2000 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, 講師 (50224499)
Keywords	非線形分散型方程式 / 実解析性 / 適切性 / Euler方程式 / 解の正則性条件 / Serrin条件Besov空間 / 渦度
Research Abstract	研究実績は以下のとおり. 研究代表者の小川は研究分担者の加藤と共に,非線型分散系の方程式についてその特異点が時間が経過した後に、時間空間両方向につき、実解析的となることを証明した。これはまずKdV方程式について初期条件を現在その時間局所的な適切性が保証されている、H^<-3/4>よりもわずかに強い正則性の下で特異点が原点のみの一点の場合に限り、時間が正または負となると解が時空両方向に対して実解析的となることを証明した。さらに非線形Schrodinger型方程式についてはgauge不変なべき非線形項を持つ場合はL^2からまたgauge不変性を持たない特殊な非線形項については、負の指数を持つSobolev空間の特異点が実解析的となることを示した。また小川は共同研究者の谷内靖とともに、3次元Euler方程式の解の特異性の発生のための十分条件(爆発条件)を調べ、従来知られていた渦度に対するBeale-Kato-Majdaの条件をゆるめて、Besov空間のノルムにより、条件を与えて、解が対数程度の指数を持つ場合には解が正則となることを証明した。さらに小川は研究分担者の小薗と谷内と共同で、Navier-Stokes方程式、Euler方程式、球面上への調和型熱流に対するSerrin型の正則性条件を、Besov空間における対数型のSobolev不等式を証明することにより、新たな条件に拡張した。この条件により特にNavier-Stokes方程式の弱解の渦度が、対数程度の特異性を持つ解に対しても、解の正則性が保証されることがわかった。川島は一般の双曲・楕円型連立系のある種の特異極限を論じた。この特異極限で双曲・楕円型連立系の解が対応する双曲・放物型連立系の解に収束することを、その収束の速さも込めて証明した。輻射気体の方程式系ではこの特異極限は、Boltzmann数とBouguer数の積を一定にしたままBoltzmann数を零に近づける極限に対応していることを明らかにした。

Research Products
(18 results)

All Other

All Publications (18 results)

[Publications] Keiichi kato,Takayoshi Ogawa: "Analiticity and smoothing effect for the Korteweg-de Vries equation with a single point singularity,"Math.Annalen. 316. 577-608 (2000)
[Publications] Keiichi Kato,Takayoshi Ogawa: "Analytic smoothing effect and single point singularity for the nonlinear Schr" odinger equations,"J.Korean Math.Soc.. 37. 100-120 (2000)
[Publications] Takayoshi Ogawa,Yasushi Taninchi: "Remarks on uniqueness and blow-up criterion to the Euler equations in the generalized Besov spaces."J.Korean Math.Soc.. 37. 1021-1029 (2000)
[Publications] Keiichi Kato,Takayoshi Ogawa: "Analytic smoothing effect and single point conormal regularity for the semilinear dispersive type equations,"数理解析研究所講究録. 1123. 113-123 (2000)
[Publications] S.Kawashima and S.Nishibata: "A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics."Indiana Univ.Math.J. 49. (2000)
[Publications] Y.Nikkuni and S.Kawashima: "Stability of stationary solutions to the half-space problem for the discrete Boltzmann equation with multiple collisions,"Kyushu J.Math.. 54. 233-255 (2000)
[Publications] S.Kawashima and S.Nishibata: "Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries,"Commun.Math.Phys.. 211. 183-206 (2000)
[Publications] Takeshi Wada: "Asymptotic expansion of the solution to to the nonlinear Schrodinger equation with nonlocal interaction."J.Funct.Anal.. (to appear).
[Publications] Naoyasu Kita,Takeshi Wada: "Sharp asymptotic behavior of solutions to nonlinear Schrodinger equations in one space dimension."Funkcial.Ekvac.. (to appear).
[Publications] Naoyasu Kita,Takeshi Wada: "Sharp asymptotics of the small solutions to the nonlinear Schrodinger equationsof derivative type."Differential Integral Equations. (to appear).
[Publications] Kazushi Yoshitomi: "Eigenvalue problems on domains with cracks."シンポジウム「スペクトル、散乱理論とその周辺」講究録,京都大学数理解析研究所. (to appear). (2001)
[Publications] Keiichi Kato,Patrick-Nicolas Pipolo: "Analyticity of solitary wave solutions to generalized Kadomtsev-Petviashvili equations"Proc.of Royal Soc. of Edinburgh. (掲載予定).
[Publications] Takayuki Kobayashi: "Some estimates of solutions for the equations of motion of compressible viscous fluid in an 3D exterior domain."Progress in Nonlinear Differential Equations and Their Applications,. 42. 151-158 (2000)
[Publications] Ryo Ikehata,Takayuki Kobayashi,Tokio Matsuyama: "Remark on the L_2 estimates of the density for the compressible Navier-Stokes flow in R^3,"Nonlinear.Anal. (To appear).
[Publications] Y.Kagei,M.Ruzicka,G.Thaeter: "Natural Convection with Dissipative Heating"Commun.Math.Phys.. 214. 287-313 (2000)
[Publications] Y.Kagei: "On thermal convection equations of Oberbeck-Boussinesq type with the dissipation function"京都大学数理解析研究所講究録「流体と気体の数学解析」. 1146. 1-15 (2000)
[Publications] Kazuo Ito,Yoshihito Kohsaka: "Three-phase boundary motion by surface diffusion : stability of a mirror symmetric stationary solution"Interfaces and Free Boundaries. 3. 45-80 (2001)
[Publications] Kazuo Ito,Yoshihito Kohsaka: "On three-phase boundary motion for a fourth order model-symmetric case "Free Boundary Problems Theory and Applications II, Gakuto International Series : Mathematical Sciences and Applications. 14. 207-221 (2000)

