2001 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)	小薗秀雄東北大学, 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	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Benjamin-Ono方程式の初期値問題の解がその初期値に一点のみSobolev空間H^s(s>3/2)程度の特異点を持つ場合に、対応する弱解が時間が立てば、時間、空間両方向につき実解析的となるsmoothing effectを持つことを示した。その過程で、無限連立のBenjamin-Ono型連立系の時間局所適切性を証明した。またKdV方程式とBenjamin-Ono方程式の中間的な効果を表すBenjaminのoriginal方程式に関して、その初期値問題が負の指数をも許すSobolev空間H^s(R)(s>-3/4)で時間局所的に適切となることを示した。さらに、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。分担者の川島は一般の双曲・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。また、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。分担者の隠居はVlasov-Poisson-Fokker-Planck方程式(VPFP方程式)の初期値問題に対して,重み付きソボレフ空間において不変多様体を構成し、解の時間無限大での漸近形を導出した。分担者の加藤は一般化されたKadmtsev-Petviashvili方程式の孤立波解の解析性について考察した。分担者の小林は3次元全空間における圧縮性粘性流体の運動を記述した方程式の解の漸近挙動を調べ、特別な初期値に対しては密度が時間に関して減衰することを示した。

Research Products

(25 results)

All Other

All Publications (25 results)

[Publications] H.Kozono, T.Ogawa, Y.Tanisaka: "Well-posedness for the Benjamin equations"J.Korean Math.Soc.. 38. 1205-1234 (2001)
[Publications] B.Kaikina, K.Kato, P.Naumkin, T.Ogawa: "Analytic smoothing effect for the Benjamin-Ono equations"京都大学数理解析研究所講究録. 1204. 77-84 (2001)
[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.(electric). 239. (2002)
[Publications] T.Ogawa: "Analytic smoothing effect for the Benjamin-Ono equation"京都大学数理解析研究所講究録. 1234. 113-126 (2001)
[Publications] B.Kaikina, K.Kato, P.Naumkin, T.Ogawa: "Wellposedness and analytic smoothing effect for the Benjamin-Ono equations"Publ.Res.Inst.Math.Sci.. (in press).
[Publications] S.Kawashima, S.Nishibata: "A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics"Indiana Univ.Math.J.. 50. 567-589 (2001)
[Publications] T.Iguchi, S.Kawashima: "On space-time decay properties of solutions to hyperboic-elliptic coupled systems"Hiroshima Math.J.. (2001)
[Publications] P.Exner, K.Yoshitomi: "Band gap of the Schrodinger operator with a strong delta-interactionon a periodic curve"Annals Henri Poincare. 2. 1139-1158 (2001)
[Publications] K.Yoshitomi: "Eigenvalue problems on domains with cracks"京都大学数理解析研究所講究録. 1208. 7-23 (2001)
[Publications] K.Kato, P.Pipolo: "Analiticity of solitary wave solutions to generalized Kadmtsev-Petviashvili equations"Proceedings of the Royal Society of Edinburgh. 131A. 391-424 (2001)
[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"Third World Congress of Nonlinear. Anal.. 47. 2519-2526 (2001)
[Publications] Takayuki 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. (to appear).
[Publications] Takayuki Kobayashi: "Some estimates of solutions for the equatious of motion of compressible viscous fluid in an exterior domain in R^3"J. Differential Equations. (to appear).
[Publications] Takayuki Kobayaashi, Yoshihiro Shibata: "Remark on the rate of decay of solutions to linearlized for the compressible Navier-Stokes equations"Pacific J.Math. (to appear).
[Publications] Yoshiyuki Kagei: "Invariant manifolds and long-time asymptotics for the Vlasov-Poisson-Fokker-Planck equation"SIAM J.Math.Anal.. 33. 489-507 (2001)
[Publications] H.Kozono: "On well-posedness of the Navier-Stokes equations"Adv.Math.Fluid Mech.. 207-236 (2001)
[Publications] Kozono, H: "Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains"Math.Ann.. 320. 709-730 (2001)
[Publications] Kozono, H.: "Weak solutions of the Navier-Stokes equations with test functions in the weak-L^n space"Tohoku Math.J.. 53. 55-79 (2001)
[Publications] K.Ito, Y.Kohsaka: "Three phase boundary motion by surface diffusion : stability of a mirror symmetric stationary solutio"Interfaces and Free Boundaries. 3. 45-80 (2001)
[Publications] K.Ito, Y.Kohsaka: "Three phase boundary motion by surface diffusion in triangular domain"Advances in Math.Sci.Appl.. 11. 753-779 (2001)
[Publications] K.Ito, Y.Kohsaka: "Three phase boundary motion by surface diffusion, Free Boundary Problem"数理解析研究所講究録. 1210. 43-53 (2001)
[Publications] J.Escher, Y.Giga, K.Ito: "On a limiting motion and self-intersections of curves moved by the inter-mediate surface diffusion flow"Nonlinear Analysis, T.M.A.. 47. 3717-3728 (2001)
[Publications] Takeshi Wada: "Asymptotic expansion of the solution to to the nonlinear Schrodinger equa-tion with nonlocal interaction"J.Funct.Anal.. 180. 11-30 (2001)
[Publications] Takeshi Wada: "Long Range Scattering for Time-dependent Hartree-Fock Type Equation"Nonlinear Anal.. 48. 175-190 (2002)
[Publications] Takeshi Wada, Naoyasu Kita: "Sharp asymptotics of the small solutions to the nonlinear Schrodinger equa-tions of derivative type"Duff.and Int.Eq.. 15. 367-384 (2002)

