京都大学NLPDEセミナー

2018年度のセミナーの記録

4 月 13 日: 岩渕司氏（東北大学）　　Tsukasa Iwabuchi (Tohoku University)

● 2018 年 4 月 13 日　(Fri)　15:30 ～ 17:30

講演者: 岩渕司氏（東北大学大学院理学研究科）
Tsukasa Iwabuchi (Tohoku University)
講演題目: Besov spaces on open sets with the Dirichlet boundary condition and their application to the fractional Laplacian
講演要旨: In this talk, we introduce the Besov spaces associated with the Dirichlet Laplacian on an arbitrary open set. Basic properties such as the completeness, the embedding theorem, e.t.c. are shown. As an application, we will study the semigroup generated by the Dirichlet Laplacian of fractional order.

● 2018 年 4 月 20 日　(Fri)　15:30 ～ 17:30

講演者: 中西賢次氏（京都大学数理解析研究所）
Kenji Nakanishi (Kyoto University)
講演題目: Small data scattering for the Gross-Pitaevskii equation in the 3D energy space with angular regularity
講演要旨: This talk is based on joint work with Zihua Guo and Zaher Hani. We study the asymptotic stability of plane wave solutions to the defocusing cubic nonlinear Schrödinger equation for initial perturbation with small energy in three space dimensions. It is known that the very long range interaction with the plane wave generates linear and quadratic modifications in the dispersive asymptotic evolution of the disturbance compared with the free Schrödinger equation. We prove the modified scattering for small initial data in the energy space with some angular regularity, extending the previous result with Gustafson and Tsai to a larger space, as well as the recent one by Killip, Murphy and Visan. The proof is different from them, exploiting the Strichartz estimate with angular average together with a quadratic transform. The latter is also different from the previous versions, with subtle cancellation in higher order terms.

● 2018 年 4 月 27 日　(Fri)　15:30 ～ 17:30

講演者: 寺本有花氏（九州大学大学院数理学府）
Yuka Teramoto (Kyushu University)
講演題目: On the stability of stationary solutions of artificial compressible system
講演要旨: In this talk we consider the stability of stationary solutions of the incompressible Navier-Stokes system and the corresponding artificial compressible system. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero limit of the artificial Mach number $\epsilon$ which is a singular limit. It is shown that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small $\epsilon$. The result is applied to the Taylor problem. In general, the range of $\epsilon$ for which the above mentioned stability result holds shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of $\epsilon$ can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.

● 2018 年 5 月 11 日　(Fri)　15:30 ～ 17:30

講演者: 川上竜樹氏（龍谷大学理工学部）
Tatsuki Kawakami (Ryukoku University)
講演題目: Critical Fujita exponents for semilinear heat equations with quadratically decaying potential
講演要旨: We study the existence/nonexistence of global-in-time positive solutions of the Cauchy problem of semilinear heat equations with potential $V$. It is well known that the critical Fujita exponent, which separate that the problem possesses global-in-time solutions or not, depends on the behavior of the potential $V$. In particular, the case where $V$ decays quadratically at the space infinity is on the borderline where the critical Fujita exponent can vary in $(1,\infty]$, and there are several partial results. In this talk we identify the critical Fujita exponent for the case where $V$ is a radially symmetric potential decaying quadratically. The identification of the critical Fujita exponent for this problem is a delicate issue, in particular, when the Schrödinger operator $L_V:=-\Delta+V$ on $L^2$ is critical. Indeed, in the critical case, the critical Fujita exponent is different from previous results and it depends on a new threshold number.
This talk is based on a joint work with Kazuhiro Ishige (University of Tokyo).

● 2018 年 5 月 18 日　(Fri)　15:30 ～ 17:30

講演者: 関行宏氏（九州大学大学院数理学研究院）
Yukihiro Seki (Kyushu University)
講演題目: 球面に値を取る調和写像流方程式の解の爆発について
On blow-up for harmonic map heat flow to a sphere
講演要旨: 球面に値を取る調和写像流方程式は極座標系を介してスカラー値の半線形熱方程式に帰着される．近年，有限時間に現れる解の特異性やその延長について活発に研究されている．特に Bizon--Wasserman (2015) は高次元において後方自己相似解の非存在を示し，Struwe (1988) の一般的結果と合わせて解の爆発が自己相似的とならないことを証明した．彼らの議論は背理法によるため爆発解の詳細な挙動については言及していない．本講演では藤田方程式に対して知られている手法を精密化した形で応用し，典型的な解の構成法を述べる．さらにその解の爆発率が方程式の自己相似的性から決まるオーダーとどれほどずれるかを述べ，特異点周りで詳細な各点評価について報告する．本研究の一部は P. Bienart 氏（Bonn大学）との共同研究を含む．

