京都大学 NLPDE セミナー

2012年度のセミナーの記録

日時
2012 年 4 月 13 日(金曜日) 15:30 〜 17:30 
(力学系セミナーと合同開催)
場所
京都大学 大学院理学研究科 3 号館 108 号室
講演者
千葉 逸人 氏(九州大学)
講演題目
一般化スペクトル理論とその無限次元力学系への応用
講演要旨
Gelfand tripletと呼ばれる、線形位相空間の3つ組上での線形作用素のスペクトル理論を展開する。 通常、作用素のスペクトルは、C上におけるレゾルベントの 特異点集合として定義されるが、 Gelfand tripletを導入すると、レゾルベントが複雑なRiemann面を持ちうる。 そこで、Riemann面全体を 見渡した時のレゾルベントの特異点集合を一般化スペクトルと呼ぶ。 一般化スペクトルは、普通のスペクトルと同じくらい、作用素についての 重要な情報を持っており、これを用いることで従来は見えなかった現象を捉えることができる。 講演では、これをいくつかの無限次元力学系やPDEの解の安定性と分岐理論などに応用する。


日時
2012 年 4 月 20 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
高田 了 氏(京都大学大学院理学研究科)
講演題目
Local well-posedness for the Navier-Stokes equations in the rotational framework
講演要旨
In this talk, we consider the initial value problems for the Navier-Stokes equations with the Coriolis force. We prove the local in time existence and uniqueness of the mild solution in the framework of homogeneous Sobolev spaces. Furthermore, we give an exact characterization for the time interval of its local existence in terms of the Coriolis parameter. It follows from our characterization that the existence time of the solution can be taken arbitrarily large provided the speed of rotation is sufficiently fast.


日時
2012 年 4 月 27 日(金曜日) 15:30 〜 17:45
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
Frederic Rousset 氏(IRMAR, Universite de Rennes 1)
講演題目
Transverse stability of solitary waves in dispersive PDE
講演要旨
<第1部> 15:30〜16:30
In the first talk, I will introduce the problem and present a general criterion for transverse linear instability. This criterion will be discussed on various examples like KP and Schrodinger type equations. I will also discuss the stability problem in the case of transverse periodic perturbations.
<第2部> 16:45〜17:45
In the second talk, I will focus on the study of nonlinear instability, the main example will be the water-waves system that describes the propagation of capillary-gravity waves.


日時
2012 年 5 月 11 日(金曜日) 15:30 〜 17:00
(関西確率論セミナーと合同開催)
場所
京都大学 大学院理学研究科 3 号館 552 号室
講演者
前川 泰則 氏(神戸大学)
講演題目
輸送項付き分数冪拡散方程式の基本解について
講演要旨
本講演では非圧縮性条件を満たす輸送項の伴った分数冪拡散方程式を考察する. 輸送項の速度場に対しては,拡散の度合いを表す指数から定まるスケーリングに対し て不変な関数空間を設定する.2階放物型方程式の基本解に対する古典的なNash('58) の結果を分数冪拡散方程式に拡張したKomatsu('95)の手法を基に,基本解の存在とヘ ルダー連続性,各点評価を証明する.2次元消散型準地衡流方程式への応用について も述べる.なお,本講演は三浦英之氏(大阪大学)との共同研究に基づく.


日時
2012 年 5 月 18 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
津川 光太郎 氏(名古屋大学)
講演題目
Well-posedness of the KdV equation with almost periodic initial data
講演要旨
First, we prove the local well-posedness for the Cauchy problem of Korteweg-de Vries equation in a quasi periodic function space. The function space contains functions satisfying f=f_1+f_2+...+f_N where f_j is in the Sobolev space of order s>?1/2N of a_j periodic functions. Note that f is not periodic when the ratio of periods a_i/a_j is irrational. Next, we prove an ill-posedness result in the sense that the flow map (if it exists) is not C2, which is related to the Diophantine problem. We also prove the global well-posedness in an almost periodic function space.


日時
2012 年 5 月 22 日(火曜日) 16:45 〜 18:15
場所
京都大学 大学院理学研究科 3 号館 552 号室
講演者
Gustav Holzegel 氏 (Princeton University)
講演題目
The Black Hole Stability Problem
講演要旨
The first part of the talk will be an introduction to General Relativity with an emphasis on how the subject is linked to the subjects of partial differential equations and geometry. I will discuss some of the major results (Initial Value Formulation, Stability of Minkowski space) as well as future challenges and open problems. The second part will focus on recent progress in the context of the black hole stability problem. Here I will survey the recent results and techniques regarding the wave equation on Kerr black hole spacetimes, explain their relevance for the stability problem, and, finally, discuss some of my own work on so-called ``ultimately Schwarzschildean spacetimes".


