Well-posedness and qualitative behavior of Peskin's problem of an immersed elastic filament in 2D Stokes flow
(Japanese)
[ 講演概要 ]
A prototypical fluid-structure interaction (FSI) problem is that of a closed elastic filament immersed in 2D Stokes flow, where the fluids inside and outside the closed filament have equal viscosity. This problem was introduced in the context of Peskin's immersed boundary method, and is often used to test computational methods for FSI problems. Here, we study the well-posedness and qualitative behavior of this problem.

We show local existence and uniqueness with initial configuration in the Holder space C^{1,\alpha}, 0<\alpha<1, and show furthermore that the solution is smooth for positive time. We show that the circular configurations are the only stationary configurations, and show exponential asymptotic stability with an explicit decay rate. Finally, we identify a scalar quantity that goes to infinity if and only if the solution ceases to exist. If this quantity is bounded for all time, we show that the solution must converge exponentially to a circle.

This is joint work with Analise Rodenberg and Dan Spirn.

### 2017年12月14日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室
158から128に変更されました．
I-Kun, Chen 氏 (Kyoto University)
Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain
(English)
[ 講演概要 ]
We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.

### 2017年07月13日(木)

16:00-17:30   数理科学研究科棟(駒場) 122号室

Behaviors of solutions for a singular prey-predator model and its shadow system
(JAPANESE)
[ 講演概要 ]
We study the asymptotic behavior and quenching of solutions for a two-component system of reaction diffusion equations modeling prey-predator interactions in an insular environment. First, we give the global existence of solutions to the corresponding shadow system. Then, by constructing some suitable Lyapunov functionals, we characterize the asymptotic behaviors of global solutions to the shadow system. Also, we give a quenching result for the shadow system. Finally, some global existence results and the asymptotic behavior for the original reaction diffusion system are given.

This is joint work with Jong-Shenq Guo (Tamkang Univ.) and Arnaud Ducrot (Univ. Bordeaux).

### 2017年02月16日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室
Danielle Hilhorst 氏 (CNRS / University of Paris-Sud)
Diffusive and inviscid traveling wave solution of the Fisher-KPP equation
(ENGLISH)
[ 講演概要 ]
Our purpose is to study the limit of traveling wave solutions of the Fisher-KPP equation as the diffusion coefficient tends to zero. More precisely, we consider monotone traveling waves which connect the stable steady state to the unstable one. It is well known that there exists a positive constant c* such that there does not exist any traveling wave solution if c < c* and a unique (up to translation) monotone traveling wave solution of wave speed c for each c > c*.

We consider the corresponding inviscid ordinary differential equation where the diffusion coefficient is equal to zero and show that it possesses a unique traveling wave solution. We then fix c > 0 arbitrary and prove the convergence of the travelling wave of the parabolic equation with velocity c to that of the corresponding traveling wave solution of the inviscid problem.

Further research should involve a similar problem for monostable systems.

This is joint work with Yong Jung Kim.

### 2016年10月27日(木)

16:00-17:30   数理科学研究科棟(駒場) 126号室
Fred Weissler 氏 (パリ第13大学)
Sign-changing solutions of the nonlinear heat equation with positive initial value
(ENGLISH)
[ 講演概要 ]
We consider the nonlinear heat equation with a power nonlinear source term on all of N-dimensional space. It is well known that the associated Cauchy problem is locally well-posed in a variety of function spaces, including certain Lebesgue spaces, depending on the power. In other Lebesgue spaces, it can happen that the Cauchy problem is not well-posed. In particular, there exist non-negative initial values for which no local (in time) non-negative solution exists. This can happen also for some homogeneous functions, where the homogeneity is linked to the scaling properties of the equation.

I will discuss recent work, in collaboration with T. Cazenave, F. Dickstein and I. Naumkin. We show that for a certain class of non-negative initial values which, as mentioned above, do not admit local non-negative solutions, there exist in fact local (or global) solutions which change sign. In particular, in the case of non-negative homogeneous initial data which do not admit non-negative solutions, we construct sign-changing self-similar solutions with the given initial data.

http://www.ms.u-tokyo.ac.jp/~miyamoto/Weissler-abstract.pdf

### 2015年11月05日(木)

16:00-17:30   数理科学研究科棟(駒場) 123号室

Henri Berestycki 氏 (フランス高等社会科学院(EHESS))
The effect of a line with fast diffusion on Fisher-KPP propagation (ENGLISH)
[ 講演概要 ]
I will present a system of equations describing the effect of inclusion of a line (the "road") with fast diffusion on biological invasions in the plane. Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain precise threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. I will discuss several further effects such as transport or reaction on the road. I will also discuss the influence of various parameters on the asymptotic behaviour of the invasion speed and shape. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.