● 2018 年 6 月 1 日　(Fri)　15:30 ～ 17:30

講演者: 溝口紀子氏（東京学芸大学教育学部）
Noriko Mizoguchi (Tokyo Gakugei University)
講演題目: Finite-time blowup in a Cauchy problem of parabolic-parabolic chemotaxis system
講演要旨: This talk is concerned with blowup in a parabolic-parabolic system describing chemotactic aggregation in the whole plane. Although some results on blowup in the system were given in a disk, there have been no results in the whole plane except existence of a special solution constructed by Schweyer. In the case of a disk, solutions blow up in finite time if their initial energy is less than some specific value. On the other hand, the energy diverges to $-\infty$ as time goes to $+\infty$ for any forward selfsimilar solution in the plane. This implies that one cannot expect to get a criterion for finite-time blowup using energy. For a solution $(u,v)$, $u$ and $v$ denote density of cells and of chemical substance, respectively. Let $\tau$ be the coefficient of time derivative of $v$. I will talk that there exists $M(\tau)>0$ with $M(\tau) \to \infty$ as $\tau \to \infty$ such that all solutions $(u,v)$ with initial mass of $u$ larger than $M(\tau)$ blow up in finite time.

● 2018 年 6 月 8 日　(Fri)　15:30 ～ 17:30

講演者: 上田好寛氏（神戸大学大学院海事科学研究科）
Yoshihiro Ueda (Kobe University)
講演題目: Characterization of the dissipative structure for the symmetric hyperbolic system with non-symmetric relaxation
講演要旨: In this talk, we discuss the dissipative structure for a hyperbolic system with relaxation. If the relaxation term of the system has symmetric property, Shizuta-Kawashima(1984,1985) introduced a stability condition which induces the decay estimate for the solution of the Cauchy problem. However, there are some complicated physical models which possess a non-symmetric relaxation term and we can not apply this stability condition to these models. Under this situation, our purpose of this talk is to extend the stability condition for complicated models and get the quantitative decay estimate. Furthermore, we shall explain the new dissipative structure by using the several concrete examples.

● 2018 年 6 月 15 日　(Fri)　15:30 ～ 17:30

講演者: 戍亥隆恭氏（大阪大学大学院理学研究科）
Takahisa Inui (Osaka University)
講演題目: Strichartz estimates for the damped wave equation and its application to the nonlinear problem
講演要旨: In this talk, we consider the damped wave equation (DW). The $L^p$-$L^q$ type estimate was firstly obtained by Matsumura in 1976. After his work, many researchers have obtained such type estimates. However, there are less results for the space-time estimates, which are so called Strichartz estimates. Recently, Watanabe (2017) proved the Strichartz estimates for DW in the low space-dimensional case. In this talk, I show the Strichartz estimates in the higher dimensional case. Moreover, we consider the energy critical nonlinear damped wave equation (NLDW). Precisely, we discuss the local well-posedness, the decay property, and the finite time blow-up of the solutions to NLDW.

● 2018 年 6 月 22 日　(Fri)　15:30 ～ 17:30

講演者: 石渡通徳氏（大阪大学大学院基礎工学研究科）
Michinori Ishiwata (Osaka University)
講演題目: On the soliton-resolution of time-global solutions for the energy critical heat equation
講演要旨: In this talk, we are concerned with the soliton-resolution of time-global solutions for the energy critical semilinear heat equation. Such resolution is known for e.g., radially symmetric, nonnegative time-global solutions along full-time orbits as well as along some time sequences for general time-global solutions. Previous results for radially symmetric solutions heavily rely on the intersection-comparison principle to have a precise information on the behavior of solutions. In this talk, we take a different approach which depends on the profile-decomposition of the critical Sobolev embedding introduced by Gerard together with the variational type argument. The corresponding result for finite-time blow up solutions together with the connection with types of blow-up phenomena will be also discussed.

● 2018 年 6 月 29 日　(Fri)　15:30 ～ 16:30

講演者: Yannick Sire 氏（Johns Hopkins University）
講演題目: Extremal metrics on manifolds
講演要旨: I will report on some recent results on extremal metrics in confomal classes for the spectrum of the laplace-beltrami operator or other differential operators. Extremal metrics are important since they produce minimal surfaces into spheres and encode topology and geometry of the manifold. I will also describe the case of Kahler manifolds.