2001 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.Tanisaka: "Well-posedness for the Benjamin equations"J.Korean Math.Soc.. 38. 1205-1234 (2001)

[Publications] B.Kaikina, K.Kato, P.Naumkin, T.Ogawa: "Analytic smoothing effect for the Benjamin-Ono equations"京都大学数理解析研究所 講究録. 1204. 77-84 (2001)

[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.(electric). 239. (2002)

[Publications] T.Ogawa: "Analytic smoothing effect for the Benjamin-Ono equation"京都大学数理解析研究所 講究録. 1234. 113-126 (2001)

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

[Publications] S.Kawashima, S.Nishibata: "A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics"Indiana Univ.Math.J.. 50. 567-589 (2001)

[Publications] T.Iguchi, S.Kawashima: "On space-time decay properties of solutions to hyperboic-elliptic coupled systems"Hiroshima Math.J.. (2001)

[Publications] P.Exner, K.Yoshitomi: "Band gap of the Schrodinger operator with a strong delta-interactionon a periodic curve"Annals Henri Poincare. 2. 1139-1158 (2001)

[Publications] K.Yoshitomi: "Eigenvalue problems on domains with cracks"京都大学数理解析研究所 講究録. 1208. 7-23 (2001)

[Publications] K.Kato, P.Pipolo: "Analiticity of solitary wave solutions to generalized Kadmtsev-Petviashvili equations"Proceedings of the Royal Society of Edinburgh. 131A. 391-424 (2001)

[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"Third World Congress of Nonlinear. Anal.. 47. 2519-2526 (2001)

[Publications] Takayuki 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. (to appear).

[Publications] Takayuki Kobayashi: "Some estimates of solutions for the equatious of motion of compressible viscous fluid in an exterior domain in R^3"J. Differential Equations. (to appear).

[Publications] Takayuki Kobayaashi, Yoshihiro Shibata: "Remark on the rate of decay of solutions to linearlized for the compressible Navier-Stokes equations"Pacific J.Math. (to appear).

[Publications] Yoshiyuki Kagei: "Invariant manifolds and long-time asymptotics for the Vlasov-Poisson-Fokker-Planck equation"SIAM J.Math.Anal.. 33. 489-507 (2001)

[Publications] H.Kozono: "On well-posedness of the Navier-Stokes equations"Adv.Math.Fluid Mech.. 207-236 (2001)

[Publications] Kozono, H: "Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains"Math.Ann.. 320. 709-730 (2001)

[Publications] Kozono, H.: "Weak solutions of the Navier-Stokes equations with test functions in the weak-L^n space"Tohoku Math.J.. 53. 55-79 (2001)

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

[Publications] K.Ito, Y.Kohsaka: "Three phase boundary motion by surface diffusion in triangular domain"Advances in Math.Sci.Appl.. 11. 753-779 (2001)

[Publications] K.Ito, Y.Kohsaka: "Three phase boundary motion by surface diffusion, Free Boundary Problem"数理解析研究所 講究録. 1210. 43-53 (2001)

[Publications] J.Escher, Y.Giga, K.Ito: "On a limiting motion and self-intersections of curves moved by the inter-mediate surface diffusion flow"Nonlinear Analysis, T.M.A.. 47. 3717-3728 (2001)

[Publications] Takeshi Wada: "Asymptotic expansion of the solution to to the nonlinear Schrodinger equa-tion with nonlocal interaction"J.Funct.Anal.. 180. 11-30 (2001)

[Publications] Takeshi Wada: "Long Range Scattering for Time-dependent Hartree-Fock Type Equation"Nonlinear Anal.. 48. 175-190 (2002)

[Publications] Takeshi Wada, Naoyasu Kita: "Sharp asymptotics of the small solutions to the nonlinear Schrodinger equa-tions of derivative type"Duff.and Int.Eq.. 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)

[Publications] B.Kaikina, K.Kato, P.Naumkin, T.Ogawa: "Analytic smoothing effect for the Benjamin-Ono equations"京都大学数理解析研究所講究録. 1204. 77-84 (2001)

[Publications] T.Ogawa: "Analytic smoothing effect for the Benjamin-Ono equation"京都大学数理解析研究所講究録. 1234. 113-126 (2001)

[Publications] K.Yoshitomi: "Eigenvalue problems on domains with cracks"京都大学数理解析研究所講究録. 1208. 7-23 (2001)

[Publications] K.Ito, Y.Kohsaka: "Three phase boundary motion by surface diffusion, Free Boundary Problem"数理解析研究所講究録. 1210. 43-53 (2001)