流体力学方程组中若干奇异极限问题的研究

流体力学中的奇异极限问题是数学物理学家关心的核心问题，也是偏微分方程中的重要研究方向。本项目拟研究两个奇异极限问题：不可压缩Navier-Stokes方程的粘性消失极限和水波方程的modulation极限，其主要内容包括：.（1）对于带Navier friction边界条件的2维不可压缩Navier-Stokes方程，在剪切流附近，研究边界层展开在Sobolev空间中的有效性。.（2）考虑圆环上的稳态Navier-Stokes方程，对于一类特殊的Dirichlet边界条件，研究稳态边界层展开在Sobolev空间中的有效性。.（3）对于2d有限深水波方程，严格验证非线性Schrodinger方程的有效性。

The singular limit problem in hydrodynamical equations has been a key problem for mathematical physicists and an important research direction in PDE. This project intends to study two singular limit problems: the vanishing viscosity limit of the Navier-Stokes equations and the modulation limit of the water wave equations，and the main objective contains:.(1)Consider the incompressible Navier-Stokes equations with Navier friction boundary condition, we study the validity of Prandtl expansion in Sobolev space in the neighborhood of shear flow..(2)Consider the steady Navier-Stokes equations in ring, we justify the validity of steady Prandtl expansion in Sobolev space for some special Dirichlet boundary condition..(3)Consider the 2d water wave equations with finite depth, we justify the validity of nonlinear Schrodinger equations .