Boundary value problems for hyperbolic systems from fluid-mechanics and electromagnetics arising as the limit of singular perturbation

流体力学和电磁学双曲系统的边值问题作为奇异摄动的极限而出现

• 批准号: 13640173
• 负责人: YANAGISAWA Taku
• 金额: $ 0.96万
• 依托单位: Nara Women's 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 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640173/
• 关键词: singular perturbation; vanishing viscosity limit; Navier-Stokes equations; Prandtl equation; vorticity; ホッジ分解; 国際情報交換; 中国; 対称双曲性; 圧縮性オイラー方程式; 圧縮性ナビエ・ストークス方程式; ブラントール方程式; 非圧縮性ナビエ・ストークス方程式

For the purpose of building the mathematical framework to investigate the boundary value problems for hyperbolic systems as the limit of singular perturbation, we have shown the following mathematical results through the consideration of concrete problems appearing in the fluid-mechanics.1)."Vanishing viscosity limit for the initial boundary value problems of the compressible Navier-Stokes equations in a domain with the boundary"We study the existence theorem for the initial boundary value of the Prandtl equation which appears as the first term of the boundary expansion of asymptotic solution to the compressible Navier-Stokes equations. By taking the Fokker-Planck type equation as the linearized equation, we can show the estimate with the improvement in the order of regularity. However, it is also observed that there should occur the phenomenon of the "loss of derivatives" for this linearized problem. Hence it is so far most likely to be difficult to show the existence theorem for the Prandtl equation in the Sobolev spaces.2)."On the relation of the smoothness of the solutions of the 3-D Navier-Stokes equations in a bounded domain with the vorticity"We have shown a new a-priori estimate of the solutions to the 3-D Navier-Stokes equations in a bounded domain which reveals that the maximum norm of the vorticity controls the smoothness of the solutions. Further we presented a generalized Biot-Savart law on a bounded domain with the estimates of the Green's matrix of the Laplace operator, which was used in the proof of the new a-priori estimate stated above.

为了建立研究作为奇摄动极限的双曲型方程组边值问题的数学框架，我们通过对流体力学中具体问题的考虑，得到了如下数学结果：1）.“可压缩Navier-Stokes方程初边值问题的粘性消失极限“研究了可压缩Navier-Stokes方程渐近解的边界展开式的第一项Prandtl方程初边值的存在性定理。通过将Fokker-Planck型方程作为线性化方程，我们可以证明估计值在正则性量级上的改进。然而，它也被观察到，应该发生的现象的“损失的衍生物”的线性化问题。因此，到目前为止，很可能很难证明Sobolev空间中的普朗特方程的存在性定理。“关于有界区域中三维Navier-Stokes方程解的光滑性与涡量的关系“我们给出了有界区域中三维Navier-Stokes方程解的一个新的先验估计，揭示了涡量的最大范数控制解的光滑性。进一步给出了有界区域上的广义Biot-Savart定律，并给出了拉普拉斯算子的绿色矩阵的估计，用于证明上述新的先验估计.