2006 Fiscal Year Final Research Report Summary
Research on global solutions in time and their asymptotic behaviors for viscous and relaxation models of conservation laws
| Project/Area Number |
15340043
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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 |
Basic analysis
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| Research Institution | Osaka University |
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Principal Investigator |
MATSUMURA Akitaka Osaka University, Graduate School of Information Science and Technology, Professor, 大学院情報科学研究科, 教授 (60115938)
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| Co-Investigator(Kenkyū-buntansha) |
HAYASHI Nakao Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (30173016)
KOTANI Shin'ichi Osaka University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (10025463)
ODANAKA Shinji Osaka University, Cybermedia Center, Professor, サイバーメディアセンター, 教授 (20324858)
NISHIHARA Kenji Waseda University, Dept.of Politics and Economics, Professor, 政治経済学部, 教授 (60141876)
NISHIBATA Shinya Tokyo Institute of Technology, Graduate School of Information Science and Engineering, Associate Professor, 大学院情報理工学研究科, 助教授 (80279299)
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| Project Period (FY) |
2003 – 2006
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| Keywords | conservation law / asymptotic stability / boundary layer / viscous shock wave / viscous contact wave / semiconductor / quantum fluid. dynamical model / dissipative wave equation |
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| Research Abstract |
1.Initial boundary value problems on the half space to one-dimensional isentropic models for compressible fluid are investigated. In a case where the fluid inflows on the boundary, it is proved that the superposition of the boundary layer solution and viscous shock wave (or rarefaction wave) is asymptotically stable under some smallness conditions. In a case where the fluid outflows on the boundary, the existence of the boundary layer solution and its asymptotic stability are also proved. Furthermore, an initial boundary value problem on the half space to a one-dimensional ideal gas model (3 by 3 system) is investigated, and then the asymptotic stability of the viscous contact wave is proved. The asymptotic stability of the viscous contact wave for the Cauchy problem is also proved, if the integral of the initial perturbation is zero. Among multi-dimensional problems, it is proved that the spherically symmetric solution of an isothermal model of the compressible Navier-Stokes equation
… More
globally exists in time and tends toward its stationary solution.
2.Initial boundary value problems for a one-dimensional model which describes the movement of electrons in semiconductors are investigated. It is proved that even for any large doping profile the stationary solution exits and is asymptotically stable. As for the multi-dimensional models, an iterative algorithm of numerical computation with high resolution to simulate the stationary solutions is developed.
3.Asymptotic behavior of solutions of a one-dimensional model of compressible viscous fluid in porous media is investigated. It is proved that the solution tends toward a diffusion wave of a parabolic equation, and the precise decay rate of asymptotics to the diffusion wave is also obtained.
4.Asymptotic behavior of solutions of nonlinear dispersive equations and dissipative equations with a critical nonlinearity is investigated. It is proved that the solution of dissipative wave equation tends toward a self-similar solution of a heat equation. As for the dispersive equations, a new result on how the resonance phenomena between the characteristic frequency number of the linearized equation and that of nonlinear term influences the asymptotic behavior of the solutions. Less
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