名古屋微分方程式セミナー 2019年度

セミナー世話人：杉本充　菱田俊明　加藤淳　寺澤祐高

In this talk we discuss the gradient estimates for heat equation in an exterior domain. Our results describe the time decay rates of the derivatives of solutions to the heat equation. As an application, we also consider the fractional Leibniz rule for the Dirichlet or Neumann Laplacian on the exterior domain. This is based on the joint work with Vladimir Georgiev (University of Pisa).

Motivated by the two-dimensional nucleation in crystal growth phenomena, we consider the initial-value problem of the level-set mean curvature flow equation with discontinuous source terms. We discuss uniqueness and existence of viscosity solutions and study the asymptotic shape of solutions. A game-theoretic representation of solutions is also established. Applying this formula, we study the asymptotic speed of solutions. This talk is based on a joint work with K. Misu (Hokkaido University).

In this talk, we consider the local well-posedness for higher order Benjamin-Ono type equations, especially fourth order equations. The proof is based on the energy method with correction terms. Our equations have at most three derivatives in nonlinear terms, so that we need to cancel out all derivative losses by introducing correction terms into the energy. We also employ the Bona-Smith approximation technique in order to show the continuity of the flow and the continuous dependence.

We analyze the asymptotic behavior of solutions to wave equations with the strong damping term. When we impose additional weighted $L^1$ conditions on the initial data, a lower bound for the $L^2$ difference between the solution and the leading term can be obtained.

プラズマの運動は Euler-Poisson 方程式を用いて記述される. 一般的な状況では, 電子及び正イオンは, それぞれの連続の式, 運動方程式に従うため, 二流体方程式が用いられる. 一方, 特殊な状況下では, 電子密度は Boltzmann の関係式に従うとされる. この関係式の利点は, 電子密度を電位の関数として表現できる点にあり, プラズマの運動は, 正イオンに関する一流体方程式で記述できる. また, この関係式は, 二流体方程式において電子と正イオンの質量比を形式的に零とすれば得られる. 本講演では, この極限操作を数学的に正当化する.

We consider a Cauchy problem of the Boltzmann equation near the equilibrium without angular cutoff. In the known literature, $L^2$-based Sobolev and Besov spaces are used. In this talk, we will use the Wiener space, which is the set of functions whose Fourier coefficients absolutely converge. We will show the unique existence of a global solution for small data in this space. Also, we will see that the proof can be applied to a Cauchy problem of the Landau equation, which is closely related to the Boltzmann equation. This talk is based on a joint work with Renjun Duan (the Chinese University of Hong Kong), Shuangqian Liu (Jinan University), and Robert M. Strain (University of Pennsylvania).

講演では, 局所的に摂動された2層媒質からなる3次元空間における波動伝播の定常散乱問題に関する結果を報告する. 具体的には同媒質中での定常波動方程式の解の空間遠方での漸近解析とそれに基づく散乱振幅の導出である. 研究遂行上で技術的に工夫を要する点は, 屈折・透過波の存在から通常のフーリエ解析, 特に定常(停留)位相の方法の直接適用が困難であることから, 定常位相の方法などの基本的な道具立ての見直しを行うことである. 講演ではこの点についても極力語りたい. また, 合わせて本研究の拡張でもあり動機でもある自由境界3次元半空間での弾性波動との絡みについても言及したい. なお, この講演は磯崎洋氏 (筑波大名誉教授) と渡邊道之氏 (新潟大) との共同研究に基づく.

We consider asymptotic behavior (at the infinity) of slowly decaying positive solutions of quasilinear ordinary differential equations with critical coefficient functions. Equations under consideration are generalizations of those which are satisfied by radially symmetric solutions of pseudo-Laplace equations. When the coefficient functions do not have critical behavior, slowly decaying solutions all have algebraic decays. On the other hand, slowly decaying solutions all have logarithmic decays when the coefficient functions have critical behavior.

$\mathcal{R}$ 有界作用素の理論は線形放物型方程式系の初期値・境界値問題に関する最近理論であり, あまりなじみはないかもしれないが, 解析半群の理論の上位構造であり準線形放物型方程式系を解くための基本原理である. この講義では初期値・境界値問題のみ扱うが, 周期解についても自動的に扱える.

1. $\mathcal{R}$ 有界作用素の概念と作用素値フーリエ掛け算作用素
2. 最大正則性原理、解析半群
3. 自由境界問題の導出と線形化問題
4. 自由境界問題の時間局所解の一意存在
5. 線形化問題の解の時間減衰定理と自由境界問題の時間大域解の一意存在

We consider a semilinear heat equation without the self-similar structure. By focusing on some quasi-scaling property and its invariant integral, we develop a classification theory for the existence and nonexistence of local in time solutions, and then we discuss the existence of global in time solutions for small initial data. We also study the nonexistence of global in time solutions for nonnegative initial data. These results gives a generalization of the Fujita exponent for a semilinear heat equation with general nonlinearity. This talk is based on a joint work with N. Ioku (Ehime University).

非線形波動方程式の初期値問題を考察し, 以下の点を解説する.

1. 波動方程式の導出と解表示
2. 有限伝播性とホイへンスの原理
3. エネルギー評価
4. 非線形問題への応用
5. 局所エネルギーの減衰評価

We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banach's fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the $L^2(\mathbb{R}^d)$-framework.

