Study of the stability of motions of incompressible fluids and the well-posedness of equations that govern their flow

研究不可压缩流体运动的稳定性和控制其流动的方程的适定性

• 批准号: 13440055
• 负责人: WATANABE Kiyoshi
• 金额: $ 9.73万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440055/
• 关键词: incompressible fluids; d'Alembert's paradox; viscous fluids; asymptotic behavior; heat kernel; decay rates; ideal fluids; Navier-Stokes flows; 非圧縮流体の方程式; 初期値境界値問題; 安定性; 漸近挙動; 渦輪; 結び目; 特異摂動

One of the investigator Tetsuro Miyakawa studied the asymptotic profiles of incompressible viscous flows in the whole space, in the half-space, or in exterior domains. As a result, he showed the following : The asymptotic behavior of flows depends on decaying properties of the initial velocities. However, under the boundary condition that the fluids rest at infinity, the first-order asymptotics of the velocities are given by the heat kernel or its first-order derivative. In particular, as for the flows that decay fast, the first-order derivative of the heat kernel appears in the first-order asymptotics. He also proved that upper bounds of decay rates for flows are actually the same as the ones for their first-order asymptotics. For instance, as for the flows in exterior domains, the net force exerted by the fluids must be zero, if the upper bounds of decay rates are attained by them. This fact corresponds to d'Alembert's paradox for ideal fluids. This study first showed that there exist corresponding solutions also for the equations of viscous fluids that were introduced in order to avoid the paradox. D'Alembert's paradox for ideal fluids arises under general boundary conditions. On the other hand, the paradox for viscous fluids seems to need special symmetry. This may imply the advantage of the theory for viscous fluids in comparison with the theory for ideal fluids.

研究者之一Tetsuro Miyakawa研究了不可压缩粘性流在整个空间、半空间或外部区域中的渐近分布。结果，他证明了：流动的渐近行为取决于初速度的衰减特性。然而，在边界条件下，流体静止在无穷远，速度的一阶渐近给出的热核或其一阶导数。特别地，对于快速衰减的流动，热核的一阶导数出现在一阶渐近性中。他还证明了流的衰减率的上界实际上与其一阶渐近性的上界相同。例如，对于外部区域中的流动，如果流体达到衰减率的上界，则流体施加的净力必须为零。这一事实对应于理想流体的达朗贝尔悖论。这项研究首先表明，存在相应的解决方案，也为粘性流体的方程，为了避免悖论。理想流体的达朗贝尔悖论在一般边界条件下出现。另一方面，粘性流体的佯谬似乎需要特殊的对称性。这可能意味着与理想流体理论相比，粘性流体理论的优势。

项目成果

Teturo Miyakawa: "Notes on space-time decay properties of nonstationary incompressible Navier-Stokes flow in R^n"Funkcialaj Ekvacioj. 45. 271-289 (2002)

Teturo Miyakawa：“关于 R^n 中非平稳不可压缩纳维-斯托克斯流的时空衰变特性的注释”Funkcialaj Ekvacioj。

Teturo Miyakawa: "On optimal decay rates for weak solutions to the Navier-Stokes equations in R^n"Mathematica Bohemica. 126. 443-455 (2001)

Teturo Miyakawa：“关于 R^n 中纳维-斯托克斯方程弱解的最佳衰减率”Mathematica Bohemica。

Tetsuro Miyakawa: "Notes on space-time decay properties of nonstationary incompressible Navier-Stokes flows in R^n"Funkcialaj Ekvacioj. 45(印刷中). (2002)

Tetsuro Miyakawa：“R^n 中非平稳不可压缩纳维-斯托克斯流的时空衰变特性的注释”Funkcialaj Ekvacioj 45（出版中）。

Yoshiko Fujigaki: "Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space"SIAM Journal on Mathematical Analysis. 33. 523-544 (2001)

Yoshiko Fujigaki：“整个空间中非平稳不可压缩纳维-斯托克斯流的渐近轮廓”SIAM 数学分析杂志。

Yasutaka Nakanishi, Yoshiyuki Ohyama: "Delta link homotopy for two component links, II"Journal of Knot Theory and Its Ramifications. 11. 353-362 (2002)

Yasutaka Nakanishi、Yoshiyuki Ohyama：“两个分量连杆的 Delta 连杆同伦，II”结理论及其分支杂志。