セミナー世話人：杉本充 菱田俊明 津川光太郎 加藤淳
In this talk, we consider the existence of positive solutions to the semilinear elliptic equation involving Sobolev-Hardy critical terms. In particular, we investigate the equation having two different kinds of non-compact terms, that is, the Sobolev critical term and the Sobolev-Hardy critical term. Of course, the corresponding energy functional can not satisfy the Palais-Smale condition in general. However, we will show that there exists a threshold number such that if the min-max value can be taken strictly smaller than it, the min-max value becomes a critical value with a suitable bounded domain. More precisely, it turns out that the threshold number can be characterized by a least-energy positive solution on the half space.
非線形分散型方程式は散乱・爆発・孤立波など様々な大域挙動の解を持つ事が知られている。 本研究の目標は一般解の大域挙動を分類し、また初期値からそれを予測する事である。 この講演では、集約性の非線形クラインゴルドン方程式について、基底状態（正値定常解）の エネルギーを少し超えるまでの全ての解が、基底状態から生成する進行波解の周りの 中心安定多様体・中心不安定多様体による分割で、９通りの大域挙動へ分かれることを示す。 特にその内の二つは散乱から爆発へ（及びその逆）の遷移を示す解の開集合であり、 このような解の存在はこれまで証明されていなかった。 また、中心（不）安定多様体は励起状態の進行波解に対しても構成できる。この講演は Wilhelm Schlag との共同研究に基づくものである。
We discuss diffusion-approximation for a random nonlinear Schrodinger equation. Continuity of solutions with respect to Brownian paths is the key property and this property is proved by a PDE method in the construction of the fundamental solution.
本講演の目的は、多変数ウェーブレット変換（1995，講演者） と古典的なラドン変換を組み合わせて、新たな逆変換公式を与 えることである。これは、Donoho-Candes（1998）によって 与えられたリジレット変換による公式の自然な局所化になって いる。局所リジレット空間の定義と特徴付けが与えられる。
|u|^{α-1}u 型の非線形項を持つ1次元シュレディンガー方程式を考える。 ここで1<α<5とする。初期値がL^p-空間に属するときにこのコーシー問題 の大域解を構成できるか？という問題に興味がある。本講演では主にp>2の場合の結果について述べる。この場合においては、初期値のフーリエ変換がL^{p’}-空間に属しているとき、 "admissibleではない" 一般化されたStrichartz評価を用いて 特定の範囲にあるpに対して大域解を構成することができる。堤正義氏との共同研究。
In this talk, we consider the existence of positive solutions to the semilinear elliptic equation involving Sobolev-Hardy critical terms. In particular, we investigate the equation having two different kinds of non-compact terms, that is, the Sobolev critical term and the Sobolev-Hardy critical term. Of course, the corresponding energy functional can not satisfy the Palais-Smale condition in general. However, we will show that there exists a threshold number such that if the min-max value can be taken strictly smaller than it, the min-max value becomes a critical value with a suitable bounded domain. More precisely, it turns out that the threshold number can be characterized by a least-energy positive solution on a half space.
We discuss the Cauchy problems for Navier-Stokes equations in (homogeneous) weak Herz spaces. In particular, we construct the global solutions with small initial data, and prove the uniqueness of global solutions with large data. Also, we give several embedding relations of weak Herz spaces into Besov spaces.
In this talk, we consider the propagation of singularities for the Schr\"odinger equation of a free particle and with some potential including the one of harmonic oscillator. We determine the positions of the singularities of the solutions from the information of the initial data.
We shall present recent progress in the understanding of the spectral and subelliptic properties of non-elliptic quadratic operators. We shall then explain how these results allow to describe the spectral and pseudospectral properties for some classes of semiclassical non-selfadjoint pseudodifferential operators with double characteristics in a neighborhood of their doubly characteristic sets.
We give a decomposition formula of formal solution of the Cauchy problem for a quasi homogeneous partial differential equation with constant coefficients in two dimensional complex plane. The decomposition formula is given in a form associated with the factorization of the operator which is similar with decomposition of solution of an ordinary differential equation with constant coefficients. This is a joint work with M. Miyake.
We consider the Navier-Stokes equations in a smooth domain, which is unbounded and has even unbounded boundary. In such domains, the Helmholtz-decomposion fails to hold in L^q in general, so in particular the Stokes operator cannot be well-defined. However, as pointed out by Farwig, Kozono, Sohr in 2007, if we consider spaces of functions having local L^q behaviour and L^2 decay, the Stokes operator can be defined and has maximal regularity. In the talk we will construct very weak solutions to the Navier-Stokes equations using those modified L^q spaces. These solutions can be used to prove local regularity properties of weak solutions in the sense of Leray and Hopf.