We consider the stochastic quantization of the quantum field model with exponential interactions on the two-dimensional torus, which is called Hoegh-Krohn model. The model has been studied by Dirichlet forms. In this talk, we study the model by singular stochastic differential equations, which is recently developed. By the method, we construct the time-global solution and the invariant probability measure of the stochastic quantization, and see the relation to the process obtained by quasi-regular Dirichlet forms. This is a joint work with Masato Hoshino and Hiroshi Kawabi.

We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).

In this talk, we consider the Cauchy problem for semi-linear heat equations with exponential nonlinearity. The main purpose of this talk is to prove the existence of solutions lying on the borderline between global existence and blow-up infinite time. The existence has been shown for semi-linear heat equations with power type nonlinearity. We explain the main strategy to prove the existence. By using the definition of exponential function, we approximate the solution to exponential type equation by that of power type equation. Then we can use directly the knowledge for power type equation.

We discuss blow-up behavior for a semilinear heat equation with Sobolev supercritical power nonlinearity, focusing on the cases where the power is determined by Lepin or Joseph-Lundgren exponents. Based on matched asymptotic expansions and a priori estimates, I will introduce new examples of type II blow-up solutions satisfying various local-in-space estimates. In particular, several kinds of blow-up rates appear. They all differ from the ones in the previous studies. The construction also improves known results on classification of radial blow-up solutions.

The classical problem of a viscous flow past a fixed rigid obstacle is modeled by the steady incompressible Navier-Stokes equations. In this problem, it is known that the asymptotic behavior of the velocity field is described by the linearized problem around the velocity at infinity. The aim of this talk is to present the asymptotic behavior of the vorticity, which is surprisingly not characterized by the previous linearization in two dimensions, but by the linearization around a harmonic flow. If time permits, I will make a link to the Leray problem for large values of the fluxes, where a similar harmonic flow is also the main problem. This is joint work with Peter Wittwer.

$S_{0,0}$ 型クラスに対する双線形擬微分作用素の $L^2 \times L^2$ から $L^r$ への有界性について考察する. 宮地-冨田 (2013) は双線形ヘルマンダー・クラスに対して, $L^2 \times L^2$ から $L^1$ への有界性を示している. 本講演では, そのクラスをより広いシンボルクラスへと拡張し, 同様の有界性が成り立つこと, さらには, 写り先の空間は $L^1$ でなくともよい, ということについて報告したい. なお, 本講演は東京女子大学の宮地晶彦氏, 大阪大学の冨田直人氏との共同研究に基づく.

本講演では, 外部領域における Boussinesq 方程式の時間周期解の構成について述べる. Boussinesq 方程式は, 熱対流を記述した方程式で, Navier-Stokes 方程式と熱方程式の連立系である. 3 次元外部領域において, Yamazaki (2000) が Navier-Stokes 方程式の時間周期的な mild solutioon を構成した. 講演者は, 3 次元外部領域における Boussinesq 方程式の時間周期的な mild solution を Yamazaki の方法により構成した. またその際, 重力と温度の積による浮力を $L^1$ ノルムで評価した.

3 次元 Euclid 空間内の滑らかでコンパクトな曲面を境界に持つ外部領域において, $L^{r}$-ベクトル場の de Rham-Hodge-Kodaira 型分解定理を考察する. ベクトル場の境界条件は, 境界に接するものと直交するものの 2 種類を対象とする. まず最初に, これらの調和ベクトル場の空間が, 共に有限次元であることを示す. 有界領域の場合と異なり, 次元数は可積分指数 $r$ によって異なることも明らかにする. その際, 外部領域に固有の位相不変量である Betti 数を定義し, 次元数との関連についても言及する. 次に任意の $L^{r}$-ベクトル場が, 調和部分とベクトルポテンシャル, スカラーポテンシャルのそれぞれの回転と勾配の和で表現できることを証明する. ただし, その分解の一意性, すなわち直和分解の正当性については, 調和部分の境界条件と可積分指数 $r=3$ を閾値として分類がなされる. 本講演の内容は, M. Hieber 教授 (Darmstadt 工科大, ドイツ)，A. Seyfert 博士 (同工科大)，清水扇丈教授 (京都大), 柳澤卓教授 (奈良女子大) との共同研究に基づくものである.

In this talk we discuss dynamical stability of multiply covered circles under the surface diffusion flow. To this end we first establish a general form of the isoperimetric inequality for immersed closed curves under rotational symmetry, and then apply it to obtaining a certain class of rotationally symmetric initial curves from which solutions to the surface diffusion flow exist globally-in-time and converge to multiply covered circles. This talk is based on a joint work with Prof. Okabe at the Tohoku University.

In this talk, we consider Dirichlet forms given by two-dimensional space-time quantum fields with interactions of exponential type, called ${\exp}(\Phi)_{2}$-measures (i.e., Hoegh-Krohn's model) in Euclidean quantum field theory, in finite volume. We prove strong uniqueness of the corresponding Dirichlet operator and construct a unique solution of the modified-stochastic quantization equation under suitable conditions on the charge constant and the regularization parameter. This talk is based on a joint work with Sergio Albeverio, Stefan Mihalache and Michael Röckner.