日時
2012 年 5 月 25 日(金曜日) 15:30 〜 18:00
(15:30-16:30 第1部,17:00-18:00 第2部)
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
Franco Flandoli 氏(University of Pisa)
講演題目
The effect of noise on uniqueness and singularities of differential equations and the zero-noise limit problem
講演要旨
The general theme of these two lectures is the regularization introduced by noise in ordinary and partial differential equations. The final examples we have in mind arise from fluid dynamics.

The first lecture will be devoted to a review of definitions and results of uniqueness for SDEs with additive noise and non smooth drift with emphasis on the fact that the deterministic equations with the same drift may have non unique solutions. We sketch the proof of the construction of a stochastic flow of diffeomorphisms in the case of Holder continuous drift. We also describe the zero-noise limit problem when the limit deterministic equation is not well posed and we recall a known result in dimension 1.

The second lecture is devoted to examples of SPDEs where similar regularization occurs. The role of bilinear multiplicative noise is discussed. At the PDE level the regularization due to noise may appear both for the problem of uniqueness and for the problem of singularities. We show in particular examples of linear transport equations and linear vector advection equations where noise prevents singularities which otherwise would emerge for the corresponding deterministic PDE. We give also an example where we may control the zero-noise limit.

Part of the literature related to these lecture is:

F. Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Saint Flour summer school lectures 2010, Lecture Notes in Mathematics n. 2015, Springer, Berlin 2011.

F. Flandoli, M. Gubinelli, E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1, 1--53.


日時
2012 年 6 月 1 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
岩渕 司 氏(中央大学)
講演題目
Ill-posedness for the nonlinear Schrodinger equations in one space dimension
講演要旨
空間1次元および空間2次元の非線形シュレディンガー方程式の初期値問題を扱い、 初期値問題の非適切性をSobolev空間あるいはBesov空間の枠組みで考え、 特に解の初期値連続依存性が一般には成り立たないことを示す。 既存の結果では、Bejenaru-Tao (2006) が背理法によって初期値連続依存性の不成立を示したが、 本発表では解のノルムを直接評価する方法を用いる。 尚、本研究は小川卓克氏(東北大学)との共同研究に基づくものである。


日時
2012 年 6 月 8 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
三浦 英之 氏(大阪大学)
講演題目
Asymptotics of small exterior Navier-Stokes flows with nonhomogeneous boundary data
講演要旨
We consider the 3D incompressible Navier-Stokes flows in an exterior domain with small boundary data which do not necessarily decay in time. We prove that the spatial asymptotics of a time periodic flow is given by a Landau solution. We next show that if the boundary datum is time-periodic and the initial datum is asymptotically homogeneous with order -1, the solution converges to the sum of a time-periodic vector field and a forward self-similar vector field as time goes to infinity. This is joint work with Kyungkuen Kang and Tai-Peng Tsai.


日時
2012 年 6 月 15 日(金曜日) 15:30 〜 16:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
Luc Molinet 氏 (Université François-Rabelais de Tours)
講演題目
Dispersive limit from the Kawahara to the KdV equation
講演要旨
We will discuss the limit behavior of the solutions to the Kawahara equation
$ u_t + u_{3x} + \varepsilon u_{5x} + u u_x = 0, \varepsilon>0 $
as $\varepsilon \to 0$. In this equation, the terms $u_{3x}$ and $\varepsilon u_{5x}$ do compete together and do cancel each other at frequencies of order $1/\sqrt{\varepsilon}$. This prohibits the use of a standard dispersive approach for this problem. Nevertheless, by combining different dispersive approaches according to the range of spaces frequencies, we will see that the solutions to this equation converge in $C([0,T];H^1(R))$ towards the solutions of the KdV equation for any fixed $T>0$.


日時
2012 年 6 月 19 日(火曜日) 15:00 〜 17:00
場所
京都大学 大学院理学研究科 3 号館 552 号室
(通常と曜日・時刻・会場が異なります.会場が多少わかりにくいのでご注意ください.)
講演者
眞崎 聡 氏 (学習院大学)
講演題目
On minimal non-scattering solution for focusing mass-subcritical NLS equation
講演要旨
We consider time global behavior of solutions to focusing mass-subcritical NLS equation in framework of weighted $L^2$ space. We prove that there exists an initial data such that (i) corresponding solution does not scatter (non-scattering data); (ii) with respect to a certain scale-invariant quantity, this attains minimum value in all non-scattering data. Here, we call a solution with the above data as a minimal non-scattering solution. In mass-critical and -supercritical cases, it is known that the ground states are this kind of minimal non-scattering solutions. However, in this case, we can show that the non-scattering solution is NOT a standing wave solution such as ground state or excited state.