● 2018 年 7 月 13 日　(Fri)　15:30 ～ 17:00

講演者: Jalal Shatah 氏（New York University）
講演題目: Wave turbulence for the nonlinear Schrödinger equation
講演要旨: Wave turbulence describes the out-of-equilibrium statistical mechanics of weakly nonlinear waves. It was first put forward by Peierls in 1928, in the context of solid state physics, and then by Hasselman in 1962, in the context of water waves. The theory was invigorated and investigated by Zakharov and his school in the 70's. The theory of weak turbulence proposed by Zakharov relies on the derivation of a kinetic wave equation. In this talk we will derive the kinetic equation for the nonlinear Schrödinger equation and discuss the time scale where the kinetic equation is a valid approximation.

● 2018 年 7 月 20 日　(Fri)　15:30 ～ 17:30

講演者: 木下真也氏（名古屋大学大学院多元数理科学研究科）
Shinya Kinoshita (Nagoya University)
講演題目: The Cauchy problem for the 2D Zakharov-Kuznetsov equation
講演要旨: 本講演では，2 次元のザハロフクズネツォフ方程式の初期値問題について考える．ザハロフクズネツォフ方程式の特徴的な点はよく知られたシュレディンガー方程式などとは異なり，線形部の相関数が球対称とならないことである．このことが非線形項の評価において困難さを産み，正則性が低い空間での適切性の研究を阻害していた．このたび，超曲面上の合成積評価と呼ばれる評価を利用して危険な非線形項相互作用を捉え評価し，Whitney 型の空間分割と組み合わせることで既存の非線形評価式を改良することができたので紹介する．

● 2018 年 10 月 19 日　(Fri)　15:30 ～ 17:30

講演者: 小薗英雄氏（早稲田大学理工学術院・東北大学数理科学連携センター）
Hideo Kozono (Waseda University / Tohoku University)
講演題目: Harmonic vector fields in $L^r$ on 3D exterior domains
講演要旨: In this talk, we characterize the space of harmonic vector fields in $L^r$ on the 3D exterior domain with smooth boundary. There are two kinds of boundary conditions. One is such a condition as the vector fields are tangential to the boundary, and another is such one as those are perpendicular to the boundary. In bounded domains both harmonic vector spaces are of finite dimensions and characterized in terms of topologically invariant quantities which we call the first and the second Betti numbers. These properties are closely related to characterization the null spaces of solutions to the elliptic boundary value problems associated with the operators div and rot. We shall show that, in spite of lack of compactness, spaces of harmonic vector fields in $L^r$ on the 3D exterior domain are of finite dimensions and characterized similarly to those in bounded domains. It will be also clarified a significant difference between interior and exterior domains in accordance with the integral exponent $1 < r < \infty$. This is based on the joint work with Profs. Matthias Hieber, Anton Seyferd, Senjo Shimizu and Taku Yanagisawa.

● 2018 年 10 月 26 日　(Fri)　15:30 ～ 17:30

講演者: 千頭昇氏（大阪大学大学院基礎工学研究科）
Noboru Chikami (Osaka University)
講演題目: Global well-posedness of the compressible Navier-Stokes-Korteweg system
講演要旨: 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 viscous fluids. This enables us to apply Banach's fixed point theorem to show the existence of global solution. If time allows, we also comment on the decay estimates of the solution. This talk is based on a joint work with T. Kobayashi (Osaka University).

● 2018 年 11 月 2 日　(Fri)　15:30 ～ 17:30

講演者: 山崎陽平氏（広島大学）
Yohei Yamazaki (Hiroshima University)
講演題目: Center stable manifolds around line solitary waves of the Zakharov--Kuznetsov equation
講演要旨: In this talk, we consider center stable manifolds of unstable line solitary waves for the Zakharov--Kuznetsov equation on a cylindrical space and show the stability of the unstable line solitary waves for initial datum on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. To construct the center stable manifolds, we apply Hadamard's graph transform approach by Nakanishi--Schlag'12. To treat the nonlinear term of the Zakharov--Kuznetsov equation, we use the bilinear estimate on Fourier restriction spaces by Molinet--Pilod'15. Since generalized eigenfunctions of the dual operator of the linearized operator of the Zakharov--Kuznetsov equation around line solitary waves with respect to the 0 eigenvalue is not L^2 function, we can not show directly the estimate to get a contraction map on a set of graphs by using the mobile distance which was introduced by Nakanishi--Schlag'12. Modifying the mobile distance by Nakanishi--Schlag'12, we construct a contraction map on the graph space.

