第35回セミナー(臨時) |
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日 時: |
平成29年(2017年)1月12日(木)15:20~17:30 |
場 所: |
大阪府立大学 中百舌鳥キャンパス 数理工学科B9棟111号教室 |
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15:20-16:20 |
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講 演 者: |
壁谷 喜継 氏 (大阪府立大学) |
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タ イ ト ル: |
Positive Schr\"odinger operators with the inverse square potential
and related parabolic problems |
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アブストラクト: |
We consider positive radial solutions to $-\Delta u + V(|x|)u=0$ in ${\mathbb
R}^N$ with radial potential $V$ which may have singularity at the origin
and decays in the inverse square way at infinity. We consider properties
of the operator $H:=-\Delta +V$ and give the precise behavior of solutions
according to the properties of $H$. If $V$ is bounded near the origin,
M. Murata studied properties of the operator $H$ intensively. We here generalize
his results to potentials with a singularity at the origin. If I have time,
I will discuss properties of the corresponding parabolic problems. This
talk is based on the joint work with Professors Kazuhiro Ishige (Tohoku
University, Japan) and El Maati Ouhabaz (Universit\'e de Bordeaux, France). |
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16:30-17:30 |
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講 演 者: |
Vitaly Moroz 氏 (University of Swansea) |
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タ イ ト ル: |
Choquard equations: a survey |
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アブストラクト: |
Choquard equation is a stationary nonlinear Schrodinger type equation
where the nonlinearity is coupled with an attractive nonlocal Coulombic term.
We present a survey of old and recent results on the Choquard equation.
This is a joint work with Jean Van Schaftingen (Louvain-la-Neuve, Belgium). |
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第34回セミナー |
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日 時: |
平成28年(2016年)12月10日(土)14:00~17:30 |
場 所: |
大阪市立大学(杉本キャンパス)理学部E棟数学講究室(E408号室) |
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14:00-15:00 |
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講 演 者: |
三浦 達哉 氏 (東京大学大学院 数理科学研究科) |
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タ イ ト ル: |
Elastic curves and phase transitions |
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アブストラクト: |
We consider the minimizing problem of the modified total squared curvature defined for planar curves with clamped open ends.
This minimizing problem invokes a boundary value problem concerning Euler's elastica.
In this talk, I will indicate a new theoretical connection
between such an elastic problem and the theory of phase transitions.
Regarding the higher order term as a singular perturbation,
we obtain an asymptotic expansion of our energy as the Modica-Mortola theorem,
which is sharp enough to control the asymptotic shape of minimizing curves topologically and analytically. |
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15:15-16:15 |
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講 演 者: |
仙葉 隆 氏 (福岡大学 理学部) |
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タ イ ト ル: |
Global existence and boundedness of solutions to chemotaxis systems with general sensitivity |
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アブストラクト: |
PDF 参照 |
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16:30-17:30 |
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講 演 者: |
Catherine Bandle 氏 (University of Basel) |
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タ イ ト ル: |
Stability of periodic solutions of semilinear parabolic problems |
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アブストラクト: |
The stability of periodic solutions is related to the principal eigenvalue of the linearized problem.
We discuss several techniques to estimate this eigenvalue
and extend some results of Casten and Holland and Hess to problems on Riemannian manifolds.
A major tool is the Bochner-Weitzenbock formula. |
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第33回セミナー |
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日 時: |
平成28年(2016年)11月19日(土)14:00~17:30 |
場 所: |
大阪府立大学 中百舌鳥キャンパス 数理工学科B9棟111号教室 |
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14:00-15:00 |
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講 演 者: |
川上竜樹氏(大阪府立大学・数理工学科) |
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タ イ ト ル: |
Existence of mild solutions for the Hamilton-Jacobi equation
with critical fractional viscosity in the Besov spaces |
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アブストラクト: |
In this talk we consider the Cauchy problem
for the Hamilton-Jacobi equation with critical dissipation.
We show that,
for sufficiently small initial data which belongs to the critical Besov space,
there exists a global-in-time mild solution.
Furthermore,
thanks to the Duhamel formula,
we prove that global-in-time solutions with relevant decay estimates
behave asymptotically like suitable multiplies of the Poisson kernel.
This talk is based on the joint work with Tsukasa Iwabuchi (Tohoku University). |
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15:15-16:15 |
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講 演 者: |
隠居 良行 氏 (九州大学 数理学研究院) |
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タ イ ト ル: |
Large time behavior of solutions to the compressible Navier-Stokes equations
in an infinite layer under slip boundary condition |
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アブストラクト: |
In this talk I will consider large time behavior of solutions to the compressible
Navier-Stokes equations in a 2D infinite layer under slip boundary condition.
