2003 Fiscal Year Annual Research Report

自己共役作用素の散乱理論と特異摂動に関する研究

Research Project

Project/Area Number	14540183
Research Institution	Gakushuin University
Principal Investigator	渡辺一雄学習院大学, 理学部, 助手 (90260851)
Co-Investigator(Kenkyū-buntansha)	下村明洋学習院大学, 理学部, 助手 (00365066) 水谷明学習院大学, 理学部, 教授 (80011716) 藤原大輔学習院大学, 理学部, 教授 (10011561) 小川卓克九州大学, 大学院・数理学研究院, 助教授 (20224107) 門脇光輝愛媛大学, 工学部, 助教授 (70300548)
Keywords	Maxwell / Schrodinger / 自己共役作用素 / Navier-Stokes / 特異摂動 / Besov空間 / 重調和 / 消散型
Research Abstract	渡辺一雄 Maxwell, Stokes, Navier-Stokes方程式の解の正則性について調べた.界面が現われる場合でも,解の法線成分の正則性が良くなることを示した.また,解の接成分の特異性がない場合には,解は正則性を持つことを示した. 消散項を持つ微分方程式の研究を行った.解の時間減衰は,その方程式に対応する作用素のスペクトルが重要となっているが,実際にスペクトルのみから判定することが困難である.その具体的なものとしていくつかの例を与え,これからの研究の第一歩とした. 水谷明凸とは限らぬ2次元多角形領域上の重調和Dirichlet問題に対して、有限要素法による近似を考える.その有限要素解の最適な収束の速さが、多角形領域の最大内角に応じて決まることを示した. 下村明洋非線形発展方程式の代表的な例の一つである非線形Schrodinger方程式や,Schrodinger方程式と波動型方程式の連立系(Klein-Gordon-Schrodinger方程式系やMaxwell-Schrodinger方程式等)の解の時刻無限大に於ける漸近挙動を非線形散乱理論の枠組みで研究した.短距離型と長距離型の丁度境目に相当する散乱理論を扱った.これらの方程式(系)に対する波動作用素(長距離型の場合は修正波動作用素)の存在を示した. 小川卓克有界領域におけるEuler方程式の初期値境界値問題に対する正則性、一意性条件をYudovichの与えた函数のクラスよりも広い渦度に特異点を許すあるBMOに近い函数空間の条件として与えた.そのために、Brezis-Gallouet型不等式の境界条件を含む形を有界領域における楕円型評価理論を用いて証明した。

Research Products

(15 results)

All Other

All Publications (15 results)

[Publications] T.Kobayashi, T.Suzuki, K.Watanabe: "Interface regularity for the Maxwel and Stokes systems"Osaka J.Math. 40. 925-943 (2003)
[Publications] A.Shimomura: "Modified wave operators for the coupled Wave-Schrodinger equations in three space dimensions"Discrete Contin.Dyn.Syst.. 9. 1571-1586 (2003)
[Publications] A.Shimomura: "Modified wave operators for Maxwell-Schrodinger equations in three space dimensions"Ann.Henri Poincare. 4. 661-683 (2003)
[Publications] A.Shimomura: "Wave operators for the coupled Klein-Gordon-Schrodinger equations in two space dimensions"Funkcial.Ekvac.. 47(発行予定). (2004)
[Publications] Y.Nakamura, A.Shimomura: "Local well-posedness and smoothing effects of strong solutions for nonlinear Schrodinger equations with potentials and magnetic fields"Hokkaido Math.J. (to appear)(発行予定). (2004)
[Publications] A.Shimomura: "Scattering theory for Zakharov equations in three space dimensions with large data"Commun.Contemp.Math.. (to appear)(発行予定). (2004)
[Publications] A.Shimomura: "Long range scattering for nonlinear Schrodinger equations in one and two space dimensions"Differential Integral Equations.. (to appear)(発行予定). (2004)
[Publications] A.Shimomura: "'T.Cazenave and A.Haraux, An Introduction to Semilinear Evolution Equations'の書評"雑誌数学. (印刷中)(発行予定). (2004)
[Publications] A.Shimomura: "Scattering theory for the Zakharov equations in three space dimensions"京都大学数理解析研究所講究録. (in press). (2004)
[Publications] T.Ogawa, Y.Taniuchi: "A note on blow-up criterion to the 3-D Euler Equations in a bounded domain"J.Math.Fluid Mech.. 5. 17-23 (2003)
[Publications] T.Ogawa, Y.Taniuchi: "On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain"J.Differential Equations. (in press). (2003)
[Publications] M.Kurokiba, T.Ogawa: "Finite time blow-up of the solution for the nonlinear parabolic equation of the drift diffusion type"Diff.Integral Equations. 16. 427-452 (2003)
[Publications] H.Kozono, T.Ogawa, Y.Taniuchi: "Navier-Stokes equations in the Besov space near L^∞ and BMO"Kyushu J.Math.. 57. 303-324 (2003)
[Publications] Y.Taniuchi, T.Ogawa: "The limiting uniqueness criterion by vorticity to Navier-Stokes equations in Besov spaces"Tohoku Math.J.. 56. 65-77 (2004)
[Publications] T.Ogawa, T.Yokota: "Uniqueness and inviscid limit to the complex Ginzburg-Landau equation in two dimensional general domain"Comm.Math.Phys.. 245. 105-121 (2004)

