不可压缩流体边界层若干问题的研究

The vanishing viscosity limit of incompressible fluid is one of the hot and difficult problems in the field of fluid mechanics. In this project, we study the well-posedness theory of boundary layer equation, boundary layer expansion and vanishing dissipation limit for the incompressible fluid. First, We study the well-posedness of boundary layer equations and boundary layer expansion problem for inhomogeneous incompressible fluid. Since the inhomogeneous incompressible boundary layer equations are hyperbolic and degenerate parabolic coupled equations, we need to not only deal with the loss of tangential derivatives of horizontal velocity, but also establish appropriate estimates for density. Secondly, we establish the well-posedness results of boundary layer equations and boundary layer expansion for the incompressible magnetohydrodynamic fluids in three-dimensional flat or curved boundary regions, and explore the effect of magnetic field on the boundary layer stability. Due to the bending effect of the boundary and the singularity caused by small viscosity, we need to find the basic space that can be compatible with these two elements to study the boundary layer expansion problem. Finally, we study the vanishing dissipation limit problem for the full incompressible Navier-Stokes equations, and explore the influence of obstacle disappearance and region expansion on this problem. Through the study of the content involved in this project, We hope to achieve some good research results on the limit of vanishing viscosity for incompressible fluid.

不可压缩流体的粘性消失极限问题是流体力学领域的热点和难点问题之一。本项目拟研究不可压缩流体的边界层方程适定性理论、边界层展开和耗散消失极限问题。首先，我们研究非均匀不可压缩流体的边界层方程适定性和边界层展开问题。考虑到非均匀不可压缩边界层方程组为双曲和退化抛物的耦合方程组，我们不仅需要处理好水平速度切向导数丢失问题，而且还要对密度建立合适的估计。其次，在三维平坦或弯曲边界的区域中建立不可压缩磁流体的边界层方程适定性和边界层展开式，探索磁场对边界层的稳定性效应。由于区域边界的弯曲效应和小粘性引起奇性，所以我们需要寻找能够兼容这二者的基本空间来研究边界层展开问题。最后，研究完全不可压缩 Navier-Stokes 方程组的耗散消失极限问题，并探索障碍消失和区域扩张对其产生的影响。通过本项目所涉及内容的研究，我们希望能在不可压缩流体的粘性消失极限方面取得一些好的研究成果。

不可压缩流体的粘性消失极限问题是流体力学领域的热点和难点问题之一。本项目拟研究不可压缩流体的边界层方程适定性理论、边界层展开和耗散消失极限问题。首先，我们研究非均匀不可压缩流体的边界层方程适定性和边界层展开问题。考虑到非均匀不可压缩边界层方程组为双曲和退化抛物的耦合方程组，我们不仅需要处理好水平速度切向导数丢失问题，而且还要对密度建立合适的估计。其次，在三维平坦或弯曲边界的区域中建立不可压缩磁流体的边界层方程适定性和边界层展开式，探索磁场对边界层的稳定性效应。由于区域边界的弯曲效应和小粘性引起奇性，所以我们需要寻找能够兼容这二者的基本空间来研究边界层展开问题。最后，研究完全不可压缩 Navier-Stokes 方程组的耗散消失极限问题，并探索障碍消失和区域扩张对其产生的影响。通过本项目所涉及内容的研究，我们希望能在不可压缩流体的粘性消失极限方面取得一些好的研究成果。