2003 Fiscal Year Annual Research Report

臨界型非線形偏微分方程式の解の特異性と正則性の研究

Research Project

Project/Area Number	15340056
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) 三沢正史熊本大学, 理学部, 助教授 (40242672)
Keywords	Navier-Stokes方程式 / 調和写像流 / Euler方程式 / 解の正則性条件 / Besov空間 / 対数型臨界Sobolev不等式 / 半導体素子方程式 / 複数Ginzburg-Landau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方程式の弱解の一意性判定条件、3次元Euler方程式の解の爆発判定条件の精密化を行った。さらに有界領域のような境界を持つ場合に、これらの不等式を拡張して、Euler方程式の爆発判定条件を拡張した。また同様の不等式をLizorkin-Triebel空間のsemi-normを用いて拡張することにより、BMO-L^∞型の臨界不等式の最良形を得ることができた。これから、調和写像熱流の解に対してある正則性条件を示すことができた。また協力者の黒木場正城と共同で、半導体素子設計のための基礎放物-楕円型方程式系に対する解の適切性を示し、さらに臨界状況における、解の有限時間内の爆発を証明した。特に初期値に対する敷居値条件8πを精密に求めた。また脇力者の横田智巳とともに、2次元一般領域における、複素係数を持つGinzburg-Landau型方程式の初期値境界値問題を考えて、その弱解の一意性と、粘性係数にあたる係数の実部を0に近づけたときのいわゆる粘性消滅極限を議論し、従来得られていなかった2次元一般領域の問題に対して弱解が対応する非線形Schrodinger型方程式の解に強収束することを示した。分担者の隠居は熱対流現象を記述するモデル方程式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極限を考察した. 分担者の三沢は変数係数時間発展p調和作用素の線形化放物型作用素に対するSchauder評価の証明とそれによる古典解の構成.、BMO関数の導関数を外力とし,VMO関数を係数とする時間発展p調和方程式の解の空間一階導関数に対するL^q評価の証明、及びL^1関数を外力とするp-調和作用素に対する弱解の構成についての研究を行った。

Research Products

(17 results)

All Other

All Publications (17 results)

[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: "A note on blow-up criterion to the 3-D Euler Equations in a bounded domain,"J.Math.Fluid Mech.. 5. 17-23 (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 a nonlinear parabolic equation of 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, T.Yokota: "Uniqueness and inviscid limits to the complex Ginzburg-Landau equation in two dimensional general domain,"Comm.Math.Phys.. 245. 105-121 (2004)
[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. Suppl. Vol. 469-476 (2003)
[Publications] S.Kawashima, S.Nishibata, P.Zhu: "Asymptotic stability of the stationary solution to the 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.Rat.Mech.Anal.. 170. 297-329 (2003)
[Publications] S.Kawashima, Y.Tanaka: "Stability of rarefaction waves for a model system of a radiating gas,"Kyushu J.Math.. (in press). (2003)
[Publications] Y.Kagei: "A limit problem in natural convection, to appear in Nonlinear Differential Equations and Applications,"Nonlinear Differential Equations and Applications. (in press). (2003)
[Publications] N.Kita: "Sharp L^r asymptotics of the small solutions to the nonlinear Schrodinger equations"Nonlinear Analysis. 52. 1365-1377 (2003)
[Publications] T.Kobayashi, T.Suzuki, K.Watanabe: "Interface regularity for Maxwell and Stokes systems"Osaka J.Math.. 40-4. 925-944 (2003)
[Publications] M.Misawa: "Existence of classical solution for linear parabolic system."Comment.Math.Univ.Carolin.. in press. (2004)
[Publications] M.Misawa: "L^q estimates of gradients for evolutional p-Laplacian systems,"Ark.Mat.. in press. (2004)
[Publications] M.Misawa: "Existence for a Cauchy-Dirichlet problem for evolutional p-Laplacian systems"Applicationes Math.. in press. (2004)

2003 Fiscal Year Annual Research Report

臨界型非線形偏微分方程式の解の特異性と正則性の研究

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] 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: "A note on blow-up criterion to the 3-D Euler Equations in a bounded domain,"J.Math.Fluid Mech.. 5. 17-23 (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 a nonlinear parabolic equation of 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, T.Yokota: "Uniqueness and inviscid limits to the complex Ginzburg-Landau equation in two dimensional general domain,"Comm.Math.Phys.. 245. 105-121 (2004)

[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. Suppl. Vol. 469-476 (2003)

[Publications] S.Kawashima, S.Nishibata, P.Zhu: "Asymptotic stability of the stationary solution to the 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.Rat.Mech.Anal.. 170. 297-329 (2003)

[Publications] S.Kawashima, Y.Tanaka: "Stability of rarefaction waves for a model system of a radiating gas,"Kyushu J.Math.. (in press). (2003)

[Publications] Y.Kagei: "A limit problem in natural convection, to appear in Nonlinear Differential Equations and Applications,"Nonlinear Differential Equations and Applications. (in press). (2003)

[Publications] N.Kita: "Sharp L^r asymptotics of the small solutions to the nonlinear Schrodinger equations"Nonlinear Analysis. 52. 1365-1377 (2003)

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

[Publications] M.Misawa: "Existence of classical solution for linear parabolic system."Comment.Math.Univ.Carolin.. in press. (2004)

[Publications] M.Misawa: "L^q estimates of gradients for evolutional p-Laplacian systems,"Ark.Mat.. in press. (2004)

[Publications] M.Misawa: "Existence for a Cauchy-Dirichlet problem for evolutional p-Laplacian systems"Applicationes Math.. in press. (2004)

小川卓克九州大学, 大学院・数理学研究院, 助教授 (20224107)