名古屋微分方程式セミナー 2014年度

We consider a phase field model for the flow of two partly miscible incompressible, viscous fluids of non-Newtonian (power law) type. In the model it is assumed that the densities of the fluids are equal. We prove existence of weak solutions for general initial data and arbitrarily large times with the aid of a parabolic Lipschitz truncation method, which preserves solenoidal velocity fields and was recently developed by Breit, Diening, and Schwarzacher. The talk is based on a joint work with Helmut Abels (Regensburg) and Lars Diening (Munich).

We show some a priori estimates for the resolvent Stokes equations and prove that the Stokes operator generates an analytic semigroup on spaces of bounded functions for bounded and exterior domains with smooth boundaries. The proof is based on a localization technique for elliptic operator due to K. Masuda and H. B. Stewart. This talk is based on a joint work with Y. Giga (U. Tokyo) and M. Hieber (TU Darmstadt).

本講演では, 次元 4 次元以上の臨界空間における Zakharov system の初期値問題について考える. Ginibre-Tsutsumi-Velo はフーリエ制限ノルム法を用いることにより, 劣臨界空間での適切性を示した. 一方, 臨界空間の場合には, この手法で必要となる双線形評価を示すのが難しい. 本研究では, 上記の手法を精密化した U ², V ² 型フーリエ制限ノルム法を用いる. ただし, シュレディンガー方程式の解空間として U ² 型よりも広く V ² 型よりも狭い空間が必要となるため, V ² 型空間と時空ルベーグ空間の共通部分を用いるという工夫した. これにより, 臨界空間での適切性の結果が得られる. なお, 本研究は名古屋大学の津川光太郎氏との共同研究である.

本講演では, 放物型方程式の最大正則性と, それに関連の深いマルチンゲール の枠組みでの調和解析に関する最近の発展を自身の結果を交えて述べたい. まず, 最初に放物型方程式の最大正則性とは何かということを説明し, それと関連の深いフーリエ乗算作用素の有界性について述べる. これらに関する一連の理論は, 放物型方程式, 特に, ナヴィエ・ストークス方程式や, それの一般化であ る偏微分方程式の解の存在を示す際に有効に用いられるが, その分野における自身の貢献を述べる. 次に, 有限とは限らない測度空間でのマルチンゲールの枠組みを導入し, その枠組みにおける正作用素や極大作用素の重み付き評価に関する自身の貢献を述べる. 最後に両方の分野に関連する, 今後の課題について述べる. なお本講演の前半は, Helmut Abels 氏 (マックス・プランク応用数学研究所) との共同研究に基づき, 後半は田中仁氏 (東大数理) との共同研究に基づく.

本講演では 3 次の非線形項をもつ空間 3 次元の Klein-Gordon 方程式について考える. この方程式は基底状態解 (ground state) をもつことが知られている. Ibrahim, Masmoudi, Nakanishi は Kenig-Merle の手法を応用することにより, 実数値解において基底状態解よりエネルギーが小さい解は散乱するか, 爆発するかのいずれかであることが示された. またどちらになるかをある汎関数に初期値を代入したときの符号によって分類した.
　本講演ではこれの複素数値解への拡張について考える. 複素数においてはチャージを考えることにより, 定在波以下の解に対して散乱するか, 爆発するかのいずれかであることを示すことができ, さらにどちらになるかをある汎関数に初期値を代入したときの符号によって分類できた. 今回はこの結果を紹介する.

We consider the eigenvalues of the Laplace-Beltrami operator on a spherical cap under the Neumann condition. All the eigenvalues of the Laplace-Beltrami operator on the unit sphere are well-known. However, if we remove a small region from the unit sphere, then how are the eigenvalues affected by such a domain perturbation? We give an answer to this question and show how the eigenvalues behave as the region becomes smaller and smaller. Also, as related topics, we discuss nonlinear problem on a spherical cap from the bifurcation theoretic point of view. The ingredients of the talk is based on the joint works with Professors C. Bandle (Univ. Basel), T. Kawakami (Osaka Prefecture Univ.), A. Kosaka (Osaka City Univ.) and H. Ninomiya (Meiji Univ.).

This talk is based on my recent joint work with N. Visciglia, Pisa University. We revisit the classical approach by Brézis-Gallouët to prove global well-posedness for nonlinear evolution equations.

非線形シュレディンガー方程式のコーシー問題を考える. 非線形項として pure-power 型もしくは Hartree 型を与える. 初期値が負の可微分性をもつ場合の適切性に関してはほとんど結果が得られていなかったが, Hidano (2008), Fang-Wang (2011), Cho-Hwang-Ozawa (2013) らによって球対称性, もしくはある程度の角度正則性を仮定すれば適切性が得られることがわかっている. 本講演ではその手法と同様の手法によって, 既存の結果より非線形項の条件を緩和することができたことを紹介する.

We consider the initial value problem for nonlinear wave equation with weighted functions in one space dimension. Kubo & Osaka & Yazici (2013) showed that the solution exists globally in time if the initial data are odd functions. On the other hand, they showed that the solution blows up in finite time if the initial data are not odd functions. Howerver, the lifespan, which is the maximal existence time of solutions was not clarified in their result. Our aim in this talk is to show the sharp upper and lower bounds of the lifespan in such case.

The Dirac operator on Riemannian spin manifolds was introduced by Atiyah and Singer. In this talk, I would like to explain some results on resonances (poles of the resolvent operator) of the Dirac operator on hyperbolic and parabolic cylinders. Some parts of the contents are common in the previous talk at Himeji, February 20. However I expect to explain some preliminaries on spin geometry, and how to derive the expression of the Dirac operator this time.