日時
2012 年 6 月 22 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
上田 好寛 氏 (神戸大学)
講演題目
Decay structure for symmetric hyperbolic systems with non-symmetric relaxation
講演要旨
In this talk, we consider the initial value problem for symmetric hyperbolic systems. When the systems satisfy the Shizuta-Kawashima condition, we can obtain the asymptotic stability result and the explicit rate of convergence. There are, however, some physical models which do not satisfy the Shizuta-Kawashima condition (cf. Timoshenko system, Euler-Maxwell system). Moreover, it had already known that the dissipative structure of these systems is weaker than the standard type. Our purpose of this talk is to construct a new condition which include the Shizuta-Kawashima condition, and to analyze the weak dissipative structure.


日時
2012 年 6 月 29 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学・、究科 3 号館 251 号室
講演者
Sanghyuk Lee 氏 (Seoul National University)
講演題目
On space time estimates for the Schrödinger operator
講演要旨
We discuss about the mixed-norm space-time estimates for the Schrödinger operator including their relation to Fourier restriction estimate and recent developments related to multilinear restriction estimates.


日時
2012 年 7 月 13 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
和田出 秀光 氏 (岐阜大学)
講演題目
Remarks on logarithmic Hardy inequality in critical Sobolev-Lorentz spaces
講演要旨
The classical Hardy inequality is a weighted inequality for the lower order Sobolev space. On the other hand, the logarithmic Hardy type inequality is known for the critical Sobolev space. In this talk, we give a variant of the logarithmic Hardy inequality in terms of the critical Sobolev-Lorentz space. In particular, we can establish the logarithmic Hardy inequality for the weak critical Sobolev space. At the same time, we investigate the optimality for the exponents appearing in the inequality. This is a joint work with Professor Tohru Ozawa in Waseda University and Professor Shuji Machihara in Saitama University.


日時
2012 年 7 月 27 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
小薗 英雄 氏 (早稲田大学)
講演題目
Stationary Navier-Stokes equations in multi-connected domains
講演要旨
In multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stokes equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leray's inequality and the topological type of the domain is also clarified. This talk is based on the joint work with Prof.Taku Yanagisawa at Nara Women University.


日時
2012 年 9 月 14 日(金曜日) 15:30 〜 17:45
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
Søren Fournais 氏 (Aarhus University)
講演題目
Part I (15:30 〜 16:30) : On the third critical field in Ginzburg-Landau theory
Part II (16:45 〜 17:45) : Semiclassics in self-generated magnetic fields
講演要旨
[Part I] This talk concerns the theory of superconductivity in the Ginzburg-Landau model. It is a classical result that when a superconducting material is submitted to a sufficiently strong magnetic field, the field will completely penetrate the sample and the material will loose its superconducting properties. The value of the external magnetic field strength for which such a transition takes place is called the third critical field. We will discuss problems concerning the precise definition of this critical field and both positive and negative results on its calculation. This is joint work with B. Helffer and M. Persson.
[Part II] Consider a gas of non-interacting electrons in an external electric potential and a magnetic field. The novelty is that we consider the magnetic field as self-generated, i.e. we include the classical field energy of the magnetic field and minimize over the total system (electrons)+(field). Under appropriate assumptions on the electric potential, this combined system will be stable, i.e. its energy will be bounded from below. We further study the system in a semi-classical limit. The results depend on the magnitude of the constant in front of the magnetic field energy. For certain regimes, one can obtain leading order (Weyl) asymptotic formula and even higher order semiclassics. This is joint work with L. Erdös and J.P. Solovej.


