名古屋微分方程式セミナー 2015年度

変数係数の摩擦項を持つ消散型波動方程式に対し, 時間無限大における解の漸近形について考察する. 本講演では, 摩擦項に対する適当な条件の下で, 解の時間無限大における漸近形が対応する熱方程式の熱核で 与えられることを示す. 証明は Gallay-Raugel (1998 年) による自己相似変換を用いる手法による.

Marcel Riesz constructed a fundamental solution of the wave equation as the analytic continuation of powers of the Lorentzian quadratic form, and explained the Huygens principle of the differential equation in connection with the poles of some gamma factors. Gindikin developed Riesz's idea in the theory of homogeneous cones, and constructed fundamental solutions of various differential operators with large symmetry. In this talk, we discuss a further generalization of Riesz-Gindikin theory, starting from new Laplace transform formula on a convex cone inspired by　statistics.

In this talk we discuss the L ² asymptotic stability of the time-global solutions to the Navier-Stokes equations in a two-dimensional exterior domain, when the global solutions possess at most scale-critical decays in space and time. In such a situation we are faced with a serious difficulty mainly coming from the lack of the Hardy-type inequality, which is specific to the two-dimensional case. We will overcome this difficulty by firstly establishing a logarithmic growth estimate for the L ² norm of the disturbances. Our approach provides a global L ² stability for small global solutions obtained by Kozono and Yamazaki (1995) in the weak L ² space. In the latter part of the talk we will also discuss the stability of a special stationary solution decaying in a scale-critical order.

We introduce a new kind of convolution, which is a sort of parabolic version of the classical supremal convolution of convex analysis. This operation allow us to compare solutions of different parabolic problems in different domains. As examples of applications of our main result, we study the parabolic concavity of solutions to parabolic boundary value problems, analyzing in particular the case of heat equation with an inhomogeneous term and with a nonlinear reaction term. We also apply our technique to the study of the dead core problem obtaining new results about necessary conditions for the existence of a dead core and estimates of the dead core time, proving some optimality of the ball.

P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan established method of solution of homogenization problem for Hamilton-Jacobi equations whose Hamiltonians is coercive. However, their method is not valid for non-coercive Hamiltonians. We will treat a certain non-coercive Hamiltonians which is motivated by crystal growth, and explain that a result for non-coercive Hamiltonian is essentially different from coercive Hamiltonians.

多重線形フーリエマルチプライヤーに対する Coifman-Meyer の定理は, 今日ではこの分野での基本的な結果としてよく知られている. Coifman-Meyer の定理がとらえているマルチプライヤーは, 0 次斉次の関数であり, その特異性は原点のみに想定されている. 一方, Muscalu は flag paraproduct と呼ばれる, マルチプライヤーをかけ合わせ, 原点以外にも特異性を持つ多重線形フーリエマルチプライヤー作用素の有界性を, 時間周波数解析の手法を用いて得ている. 最近, Germain-Masmoudi-Shatah は, flag paraproduct を非線形偏微分方程式に応用し, また Muscalu の結果の一部ではあるが, flag paraproduct の有界性に対して Littlewood-Paley 理論を基本とするシンプルな別証明を与えた. この講演では, Germain-Masmoudi-Shatah のアイデアで, Muscalu の結果が完全にカバー出来ること, また Muscalu の結果を拡張出来ることをご報告したい. なお, この話題は宮地晶彦氏(東京女子大)との共同研究である.

In this talk we will give an overview of recent research on pseudo-differential operators on spaces with addition structures: for example on compact or nilpotent Lie groups, or spaces equipped with a fixed system of eigenfunctions for a given operator. We will make an overview of results and applications in related areas such as harmonic analysis or the theory of partial differential equations.

Wiener amalgam spaces are modern spaces introduced by H.G. Feichtinger in time-frequency analysis. In this talk, we will discuss the inclusion properties of these spaces with L ^p-Sobolev spaces. We will also show sharpness of the estimates arising in this inclusion. For applications, we have boundedness of unimodular Fourier multipliers, some Littlewood-Paley type estimates, and L ^p boundedness of pseudo-differential operators with symbols coming from modulation spaces M ^∞,1(R ²ⁿ).

