2022 Fiscal Year Final Research Report
To global analysis for solutions of nonlinear partial differential equations tems
| Project/Area Number |
20K03699
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2023-03-31
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| Keywords | 非線形編微分方程式系 / 熱対流問題 / Rayleigh-Benard 熱対流 / route to chaos / 擬圧縮性近似法 / 粘性流体の自由表面問題 |
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| Outline of Final Research Achievements |
Analysis on global behaviors of solutions according to change of physical parameters in the heat convection problems. When the Rayleigh number (the difference of temperature between the top surface and the bottom ) increases far across the critical number in the Rayleigh-Benard heat convection, the present analysis has no theory for the solutions. We made numerical simulations of the roll-type solutions for big Rayleigh numbers and found a route to chaos along bigger Rayleigh numbers.
There is a Chorin's method which makes pseudo-compressible approximations for the solution of incompressible viscous fluids. We made a justification of the method for the stationary solutions, stationary bifurcations and Hopf-bifurcation of Navier-Stokes equations and Rayleigh-Benard equations.
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| Free Research Field |
非線形偏微分方程式系の解析
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| Academic Significance and Societal Importance of the Research Achievements |
熱対流問題で上下の温度差が大きくなるに従って、熱伝導解から定常対流解に分岐し、パターンを形成した後、周期解分岐を起こし、周期倍分岐、周期４倍分岐を経て徐々に chaos 解に遷移する道を見つけた。長期的な予想の困難さの例になる。
非圧縮性粘性流体の解析をその擬圧縮性近似の方法によって行える事の正当性を示した。
非圧縮性条件の困難を避ける一つの方法である。
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