日時
2012 年 10 月 12 日(金曜日) 15:30 〜 17:00  【関西確率論セミナーと共催】
場所
京都大学 大学院理学研究科 3 号館 552 号室
講演者
曽我 幸平 氏 (早稲田大学)
講演題目
差分法が有する拡散効果の確率論的特徴付けとその応用
講演要旨
1階双曲型PDEに粘性項を付し粘性係数を0にする極限によって、双曲型PDEの解を放物型PDEの解で近似できる。粘性項をブラウン運動で特徴づけることによって、この極限を大数の法則に基づいて確率論的に論ずる方法は知られている。本講演では、これに類似した考え方が1階双曲型PDEの差分近似でも可能であることを報告する。1階双曲型PDEの差分化方程式は、粘性項に対応する項を付けなくても、ある種の拡散効果を有する。これをあるランダムウォークで特徴付け、双曲型スケーリング極限を取ることによって、差分近似の収束を大数の法則に基づいて論ずる。この考え方を一般的な非線形双曲型保存則方程式の差分近似に適用し、従来の方法では困難であった時間大域的な安定性/漸近挙動/収束証明/誤差評価/特性曲線の近似などを示す。この方法では、PDEの解だけでなく、これに付随した特性曲線の近似も可能なため、両者の関係を論ずる力学系理論(弱KAM理論)への応用価値がある。時間が許せばこの点についても触れたい。


日時
2012 年 10 月 19 日(金曜日) 15:30 〜 17:00
場所
京都大学 大学院理学研究科 3 号館 251 号室
講・猿メ
Dongho Chae 氏 (Chung-Ang University)
講演題目
On the blow-up problem for the Euler equations and the Liouville type results for the fluid equations
講演要旨
In the first part of the talk we discuss some new observations on the blow-up problem in the 3D Euler equations. We consider the scenarios of the self-similar blow-ups and the axisymmetric blow-up. For the self-similar blow-up we prove a Liouville type theorem for the self-similar Euler equations. For the axisymmetric case we show that some uniformity condition for the pressure is not consistent with the global regularity. In the second part we present Liouville type theorems for the steady Navier-Stokes equations for both of the incompressible and the compressible cases. In the time dependent case we prove that some pressure integrals have definite sign unless the solution is trivial.


日時
2012 年 10 月 26 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
和田 健志 氏 (熊本大学)
講演題目
Smoothing effects for Schrödinger equations with electro-magnetic potentials and applications to the Maxwell-Schrödinger Equations
講演要旨
We consider Schrödinger equations in $\R^{1+2}$ with electro-magnetic potentials. The potentials belong to $H^1$, and typically they are time-independent or determined as solutions to inhomogeneous wave equations. We prove Kato type smoothing estimates for solutions. We also apply this result to the Maxwell-Schrödinger equations in the Lorenz gauge and prove unique solvability of this system in the energy space.


日時
2012 年 11 月 6 日(火曜日) 15:00 〜 17:15
場所
京都大学 大学院理学研究科 3 号館 552 号室
講演者
Jean-Claude Saut 氏 (Université Paris-Sud)
講演題目
Part I (15:00 〜 16:00) : Well-posedness of the Euler-Poisson system and rigorous justification of the Zakharov-Kuznetsov equation
Part II (16:15 〜 17:15) : Initial boundary value problem and control of the Zakharov-Kuznetsov equation
講演要旨
【Part I】
We consider the rigorous derivation of the Zakharov-Kuznetsov (ZK) equation from the Euler-Poisson (EP) system for uniformly magnetized plasmas. We first provide a proof of well-posedness of the Cauchy problem for the EP system in long time, in dimensions two and three. Then we prove that the long wave small amplitude limit is described by the ZK equation. This is done first in the case of a cold plasma; then we show how to extend the result in presence of the isothermal pressure term with uniform estimates when this latter tends to zero.
【Part II】
Imposing suitable boundary conditions yields strong dissipative effects on dispersive equations. We will develop this scenario for the Zakharov-Kuznetsov equation posed on some semi-bounded or bounded domains. The results will be applied to various control problems for the Zakharov-Kuznetsov equation.


日時
2012 年 11 月 13 日(火曜日) 15:30 〜 17:00
場所
京都大学 大学院理学研究科 3 号館 552 号室
講演者
Viktor I. Burenkov 氏 (Cardiff University)
講演題目
Sharp spectral stability estimates for uniformly elliptic differential operators
講演要旨
こちら


日時
2012 年 11 月 16 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
大縄 将史 氏 (早・ッc大学)
講演題目
Asymptotic stability of boundary layers in plasma physics with fluid-boundary interaction
講演要旨
We study the asymptotic stability of a boundary layer called sheath, which appears over a material in contact with plasma. Two types of boundary conditions on the electrostatic potential are considered depending on the physical situation. In our previous works with Prof. Shinya Nishibata and Prof. Masahiro Suzuki at Tokyo Institute of Technology, we formulated the sheath by a monotone stationary solution to the Euler-Poisson system over a half space under Bohm's criterion and proved its asymptotic stability with a boundary condition which fixes potential value on the wall. In this talk, we take into account the accumulation of charged particles on the boundary due to the flux from the inner fluid, which changes potential gradient on the boundary and further influences the entire fluid. Our main results claim the asymptotic stability of the stationary solution also with this fluid-boundary interaction.


