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Kobayashi Takayuki
小林　孝行
Kobayashi Takayuki
小林　孝行
Institutional repository
Graduate School of Engineering Science　Department of Systems Innovation, Professor
Institutional repository

Research History 1

  1. 2014/03 - Present
    Osaka University　Graduate School of Engineering Science　Professor

Professional Memberships 1

  1. 日本数学会

Research Areas 1

  1. Natural sciences / Basic analysis /

Papers 31

  1. Resolvent Estimates for a Compressible Fluid Model of Korteweg Type and Their Application

    T. Kobayashi, M. Murata, H. Saito

    Journal of Mathematical Fluid Mechanics　Vol. 24　No. 1　2022/02　Research paper (scientific journal)

    Publisher: Springer Science and Business Media LLC

  2. Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

    T. Kobayashi, M. Misawa, K. Tsuda

    Mathematics　Vol. 9(6)　No. 683　p. 1-20　2021/03/22　Research paper (scientific journal)

    Publisher: MDPI AG

  3. The global well-posedness of the compressible fluid model of Korteweg type for the critical case

    T. Kobayashi and M. Murata

    Differential and Integral Equation　Vol. 34　No. 5-6　p. 245-264　2021　Research paper (scientific journal)

  4. Time decay estimate with diffusion wave property and smoothing effect for solutions to the compressible Navier-Stokes-Korteweg system

    T. Kobayashi and K. Tsuda

    Funkcialaj Ekvacioj　Vol. 64　p. 163-187　2021　Research paper (scientific journal)

  5. Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition

    T. Kobayashi and Y. Tsuda

    Asymptotic Analysis　Vol. 121　No. 2　p. 195-217　2021　Research paper (scientific journal)

  6. Global well-posedness and time-decay estimates of the compressible Navier-Stokes-Korteweg system in critical Besov spaces

    N. Chikami and T. Kobayashi

    J. Math. Fluid Mech.　Vol. 21　No. 2　2019　Research paper (scientific journal)

  7. Time-space L2-boundedness for the 2D Navier-Stokes equations and hyperbolic Navier-Stokes

    T. Kobayashi , M. Misawa and K. Nakamura

    Tsukuba J. Mathematics　Vol. 43　No. 2　p. 223-237　2019　Research paper (scientific journal)

  8. On a local energy decay estimate of solutions to the hyperbolic type Stokes equations

    T. Kobayashi, T. Kubo and K. Nakamura

    J. Differential Equations　Vol. 263　No. 10　p. 6061-6081　2018　Research paper (scientific journal)

  9. Weighted Lp - Lq estimates of stokes semigroup in half-space and its application to the Navier-Stokes Equation

    T. Kobayashi, T. Kubo

    Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday　Vol. none　p. 337-349　2016　Research paper (international conference proceedings)

    Publisher: Springer Verlag

  10. Weighted estimate of Stokes semigroup in unbounded domains

    T. Kobayashi and T. Kubo

    Advanced Studies in Pure Mathematics, Nonlinear Dynamics in Partial Differential Equations　Vol. 64　p. 427-435　2014　Research paper (scientific journal)

  11. Fluid mechanical approximation to the degenerated drift-diffusion system from the compressible Navier-Stokes-Poisson system

    T. Kobayashi, T. Ogawa

    Indiana University Mathematics Journal　Vol. 62　No. 3　p. 1021-1054　2013　Research paper (scientific journal)

    Publisher: Department of Mathematics, Indiana University

  12. $L^2$ boundedness for the 2D exterior problems for the semilinear heat and dissipative wave equations

    T. Kobayashi and M. Misawa

    Rims Kokyuroku Bessatsu　Vol. B42　p. 1-11　2013　Research paper (scientific journal)

  13. Weighted Lp-Lq-estimates of the Stokes semigroup in some unbounded domains

    T. Kobayashi and T. Kubo

    Tsukuba J. Mathematics　Vol. 37　No. 2　p. 179-205　2013　Research paper (scientific journal)