● 2018 年 11 月 9 日　(Fri)　15:30 ～ 16:30

講演者: Robert L. Jerrard 氏（University of Toronto）
講演題目: Interface dynamics in semilinear wave equations
講演要旨: Numerous results of many authors have established deep connections between the parabolic Allen-Cahn equation and motion by mean curvature. We will discuss the little that is known about similar relations between semilinear wave equations and a natural hyperbolic analogue of the mean curvature flow. In particular, we will review some older results and possibly relevant geometric measure theory constructions, and we will describe a new, constructive approach to these problems (joint work with M del Pino and M Musso).

● 2018 年 11 月 16 日　(Fri)　13:30 ～ 14:30 / 17:00 ～ 18:00

講演者: Leonardo Tolomeo 氏（University of Edinburgh / MIGSAA）
講演題目: 【13:30--14:30】 Global well-posedness of the two-dimensional cubic stochastic nonlinear wave equation
【17:00--18:00】 Ergodicity for stochastic dispersive equations
講演要旨: 　【13:30--14:30】
In this talk, we consider the Cauchy problem for the defocusing cubic stochastic nonlinear wave equation (SNLW) with additive space-time white noise forcing, posed in two spatial dimensions, both on the torus and on the Euclidean space. Because of the roughness of the forcing, we will use a modified version of the I-method to get global well posedness for this equation on the torus. Furthermore, we will discuss how to combine this argument with finite speed of propagation to get the same result on R2.

　【17:00--18:00】
In this talk, we study the long time behaviour of some stochastic partial differential equations (SPDEs). After introducing the notions of ergodicity, unique ergodicity and convergence to equilibrium, we will discuss how these have been proven for a very large class of parabolic SPDEs. We will then shift our attention to dispersive SPDEs, where the general strategy for the parabolic case fails. We will describe this failure for the wave equation on the 1-dimensional torus and present a recent result that settles unique ergodicity even in this case.

● 2018 年 11 月 30 日　(Fri)　15:30 ～ 16:30

講演者: Chao-Jiang Xu 氏（Rouen University）
講演題目: Some recent progress on the mathematical analysis of the Prandtl boundary layer equation
講演要旨: In 1904, Prandtl said that, in fluid of small viscosity, the behavior of fluid near the boundary is completely different from that away from the boundary. Away from the boundary part can be almost considered as ideal fluid, but the near boundary part is deeply affected by the viscous force and is described by Prandtl boundary layer equation which was firstly derived formally by Prandtl. From the mathematical point of view, the well-posedness and justification of the Prandtl boundary layer theory don’t have satisfactory theory yet. In this talk, we present some recent progress on the mathematical analysis of the Prandtl boundary layer equation. By using energy method, we study the well-posedness of Cauchy problem and the smoothness effect of solutions for Prandtl equations in Sobolev space.

　[1] R. Alexandre, Y.G. Wang, C.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc. 28 (2015), 745-784.
　[2] W.-X. Li, D. Wu and C.-J. Xu, Gevery class smoothing effect for the Prandtl equation, SIAM J. Math. Anal. 48 (2016), 1672-1726.
　[3] C.-J. Xu and X. Zhang, Long time well-posdness of the Prandtl equations in Sobolev space, J. Differential Equations 263 (2017), 8749-8803.
　[4] W.-X. Li, V.-S. Ngo and C.-J. Xu, Boundary layer analysis for the fast horizontal rotating fluid, Preprint.

● 2018 年 12 月 7 日　(Fri)　15:30 ～ 16:30

講演者: Weiping Yan 氏（Xiamen University）
講演題目: Nonlinear stability of infinite energy blowup solutions for the three dimensional incompressible Navier-Stokes equations
講演要旨: In this talk, we first introduce two family of new explicit finite time blowup solutions for the three dimensional incompressible Navier-Stokes equations. One family of those solutions admit the smooth initial data and infinite energy. After that, we prove those finite time blowup solutions are Lyapunov nonlinear stability in bounded domain with smooth boundary and Dirichlet boundary condition. This result tells us the three dimensional incompressible Navier-Stokes equations in bounded domain with smooth boundary and Dirichlet boundary condition admits a family of stable blowup solutions with large smooth initial data and finite energy.

