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

为了建立研究作为奇异摄动极限的双曲系统边值问题的数学框架，我们通过考虑流体力学中出现的具体问题，给出了以下数学结果。本文研究了Prandtl方程初边值作为可压缩Navier-Stokes方程渐近解的边界展开的第一项的存在性定理。通过将Fokker-Planck型方程作为线性化方程，我们可以用改进的规则阶来表示估计。然而，我们也观察到，对于这种线性化的问题，会出现“导数损失”的现象。因此，到目前为止，在Sobolev空间中证明普朗特方程的存在性定理很可能是困难的。“关于三维Navier-Stokes方程在有界域中解的光滑性与涡度的关系”我们给出了三维Navier-Stokes方程在有界域中解的一个新的先验估计，揭示了涡度的最大范数控制着解的光滑性。在此基础上，利用拉普拉斯算子的格林矩阵的估计给出了有界域上的广义Biot-Savart定律，并将其用于证明上述新的先验估计。

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.