日時
2012 年 12 月 14 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
中村 誠 氏 (東北大学)
講演題目
Energy solutions for dissipative wave equations with weighted nonlinear terms
講演要旨
The Cauchy problem for dissipative wave equations with weighted nonlinear terms is considered. The nonlinear terms are power type with a singularity at the origin of Coulomb type. The local and global solutions are shown in the energy class by the use of the Caffarelli-Kohn-Nirenberg inequality. The exponential type nonlinear terms are also considered in the critical two-spatial dimensions.


日時
2012 年 12 月 21 日(金曜日) 15:00 〜 17:00
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
利根川 吉廣 氏 (北海道大学)
講演題目
A local regularity theorem on varifold mean curvature flow
講演要旨
A family of k-dimensional surfaces in the n-dimensional Euclidean space (or more generally in a Riemannian manifold) is called the mean curvature flow (MCF) if the velocity of motion is equal to its mean curvature. One can define a weak notion of MCF using a language of geometric measure theory called varifold. Recently we proved a general local epsilon-regularity theorem in this setting, which shows among other things almost everywhere smoothness for the so called unit density MCF in Riemannian manifold for any codimensions. The Allard regularity theorem on generalized minimal submanifolds turns out to be a special case of our theorem. I will spend most time explaining the background materials and main results, and sketch the outline of proof.


日時
2013 年 1 月 11 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
柴田 徹太郎 氏 (広島大学)
講演題目
Inverse and direct bifurcation problems for nonlinear elliptic equations
講演要旨
We consider the semilinear elliptic equations which are motivated by logistic equation of population dynamics. The purpose of this talk is to consider the inverse and direct bifurcation problem. For the direct problems, we establish precise asymptotic formulas for the bifurcation curves in $L^q$-framework. As for inverse problems, taking some asymptotic properties of bifurcation curves into account, we propose new concept of inverse bifurcation problem. This problem, in some sense, corresponds to the linear inverse eigenvalue problems, which determine unknown potential from the information about eigenvalues.


日時
2013 年 1 月 18 日(金曜日) 15:00 〜 17:00  【京都力学系セミナーとの共催】
場所
京都大学 大学院理学研究科 6 号館 609 号室
講演者
Yancong Xu 氏 (Hangzhou Normal University)
講演題目
Snakes and isolas in non-reversible conservative systems
講演要旨
Reversible variational partial differential equations such as the Swift?Hohenberg equation can admit localized stationary roll structures whose solution branches are bounded in parameter space but unbounded in function space, with the width of the roll plateaus increasing without bound along the branch: this scenario is commonly referred to as snaking. In this work, the structure of the bifurcation diagrams of localized rolls is investigated for variational but non-reversible systems, and conditions are derived that guarantee snaking or result in diagrams that either consist entirely of isolas.


日時
2013 年 1 月 25 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
柴田 良弘 氏 (早稲田大学)
講演題目
On the $R$-sectoriality of the Stokes operators in a general domain
講演要旨
I will talk about the $R$-sectoriality of the Stokes operators, which implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity at the same time. The notion of $R$-sectoriality is related to the $R$-bound of the operator families, which plays an important role to use the Weis operator valued Fourier multiplier theorem. The main assumption on the domain is the unique solvability of weak Dirichlet-Neumann problem. According to our theory, a local in time existence of the Navier-Stokes equations follows from the unique solvability of weak Dirichlet-Neumann problem.


日時
2013 年 2 月 1 日(金曜日) 15:30 〜 17:30
場所
京都大学 大学院理学研究科 3 号館 251 号室
講演者
阿部 健 氏 (東京大学)
講演題目
The $L^{\infty}$-Stokes semigroup in exterior domains
講演要旨
Analyticity of the Stokes semigroup is well understood on $L^{p}$ space for various kinds of domains including bounded and exterior domains with smooth boundaries. The situation is different for the case $p=\infty$ since the Helmholtz projection is not bounded on $L^{\infty}$ anymore. In this talk, we give an a priori $L^{\infty}$-estimate of the Stokes flow by a blow-up argument for the analyticity of the semigroup on $L^{\infty}$. This talk is based on a joint work with Professor Yoshikazu Giga.