● 2018 年 12 月 14 日　(Fri)　15:30 ～ 17:30

講演者: 瓜屋航太氏（岡山理科大学）
Kota Uriya (Okayama University of Science)
講演題目: Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions
講演要旨: We consider the asymptotic behavior of the complex-valued solution to the quadratic nonlinear Klein-Gordon equation in two dimensions. As in the real-valued case, we construct a solution which tends to a free solution with a logarithmic phase correction. However, the phase correction is determined by not only the nonlinearity but also the ratio of the amplitude of two waves reflecting the fact that asymptotic behavior of the solution to the free Klein-Gordon equation consists of two waves. This is a joint work with J. Segata (Tohoku University) and S. Masaki (Osaka University).

● 2018 年 12 月 21 日　(Fri)　15:30 ～ 17:30

講演者: 杉山裕介氏（滋賀県立大学）
Yuusuke Sugiyama (University of Shiga Prefecture)
講演題目: Loss of strict hyperbolicity in finite time for 1D quasilinear wave equations
講演要旨: 本発表では，まず 1 次元双曲型保存系と同値な準線形波動方程式の初期値問題を考察する．伝播速度が未知関数に依存するこのモデルにおいては，伝播速度が初期時刻で「適当な正定数以上である」（方程式の非退化性）という条件を仮定する．この条件によって，方程式が「strictly hyperbolic」となり一意に解くことができる．しかしながら，時刻 0 で方程式が退化していなくても，有限時間で「方程式の退化」が起こることがある．一般に，「strict hyperbolicity」が失われた方程式においては，いわゆる「persistence of regularity」が保証されない．この講演では，まず「方程式の退化」が起こるための条件を与える．退化する点での正則性に関する最近の研究結果，関連する方程式の結果などを述べ，証明の概要を説明する．

● 2019 年 1 月 11 日　(Fri)　15:30 ～ 17:30

講演者: 岡部考宏氏（大阪大学大学院基礎工学研究科）
Takahiro Okabe (Osaka University)
講演題目: Remark on the local solvability of the Navier-Stokes flow in the weak $L^n$ space
講演要旨: We consider the incompressible Navier-Stokes equations on the whole space $R^n$, $n\geq 3$. Especially, we consider the Cauchy problem in the frame work of the weak Lebesgue space $L^{n,\infty}(R^n)$ with scale invariant forces in $BC([0,T);L^{n/3, \infty}(R^n))$. It is well known that $C^{\infty}_0(R^n)$ is not dense in $L^{n,\infty}(R^n)$ and that the Stokes semigroup is not strongly continuous at $t=0$. For this difficulty, the existence of local mild solutions and its uniqueness are not yet completely clarified in $L^{n,\infty}(R^n)$. Firstly we consider the local existence of weak mild solutions of (N-S) with restriction of the singularity of initial data and external forces. Due to the local solution, we may apply it to the uniqueness theorem within the weak mild solutions in the class $BC([0,T);L^{n,\infty}(R^n))$. As an application of the uniqueness theorem, we are able to investigate the regularity of weak mild solutions by the singularity of initial data.

● 2019 年 1 月 18 日　(Fri)　15:30 ～ 17:30

講演者: 張龍傑氏（東京大学）
Longjie Zhang (University of Tokyo)
講演題目: Curvature flow with driving force on fixed boundary points
講演要旨: In this research, we consider the curvature flow with driving force on fixed boundary points in the plane. We give a general local existence and uniqueness result of this problem with $C^2$ initial curve. For a special family of initial curves, we classify the solutions into three categories. Moreover, in each category, the asymptotic behavior is given.

● 2019 年 1 月 25 日　(Fri)　15:30 ～ 17:30

講演者: 杉山由恵氏（大阪大学大学院情報科学研究科）
Yoshie Sugiyama (Osaka University)
講演題目: On the ε-regularity theorem for the Keller-Segel systems and its application to the analysis of singular sets
講演要旨: 　移流拡散方程式に分類される Keller-Segel 方程式系は多くのパラメータを有し，その取り方によって半線形型，退化型，特異型が現れる豊富な構造を内在している．特に，初期条件が小さい場合には時間大域解が存在し，初期条件が大きいときには，初期凝集に依存してその解は有限時刻で爆発し得る．
　本講演では，半線形型をした放物-楕円型 Keller-Segel 方程式系に焦点を絞り，測度値関数まで解のクラスを広げて“適切性”を取り扱う．また，特異点の存在を論じ，この特異点集合の有する位相的構造に関して最近得られた研究成果を報告する．尚，本研究は J. Choi （KIAS，韓国）と三浦正成氏（大阪大学）との共同研究に基づく．

別の年度

〒606-8502 京都市左京区北白川追分町
京都大学大学院理学研究科数学教室（理学研究科 3 号館）

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京都大学 NLPDE セミナー

2018年度のセミナーの記録