2003 Fiscal Year Final Research Report Summary
Boundary value problems for hyperbolic systems from fluid-mechanics and electromagnetics arising as the limit of singular perturbation
| Project/Area Number |
13640173
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Nara Women's University |
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Principal Investigator |
YANAGISAWA Taku Nara Women's University, Grasuate School of Humanities and Sciences, Associate Professor, 大学院・人間文化研究科, 助教授 (30192389)
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| Project Period (FY) |
2001 – 2003
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| Keywords | singular perturbation / vanishing viscosity limit / Navier-Stokes equations / Prandtl equation / vorticity |
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| Research Abstract |
For the purpose of building the mathematical framework to investigate the boundary value problems for hyperbolic systems as the limit of singular perturbation, we have shown the following mathematical results through the consideration of concrete problems appearing in the fluid-mechanics.
1)."Vanishing viscosity limit for the initial boundary value problems of the compressible Navier-Stokes equations in a domain with the boundary"
We study the existence theorem for the initial boundary value of the Prandtl equation which appears as the first term of the boundary expansion of asymptotic solution to the compressible Navier-Stokes equations. By taking the Fokker-Planck type equation as the linearized equation, we can show the estimate with the improvement in the order of regularity. However, it is also observed that there should occur the phenomenon of the "loss of derivatives" for this linearized problem. Hence it is so far most likely to be difficult to show the existence theorem for the Prandtl equation in the Sobolev spaces.
2)."On the relation of the smoothness of the solutions of the 3-D Navier-Stokes equations in a bounded domain with the vorticity"
We have shown a new a-priori estimate of the solutions to the 3-D Navier-Stokes equations in a bounded domain which reveals that the maximum norm of the vorticity controls the smoothness of the solutions. Further we presented a generalized Biot-Savart law on a bounded domain with the estimates of the Green's matrix of the Laplace operator, which was used in the proof of the new a-priori estimate stated above.
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