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2001 Fiscal Year Final Research Report Summary

Study on the fundamental solutions to the equations of radiating gases and its applications

Research Project

Project/Area Number 11440049
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

KAWASHIMA Shuichi  Kyushu. Univ., Grad. Sch. Math., Prof., 大学院・数理学研究院, 教授 (70144631)

Co-Investigator(Kenkyū-buntansha) OGAWA Takayoshi  Kyushu. Univ., Grad. Sch. Math., Associate Prof., 大学院・数理学研究院, 助教授 (20224107)
KAGEI Yoshiyuki  Kyushu. Univ., Grad. Sch. Math., Associate Prof., 大学院・数理学研究院, 助教授 (80243913)
YOSHIKAWA Atsushi  Kyushu. Univ., Grad. Sch. Math., Prof., 大学院・数理学研究院, 教授 (80001866)
KOBAYASHI Takayuki  Kyushu Ins. Tech., Fac. Eng., Associate Prof., 工学部, 助教授 (50272133)
NISHIBATA Shinya  Tokyo Ins. Tech., Grad. Sch. Inf. Sci. Eng., Associate Prof., 大学院・情報理工学研究科, 助教授 (80279299)
Project Period (FY) 1999 – 2001
Keywordshyperbolic-elliptic system / hyperbolic-parabolic system / fundamental solution / pointwise estimate / global solutions / asymptotic slability / nonlinear waves / singular limit
Research Abstract

We study the stability of nonlinear waves for hyperbolic-elliptic coupled systems in radiation hydrodynamics and related equations.
1. By using the Fourier transform, we give a representation formula for the fundamental solutions to the linearized systems of hyperbolic-elliptic coupled systems and verify that the principal part of the fundamental solutions is given explicitly in terms of the heat kernel. Also, we obtain the sharp pointwise estimates for the error terms.
2. We obtain the pointwise decay estimate of solutions to the hyperbolic-elliptic coupled systems by using the representation formula for the fundamental solution and the corresponding estimates. Furthermore, we prove that the solution is asymptotic to the superposition of diffusion waves which propagate with the corresponding characteristic speeds.
3. We discuss a singular limit of the hyperbolic-elliptic coupled systems. We prove that at this limit, the solution of the hyperbolic-elliptic coupled system converges to that of the corresponding hyperbolic-parabolic coupled system.
4. We show the existence of stationary solutions to the discrete Boltzmann equation in the half space. It is proved that the stationary solution approaches the far field exponentially and is asymptotically stable for large time.
5. We study the asymptotic behavior of nonlinear waves for the isentropic Navier-Stokes equation in the half space. For the out-flow problem, we prove the asymptotic stability of nonlinear waves such as (1)stationary wave, (2)rarefaction wave, and (3)superposition of stationary wave and rarefaction wave.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] Shuichi Kawashima: "Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries"Commun. Math. Phys.. 211. 183-206 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shuichi Kawashima: "A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics"Indiana Univ. Math. J.. 50. 567-589 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Tatsuo Iguchi: "On space-time decay properties of solutions to hyperbolic-elliptic coupled systems"Hiroshima Math. J.. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Keiichi Kato: "Analyticity and smoothing effect for the Korteweg-de Vries equation with a single point singularity"Math. Annalen. 315. 577-608 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yoshiyuki Kagei: "Invariant manifolds and long-time asymptotic for the Vlasov-Poisson-Fokker-Plank equation"SIAM J. Math. Anal.. 33. 489-507 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shinya Nishibata: "Large time behavior of solutions to the Cauchy problem for one-dimensional thermoelastic system with dissipation"J. Inequal. Appl.. 6. 167-189 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Shuichi Kawashima: "Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries"Commun. Math. Phys.. 211. 183-206 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shuichi Kawashima: "A singular limit for hyperbolic-elliptic coupled systems in radiation hydrodynamics"Indiana Univ. Math. J.. 50. 567-589 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Tatsuo Iguchi: "On space-time decay properties of solutions to hyperboic-elliptic coupled systems"Hiroshima Math. J.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Keiichi Kato: "Analyticity and smoothing effect for the Korteweg-de Vries equation with a single point singularity"Math. Annalen.. 315. 577-608 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yoshiyuki Kagei: "Invariant manifolds and long-time asymptotics for the Vlasov-Poisson-Fokker-Planck equation"SIAM J. Math. Anal.. 33. 489-507 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shinya Nishibata: "Large time behavior of solutions to the Cauchy problem for one-dimensional thermoelastic system with dissipation"J. Inequal. Appl.. 6. 167-189 (2001)

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

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Published: 2003-09-17  

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