2003 Fiscal Year Annual Research Report

自己共役作用素の散乱理論と特異摂動に関する研究

Principal Investigator

渡辺 一雄 学習院大学, 理学部, 助手 (90260851)

Research Products

[Publications] T.Kobayashi, T.Suzuki, K.Watanabe: "Interface regularity for the Maxwel and Stokes systems"Osaka J.Math. 40. 925-943 (2003)

[Publications] A.Shimomura: "Modified wave operators for the coupled Wave-Schrodinger equations in three space dimensions"Discrete Contin.Dyn.Syst.. 9. 1571-1586 (2003)

[Publications] A.Shimomura: "Modified wave operators for Maxwell-Schrodinger equations in three space dimensions"Ann.Henri Poincare. 4. 661-683 (2003)

[Publications] A.Shimomura: "Wave operators for the coupled Klein-Gordon-Schrodinger equations in two space dimensions"Funkcial.Ekvac.. 47(発行予定). (2004)

[Publications] Y.Nakamura, A.Shimomura: "Local well-posedness and smoothing effects of strong solutions for nonlinear Schrodinger equations with potentials and magnetic fields"Hokkaido Math.J. (to appear)(発行予定). (2004)

[Publications] A.Shimomura: "Scattering theory for Zakharov equations in three space dimensions with large data"Commun.Contemp.Math.. (to appear)(発行予定). (2004)

[Publications] A.Shimomura: "Long range scattering for nonlinear Schrodinger equations in one and two space dimensions"Differential Integral Equations.. (to appear)(発行予定). (2004)

[Publications] A.Shimomura: "'T.Cazenave and A.Haraux, An Introduction to Semilinear Evolution Equations'の書評"雑誌数学. (印刷中)(発行予定). (2004)

[Publications] A.Shimomura: "Scattering theory for the Zakharov equations in three space dimensions"京都大学数理解析研究所講究録. (in press). (2004)

[Publications] T.Ogawa, Y.Taniuchi: "A note on blow-up criterion to the 3-D Euler Equations in a bounded domain"J.Math.Fluid Mech.. 5. 17-23 (2003)

[Publications] T.Ogawa, Y.Taniuchi: "On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain"J.Differential Equations. (in press). (2003)

[Publications] M.Kurokiba, T.Ogawa: "Finite time blow-up of the solution for the nonlinear parabolic equation of the drift diffusion type"Diff.Integral Equations. 16. 427-452 (2003)

[Publications] H.Kozono, T.Ogawa, Y.Taniuchi: "Navier-Stokes equations in the Besov space near L^∞ and BMO"Kyushu J.Math.. 57. 303-324 (2003)

[Publications] Y.Taniuchi, T.Ogawa: "The limiting uniqueness criterion by vorticity to Navier-Stokes equations in Besov spaces"Tohoku Math.J.. 56. 65-77 (2004)

[Publications] T.Ogawa, T.Yokota: "Uniqueness and inviscid limit to the complex Ginzburg-Landau equation in two dimensional general domain"Comm.Math.Phys.. 245. 105-121 (2004)

渡辺一雄学習院大学, 理学部, 助手 (90260851)