We consider the derivation of the conservation law for nonlinear Schrödinger equations with non-vanishing boundary conditions at spatial infinity. For usual nonlinear Schrödinger equations with power nonlinearities, Ozawa (2006) derives conservation laws for a time local solution without approximating procedure. In this talk, combining the idea of Ozawa (2006) with decomposing the nonlinearity by applying the method for the decomposition of Schrödinger operator in Gérard (2006), we show that the conservation law is derived in a way independent of approximating procedure. As an application for the result, we remove some of the technical assumptions for the nonlinearity necessary to construct a time global solution.

In this talk we discuss the space of solenoidal vector fields in an unbounded domain Ω⊂R ⁿ, n≥2, whose boundary is given as a Lipschitz graph. It is shown that, under suitable functional setting, the space of solenoidal vector fields is isomorphic to the n−1 product space of the space of scalar functions. This result reveals a generic structure for the Stokes operator and the associated semigroup. Our result also covers the whole space case Ω=R ⁿ. This talk is based on a joint work with Yasunori Maekawa (Tohoku University).

本講演では Coriolis 力のある Navier-Stokes 方程式について, 回転軸を時間変動させた時の Sobolev 空間 H ^s (s>1/2) における適切性を紹介する. また鍵となる評価として, Van der Corput's lemma を用いて Coriolis 力に対する線形化方程式から時間減衰評価を示す.

In this talk we are interested in the radial solutions of semilinear wave equations with a supercritical nonlinearity, in dimension 3, and with initial data lying in the critical Sobolev space. It is well-known that one can construct solutions on a short time interval. Then next question is: what is the asymptotic behavior of these solutions? In other words do the solutions behave like free solutions or is there blow-up? In the latter case, the standard blow-up criterion says that a Strichartz-type norm must explode. In this talk we upgrade the standard blow-up criterion. We prove that if there is blow-up, then the critical Sobolev norm must also explode. (in collaboration with Thomas Duyckerts.)

We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in the whole space. We focus on the so-called critical Besov regularity framework. After recasting the whole system in Lagrangian coordinates, and working with ``the total energy along the flow’’ rather than with the temperature, we discover that the system may be solved by means of Banach fixed point theorem in a critical functional framework whenever the space dimension is greater than two. Back to Eulerian coordinates, this allows to improve the range of the Lebesgue exponent for which the system is locally well-posed, compared to previous results. This is a joint work with Raphaël Danchin (UPEC, LAMA).

非コンパクト型対称空間上のラプラシアンに対する散乱理論について紹介します. ラプラシアンのレゾルベントに対する極限吸収原理, レゾルベントと Poisson 作用素に対する無限遠での漸近展開, Helmholtz 方程式の解の特徴づけ等の散乱理論における基本的な定理について解説します. また, ランク１の場合に散乱行列と呼ばれるユニタリ作用素のスペクトルの性質について, 詳しく述べたいと思います.

幾何学的測度論で重要な基本概念であるハウスドルフ測度や可算修正可能集合について学び, それらを用いた枠組みで, 一般化された極小曲面の満たす性質についての概観を得ることを到達目標とする. 概ね以下のような流れで講義を進める.
　1. ハウスドルフ測度, 測度論の諸定理
　2. 可算修正可能集合, 概接空間
　3. 一般化された極小曲面, 単調性公式
　4. Allard の正則性定理
　5. varifold の理論

We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential. We prove that for a given small asymptotic profile, there exists a solution to the cubic nonlinear Schrödinger equation with delta potential which converges to given asymptotic profile in L² as t → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential.

We consider the initial-boundary value problem of the Navier-Stokes equations in a space of bounded functions. The L^∞-theory of the Navier-Stokes equations is important in order to investigate behavior of solutions near a blow-up time. So far, L^∞-type results are available for the whole space and a half space. In this talk, we consider the initial-boundary problem subject to the non-slip boundary condition and establish a blow-up rate for a certain class of domains including bounded and exterior domains.

非線形 Schrödinger 方程式の物理的背景や標準的な解法を学びつつ, 確率効果の付加による変化の一例に触れることを目的とする.

We study the k-summability of divergent formal solutions for the Cauchy problem of linear partial differential equations of first order with respect to t whose coefficients are polynomials in t. Our problem is reduced to k-summability of formal solutions for linear ordinary differential equations associated with the Cauchy problem. We employ the method of successive approximation in order to construct the formal solutions and to obtain the k-summability property. The talk is based on a joint work with Masatake Miyake (Professor Emeritus, Nagoya University).

本講演では, 空間 1 次元に於ける絶対値冪乗型の非線型項を伴う半相対論的方程式に対する初期値問題を考え, 特定の初期値に対する時間大域解の非存在に就いて紹介する. 特に, 質量が 0 である半相対論的方程式は非局所的な作用素である分数羃 Laplacian を伴うが, 半相対論的方程式を波動方程式に帰着し, test function method を用いる事で時間大域解の存在を否定する手法に就いて解説する. 猶, 本講演は早稲田大学の小澤徹氏との共同研究に基づく.

The boundary triples provide a powerful tool for the study of self-adjoint extensions of symmetric operators. This approach allows one to present the self-adjoint extensions as abstract boundary value problems and to reduce their study to the analysis of some operator-valued Herglotz functions which generalize the notion of the Dirichlet-to-Neumann map. The talk will give a review of the machinery and present some applications to the study of special classes of differential operators appearing in various domains of mathematical physics. A special attention will be given to recent applications such as quantum graphs.