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2003 Fiscal Year Final Research Report Summary

Mathematical Analysis to Nonlinear Partial Differential Equations in Mathematical Science and Its Applications

Research Project

Project/Area Number 13440053
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionOsaka University

Principal Investigator

SUZUKI Takashi  Graduate School of Engineering Science, Division of Mathematical Science, Professor, 大学院・基礎工学研究科, 教授 (40114516)

Co-Investigator(Kenkyū-buntansha) ODANAKA Shinji  Cyber Media Center, Professor, サイバーメディアセンター, 教授 (20324858)
MATSUMURA Akitaka  Graduate School of Information and Technology, Department of Information Mathematics, Professor, 大学院・情報科学研究科, 教授 (60115938)
KOTANI Shin'ichi  Graduate School of Science, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (10025463)
WADA Takashi  Graduate School of Science, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助手 (70294139)
SUGIMOTO Mitsuru  Graduate School of Science, Department of Mathematics, Associate Professor, 大学院・理学研究科, 助教授 (60196756)
Project Period (FY) 2001 – 2003
Keywordsquantized blowup mechanism / self-interacting particles / chemotaxis / self-dual gauge field theory / Chem-Simons-Higgs theory / dual variation / tumour growth equation / parallel optimization
Research Abstract

We have established the quantized blowup mechanism for the non-equilibrium mean field of many self interacting. First, we formulate the equilibrium state as a nonlinear eigenvalue problem with non-local tern. Then, the above mechanism is observed in this level, and thus, the problem arises as a story of nolinear quantum mechanics. We have settled down this long standing problem from the study of the weak solution arising in the backward self similar transformation, employing the method of second moment and forward self similar transformation. Furthermore, we observed that the type I blowup point has high emergence in the sense of Kauffman. To examine actual existence of such a blowup point, we evolved a numerical scheme provided with the conservation of mass, decrease of the fee energy, and the positivity of the solution. Next, we noticed that thus equilibrium state shares the same mathematical structure in the problem of self dual gauge field theory concerned with the super-conducting … More or the string theory. In use of the mathematical technique developed in the study of chemotaxis, such as the compensated compactness via the symmetiization or the theory of dual variation, first, we clarified the blowup mechanism in the Abelian-Higgs theory with vortices, in particular, quantization shift and the location of the blowup points. Next, we apply the method to SU (3)Toda system and obtained the saddle type solution. Furthermore, we constructed the periodic solution in Chern-Simons-Higgs theory, in use of the gluing method of Nolasco. From the view point of the formation of the field, we took up the system of tumour growth and obtained the solution provided with the profile expected in mathematics in medicine, globally in tune. To study other mean field theories following the story of nonlinear quantum mechanics, we provided the general theory of dual variation, and applied the study on the Euler-Poisson equation concerning the formation of nebula, and the phase separation model. of Penrose-Fife. To solve many under determined problems, we proposed the method of parallel optimization, developed the technique of clustering, and applied it to the data analysis of magnetoencephalography. We studied the profile of the interface between different media or in the. context of the free boundary problem such as the interface vanishing in the Maxwell system or the Stokes system and the boundary trial method for the problem of filtration. Less

  • Research Products

    (47 results)

All Other

All Publications (47 results)