  14. DECAY PROPERTY FOR THE LINEAR WAVE EQUATIONS IN TWO-DIMENSIONAL EXTERIOR DOMAINS

    Masashi Misawa, Shoko Okamura, Takayuki Kobayashi

    DIFFERENTIAL AND INTEGRAL EQUATIONS　Vol. 24　No. 9-10　p. 941-964　2011/09　Research paper (scientific journal)

  15. Weak solutions to the Navier-Stokes-Poisson equation equations

    T. Kobayashi and T.Suzuki

    Adv. Math. Sci. Appl.　Vol. 18　No. 1　p. 141-161　2008　Research paper (scientific journal)

  16. Interface vanishing for solutions to Maxwell and Stokes systems

    Takayuki Kobayashi, Takashi Suzuki, Kazuo Watanabe

    JOURNAL OF MATHEMATICAL FLUID MECHANICS　Vol. 8　No. 3　p. 382-397　2006/08　Research paper (scientific journal)

  17. Asymptotic behavior of solutions to the compressible Navier-Stokes equations on the half space

    Y.Kagei and T. Kobayashi

    Arch. Rat. Mech. Anal.　Vol. 177　p. 231-330　2005　Research paper (scientific journal)

  18. Interface regularity for Maxell and Stokes systems

    T Kobayashi, T Suzuki, K Watanabe

    OSAKA JOURNAL OF MATHEMATICS　Vol. 40　No. 4　p. 925-943　2003/12　Research paper (scientific journal)

  19. Some estimates of solutions for the equations of motion of compressible viscous fluid in the three-dimensional exterior domain

    T Kobayashi

    JOURNAL OF DIFFERENTIAL EQUATIONS　Vol. 184　No. 2　p. 587-619　2002/09　Research paper (scientific journal)

  20. L-2 and L-infinity estimates of the solutions for the compressible Navier-Stokes equations in a 3D exterior domain

    T. Kobayashi

    PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES　Vol. 38　No. 1　p. 211-225　2002/02　Research paper (scientific journal)

  21. Remark on the rate of decay of solutions to linearlized for the compressible Navier-Stokes equations

    T. Kobayashi and Y. Shibata

    Pacific J. Math.　Vol. 207　No. 1　p. 199-234　2002　Research paper (scientific journal)

  22. On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in R3

    Y. Kagei and T. Kobayashi

    Arch. Rat. Mech. Anal.　Vol. 207　No. 1　p. 89-159　2002　Research paper (scientific journal)

  23. Remark on the L-2 estimates of the density for the compressible Navier-Stokes flow in R-3

    R Ikehata, T Kobayashi, T Matsuyama

    NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS　Vol. 47　No. 4　p. 2519-2526　2001/08　Research paper (scientific journal)

  24. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R-3

    T Kobayashi, Y Shibata

    COMMUNICATIONS IN MATHEMATICAL PHYSICS　Vol. 200　No. 3　p. 621-659　1999/02　Research paper (scientific journal)

  25. On global motion of a compressible baro-tropic viscous fluid with boundary slip condition

    T. Kobayashi and W. M. Zajaczhowski

    Applicationes Mathematiae　Vol. 26　No. 2　p. 159-194　1999　Research paper (scientific journal)

  26. On the Oseen equation in the three dimensional exterior domains

    T Kobayashi, Y Shibata

    MATHEMATISCHE ANNALEN　Vol. 310　No. 1　p. 1-45　1998/01　Research paper (scientific journal)

  27. On the local energy decay approach to some fluid flow in an exterior domain

    W. Dan, T. Kobayashi and Y. Shibata

    Lecture Notes in Num. Appl. Anal. (Kinokuniya)　Vol. 16　p. 1-51　1998　Research paper (scientific journal)

  28. On the local energy decay of higher derivatives of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R-3

    T. Kobayashi

    PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES　Vol. 73　No. 7　p. 126-129　1997/09　Research paper (scientific journal)

  29. On a local energy decay of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3

    T. Kobayashi

    Tsukuba J. Math.　Vol. 21　No. 3　p. 629-670　1997　Research paper (scientific journal)