It is shown that if the initial data is sufficiently small, the global
solution uniquely exists and the large time behavior of the solution is
described by a superposition of one-dimensional nonlinear diffusion waves.
I will also consider the problem in a cylinder. This talk is based on a
joint work with Shota Enomoto and Abulizi Aihaiti (Kyushu University). |
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16:30-17:30 |
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講 演 者: |
柴田 良弘 氏 (早稲田大学 理工学術院) |
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タ イ ト ル: |
On a free boundary value problem for the Navier-Stokes equations in an
exterior domain. |
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アブストラクト: |
In this talk, I would like to talk about mainly global well-posedness of
the free boundary problem for the Navier-Stokes equations in an exterior
domain. This talk is motivated by some cavitation problem. First, I will
give the formulation of the problem and modelling of the equations. And
then, I will talk about the local well-posedness theorem which is proved
based on the maximal $L_p$-$L_q$ regularity for the Stokes equations with
free boundary condition. The result is stated in the very much general
situation which covers the many physical background. And then, I will talk
about the global well-posedness in the exterior domain without surface
tension. Main ingredient is how to combine the maximal $L_p$-$L_q$ maximal
regularity and $L_p$-$L_q$ decay estimate for the Stokes equations with
free boundary condition. Since global well-posedness of quasi-linear parabolic
equations has not been yet treated very well, this talk will give some
cruciall ideas not only for the study of the Navier-Stokes equations but
also study of quasilinear parabolic equations. |
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第32回セミナー |
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日 時: |
平成28年(2016年)10月22日(土)14:00~17:30 |
場 所: |
大阪市立大学(杉本キャンパス)理学部E棟数学講究室(E408号室) |
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14:00-15:00 |
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講 演 者: |
古場 一 氏 (大阪大学 基礎工学研究科) |
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タ イ ト ル: |
On derivation of compressible fluid systems and diffusion equations |
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アブストラクト: |
We consider compressible fluid flow in a moving domain and on an
evolving surface from an energetic point of view. Applying our
energetic variational approach and thermodynamic theory, we derive
several governing equations for the motion of compressible fluid in a
moving domain and on an evolving surface. Moreover, we derive
diffusion equations in a moving domain and on an evolving surface from
an energetic point of view. |
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15:15-16:15 |
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講 演 者: |
砂川 秀明 氏 (大阪大学 理学研究科) |
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タ イ ト ル: |
The lifespan of small solutions to cubic derivative nonlinear
Schrodinger equations in one space dimension |
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アブストラクト: |
This talk is based on a joint work with Yuji Sagawa (Osaka University).
We consider the initial value problem for cubic derivative nonlinear
Schrodinger equations in one space dimension. We provide a detailed
lower bound estimate for the lifespan of the solution, which can be
computed explicitly from the initial data and the nonlinear term.
This is an extension and a refinement of the previous work
[H.Sunagawa: Osaka J.Math. (2006)] in which the gauge-invariant
nonlinearity was treated. |
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16:30-17:30 |
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講 演 者: |
高村 博之 氏 (公立はこだて未来大学 システム情報科学部) |
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タ イ ト ル: |
スケール不変な半線形消散波動方程式の優藤田指数における解の爆発 |
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アブストラクト: |
時間変数を係数に持つ消散項がある波動方程式は、線形自由な場合に、 その減衰オーダーによって本質的に消散項が効いている場合とそうで無い 場合に分かれる。時間変数に関して1次のオーダーで減衰するスケール
不変な場合はその境目になっており、時間変数を除いた正定数の大きさが 効いてきて1との大小によって状況が分かれる。これは半線形項が付いて いてもその分類が変わらないと思われていた。つまり、定数が1より
大きいときは消散項の影響が効いて半線形熱方程式と同様に藤田指数が 臨界になり、逆に1より小さいと藤田指数より大きくても解の爆発が 起こるという予想である。この予想は若杉勇太氏('14)とM.D'Abbico氏('15)
の仕事から得られる。解析対象の方程式は、Liouville変換によって質量項 に2次の時間減衰を持つKlein-Gordon型方程式になるが、上述の定数が2
の場合にのみ質量項が消えるという特徴がある。その特殊な場合に、 半線形波動方程式に現れる臨界指数であるStrauss指数に深く関連した指数 が臨界になることを、M.D'Abbico氏&S.Lucente氏&M.Reissig氏('15)が
波動方程式を解析するテクニックをそのまま適用して証明した。本講演では、 未知関数の導関数に対してある変換を施すことによって質量項が消えて いない場合でも解析可能になり、藤田指数を超えてStrauss指数に深く関連
した指数まで解の爆発が起きる定数の境目が1でも2でもなく、空間次元nに 関して1次の増大度を持つことを紹介したい。本講演の内容はLai Ning-An氏
(中国麗水大学)と若狭恭平氏(室蘭工業大学)との共同研究による。 |
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第31回セミナー(臨時) |
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日 時: |
平成28年(2016年)6月16日(木)13:30~18:00 |
場 所: |
大阪市立大学(杉本キャンパス)理学部E棟数学講究室(E408号室) |
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13:30-14:20 |
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講 演 者: |
渡辺 達也 氏(京都産業大学理学部数理科学科) |
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タ イ ト ル: |
Stability of gauged standing waves |
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アブストラクト: |