α-modulation 空間 (0 ≤ α ≤ 1) は Gröbnerによって導入された関数空間であり, modulation 空間は α = 0 の特別な場合である. 今回は α-modulation 空間と L ^p-Sobolev 空間 (1 ≤ p ≤ ∞) との包含関係について述べる. その後, 局所 Hardy 空間 h ^p (0 < p ≤ 1) との包含関係についても述べる.

複素幾何において最も重要な非線形偏微分方程式の 1 つである複素 Monge-Ampère 方程式について非専門家向けの解説を行います. Aubin, Yau の古典的な結果から, より最近の話題である退化した Monge-Ampère 方程式の研究, 幾何学的量子化との関係について紹介します.

1/x という関数は反比例のグラフとしてお馴染であるが実は中々奥が深い. この関数は x = 0 が特異点であるが, 奇関数であることから特異性がうまく打ち消し会う. このような関数を積分核にもつ積分を特異積分といい, 解析学の様々なところで自然な形で現れる. この特異積分の最も基本的なヒルベルト変換の基本性質について講義する. 参考書 [薮田公三, 特異積分, 出版年 2010, 岩波書店] が自力で読めるための基礎知識を身につけることを目標とする. 具体的内容は次のようなものである.

1. 主値積分(積分の定義)
2. L ^p 空間の復習
3. たたみこみ (合成積) の性質 (ヤングの不等式)
4. ヒルベルト変換の定義と基本性質
5. リプシッツ空間上のヒルベルト変換 (特異点の取り扱い方)
6. ヒルベルト変換の L ² 有界性 (ほとんど直交するという考え方)
7. ヒルベルト変換の L ^p 有界性 (Calderón-Zygmund の理論)
　2 進最大関数, 弱 L ¹ 空間, 補間定理

In this talk we consider Schrödinger operators on periodic graphs. We show that the spectra of the Laplacian is absolutely continuous via the locally conjugated operator theory and that it's stable under multiplicative perturbations with a certain decay at infinity.

We first introduce the Helmholtz-Weyl decomposition in L ^r-vector fields. The variational inequality associated with the operators div and rot plays an important role for the proof. Simultaneously, we show that the generalized Lax-Milgram theorem in Banach spaces are useful to show the existence of weak solutions to the the elliptic system of boundary value problems. Next, we apply such a decomposition theorem to the problem on the stationary Navier-Stokes equations in multi-connected domains with an inhomogeneous boundary condition. It turns out that the set of non-trivial solutions to the stationary Euler equations is closely related to solvability of our problem. Finally, we investigate the relation between topological invariance of the domain and the Leray-Fujita inequality which is known as a sufficient condition for the existence of solutions. This is a joint work with Prof. Taku Yanagiswa at Nara Women University.

本講演ではスケール不変性を持つ臨界 Hardy の不等式について考える. 既存の臨界 Hardy の不等式は rearrangement invariant 空間の枠組みで最良な埋め込みを与えることが知られているが, 対数補正項の形に起因してスケール不変性を持たない. 本講演では, 伸縮に関する不変性を持つ臨界 Hardy 不等式, および冪乗型スケール不変性を持つ臨界 Hardy 不等式の二種を示し, 対応する変分問題の最小化関数が存在しないことについて述べる. さらに, 形式的最小化関数からの距離を用いて不等式が持つ剰余項を特徴付ける.

We consider the convergence rate in time to solutions of the Cauchy problem for the viscous conservation law with a nonlinear viscosity where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval. The proof is given by time-weighted energy methods under the use of the precise asymptotic properties of the interactions between the nonlinear waves.

Combining the WKB (Wentzel-Kramers-Brillouin) method and the Borel resummation method, A. Voros initiated the exact WKB analysis. Now the method is known as a powerful tool to study global properties (monodromy / Stokes structures) of solutions of differential equations with a large parameter. The aim of this talk is to give an introduction to this theory. I’ll also explain some connections to other topics such as cluster algebras, spectral networks, topological recursion, and so on.

We will consider the large time behavior of the strong solutions of the compressible Navier-Stokes equation in the whole space around the motionless state. It was shown by Kawashima-Matsumura-Nishida ('79) and Hoff-Zumbrun ('95) that the perturbation of the motionless state is time-asymptotic to the solution of the linearized problem. In this talk we will give the second-order asymptotics of strong solution.