  30. ON A GLOBAL IN TIME EXISTENCE THEOREM OF SMOOTH SOLUTIONS TO A NONLINEAR-WAVE EQUATION WITH VISCOSITY

    T KOBAYASHI, H PECHER, Y SHIBATA

    MATHEMATISCHE ANNALEN　Vol. 296　No. 2　p. 215-234　1993/06　Research paper (scientific journal)

  31. ABSTRACT BESOV SPACE APPROACH TO THE NONSTATIONARY NAVIER-STOKES EQUATIONS

    T KOBAYASHI, T MURAMATU

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES　Vol. 15　No. 9　p. 599-620　1992/12　Research paper (scientific journal)

Misc. 14

  1. L2 boundedness of the solutions to the 2D Navier-Stokes equations and hyperbolic Navier-Stokes

    T. Kobayashi

    RIMS kokyuroku　Vol. 1971　p. 69-75　2015

  2. Weak solutions to the equations of Magnet-Hydrodynamics

    T. Kobayashi and T. Suzuki

    Proceedings of the National Institute for Mathematical Science　Vol. 2　p. 175-180　2007

  3. Weak Solutions to the Navier-Stokes-Poisson Equations(Mathematical Analysis in Fluid and Gas Dynamics : A conference in honor of Professor Tai-Ping Liu on his 60th Birthday)

    KOBAYASHI Takayuki, SUZUKI Takashi

    RIMS Kokyuroku　Vol. 1495　p. 1-7　2006/05

    Publisher: Kyoto University

  4. 星雲方程式の弱解について (波の非線形現象の数理とその応用)

    小林 孝行, 鈴木 貴

    数理解析研究所講究録　Vol. 1430　p. 76-80　2005/05

    Publisher: 京都大学

  5. Weak solutions to the equations of Magnet-Hydrodynamics

    T. Kobayashi, T. Suzuki

    RIMS kokyuroku　Vol. 1430　p. 76-81　2005

  6. Weak solutions to the Navier-Stokes Poisson equations

    T. Kobayashi, T. Suzuki

    GAKUTO Internatinal Ser. Math. Sci. Appl　Vol. 22　p. 173-178　2005

  7. Asymptotic behavior of solutions to the compressible Navier-Stokes equations on the half space

    T. Kobayashi, Y. Kagei

    Regularity and other aspects of the Navier-Stokes equations Banach Center Publications, Institute of Mathematics Warszawa　Vol. 70　p. 131-138　2005

  8. Interface Vanishing for Solutions to Maxwell and Stokes System

    T. Kobayashi, T. Suzuki, K. Watanabe

    RIMS kokyuroku　Vol. 1368　p. 65-71　2004

    Publisher: Kyoto University

  9. On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in R3

    Y. Kagei, T. Kobayashi

    RIMS kokyuroku　Vol. 1322　p. 40-45　2003

    Publisher: Kyoto University

  10. Some estimates of solutions for the equations of motion of compressible viscous fluid in an 3D exterior domain

    T. Kobayashi

    Progress in Nonlinear Differential Equations and Their Applications, Birkh\"{a}user　Vol. 42　p. 151-158　2000

  11. Existence and asymptotic behavior for solutions of the equations of motion of compressible viscous

    T. Kobayashi

    RIMS kokyurok　Vol. 1123　p. 107-112　2000

  12. On the exterior problem of compressible Navier-Stokes equation

    T. Kobayashi

    RIMS kokyuroku　Vol. 1996　p. 170-189　1996

  13. Exterior problem for the Navies-Stokes equations

    T. Kobayashi, Y. Shibata

    RIMS kokyuroku　Vol. 913　p. 185-190　1995

    Publisher: 京都大学

  14. Abstract Besov space approach to the non-stationary Navier-Stokes equations

    T. Kobayashi, T. Muramatu

    RIMS kokyuroku　Vol. 785　p. 76-94　1992

Publications 3

  1. 応用数理 : 基礎・モデリング・解法

    太田雅人, 鈴木貴,小林孝行,土屋卓也

    培風館　2015/12

    ISBN: 9784563011567

  2. The proceedings on Mathematical Fluid Dynamics and Nonlinear Wave

    Mathematical Fluid Dynamics and Nonlinear Wave (Conference), T. Kobayashi, S. Shimizu, Y. Enomoto, N. Yamaguchi and T. Kubo

