2002 Fiscal Year Annual Research Report

非線形偏微分方程式の解の特異点に対する漸近解析

Research Project

Project/Area Number	11440057
Research Institution	KYUSHU UNIVERSITY
Principal Investigator	小川卓克九州大学, 大学院・数理学研究院, 助教授 (20224107)
Co-Investigator(Kenkyū-buntansha)	小薗英雄東北大学, 大学院・理学研究科, 教授 (00195728) 隠居良行九州大学, 大学院・数理学研究院, 助教授 (80243913) 川島秀一九州大学, 大学院・数理学研究院, 教授 (70144631) 小林孝行九州工業大学, 工学部, 助教授 (50272133) 加藤圭一東京理科大学, 理学部一部, 助教授 (50224499)
Keywords	Navier-Stokes方程式 / 調和写像流 / 適切性 / Euler方程式 / 解の正則性条件 / Serrin条件Besov空間 / 対数型臨界Sobolev不等式 / 半導体素子方程式
Research Abstract	研究実績は以下のとおり. 研究代表者の小川は研究協力者の谷内と共同で臨界型の対数形Sobolevの不等式(Brezis-Gallouetの不等式)を斉次,非斉次Besov空間に拡張した。またそれを用いて非圧縮性Navier-Stokes方程式、Euler方程式、及び球面上への調和写像流の解の正則延長のための十分条件をこれまでに知られているSerrin型の条件よりも拡張した。また同様の不等式の最良形をLizorkin-Triebel空間のsemi-normを用いて導いた。このことにより、同様な正則性条件を調和写像熱流に対して示すことができた。また同じく協力者の黒木場と共同で、半導体素子設計のための基礎方程式系に対する解の適切性と臨界状況における、解の有限時間内の爆発を証明した。分担者の川島は輻射気体の方程式系を含む一般の双曲・楕円型連立系の解の漸近挙動を、基本解に基づく手法で詳細に調べた。双曲・放物型連立系に対するLiu-Zengの結果の類似版である。双曲・楕円型連立系の線形化系の基本解をFourier変換により表示し、その主要部分が対応する双曲・放物型連立系の基本解の主要部分と一致すること、すなわち熱核を用いて表されることを確認した。分担者の隠居は分担者の小林と共同で3次元半空間上の圧縮性Navier-Stokes方程式の初期値境界値問題の密度が一定な静止状態を表す定常解の安定性を考察し撹乱の時間無限大でのL^2ノルムの時間減衰オーダーの最良のものを求めた. 分担者の伊藤は中間的表面拡散流方程式に対して,拡散係数が無限大になるときの解の挙動,及び,解の自己交差を示し、表面拡散流方程式に対しては,その解曲面の凸性を必ずしも保存しないことを示した. 分担者の北は分担者の和田と共同で微分型非線形Schrodinger方程式をゲージ変換により解の高次の漸近展開を求める手法を与えた.

Research Products

(20 results)

All Other

All Publications (20 results)

[Publications] H.Kozono, T.Ogawa, Y.Taniuchi: "The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations"Math. Z. 242. 251-278 (2002)
[Publications] E.Kaikina, K.Kato, P.Naumkin, T.Ogawa: "Wellposedness and analytic smoothing effect for the Benjamin-Ono equations"Publ. Res. Inst. Math. Sci.. 38. 651-691 (2002)
[Publications] T.Ogawa: "Analytic smoothing effect for the Benjamin-Ono equation"Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics. 113-126 (2002)
[Publications] Y.Goto, K.Ishii, T.Ogawa: "Approximation Scheme of the Mean Curvature Flow by the Bence-Merriman-Osher algorithm"数理解析研究所講究録. 1234. 145-156 (2002)
[Publications] T.Ogawa: "Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow"SIAM J. Math. Anal.. (in press). (2002)
[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.. (in press). (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. (in press). (2003)
[Publications] H.Kozono, T.Ogawa, Y.Taniuchi: "Navier-Stokes equations in the Besov space near L^∞ and BMO"Kyushu J. Math.. (in press). (2003)
[Publications] T.Iguchi, S.Kawashima: "On space-time decay properties of solutions to hyperbolic-elliptic coupled systems"Hiroshima Math. J.. 32. 229-308 (2002)
[Publications] Y.Kagei, T.Kobayashi: "On Large-time behavior of solutions to the compressible Nevier-Stokes Equations in the half space in R^3"Arch. Rational Mech. Anal.. 165. 89-159 (2002)
[Publications] T.Kobayashi: "L_2 and L_∞ estimates of the solutions for the compressible Navier-Stokes Equations in a 3D exterior domain in R^3"Publ. RIMS. Kyoto Univ.. 38. 211-225 (2002)
[Publications] T.Kobayashi: "Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R^3"J. Differential Equations. 184. 587-619 (2002)
[Publications] T.Kobayahi, Y.Shibata: "Remark on the rate of decay of solutions to linearlized for the compressible Navier-Stokes equations"Pacific J. Math. 207, No. 1. 199-234 (2002)
[Publications] T.Kobayashi, T.Suzuki, K.Watanabe: "Interface regularity for Maxwell and Stokes systems"Osaka J. Math.. (in press).
[Publications] J.Escher, Y.Giga, K.Ito: "On a Limiting motion and self-intersections of curves moved by the intermediate surface diffusion flow"Nonlinear Analysis T. M. A.. 47. 3717-3728 (2002)
[Publications] K.Ito: "Loss of convexity of compact hypersurfaces moved by surface diffusion"Comm. Appl. Anal.. 6. 105-126 (2002)
[Publications] N.Kita, T.Wada: "Sharp Asymptotic Behavior of Solutions to Nonlinear Schrodinger Equations in One Space Dimension"Funkcialaj Ekvacioj. 45. 53-69 (2002)
[Publications] N.Kita: "Sharp L^γ asymptotics of the solution to the nonlinear schrodinger equations"Nonlinear Analysis. 52. 1365-1377 (2003)
[Publications] N.Kita, T.Wada: "Sharp asymptotics of the small solutions to the nonlinear Schrodinger equations of derivative type"Differential Integral Equations. 15. 367-384 (2002)

