2003 Fiscal Year Annual Research Report

熱対流方程式系の数学解析

Research Project

Project/Area Number	14340057
Research Institution	Kyushu University
Principal Investigator	隠居良行九州大学, 大学院・数理学研究院, 助教授 (80243913)
Co-Investigator(Kenkyū-buntansha)	小林孝行佐賀大学, 理工学部, 助教授 (50272133) 小川卓克九州大学, 大学院・数理学研究院, 助教授 (20224107) 川島秀一九州大学, 大学院・数理学研究院, 教授 (70144631) 井口達雄東京工業大学, 大学院理工学研究科, 助教授 (20294879)
Keywords	Oberbeck-Boussinesq方程式 / Navier-Stokes方程式 / 安定性・分岐 / 漸近牽動
Research Abstract	隠居は,熱対流現象を記述するOberbeck-Boussinesq方程式の導出を数学的に正当化するための第一段階として、ある種の非斉次非圧縮Navier-Stokes方程式系の弱解の時間大域存在を示し,そのOberbeck-Boussinesq極限を,特に密度の収束に焦点をおいて考察した. また,隠居は小林と共同で,半空間における圧縮性Navier-Stokes方程式の密度が一定な静止状態を表す定常解の漸近安定性について以前の研究よりさらに詳細な解析を行った.ここでは,線形化問題の解公式を先行結果よりも取り扱いやすい形で提示し,それによって線形化問題の解の漸近挙動の詳細な解析を行うことができるようになり,振動積分の評価を注意深く行うことによって解の減衰評価として最良のものを得ることができた.ここで行った線形化問題の詳細な解析とエネルギー法とを組み合わせて,密度が一定な静止状態を表す定常解に対する撹乱の時間無限大における減衰の速さを評価し,また,全空間における初期値問題では現れない半空間問題に特有の非線形相互作用が起きることを示唆する評価を得た. 小林は,Maxwell方程式、Stokes方程式およびNavier-Stokes方程式の解の局所的な界面正則性を調べた. 川島は,粘性的保存則および圧縮性Navier-Stokes方程式の半空間における定常波解の漸近安定性を示した. 井口は,空間周期的な水底上の定常水面波の分岐について調べ,現実にどの分岐パターンが起こるのかということについて,水底を表す関数のフーリエ係数によって分類を行なった.また,双曲型保存則系について,既存の研究よりも一般のFluxを持つ系に対する初期値問題を考察し,エントロピー解の構成を行った. 小川は,Euler方程式およびNavier-Stokes方程式の解の正則性および一意性条件を、流速の渦度に対するBesov normやBMO normによる条件により与えた.また,Navier-Stokes方程式の強解を有界平均振動のクラスに近いあるBesov空間でparaproduct公式とBesov空間における対数型消散評価を元に構成した.

Research Products

(14 results)

All Other

All Publications (14 results)

[Publications] Y.Kagei, M.Ruzicka, G.Thaeter: "A limit problem in natural convection"Nonlinear Diff. Eq. Appl.. (to appear).
[Publications] T.Kobayashi, T.Suzuki, K.Watanabe: "Interface regularity for Maxwell and Stokes systems"Osaka J. math.. 40/4. 925-944 (2003)
[Publications] S.Kawashima, S.Nishibata, M.Nishikawa: "Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane"Discrete and Continuous Dynamical Systems. (supplement). 469-476 (2003)
[Publications] S.Kawashima, S.Nishibata, P.Zhu: "Asymptotic stability of the stationary solution to compressible Navier-Stokes equations in the half space"Commun.Math.Phys.. 240. 483-500 (2003)
[Publications] S.Kawashima, Y.Nikkuni, S.Nishibata: "Large-time behavior of solutions to hyperbolic-elliptic coupled systems"Arch. Rational Mech. Anal.. 170. 297-329 (2003)
[Publications] S.Kawashima, Y.Tanaka: "Stability of rare faction waves for a model system of a radiating gas"Kyushu J. Math.. (to appear).
[Publications] T.Iguchi: "On steady surface waves over a periodic bottom : relations between the pattern of imperfect bifurcation"Move Motion. 37. 219-239 (2003)
[Publications] T.Iguchi, P.LeFloch: "Existence theory for hyperbolic systems of conservation laws with general flux-functions"Arch.Rational Mech. Anal.. 168. 165-244 (2003)
[Publications] T.Ogawa: "Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow"SIAM J. Math.Anal.. 34. 1317-1329 (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. 190. 36-63 (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] T.Ogawa, Y.Taniuchi: "The limiting uniqueness criterion by vorticity to Navier-Stokes equations in Besov spaces"Tohoku Math. J.. 56. 65-77 (2004)
[Publications] T.Ogawa, Y.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

熱対流方程式系の数学解析

Principal Investigator

隠居 良行 九州大学, 大学院・数理学研究院, 助教授 (80243913)

Research Products

[Publications] Y.Kagei, M.Ruzicka, G.Thaeter: "A limit problem in natural convection"Nonlinear Diff. Eq. Appl.. (to appear).

[Publications] T.Kobayashi, T.Suzuki, K.Watanabe: "Interface regularity for Maxwell and Stokes systems"Osaka J. math.. 40/4. 925-944 (2003)

[Publications] S.Kawashima, S.Nishibata, M.Nishikawa: "Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane"Discrete and Continuous Dynamical Systems. (supplement). 469-476 (2003)

[Publications] S.Kawashima, S.Nishibata, P.Zhu: "Asymptotic stability of the stationary solution to compressible Navier-Stokes equations in the half space"Commun.Math.Phys.. 240. 483-500 (2003)

[Publications] S.Kawashima, Y.Nikkuni, S.Nishibata: "Large-time behavior of solutions to hyperbolic-elliptic coupled systems"Arch. Rational Mech. Anal.. 170. 297-329 (2003)

[Publications] S.Kawashima, Y.Tanaka: "Stability of rare faction waves for a model system of a radiating gas"Kyushu J. Math.. (to appear).

[Publications] T.Iguchi: "On steady surface waves over a periodic bottom : relations between the pattern of imperfect bifurcation"Move Motion. 37. 219-239 (2003)

[Publications] T.Iguchi, P.LeFloch: "Existence theory for hyperbolic systems of conservation laws with general flux-functions"Arch.Rational Mech. Anal.. 168. 165-244 (2003)

[Publications] T.Ogawa: "Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow"SIAM J. Math.Anal.. 34. 1317-1329 (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. 190. 36-63 (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] T.Ogawa, Y.Taniuchi: "The limiting uniqueness criterion by vorticity to Navier-Stokes equations in Besov spaces"Tohoku Math. J.. 56. 65-77 (2004)

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

隠居良行九州大学, 大学院・数理学研究院, 助教授 (80243913)