In this talk, we are interested in the existence and the stability of gauged
standing waves, which are standing waves for Klein-Gordon equation or Schrodinger
equation coupled with the Maxwell equation. We show that the gauged standing
wave is orbitally stable when the problem has a quardratic nonlinearity
and the coupling constant is small. This talk is based on joint works with
Mathieu Colin (University of Bordeaux). |
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ショートトークセッション |
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14:40-16:40 |
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(1)講 演 者: |
小坂 篤志 氏(大阪市立大学数学研究所専任研究所員) |
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タ イ ト ル: |
Point condensation phenomena of semilinear Neumann problems in a
non-smooth domain |
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アブストラクト: |
In this talk we consider profiles of solutions to semilinear Neumann
problems. First Lin, Ni and Takagi investigated a least-energy solution
to this type of problems, and they proved that point condensation
phenomena occurs by singular perturbation. Later Ni and Takagi proved
that a least-energy solution concentrates at the boundary point at which
the mean curvature attains its maximum.
In those precedent studies, it is assumed that the boundary is smooth.
On the other hand, we consider the case that the boundary is non-smooth.
In this case the open angle at the non-smooth point plays a similar role
to the mean curvature in the precedent studies. |
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(2)講 演 者: |
山崎 陽平 氏(大阪市立大学数学研究所専任研究所員) |
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タ イ ト ル: |
Stability for line solitary waves of Zakharov--Kuznetsov equation |
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アブストラクト: |
We consider the stability for line solitary waves of the two dimensional Zakharov--Kuznetsov equation. The orbital and asymptotic stability of the one soliton of Korteweg--de Vries equation on the energy space has been proved by Benjamin, Pego--Weinstein and Martel--Merle. We regard the one soliton of Korteweg--de Vries equation as a line solitary wave of Zakharov--Kuznetsov equation.
In this talk, we show the asymptotic stability for line solitary wave of Zakharov--Kuznetsov equation by using the argument of Martel and Merle, a Liouville type theorem and a corrected virial type estimate. |
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(3)講 演 者: |
佐野 めぐみ 氏(大阪市立大学後期博士課程2年) |
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タ イ ト ル: |
On the generalized critical Hardy inequality with the optimal constant |
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アブストラクト: |
On a bounded domain $\Omega$, we consider the minimization problem associated
with the optimal constant of generalized critical Hardy inequalities
in the boundary singularity case and other cases.
Especially, in the boundary singularity case, the validity of the inequality
depends on the sharpness of the corner of $\Omega$.
We also reveal the explicit optimal constant and minimizer of the inequalities on balls. |
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(4)講 演 者: |
橋詰 雅斗 氏(大阪市立大学後期博士課程2年) |
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タ イ ト ル: |
On the elliptic equation with the Hardy-Sobolev critical exponent |
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アブストラクト: |
In this talk, we consider the existence, the non-existence,
and the asymptotic behavior of the least-energy
solutions of an elliptic equation with the Hardy-Sobolev critical exponent.
In particular, we investigate the impact of the mean curvature at origin on theexistence of the least-energy solutions in the boundary singularity case. |
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17:00-17:50 |
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講 演 者: |
Zhi-Qiang Wang 氏(天津大学・ユタ州立大学) |
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タ イ ト ル: |
Synchronization and segregation in coupled nonlinear Schrodinger equations |
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アブストラクト: |
We discuss work on existence and qualitative property of positive solutions for coupled nonlinear Schrodinger equations.
Depending upon the system being attractive or repulsive synchronized or segregated type solutions can be constructed.
We also report recent work on the effect of mixed couplings for which coexistence of synchronization and segregation may occur.
Some interesting open questions are raised through the talk. |
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第30回セミナー |
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日 時: |
平成28年(2016年)5月21日(土)14:00~17:30 |
場 所: |
大阪市立大学(杉本キャンパス)理学部E棟数学講究室(E408号室) |
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14:00-15:00 |
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講 演 者: |
前川 泰則 氏 (京都大学 理学研究科) |
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タ イ ト ル: |
On viscous incompressible flows around a rotating obstacle in two dimensions |
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アブストラクト: |
In this talk we consider the two-dimensional stationary Navier-Stokes equations
describing the viscous incompressible flows around a rotating obstacle.