2002 Fiscal Year Annual Research Report

非線形偏微分方程式の解の特異点に対する漸近解析

Principal Investigator

小川 卓克 九州大学, 大学院・数理学研究院, 助教授 (20224107)

Research Products

[Publications] H.Kozono, T.Ogawa, Y.Taniuchi: "The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations"Math. Z. 242. 251-278 (2002)

[Publications] E.Kaikina, K.Kato, P.Naumkin, T.Ogawa: "Wellposedness and analytic smoothing effect for the Benjamin-Ono equations"Publ. Res. Inst. Math. Sci.. 38. 651-691 (2002)

[Publications] T.Ogawa: "Analytic smoothing effect for the Benjamin-Ono equation"Tosio Kato's Method and Principle for Evolution Equations in Mathematical Physics. 113-126 (2002)

[Publications] Y.Goto, K.Ishii, T.Ogawa: "Approximation Scheme of the Mean Curvature Flow by the Bence-Merriman-Osher algorithm"数理解析研究所講究録. 1234. 145-156 (2002)

[Publications] T.Ogawa: "Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow"SIAM J. Math. Anal.. (in press). (2002)

[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.. (in press). (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. (in press). (2003)

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

[Publications] T.Iguchi, S.Kawashima: "On space-time decay properties of solutions to hyperbolic-elliptic coupled systems"Hiroshima Math. J.. 32. 229-308 (2002)

[Publications] Y.Kagei, T.Kobayashi: "On Large-time behavior of solutions to the compressible Nevier-Stokes Equations in the half space in R^3"Arch. Rational Mech. Anal.. 165. 89-159 (2002)

[Publications] T.Kobayashi: "L_2 and L_∞ estimates of the solutions for the compressible Navier-Stokes Equations in a 3D exterior domain in R^3"Publ. RIMS. Kyoto Univ.. 38. 211-225 (2002)

[Publications] T.Kobayashi: "Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R^3"J. Differential Equations. 184. 587-619 (2002)

[Publications] T.Kobayahi, Y.Shibata: "Remark on the rate of decay of solutions to linearlized for the compressible Navier-Stokes equations"Pacific J. Math. 207, No. 1. 199-234 (2002)

[Publications] T.Kobayashi, T.Suzuki, K.Watanabe: "Interface regularity for Maxwell and Stokes systems"Osaka J. Math.. (in press).

[Publications] J.Escher, Y.Giga, K.Ito: "On a Limiting motion and self-intersections of curves moved by the intermediate surface diffusion flow"Nonlinear Analysis T. M. A.. 47. 3717-3728 (2002)

[Publications] K.Ito: "Loss of convexity of compact hypersurfaces moved by surface diffusion"Comm. Appl. Anal.. 6. 105-126 (2002)

[Publications] N.Kita, T.Wada: "Sharp Asymptotic Behavior of Solutions to Nonlinear Schrodinger Equations in One Space Dimension"Funkcialaj Ekvacioj. 45. 53-69 (2002)

[Publications] N.Kita: "Sharp L^γ asymptotics of the solution to the nonlinear schrodinger equations"Nonlinear Analysis. 52. 1365-1377 (2003)

[Publications] N.Kita, T.Wada: "Sharp asymptotics of the small solutions to the nonlinear Schrodinger equations of derivative type"Differential Integral Equations. 15. 367-384 (2002)

小川卓克九州大学, 大学院・数理学研究院, 助教授 (20224107)