  • [Publications] A.Kubo: "Asymptotic behavior of the solution to a parabolic-ODE system modeling tumour growth"Differential and Integral Equations. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Suzuki: "Existence of the non-topological condensate in self-dual Chern-Simons gauge theory"Advances in Differential Equations. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Miyasita: "Non-local Gel'fand problem in higher dimension"Banch Center Publications. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Suzuki: "Non-topological condensates in self-dual Chern-Simons gauge theory"Banch Center Publications. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Senba: "Chemotactic collapse of radial solutions to J\"ager-Luckhaus system"Advances in Mathematical Science and Applications. 14(印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Naito: "Self-similar solutions to a nonlinear parabolic - elliptic system"Taiwanise Journal of Mathematics. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Suzuki: "Energy transport problem for self-gravitating particles"Taiwanise Journal of Mathematics. 8. 1-14 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Kobayashi: "Interface regularity for Maxwell, and Stokes systems"Osaka Journal of Mathematics. 40. 925-944 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Suzuki: "Mass normalization of collapses in the theory of self-interacting particles"Advances in Mathematical Science and Applications. 13. 611-623 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Suzuki: "Mass quantization of the non-equilibrium mean field to self-interacting particles"Dynamics of Continuous, Discrete and Impulsive Systems. 10. 693-720 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Kubo: "Study on the integral equation arising in electroencephalography"Advances in Mathematical Science and Applications. 13. 273-285 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Ohtsuka: "Palais-Smale sequence relative to the Trudinger-Moser inequality"Calculus of Variations. 17. 787-820 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Senba: "Blowup behavior of solutions to rescaled J\"ager-Luckhaus system"Advances in Differential Equations. 8. 787-820 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Tsuchiya: "Finite element approximation of $H$-surfaces"Mathematics of Computation. 72. 607-617 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Senaba: "Time global solutions to a parabolic-elliptic system modeling chemotaxis"Asymptotic Analhysis. 32. 63-89 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Naito: "Self-similar solutions to a parabolic system modeling chemotaxis"J.Differential Equations. 184. 386-421 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Senba: "Weak solutions to a parabolic system modeling chemotaxis"J.Functional Analysis. 191. 17-51 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Senba: "Parabolic system of chemotaxis : blowup in a finite and the infinite time"Meth.Appl.Anal.. 8. 349-368 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Senba: "Behavior of solutions to a system related to chemotaxis"Nonlinear Analysis. 47. 2551-2562 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Nagai: "Chemotactic collapse in a parabolic system of mathematical biogoly"Hiroshima Math.J.. 30. 463-497 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Senba: "Chemotacitc collapse in a parabolic-elliptic system of mathematical bilogoy"Adv.Differential Eequaitons. 6. 21-50 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Sugimoto: "A smoothing property of Schrodinger equations along the sphere"J.Anal.Math.. 89. 15-30 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Ruzhansky: "Global L^2 estimates for a class of Fourier integral operators with symbols in Besov spaces"Russian Math.Surveys. 58. 1044-1046 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Suzhansky: "A new proof of global smoothing estimates for dispersive equations"Adv.Peudo-Differential Operators. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.Huang: "Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas"Comm.Math.Phys.. 239. 261-285 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.Huang: "A gas-solid free boundary problem for compressible viscous gas"SIAM J.Math.Anal.. 34. 261-285 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.Huang: "On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary"Osaka J.Math.. 141(印刷中). (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Senba: "Applied Analysis : Mathematical Methods in Natural Science"Imperial College Press. 370 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Suzuki: "Free Energy and Self-Interacting Particles : A Mathematical Approach"Imperial College Press. 400 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kubo, A.: "Asymptotic behavior of the solution to a parabolic-ODE system modeling tumor growth"Differential and Integral Equations. (In press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Miyasita, T.: "Non-local. Gel'fand problem in higher dimension"Banach Center Publications. (In press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takahashi, F.: "Non-topological condensates in self-dual Chern-Simons gauge theory"Banach Center Publications. (In press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Senba, T.: "Chemotactic collapse of radial solutions to Jager-Luckhaus system"Adv.Math.Sci.. (In press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takahashi, F.: "Existence of non-topological Condensate in self dual Chern-Simons gauge theory"Adv.Differential Equations. 9. 539-562 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Suzuki, T.: "Energy transport problem for self gravitating particles"Taiwanise J.Math.. 8. 1-14 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kobayashi, T.: "Interface regularity for Maxwell and Stokes sytems"Osaka J.Math.. 40. 925-944 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Suzuki, T.: "Mass normalization of collapses in the theory of self-interacting particles"Adv.Math.Sci.Appl.. 13. 611-623 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Suzuki, T.: "Mass quantization of collapses in the theory of self interacting particles"Discrete and Impulsive Systems. 10. 693-720 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kubo, A.: "Study on the integral equation arising in electorencephalography"Adv.Math.Sci.. Appl.13. 273-255 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ohtsuka, H.: "Palais-Smale sequence relative to the Trudinger-Moser inequality"Canc.Vari.. 17. 273-285 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Senba, T.: "Blowup behavior of solutions to resealed Jager-Luckhaus system"Adv.Differential Equations. 8. 878-820 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Matsuzawa, Y.: "Finite element approximation of $H$-suystems"Math.Comp.. 72. 607-617 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Senba, T.: "Time global solutions to a parabolic-elliptic system modeling chemotaxis"Asymptotic Analysis. 32. 63-89 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Naito, Y.: "Self similar solutions to a parabolic system modeling chmotaxis"J.Differential Equations. 184. 386-421 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Senba, T.: "Weak solution to a parabolic system of chemotaxis"J. Functional Analysis. 191. 17-51 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Sugimoto, M.: "A smoothing property of Schrodinger equations along the sphere"J.Anal.Math.. 89. 15-30 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Huang: "Viscous shock wave and boundary layer solution to an inflow problem"Comm.Math.Phys.. 239. 261-285 (2003)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2005-04-19  

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