This talk is a survey of our joint work with Dr. Xiangdi Huang from Chinese Academy of Science on a blowup criterion of spherically symmetric classical solutions to an initial boundary value problem on a ball for an isentropic model of viscous gas. For smooth initial data away from vacuum, it is proved that the classical solution which is spherically symmetric loses its regularity in a finite time if and only if the concentration of mass forms around the center in Lagrangian coordinate system.

The purpose of this talk is to consider estimates for the term (u·∇)v on Hardy spaces with power weights. Critical weights, we will treat, are related to the optimal L ²-decay rate (n + 2)/4 for the incompressible Navier-Stokes equations on the whole space R ⁿ. Such rate was showed by Wiegner, and then the sharpness was proved by Miyakawa & Schonbek. Main tools for the proof of our result are Bogovski's formula for the divergence equation and a restricted weak type vector-valued inequality for the Hardy-Littlewood maximal operator.

フーリエ解析をキーワードに, 主に非線形シュレディンガー方程式の解の性質について, 解の存在定理に関する基本的な結果とその証明を解説する. 非線形シュレディンガー方程式に代表される非線形分散型方程式は, 波動の分散, 集約と云ったある意味で異質な構造が混合した現象を記述する数理モデルである. 講義では, フーリエ解析による関数の波数分解理論, エネルギー法による定性的解析による理論がどのように使われるか様子を紹介する. 具体的内容は次のことを紹介する.

1. 非線形シュレディンガー方程式の性質 (散乱解, 爆発解の構成)
2. 補間定理と線形シュレディンガー作用素の L ^p-L ^q 評価式
3. ストリッカーツ評価式
4. 初期値問題の適切性
5. 時空間型評価式, あるいはエネルギー法による解の構成

一般の非線形性を持つ半線形熱方程式を扱い, 解の存在・非存在について議論する. 特に解が存在するための初期値の特異性を調べ, 非線形項の増大度との関係を明らかにする. 本講演では, 冪乗型非線形熱方程式に対する自己相似変換の一般化を導入し, その変換の下でスケール不変なある積分量が定まることを紹介する. そこで定まる積分量から解の存在を許容する初期値の特異性が推察されることを説明し, 実際にそのような特異性により解の存在・非存在が分類されることを紹介する. なお本講演の内容は猪奥倫左氏（愛媛大学）との共同研究に基づく.

I will talk about one phase problem for the incompressible viscous fluid flow with surface tension. The problem is formulated by the free boundary problem for the Navier-Stokes equations. The local well-posedness is proved by the maximal L _p-L _q regularity theorem for the linearized problem and the global well-posedness is proved by using some decay properties of the C₀ semigroup associated with the linearized problem, which is analytic. I also would like to touch the problem for the global well-posedness in the exterior domain without surface tension.

The scale-function of the space is constructed based on the Einstein equation in a uniform and isotropic space. The Cauchy problem of semilinear partial differential equations are considered in Sobolev spaces. The effect of spatial variance is studied, and some dissipative properties are remarked.

空間 1 次元 Chern-Simons-Dirac 方程式系の初期値問題の適切性および非適切性をソボレフ空間で考える. 適切性の条件である初期値から解への解写像の連続依存性を議論する. そのときにソボレフ空間における関数の積評価成立のための必要十分条件と関連付けて説明したい. 本研究は信州大学岡本葵氏との共同研究である.

The Hadamard variational formula is known as a functional differential equation of the first order for perturbation of domains. The typical example is such a variation as the Green function or eigenvalues of boundary value problems on elliptic PDEs. In this talk, we address to the Stokes equations describing the motion of viscous incompressible fluids. Our purpose is to determine the topological type of 3D domains with closed surfaces as the boundary. We first introduce an explicit representation formula of the variation for eigenvalues of the Stokes equations with multiplicity given by Jimbo-Ushikoshi. Then, it turns out that if the variation of some eigenvalue vanishes for all volume preserving perturbations of domains, then the original domain is necessarily diffeomorphic to the 2-dimensional torus. This is based on the joint work with Profs S. Jimbo, Y. Teramoto and E. Ushikoshi.