1999 Fiscal Year Final Research Report Summary
An Over-all Mathematical Study of the Nonlinear Boltzmann Equation and Fluid Dynamical Equations
| Project/Area Number |
09440051
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | Yokohama National University (1999)
Tokyo Institute of Technology (1997-1998) |
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Principal Investigator |
UKAI Siji Yokohama National University, Faculty of Engineering, Professor, 工学部, 教授 (30047170)
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| Co-Investigator(Kenkyū-buntansha) |
HIRANO Norimihi Yokohama National University, Faculty of Engineering, Professor, 工学研究科, 教授 (80134815)
TAKANO Seiji Yokohama National University, Faculty of Engineering, Professor, 工学部, 教授 (90018060)
KITADA Yasuhiko Yokohama National University, Faculty of Engineering, Professor, 工学部, 教授 (70016145)
SHIOJI Naoki Yokohama National University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (50215943)
KONNO Norio Yokohama National University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (80205575)
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| Project Period (FY) |
1997 – 1999
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| Keywords | Boltzmann equation / Boltzmann-Grad limit / asymptotic analysis / shock wave / periodic solution / non-relativistic limit / Euler equation / general outflow condition |
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| Research Abstract |
Boltzmann equation :
(1) A simple existence proof of the Boltzmann-Grad limit by means of the Cauchy-Kowalevskaya theorem and the establishement of an asymptotic relation between the Boltzmann hierachy and the macroscopic fluid equation (commpressible Euler equation) by the same theorem.
(2) An existence theorem of travering (shock) wave solutions and a solvability condition for the stationary problem in the half space, both for the discrete velocity model, which are expected to make an contribution to the study of boundary and shock layer structures of the Boltzmann equation.
(3) An existence theorem of time-periodic solutions of the Boltzmann equation, being a first analysis of the nonlinear acoustics of that equation.
Macroscopic fluid dynamical equation :
(1) Non-relativistic limits of solutions of the relativisitic Euler equation. In the case of the 1D flat Minkowski space-time, time-global weak solutions are shown to converge globally in time stongly in LィイD11ィエD1, as the speed of light tens to infinity, and similary for the case of the 3D non-flat space-time, but the convergence is time-local.
(2) A time-global existence theorem of weak and srong solutions to the Stokes approximation equation for the storngly viscous commonpressible fluid flow. While the equation has a strong nonlinearity, initial data can be arbitrarily large.
(3) An existence proof of the stationary solution to the heat covection equation without the unphysical condition of the zero outflow on each component of boundaries of the domain.
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