2006 Fiscal Year Final Research Report Summary
On a stability of the solutions to the equations of the Compressible flow
| Project/Area Number |
17540167
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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 | Saga University |
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Principal Investigator |
KOBAYASHI Takayuki Saga University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (50272133)
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| Co-Investigator(Kenkyū-buntansha) |
MASAHITO Masahito Nagoya Institute of Technology, Graduate School of Engineering, professor, 大学院・工学研究科, 教授 (80205129)
KAGEI Yoshiyuki Kyushu University, Faculty of Mathematics, professor, 大学院・数理学研究院, 教授 (80243913)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Navier-Stokes equations / Navier-Stokes-Poisson equations / Compressible viscous flow / sub differential equations |
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| Research Abstract |
We consider the initial boundary value problem of the Compressible Navier-Stokes equations and show the asymptotic behavior to the solutions for the small initial data near the equilibrium states. Concerning the initial boundary value problem of the Compressible Navier-Stokes-Poisson equations in a bounded domain, we prove the existence theorem of weak solutions globally in time. Also, we consider the component-wise regularity of the solution to stationary Maxwell or Stokes systems.
Asymptotic behavior of solutions to the compressible Navier-Stokes equation on the half is considered around a given constant equilibrium. A solution formula for the linearized problem is derived, and some estimates for solutions of the linearized problem are obtained. It is shown that, as in the case of the Cauchy problem, the leading part of the solution of the linearized problem is decomposed into two parts, one behaves like diffusion waves and the other one behaves like purely diffusively. There, however
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, appear some aspects different from the Cauchy problem, especially in considering spatial derivatives. It is also shown that the solution of the linearized problem approaches in large times to the solution of the nonstationary Stokes problem in some spaces; and, as a result, a solution formula for the nonstationary Stokes problem is obtained Large time behavior of solutions of the nonlinear problem is then investigated in some by applying the results on the linearized analysis and the weighted energy method. The results indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem.
We consider the Navier-Stokes-Poisson equation describing the motion of compressible viscous isentropic gas flow under the self-gravitational force. We prove the existence of finite energy weak solutions in three dimensional bounded domain and discuss the stability of equilibrium.
We consider the component-wise regularity of the solution to stationary Maxwell or Stokes systems. We assume that there is a surface, regarded as an interface, and the solution to one of those systems is smooth except for this interface. Then, only under those assumptions, we can show that some components of solution are smooth across the interface. Namely, in the Maxwell system, the normal component of solution is always regular across the interface. In the Stokes system, on the other hand, the singularity of solution across the interface can arise only to the normal derivatives of its tangential components. Furthermore, those results are shown to be optimal. Less
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Research Products
(35 results)
