The global existence of solutions for the motion of viscous incompressible fluids
Project/Area Number  13640179 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Basic analysis

Research Institution  KYUSHU UNIVERSITY 
Principal Investigator 
KATO Hisako KYUSHU UNIVERSITY, Faculty of Mathematics professor, 大学院・数理学研究院, 教授 (00038457)

CoInvestigator(Kenkyūbuntansha) 
ITO Kazuo KYUSHU UNIVERSITY, Faculty of Mathematics Associate Prof., 大学院・数理学研究院, 助教授 (20280860)

Project Period (FY) 
2001 – 2002

Project Status 
Completed(Fiscal Year 2002)

Budget Amount *help 
¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 2002 : ¥500,000 (Direct Cost : ¥500,000)

Keywords  NavierStokes flow / nonNewtonian flow / global existence / uniqueness / velocity gradient / incompressible / vriscous / stress tensor / 解の時間大域的存在 / 解の一意存在 
Research Abstract 
This report is concerned with the initial boundary value problem for the nonstationary NavierStokes system in a bounded domain in R^3. We have found a modified NavierStokes system. By using the modified system we have shown the existence of NavierStokes flows changing to nonNewtonian flows in the following. For a given initial velocity a(x) we find a timeglobal strong solution u(x, t) which satisfies the NavierStokes system for all the time when the velocity gradient is below a positive number (a physical quantity) and satisfies a nonNewtonian system for all the time when the velocity gradient is above the number. Furthermore, we have shown that there exists T_a > 0 such that the global solution satisfies the NavierStokes system for all t∈[T_a, ∞), and the mapping a(x) → u(x, t) in [T_a, ∞) is one to one. In the physical fluid dynamics, the NavierStokes equation is formulated under the assumption that the rate of deformation of fluids is sufficiently small and therefore the viscous stress is linearly related to the rate of deformation. Since the rate of deformation depends on the velocity gradient in the fluids the NavierStokes equation seems to be representing the motion of fluids well for small velocity gradients. From such consideration, we have found out a modified equation by taking the motion of fluids for large velocity gradients also into account.

Report
(3results)
Research Products
(20results)