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

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  KEIO UNIVERSITY 
Principal Investigator 
TANI Atusi Keio Univ., Faculty of Sci.and Tech., Professor > 慶應義塾大学, 理工学部, 教授 (90118969)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
2003 – 2005

Keywords  NavierStokes equations / Euler equation / Free boundary problems / infinite sector / Contact line / Functions of bounded variation / nonlinear acoustics 
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 twodimensional evolution free boundary problems for incompressible viscous fluid we study the case
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where the free boundary and the boundary of the container has a contact line. As a series of our study on the solvability of NavierStokes equations in a container with slip boundary conditions, here we investigated the two problems : (1)Stokes equations in infinite sector, (2)NavierStokes equations in a domain with piecewise smooth boundary. Then we proved their solvability in the weighted Sobolev spaces. 3.For the onedimensional model equations of a selfgravitating viscous radiative and reactive gas we found the unique global in time solution belonging to Hoelder spaces. For this problem we used the StefanBoltzmann relation. 4.In a twodimensional infinite elastic or viscoelastic strip with a semiinfinite 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

Research Products
(12 results)