We will show the existence and the asymptotic behavior at spatial infinity
of stationary solutions when the speed of rotation is sufficiently small.
The stability of these stationary solutions is still widely open, and we
prove the local L^2 stability when the obstacle is a unit disk. A part
of this talk is based on the joint work with Mitsuo Higaki (Kyoto university)
and Yuu Nakahara (Tohoku university). |
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15:15-16:15 |
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講 演 者: |
Igor Trushin 氏 (東北大学 国際教育院) |
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タ イ ト ル: |
Inverse scattering on sun-type graphs |
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アブストラクト: |
Differential equations on graphs arise in applications as simplified
models of the propagation of waves in thin, tube-like domains. Inverse
problems of coefficient reconstruction for such differential equations
are of particular interest. We investigate inverse scattering problem
for the Sturm-Liouville (1-D Schrodinger) operator on the “sun-type
graph”, consisting of a finite number of half-lines joint with circle at
the different points. We employ major ideas of Faddeev-Marchenko
approach, deducing the main equation of inverse scattering theory, which
gives a connection between scattering data and coefficients of the
Sturm-Liouville operator. Uniqueness of reconstruction and
reconstruction procedure on the semi-infinite lines are established.
This is a joint work with Prof.K.Mochizuki. |
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16:30-17:30 |
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講 演 者: |
片山 聡一郎 氏 (大阪大学 理学研究科) |
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タ イ ト ル: |
半線形波動方程式系の大域解の漸近挙動 |
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アブストラクト: |
未知関数の導関数のみに依存するような臨界次数の
非線形項をもつ半線形波動方程式系に対する初期値問題を
空間2, 3次元で考える.
このとき, 小さな初期値に対する大域解の存在を保証する条件としては
Klainerman の null 条件が有名であるが, 近年は null 条件よりも
弱い条件下での大域解の存在結果がいくつか知られている.
本講演では, そのような弱い条件下での大域解の存在とその大域解の
漸近挙動について紹介し, いくつかの例を通じて, null 条件下での
大域解よりも複雑な挙動が起こりうることを示す. |
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第29回セミナー |
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日 時: |
平成28年(2016年)4月16日(土)14:00~17:30 |
場 所: |
大阪府立大学(中百舌鳥キャンパス)数理工学科B9棟111号教室 |
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14:00-15:00 |
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講 演 者: |
多久和 英樹 氏 (同志社大学 理工学研究科) |
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タ イ ト ル: |
2階実主要型偏微分作用素に対するカーレマン評価とその応用 |
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アブストラクト: |
大きなパラメータを持つ重み付き$L^2$評価の一種であるカーレマン評価は、偏微分方程式論において様々な応用がなされてきた。カーレマン評価を保証する作用素と相関数の間の条件は、擬凸性(pseudo
convexity)と呼ばれている。本講演では、偏微分方程式論に現れる逆問題や制御理論における問題を紹介し、従来知られてきたカーレマン評価がそれらの問題どのように使われているかを議論る。また、講演者と共同研究者との最近の研究成果についても紹介する。 |
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15:15-16:15 |
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講 演 者: |
清水 扇丈 氏 (京都大学 人間・環境学研究科) |
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タ イ ト ル: |
Local existence of isothermal compressible two-phase flows with phase transitions |
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アブストラクト: |
We consider models for compressible two-phase flows with phase transitions.These
are based on first principles, i.e., balance of mass, momentum,and energy.As
a first step, we analyze a simplified model, where the temperature is assumed
to be constant.Performing a Hanzawa transform, the problem is transformed
to a quasilinear parabolic two-phase problem with complicated transmission
conditions on the interface in a fixed domain.Then the density can be considered
as a function of the velocity and of the height function, applying the
method of characteristics. We prove maximal Lp-regularity of the corresponding
linearized problem, and then by a fixed point argument in a suitable space,
we obtain local existence for the isothermal, compressible model with phase
transitions. This is a joint work with Prof. J. Pruess (Halle, Germany). |
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16:30-17:30 |
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講 演 者: |
四ッ谷 晶二 氏 (龍谷大学 理工学研究科) |
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タ イ ト ル: |
Global bifurcation structure for a nonlocal Allen-Cahn equation |
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アブストラクト: |
We study a Neumann problem of a nonlocal Allen-Cahn equation in a finite
interval. One of main results gives a symmetry breaking (secondary) bifurcation
point on the bifurcation curve of solutions with odd-symmetry. Our proof
is based on a level set analysis for the associated integral map. A method
using the complete elliptic integrals proves the uniqueness of secondary
bifurcation point. We also show mathematical and numerical results concerning
the global bifurcation structure. This is a joint work with K. Kuto, T.
Mori, and T. Tsujikawa. |
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