We present a uniqueness theorem for almost periodic-in-time solutions to the Navier-Stokes equations in $3$-dimensional unbounded domains. Thus far, uniqueness of almost periodic-in-time solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small almost periodic-in-time solution in $BC(R;L^3_w)$ within the class of solutions which have sufficiently small $L^{\infty}( L^3_w)$-norm. In this talk, we show that a small almost periodic-in-time solution in $BC(R;L^3_w \cap L^{6,2})$ is unique within the class of all almost periodic-in-time solutions in $BC(R;L^3_w \cap L^{6,2})$. The proof of the present uniqueness theorem is based on the method of dual equations. This is a joint work with R. Farwig.
本講演では, 拡散係数が十分小さい場合の半線形熱方程式の解の爆発集合の位置について述べる. 拡散係数が小さい場合には, 拡散項に比べて非線形項の効果が強くなるため, 解は有限時間で必ず 爆発するが, 爆発集合の位置には, 熱方程式の解の時間局所的な挙動, すなわち拡散項の影響が 強く現れる. 特に, 解は初期値の最大点近くでのみ爆発し, さらに初期値に最大点が複数点ある場合でも, 最大点近くでの初期値の形状によって, 爆発集合の位置を調べることができることを示す. なお, 本講演の内容は東北大学の石毛和弘先生との共同研究によるものである.
Modulation空間は1980年頃, H.G.Feichtinger により導入されたユークリッド空間上の函数空間の1つであり, 特に, 時間周波数解析の分野において函数または超函数の時間周波数分布を測るために用いられてきた. 最近では, 擬微分作用素の有界性や偏微分方程式の解の適切性の研究などにも応用されている. 本講演では, modulation空間とその応用について, 講演者のこれまでの研究およびそれらに関連した最近の研究について解説したい.
We consider a nonsymmetric operator $A_P$ in $\{ L^2(0, \infty )\}^2$. defined by differential expression $$ (A_P u)(x)=B u^{\prime} (x) + P(x)u(x), \quad 0 < x < \infty $$ where $$ B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ \end{matrix} \right), P(x)=\left( \begin{matrix} p_{11}(x) & p_{12}(x) \\ p_{21}(x) & p_{22}(x) \\ \end{matrix} \right), $$ with the domain $$ D = \{ u(x)= \left( \begin{matrix} u_1(x) \\ u_2(x) \\ \end{matrix} \right) \in \{H^1(R_{+} )\}^2; \; u_1(0)=h u_2(0) \}. $$ An inverse problem of reconstruction of complex-valued coefficients $p_{ij}(x)$ from the scattering data of operator $A_P$ is investigated. This is a joint work with M.Yamamoto.
Part I of the talk are demonstrated the possibilities of the self similar, approximately self similar approaches to the studying of properties of reaction- diffusion systems with double nonlinearity under an action of a convective transfer and to visualization. It is proved that there exist some values of parameters when the effect of finite velocity of perturbations, localization of solution, onside localization, the effect of “wall”, blow up have place. It is proved Fujite Samarskii type global solvability of solution for the Cauchy’s problem. Kolmogorov- Fisher type estimate of the solution for the problem of biological population. For construction of self similar, approximately self similar equation the method of the nonlinear splitting (decomposition) is offered. An action of a convective transfer to evolution of studied process is analyzed. Based on the self similar, approximately self similar solutions the numerical analysis and visualization of nonlinear reaction-diffusion processes were carried out. The results of numerical experiments had showed the affectivity of this approach to studying of the nonlinear reaction diffusion processes.
Part II of the talk is devoted to asymptotical behavior of the self similar solution of the Klein-Gordon equation and quasilinear system of generalized Klein Gordon type. For construction of self similar, approximately self similar equation the method of the nonlinear splitting (decomposition) is offered. The asymptotes of blow up solutions are considered. It is established asymptotes of solution depending on values of nonlinear parameters.
We give a fairly general class of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets of the phase space. Our class of functionals is closed under addition and multiplication because we exclude some basic functionals from our class. The use of piecewise bicharacteristic paths naturally leads us to the semiclassical approximation on the phase space.
実ユークリッド空間上の Bargmann 変換とよばれるフーリエ積分作用素の像とし て特徴づけられる Segal-Bargmann 空間とよばれる再生核 Hilbert 空間に作用 する Berezin-Toeplitz 作用素の有界性等について, 相関数から定まる正準変換 や熱流を用いて得られたささやかな結果について述べる.
In this talk, we will show the dispersive and Strichartz estimates for wave equation with a potential to the initial-boundary value problem in exterior domains. For this purpose, the fundamental tool is the generalized Fourier transform, especially, the differentiability property of the resolvent of the Schr\"odinger operator and resolvent expansion in low frequencies.