1999 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 Category	Grant-in-Aid for Scientific Research (B)
Research Institution	Kyushu University
Principal Investigator	小川卓克九州大学, 大学院・数理学研究科, 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・数理学研究科, 助教授 (80243913) 川島秀一九州大学, 大学院・数理学研究科, 教授 (70144631) 小林孝行九州工業大学, 工学部, 講師 (50272133) 加藤圭一東京理科大学, 理学部第一部, 講師 (50224499)
Keywords	KdV方程式 / 正則性 / Euler方程式 / 非圧縮性粘性流体 / Besov空間 / Boltzmann方程式 / Oberbeck-Boussinesq方程式 / 漸近安定性
Research Abstract	小川は研究分担者の加藤と共同で、KdV方程式の初期値が一点だけディラック測度のような強い特異点を持つ場合に、その解が時間が時刻0以外で時間-空間両変数方向に実解析的となることを証明した。この手法を発展させ、非線形Schrodinger方程式や一般の半線形分散型方程式で非局所作用素を含まない場合に、類似の結果を証明した。また小川は谷内靖と共同で3次元Euler方程式の初期値問題の解の爆発のための新しい十分条件を一般化されたBesov空間の記法で表現した。これにより、逆に渦度が高々対数程度の特異点を持つ場合にはEuler方程式の解は、なめらかに延長できることも示した。川島は西畑伸也と共同で空間1次元の半空間における離散的Boltzmann方程式に対し、純消散的境界条件を課した場合の定常解の存在を示し、それが時間無限大で漸近安定であることを証明した。また、その定常解が空間の無限遠方で空間一様な平行状態に指数的に漸近することを示した。さらに一般の双曲・楕円型連立系のある種の特異極限を論じた。また、輻射気体の方程式系ではこの時異極限は、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。一方、隠居は無限層状領域におけるOberbeck-Boussinesq方程式のある種の定常解に対して、定常解が線形化安定性の臨界状態にあっても、初期撹乱の大きさに関係なく無条件安定であることを示した。

Research Products

(19 results)

All Other

All Publications (19 results)

[Publications] K.Kato,T.Ogawa: "Analiticity and smoothing effect for the Korteweg-de Vries equarion with a single point singularity."Math.Annalen,. (in press). (2000)
[Publications] K.Kato,T.Ogawa: "Analytic smoothing effect and single point conormal regularity for the semilinear dispersive type equation."数理解析研講究録. 1123. 113-123 (2000)
[Publications] T.Ogawa,Y.Taniuchi: "Remarks on uniqueness and blow-up criterion to the Euler equations in the generalized Besov spaces."J.Korean Math. Soc. to appear. 1123. (2000)
[Publications] S.Kawashima and S.Nishibata: "Cauchy problem for a model system of the readiating gas:Weak solutions with a jamp and classical solutions."Math.Models Meth.Appl.Sci.,. 9. (1999)
[Publications] S.Kawashima and S.Nishibata: "Existence of a stationary wave for the discrete Boltzmann equation in the half space."Commun.Math.Phys.,. (to appear). (2000)
[Publications] S.Kawashima and S.Nishibata: "A singular limit for hyperbolic-elliptic coupled systems in rediation hydro-dynamics."Indiana Univ.Math.J.,. (to appear). (2000)
[Publications] Y.Nikkumi and S.Kawashima: "Stability of stationary solutions to the half-space problem for the discrete Boltzmann equation with multiple collisions,"Kyushu J.Math.,. (to appear). (2000)
[Publications] S.Kawashima and S.Nishibata: "Stationary waves for the discrete Boltzmann equation in the half space with reflective boundariws,"Commun0Math.Phys.,. (to appear). (2000)
[Publications] Y.Kagei and W. von Wahl: "Asymptotic stability of steady flows in infinite layers of viscous incom-pressible fluids in critical cases of stability,"Indiana Univ.Math.J.,. 48. 1083-1110 (1999)
[Publications] H.Kozono: "On global strong solution of the Navier-Stokes equations in 4 and 5 di-mensional unbounded domains,"Ann. Inst. Henri Poincare Analyse Nonlineaire,. 16. 535-561 (1999)
[Publications] H.Kozono, M.Yamazaki: "Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in ezterior domains."Nonlinear Analysis,TMA,. (in press). (2000)
[Publications] H.Kozono, Y.Taniuchi: "Bilinear esitimates in BMO and the Navier-Stokes equations,"Math. Z.. (in press). (2000)
[Publications] H.Kozono: "Weak solutions of the Navier-Stokes equations with test functions in weak-L^n."Tohoku Math. J.. (in press). (2000)
[Publications] K.Kato,: "Interaction of analytic singularities for nonlinear wave equations"Comm. in Partial Differential Equations. (to appear). (2000)
[Publications] K.Kato、P.N. Pipolo: "Analyticity of solitary wave solutions to generalized Kadomtesv-Pteviashvili equations,"Proc. Roy. Soc. Edinburg. (to appear). (2000)
[Publications] T.Kobayashi and Y.Shibata,: "Decay estiates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain R^3"Commun. Math. Phys.. 200. 621-659 (2000)
[Publications] T.Kobayashi and W. Azjaczkowski: "On global motion of a compressible viscous fluid with boundary slip con-dition."Applicationes Mathematicae. 26. 159-194 (1999)
[Publications] T.Wada: "Scattering Theory for Time-dependent Hartree-Fock Type Equation,"Osala J. Math.. 36(in press). (1999)
[Publications] T.Wada: "A remark on long range scattering for the Hartree type equation,"Kyushi J. Math.. 54(in press). (1999)