2000 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] Keiichi kato,Takayoshi Ogawa: "Analiticity and smoothing effect for the Korteweg-de Vries equation with a single point singularity,"Math.Annalen. 316. 577-608 (2000)

[Publications] Keiichi Kato,Takayoshi Ogawa: "Analytic smoothing effect and single point singularity for the nonlinear Schr" odinger equations,"J.Korean Math.Soc.. 37. 100-120 (2000)

[Publications] Takayoshi Ogawa,Yasushi Taninchi: "Remarks on uniqueness and blow-up criterion to the Euler equations in the generalized Besov spaces."J.Korean Math.Soc.. 37. 1021-1029 (2000)

[Publications] Keiichi Kato,Takayoshi Ogawa: "Analytic smoothing effect and single point conormal regularity for the semilinear dispersive type equations,"数理解析研究所 講究録. 1123. 113-123 (2000)

[Publications] S.Kawashima and S.Nishibata: "A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics."Indiana Univ.Math.J. 49. (2000)

[Publications] Y.Nikkuni and S.Kawashima: "Stability of stationary solutions to the half-space problem for the discrete Boltzmann equation with multiple collisions,"Kyushu J.Math.. 54. 233-255 (2000)

[Publications] S.Kawashima and S.Nishibata: "Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries,"Commun.Math.Phys.. 211. 183-206 (2000)

[Publications] Takeshi Wada: "Asymptotic expansion of the solution to to the nonlinear Schrodinger equation with nonlocal interaction."J.Funct.Anal.. (to appear).

[Publications] Naoyasu Kita,Takeshi Wada: "Sharp asymptotic behavior of solutions to nonlinear Schrodinger equations in one space dimension."Funkcial.Ekvac.. (to appear).

[Publications] Naoyasu Kita,Takeshi Wada: "Sharp asymptotics of the small solutions to the nonlinear Schrodinger equationsof derivative type."Differential Integral Equations. (to appear).

[Publications] Kazushi Yoshitomi: "Eigenvalue problems on domains with cracks."シンポジウム「スペクトル、散乱理論とその周辺」講究録,京都大学数理解析研究所. (to appear). (2001)

[Publications] Keiichi Kato,Patrick-Nicolas Pipolo: "Analyticity of solitary wave solutions to generalized Kadomtsev-Petviashvili equations"Proc.of Royal Soc. of Edinburgh. (掲載予定).

[Publications] Takayuki Kobayashi: "Some estimates of solutions for the equations of motion of compressible viscous fluid in an 3D exterior domain."Progress in Nonlinear Differential Equations and Their Applications,. 42. 151-158 (2000)

[Publications] Ryo Ikehata,Takayuki Kobayashi,Tokio Matsuyama: "Remark on the L_2 estimates of the density for the compressible Navier-Stokes flow in R^3,"Nonlinear.Anal. (To appear).

[Publications] Y.Kagei,M.Ruzicka,G.Thaeter: "Natural Convection with Dissipative Heating"Commun.Math.Phys.. 214. 287-313 (2000)

[Publications] Y.Kagei: "On thermal convection equations of Oberbeck-Boussinesq type with the dissipation function"京都大学数理解析研究所講究録「流体と気体の数学解析」. 1146. 1-15 (2000)

[Publications] Kazuo Ito,Yoshihito Kohsaka: "Three-phase boundary motion by surface diffusion : stability of a mirror symmetric stationary solution"Interfaces and Free Boundaries. 3. 45-80 (2001)

[Publications] Kazuo Ito,Yoshihito Kohsaka: "On three-phase boundary motion for a fourth order model-symmetric case "Free Boundary Problems Theory and Applications II, Gakuto International Series : Mathematical Sciences and Applications. 14. 207-221 (2000)

小川卓克九州大学, 大学院・数理学研究院, 助教授 (20224107)

[Publications] Keiichi Kato,Takayoshi Ogawa: "Analytic smoothing effect and single point conormal regularity for the semilinear dispersive type equations,"数理解析研究所講究録. 1123. 113-123 (2000)