2005 Fiscal Year Final Research Report Summary
Mathematical analysis on the structure of solutions for the fundamental systems of equations in continuum mechanics
| Project/Area Number |
15340050
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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 | KEIO UNIVERSITY |
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Principal Investigator |
TANI Atusi Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (90118969)
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| Co-Investigator(Kenkyū-buntansha) |
KIKUCHI Norio Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (80090041)
SIMOMOURA Shun Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (00154328)
NODERA Takashi Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (50156212)
ISHIKAWA Shiro Keio Univ., Faculty of Sci.and Tech., Associate Professor, 理工学部, 助教授 (10051913)
TAKAYAMA Masahiro Keio Univ., Faculty of Sci.and Tech., Assistant, 理工学部, 助手 (90338252)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Navier-Stokes equations / Euler equation / Free boundary problems / infinite sector / Contact line / Functions of bounded variation / nonlinear acoustics |
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| Research Abstract |
Among the fundamental equations in continuum mechanics we have obtained the following results.
1.Since it is well known that the evolution problems of isentropic Euler equation admit shock waves even if the initial data are smooth, we usually try to find the solution belonging to the function spaces of bounded variations. In order to guarantee the uniqueness of the solution it is convenient to construct such a solution as a limit of the solution to the approximate equation of parabolic type. We succeeded to construct the temporally global solution in the class of functions of bounded variations to this approximate equation. Up to the present time we have had a scenario due to Nishida and Smoller to construct such a solution. However, their scenario is valid only if the density is bounded. We firstly succeeded to prove its boundedness, so that in real sense their scenario works.
2.Among the two-dimensional evolution free boundary problems for incompressible viscous fluid we study the case
… More
where the free boundary and the boundary of the container has a contact line. As a series of our study on the solvability of Navier-Stokes equations in a container with slip boundary conditions, here we investigated the two problems :
(1)Stokes equations in infinite sector,
(2)Navier-Stokes equations in a domain with piecewise smooth boundary.
Then we proved their solvability in the weighted Sobolev spaces.
3.For the one-dimensional model equations of a self-gravitating viscous radiative and reactive gas we found the unique global in time solution belonging to Hoelder spaces. For this problem we used the Stefan-Boltzmann relation.
4.In a two-dimensional infinite elastic or visco-elastic strip with a semi-infinite crack we studied the solvability to the stationary problem and determined the propagation of the crack. Moreover, we proved the weak solvability of its evolution problem.
Now the following results are preparing : (1)To construct the solution around the Gerstner's trochoidal wave and 3D domain for incompressible inviscid flow (2)Nonlinear problems in nonlinear acoustics. Less
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