### 2015年10月22日(木)

16:00-17:50   数理科学研究科棟(駒場) 002号室
2つ講演があります．
Hans-Otto Walther 氏 (ギーセン大学)
(Part I) The semiflow of a delay differential equation on its solution manifold
(Part II) Shilnikov chaos due to state-dependent delay, by means of the fixed point index
(ENGLISH)
[ 講演概要 ]
(Part I) 16:00 - 16:50
The semiflow of a delay differential equation on its solution manifold
(Part II) 17:00 - 17:50
Shilnikov chaos due to state-dependent delay, by means of the fixed point index

(Part I)
The lecture surveys recent work on initial value problems for differential equations with variable delay. The focus is on differentiable solution operators.

The lecture explains why the theory for retarded functional differential equations which is familiar from monographs before the turn of the millenium fails in case of variable delay, discusses what has been achieved in this case, for autonomous and non-autonomous equations, with delays bounded and unbounded, and addresses open problems.

[detailed abstract]
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf

(Part II)
What can variability of a delay in a delay differential equation do to the dynamics? We find a bounded delay functional $d(\phi)$, with $d(\phi)=1$ on a neighborhood of $\phi=0$, such that the equation $x'(t)=-a x(t-d(x_t))$ has a solution which is homoclinic to $0$, with shift dynamics in its vicinity, whereas the linear equation $x'(t)=-a x(t-1)$ with constant time lag, for small solutions, is hyperbolic with 2-dimensional unstable space.

The proof involves regularity properties of the semiflow close to the homoclinic loop in the solution manifold and a generalization of a method due to Piotr Zgliczynsky which uses the fixed point index and a closing argument in order to establish shift dynamics when certain covering relations hold. (Joint work with Bernhard Lani-Wayda)

[detailed abstract]
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-2.pdf

[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf

### 2015年07月16日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

ネットワーク曲率流の3重点周りの正則性について (Japanese)
[ 講演概要 ]

### 2015年06月11日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]

$u_t = \Delta u^m - \nabla \cdot (u^{q-1} \nabla v)$,
$v_t = \Delta v - v + u$.
ここで, $m \ge 1$, $q \ge 2$ とする. この問題に対する時間大域的弱解の存在については, 最初にSugiyama-Kunii (2006)によって $q \le m$ という条件が提示され, その後Ishida-Yokota (2012)によって最大正則性原理を用いたアプローチにより$q < m +2/N$ (Nは空間次元)という条件下で示された. しかし, これらの研究において, 解の時間大域的な挙動の解明という観点から重要である「解の有界性」は未解決のまま残されている. なお, $q < m +2/N$ という条件は, $m=1$, $q=2$のときに対応する通常のKeller-Segel系に対する研究から, 初期値の大きさに制限なく時間大域的弱解の存在が言える条件としては最良であると考えられる. 有界領域上のNeumann問題に対しては, Tao-Winkler (2012), Ishida-Seki-Yokota (2014)によって同様の条件の下で時間大域解の存在だけでなく解の有界性まで示されているが, Gagliardo-Nirenbergの補間不等式を繰り返し用いるために計算が複雑であり, 証明の見通しが良いとは言い難い. 本講演では, 特別な場合に対するSenba-Suzuki (2006)の方法を参考に, Ishida-Yokota (2012)による最大正則性原理を用いるアプローチに小さな修正を施すことによって, 解の有界性が容易に導かれることを示す.

### 2015年05月14日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

Strong instability of standing waves for some nonlinear Schr\"odinger equations (Japanese)
[ 講演概要 ]
デルタ関数をポテンシャルとして含む空間1次元の非線形シュレディンガー方程式を考える．この方程式の定在波解は双曲線関数を用いて具体的に書き表すことができる．そのため，方程式がスケール不変でないにも関わらず，角振動数をパラメータとする定在波解の族のエネルギーや電荷のパラメータ依存性を具体的に計算することができ，定在波解の軌道安定性と不安定性を完全に分類することができる．この講演では，軌道不安定な定在波解の近傍から出発した解が有限時間で爆発するための条件について考察する．このとき，定在波解は強不安定であるというが，今回得られた強不安定性の十分条件と軌道不安定性に関する従来の条件との関係を数値的に調べ，関連する問題を紹介する．