1999 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・数理学研究科, 助教授 (20224107)

Research Products

[Publications] K.Kato,T.Ogawa: "Analiticity and smoothing effect for the Korteweg-de Vries equarion with a single point singularity."Math.Annalen,. (in press). (2000)

[Publications] K.Kato,T.Ogawa: "Analytic smoothing effect and single point conormal regularity for the semilinear dispersive type equation."数理解析研講究録. 1123. 113-123 (2000)

[Publications] T.Ogawa,Y.Taniuchi: "Remarks on uniqueness and blow-up criterion to the Euler equations in the generalized Besov spaces."J.Korean Math. Soc. to appear. 1123. (2000)

[Publications] S.Kawashima and S.Nishibata: "Cauchy problem for a model system of the readiating gas:Weak solutions with a jamp and classical solutions."Math.Models Meth.Appl.Sci.,. 9. (1999)

[Publications] S.Kawashima and S.Nishibata: "Existence of a stationary wave for the discrete Boltzmann equation in the half space."Commun.Math.Phys.,. (to appear). (2000)

[Publications] S.Kawashima and S.Nishibata: "A singular limit for hyperbolic-elliptic coupled systems in rediation hydro-dynamics."Indiana Univ.Math.J.,. (to appear). (2000)

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

[Publications] S.Kawashima and S.Nishibata: "Stationary waves for the discrete Boltzmann equation in the half space with reflective boundariws,"Commun0Math.Phys.,. (to appear). (2000)

[Publications] Y.Kagei and W. von Wahl: "Asymptotic stability of steady flows in infinite layers of viscous incom-pressible fluids in critical cases of stability,"Indiana Univ.Math.J.,. 48. 1083-1110 (1999)

[Publications] H.Kozono: "On global strong solution of the Navier-Stokes equations in 4 and 5 di-mensional unbounded domains,"Ann. Inst. Henri Poincare Analyse Nonlineaire,. 16. 535-561 (1999)

[Publications] H.Kozono, M.Yamazaki: "Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in ezterior domains."Nonlinear Analysis,TMA,. (in press). (2000)

[Publications] H.Kozono, Y.Taniuchi: "Bilinear esitimates in BMO and the Navier-Stokes equations,"Math. Z.. (in press). (2000)

[Publications] H.Kozono: "Weak solutions of the Navier-Stokes equations with test functions in weak-L^n."Tohoku Math. J.. (in press). (2000)

[Publications] K.Kato,: "Interaction of analytic singularities for nonlinear wave equations"Comm. in Partial Differential Equations. (to appear). (2000)

[Publications] K.Kato、P.N. Pipolo: "Analyticity of solitary wave solutions to generalized Kadomtesv-Pteviashvili equations,"Proc. Roy. Soc. Edinburg. (to appear). (2000)

[Publications] T.Kobayashi and Y.Shibata,: "Decay estiates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain R^3"Commun. Math. Phys.. 200. 621-659 (2000)

[Publications] T.Kobayashi and W. Azjaczkowski: "On global motion of a compressible viscous fluid with boundary slip con-dition."Applicationes Mathematicae. 26. 159-194 (1999)

[Publications] T.Wada: "Scattering Theory for Time-dependent Hartree-Fock Type Equation,"Osala J. Math.. 36(in press). (1999)

[Publications] T.Wada: "A remark on long range scattering for the Hartree type equation,"Kyushi J. Math.. 54(in press). (1999)

小川卓克九州大学, 大学院・数理学研究科, 助教授 (20224107)