    Gakkōtosho　2015

    ISBN: 9784762504624

  3. 線形代数入門

    中原徹,片山真一,小林孝行,工藤愛知

    学術図書出版社　2008

    ISBN: 9784780601046

Presentations 54

  1. On the diffusion wave property for the solutions to the compressible Navier-Stokes -Korteweg system

    T. Kobayashi

    Fluids and Maximal Regularit. Kyoto Univ. Japan　2024/09/11

  2. Global well-posedness of the compressible Navier-Stokes-Korteweg system under critical condition

    T. Kobayashi

    International Workshop on Multi-Phase Flows: Analysis, Modelling and Numerics. Waseda Univ. Japan　2022/12/08

  3. Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition

    T. Kobayashi

    "Maximal regularity and nonlinear PDE", RIMS in Kyoto Japan　2019/03

  4. Stability problem for a constant equilibrium to the Compressible Navier-Stokes-Korteweg system

    T. Kobayashi

    "Mathematical Fluid Mechanics and Related Topics", Tokyo Inst. Tech, Tokyo Japan　2018/09

  5. Decay property for the Compressible Navier-Stokes-Korteweg system

    T. Kobayashi

    "Workshop on Mathematical Sciences",Wayamba University, Sri Lanka,　2018/08

  6. On a local energy decay estimates of solutions to the linear hyperbolic Navier Stokes equations

    T. Kobayashi

    "The Third Wayamba International Conference", Wayamba University, Sri Lanka,　2018/08

  7. On a local energy decay estimates of solutions to the Hyperbolic type Stokes equations

    T. Kobayashi

    "Conference on Mathematical Fluid Dynamics" , Bad Boll, Germany,　2018/05

  8. Time decay estimate with diffusive property for solution to the compressible Navier-Stokes system

    T. Kobayashi

    "The 5th. Mathematical Theory of Turbulence via Harmonic Analysis and Computational Fluid Dynamics" , Nagoya University VBL, Nagoya Japan　2018/03

  9. Decay property for two-dimensional compressible Navier-Stokes equations

    T. Kobayashi

    The workshop "Nonlinear Partial Differential Equations in the Aspect of Regularity and Asymptotics”, Hotel Toya Sun Palace, Cristal Hall Toya, Hokkaido, Japan　2016/11

  10. Decay estimates of the solutions to the 2D compressible Navier-Stokes equation

    T. Kobayashi

    International Conference "The Navier-Stokes Equations and Related Topics", Nagoya University, Nagoya Japan　2016/03

  11. Decay property for the 2D Hyperbolic Navier-Stokes equations and its applications

    小林孝行

    「微分方程式の総合的研究」, 京都大学　2015/12

  12. Decay estimates of the solutions to the 2D linear viscoelastic equations and its applications to the compressible Navier-Stokes equations

    T. Kobayashi

    International Workshop on the Multi-Phase Flow; Analysis, Modeling and Numerics5", Waseda University, Tokyo Japan　2015/11

  13. L2-boundedness of the solutions to the 2D compressible Navier-Stokes equations

    T. Kobayashi

    "Mathflows 2015"，The IGESA center，Porquerolles, France　2015/09

  14. Decay estimates of solutions to the 2D dissipative wave equations and Hyperbolic Navier-Stokes

    T. Kobayashi

    "International Conference on Recent Advances in Hyperbolic Partial Differential Equations", International Conference Center Hiroshima, Japan　2014/12

  15. Decay estimates of the solutions to the 2D Hyperbolic Navier-Stokes equations.

    T. Kobayashi

    "Wayamba International Conference 2014", Wayamba University of Sri Lanka, Kuliyapitiya, Sri Lanka　2014/08

  16. Decay estimates of solutions to the 2D Hyperbolic Navier-Stokes equations

    T. Kobayashi

    2014 ICM-Satellite-Cau “Mathematical Theorey of Gases and Fluids and Related Applications”, Chung-Ang University, Seoul, Korea　2014/08

  17. L2 boundedness for the 2D exterior problems for the semilinear heat and its applications

    T. Kobayashi

    International conference "Quantization blow-up and concerntration in mathematical physics view point", Sigma hall, Dep. Engineering science, Osaka Univ, Japan　2013/03