### 2015年04月23日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室
Bernold Fiedler 氏 (ベルリン自由大学)
The importance of being just late (ENGLISH)
[ 講演概要 ]
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年01月22日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室
Arnaud Ducrot 氏 (ボルドー大学)
On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data
(ENGLISH)
[ 講演概要 ]
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年07月24日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

メビウス・エネルギーの分解定理 (JAPANESE)
[ 講演概要 ]

すなわち、特異性を持つエネルギー密度の主値積分で定義される。エネルギーが主値積分で与えられるため、変分公式の計算などでは解析的にはデリケートな扱いが必要とされた。

この分解による3つの各部分のメビウス不変性は保たれており、微分幾何的にも興味深い。解析的には、この分解によってメビウス・エネルギーの変分公式とその評価が従来の計算法よりはるかに容易に求められるという利点がある。様々な関数空間上での第一・第二変分公式の評価を紹介する。

### 2014年07月03日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]

しかし、放物型方程式に関しては時間変数を固定した上での空間変数に関する解の凸性の研究が専らであった。本講演では、放物型冪凸という概念を導入し、冪凸非斉次項をもつ熱方程式の解の時空間変数による凸性について、最近の研究成果を含めて述べる。

### 2014年01月23日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Thomas Giletti 氏 (Univ. of Lorraine at Nancy)
Inside dynamics of pushed and pulled fronts (ENGLISH)
[ 講演概要 ]
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年12月12日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室

くりこみ群の方法による大気重力波の自発的放射メカニズムの解明 (JAPANESE)
[ 講演概要 ]

### 2013年11月14日(木)

16:00-17:30   数理科学研究科棟(駒場) 002号室
Danielle Hilhorst 氏 (Université de Paris-Sud / CNRS)
Singular limit of a damped wave equation with a bistable nonlinearity (ENGLISH)
[ 講演概要 ]
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年06月06日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室
Chang-Shou Lin 氏 (National Taiwan University)
The Geometry of Critical Points of Green functions On Tori (ENGLISH)
[ 講演概要 ]
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年09月20日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室
Bernold Fiedler 氏 (Free University of Berlin)
Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes
(ENGLISH)
[ 講演概要 ]
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年01月19日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室
Philippe G. LeFloch 氏 (Univ. Paris 6 / CNRS)
Undercompressible shocks and moving phase boundaries
(ENGLISH)
[ 講演概要 ]
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年11月10日(木)

15:00-16:00   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]

において中心的な課題である。細胞体積調節を記述する標準的な数理
モデルはpump-leak model と呼ばれ、数学的には常微分方程式に

### 2011年11月10日(木)

16:30-17:30   数理科学研究科棟(駒場) 128号室
この日はダブルヘッダーです。
Bernold Fiedler 氏 (Free University of Berlin)
Schoenflies spheres in Sturm attractors (ENGLISH)
[ 講演概要 ]
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年06月30日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

3次元空間における第2種超電導の巨視的モデルについて (JAPANESE)
[ 講演概要 ]
3次元空間に置かれた第2種超電導体のまわりの電磁場の巨視的なふるまいを記述するモデルについて考察する.超電導を特徴付ける非線型の電場と電流密度の関係式をマックスウェル方程式系と組み合わせることで磁場に関する時間発展の方程式を導くことができる.この非線型のオームの法則としてビーンモデルが応用上典型的に用いられるが,超電導体の内部でピン止めされた磁束が動き出すとき一般の3次元の問題では電流と平行でない電場が現れることが予測され,ビーンモデルの仮定に反してしまう.3次元のより現実的な巨視的モデルの候補としては,Double Critical-state Modelが1980年代に提案されている.本講演ではこれらの巨視的モデルを3次元の問題に採用して導いた発展方程式の解の存在と有限要素法による離散化の方法を解説し,あわせて数値計算例を提示する.

### 2011年06月09日(木)

16:30-18:00   数理科学研究科棟(駒場) 128号室

Spectral representations and scattering for Schr\\"odinger operators on star graphs (JAPANESE)
[ 講演概要 ]
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年05月26日(木)

16:00-17:30   数理科学研究科棟(駒場) 128号室

Obstacle problem of Navier-Stokes equations in thermohydraulics (JAPANESE)
[ 講演概要 ]
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.