The global existence of solutions for the motion of viscous incompressible fluids

粘性不可压缩流体运动解的整体存在性

• 批准号: 13640179
• 负责人: KATO Hisako
• 金额: $ 0.32万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640179/
• 关键词: Navier-Stokes flow; non-Newtonian 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

This report is concerned with the initial boundary value problem for the nonstationary Navier-Stokes system in a bounded domain in R^3. We have found a modified Navier-Stokes system. By using the modified system we have shown the existence of Navier-Stokes flows changing to non-Newtonian flows in the following. For a given initial velocity a(x) we find a time-global strong solution u(x, t) which satisfies the Navier-Stokes system for all the time when the velocity gradient is below a positive number (a physical quantity) and satisfies a non-Newtonian 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 Navier-Stokes 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 Navier-Stokes 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 Navier-Stokes 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.

研究了R^3有界域上非平稳Navier-Stokes系统的初边值问题。我们发现了一个改进的纳维-斯托克斯系统。通过使用改进的系统，我们已经证明了存在的纳维-斯托克斯流转变为非牛顿流。对于给定的初速度a(x)，我们找到了一个时间全局强解u(x, t)，当速度梯度低于正数（一个物理量）时，u（x, t）一直满足Navier-Stokes系统，当速度梯度大于该数时，u（x, t）一直满足非牛顿系统。进一步，我们证明了存在T_a >，使得对于所有t∈[T_a，∞)的全局解满足Navier-Stokes系统，并且在[T_a，∞)中映射a(x)→u（x, t）是一对一的。在物理流体动力学中，Navier-Stokes方程是在流体的变化率足够小的假设下建立的，因此粘性应力与变化率线性相关。由于变形速率取决于流体中的速度梯度，因此Navier-Stokes方程似乎可以很好地表示小速度梯度下的流体运动。在此基础上，我们还考虑了大速度梯度下流体的运动，得到了一个修正方程。

项目成果

Kazuo Ito: "Three phase boundary motion by surface diffusion in triangular domain"Advances in Math.Sci.Appl.. 11. 753-779 (2001)

伊藤一夫：“三角域中表面扩散的三相边界运动”Math.Sci.Appl. 进展. 11. 753-779 (2001)

Kazuo Ito: "Three phase boundary motion by surface diffusion : stability of a mirror symmetric stationary solution"Interfaces and Free Boundaries. 3. 45-80 (2001)

伊藤一雄：“表面扩散引起的三相边界运动：镜像对称固定解的稳定性”界面和自由边界。

Kazuo Ito: "Three phase boundary motion by surface diffusion : in triangular domain"Advances in Math.Sci.Appl.. 11. 753-779 (2001)

伊藤一夫：“表面扩散的三相边界运动：在三角域中”Math.Sci.Appl. 进展. 11. 753-779 (2001)

Toachin Escher: "On a limiting motion and seef-intersections for the intermediate swface diffusion flow"J.Evolution Equations. 2. 349-364 (2002)

Toachin Escher：“关于中间表面扩散流的极限运动和自相交”J.进化方程。

Hisako Kato: "The Navier-Stokes flows Changing to non-Newtonian flows"Proceedings of the ISAAC Congress, 2001, world Scientific. (in press).

加藤久子：“纳维-斯托克斯流转变为非牛顿流”ISAAC 大会记录，2001 年，世界科学。