  18. L2 boundedness for the solutions to the 2D semilinear heat equations and the 2D Navier-Stokes equations

    T. Kobayashi

    JSPS-DFG Japanese-German Graduate Externship, "The 7th Japanese-German International Workshop on Mathematical Fluid Dynamics", Waseda University Tokyo, Japan　2012/11

  19. L2 boundedness for the solutions to the 2D Navies-Stokes equations

    T. Kobayashi

    Parabolic and Navier-Stokes Equations 2012, Banach center, Bedlewo, Poland　2012/09

  20. Weighted Lp - theory for the Stokes resolvent in some unbounded domains

    T. Kobayashi

    The 4th Japanese-German International Workshop on Mathematical Fluid Dynamics, Waseda Univ. Tokyo Japan　2011/11

  21. Decay property of solutions to the $2D$ Wave equations with damping terms or strong damping terms

    T. Kobayashi

    Pohang-Kyushu workshop on Partial Differential Equations, Pohang, Korea　2011/06

  22. Weighted estimates of the solutions to the Navier-Stokes equations in half space and perturbed

    T. Kobayashi

    Workshop on Applied Mathematics Euskadi-Kyushu 2011, Basque, Spain　2011/03

  23. Weighted Lp-Lq estimates for the Stokes semigroup in some unbounded domains

    T. Kobayashi

    Partial differential equations and Mathematical Physics, Kyoto, Japan　2010/11

  24. Weighted estimates of the solutions to the Navier-Stokes equations in unbounded domains

    T. Kobayashi

    Workshop for the regularity and singularity for some nonlinear partial differential equations, Suwon, Korea.　2010/10

  25. Weighted estimates for the Stokes semigroup in half space and perturbed half-space

    T. Kobayashi

    Regularity aspects of PDE - a week for Wojciech Zajaczkowski, Bedlewo, Poland　2010/09

  26. Weak solutions to some equations of the compressible viscous flow

    T. Kobayashi

    Linear and Nonlinear Waves, No.7, Shiga, Japan　2009/11

  27. Drift-Diffusion System from Barotoropic Compressible Naveier-Stokes-Poisson system

    T. Kobayashi

    Analysis Seminar, Beijin, Chaina.　2008/11

  28. Fluid mechanical approximation to the degenerated drift- diffusion system from compressible Navier-Stokes-Poisson system

    T. Kobayashi

    Mathematical Fluid Dynamics, Darmstadt, Germany　2008/09

  29. Weighted estimates for the Stokes semigroup in an exterior domain

    T. Kobayashi

    Parabolic and Navier Stokes 2008, Bedlewo, Poland.　2008/09

  30. Fluid mechanical approximation to the degenerated drift- diffusion system from compressible Navier-Stokes-Poisson system

    T. Kobayashi

    New Aspects and Development of Mathematical Analysis in Nonlinear Phenomena, Fukuoka, Japan　2008/05

  31. The local energy decay approach to an incompressible flow in an exterior domain

    T. Kobayashi

    Navier-Stokes equations and related topics: KOSEF-JSPS joint research and seminar project, Suwon, Korea　2008/01

  32. Fluid mechanial approximation to the degenerated drift-diffusion system from compressible Navier-Stokes-Poisson system

    T. Kobayashi

    Partial Differential Equations Seminar, Suwon, Korea　2007/08

  33. Weak solution to the equations for the Magneto-Hydrodynamics

    T. Kobayashi

    NIMS International Workshop on Fluid Dynamics, Daejeon, Korea　2007/06

  34. Semigroup approach to the weak solutions for the equations of Mageto-Hydrodynamics

    T. Kobayashi

    Navier-Stokes equations and related topics(Kosef-Jsps Joint research and seminar project), Suwon, Korea.　2007/01

  35. Weak solutions to some equations of the compressible viscous flow

    T. Kobayashi

    The Sixth East Asia PDE conference, Wuhan University, China　2006/05

  36. Weak solutions to some equations of the compressible viscous flow

    T. Kobayashi

    Navier-Stokes equations and Potetioal theory, Soeul, Korea　2006/03

  37. Weak solution to some fluid dynamics equations

    T. Kobayashi

    Pusan-Kyushu Symposium on Partial Differential Equations, Pusan, Korea　2006/01

  38. Weak Solutions to the Navier-Stokes-Poisson Equations

    T. Kobayashi

    Kyoto Conference on the Navier-Stokes Equations and Their Applications, Kyoto, Japan　2006/01

  39. 半空間における圧縮性 Navier-Stokes 方程式の解の漸近挙動について

    小林孝行

    日本数学会秋季総合分科会 函数方程式論分科会 特別講演，岡山大学　2005/09

  40. Weak solutions to the Navier-Stokes-Poisson equation

    T. Kobayashi

    The Fifth East Asia PDE Conference, Osaka, Japan　2005/02

  41. Interface Vanishing for Solutions to Maxwell and Stokes Systems

    T. Kobayashi

    Tenth International Conference on Hyperbolic Problems, Osaka, Japan　2004/09

  42. Diffusion waves for the Navier-Stokes equations of compressible flow in the half space

    T. Kobayashi

    The Fourth World Congerss of Nonlinear Analysts, Orand, Florida, USA　2004/07

  43. Asymptotic behavior of solutions to the compressible Navier-Stokes equations on the half space

    T. Kobayashi

    Regularity and other qualitative aspects of the Navier-Stokes equations, Bedlewo, Poland　2003/08

  44. Interface regularity for Maxwell and Stokes Systems

    T. Kobayashi

    Workshop on Nonlinear Problems in Gauge Theory and Fluid Dynamics, Seoul, Korea　2002/12

  45. Asymptotic behavior of solutions to the compressible Navier-Stokes equation

    T. Kobayashi

    International Conference on Partial Differential Equation, Seoul, Korea　2001/12

  46. Large time behavior of solutions for the compressible Navier-Stokes equations

    T. Kobayashi

    International Conference on Frontier of Applied Analysis- Fluid Dynamics and Pattern Formation, Fukuoka, Japan.　2001/12

  47. Some estimates of solutions for the motion of compressible viscous fluid in a 3D exterior domain

    T. Kobayashi

    The Third World Congeress of Nonlinear Analysts, Catania, Sicily, Italy　2000/07

  48. On a decay rate of solutions to compressible Navier-Stokes equations

    T. Kobayashi

    Symposium on Nonlinear Partial Differential Equations， Vienna, Austria　2000/03/06

  49. 3次元外部領域における圧縮性Navier-Stokes方程式の解の漸近挙動について

    小林孝行

    日本数学会秋季総合分科会 実函数論分科会 特別講演， 広島大学　1999/09

  50. Some estimates of solutions for the equations of motion of compressible viscous fluid in 3D exterior problem

    T. Kobayashi

    The seventh International Conference on Navier-Stokes and Related Nonlinear Problems, Ferrara, Italy　1999/09

  51. Some estimates of solutions for the equations of motion of compressible viscous fluid in 3D exterior domain

    T. Kobayashi

    International Conference on Differential Equations and Related Topics, Pusan, Korea　1999/06

  52. Some estimates of solutions for the equations of motion of compressible viscous fluid in 3d exterior

    T. Kobayashi

    Fluid Dynamics: Existence, Regularity and Singularity, Warsow, Poland　1999/03

  53. Decay estimates for the compressible Navier-Stokes equations in 3D-exterior domains

    T. Kobayashi

    First International Conference on Semigroups of Operators, Theorey and Applications,California, USA.　1998/12

  54. Decay estimates for the compressible Navier-Stokes equations in 3D-exterior domains

    T. Kobayashi

    Fifth International Conference on Evolution Equations and Their Applications to Technology, Hiroshima, Japan　1996/10

Institutional Repository 1

Content Published in the University of Osaka Institutional Repository (OUKA)

  1. Interface regularity for Maxell and Stokes systems

    Kobayashi Takayuki, Suzuki Takashi, Watanabe Kazuo

    Osaka Journal of Mathematics　Vol. 40　No. 4　p. 925-943　2003/12