Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space.¶ II. Construction of the Navier-Stokes Solution
Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space.¶ II. Construction of the Navier-Stokes Solution
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DOI：
10.1007/s002200050305
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发表时间：
1998-03
影响因子：
2.4
通讯作者：
M. Sammartino;R. Caflisch
中科院分区：
文献类型：
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作者：
M. Sammartino;R. Caflisch
这是两篇关于不可压缩Navier-Stokes方程在二维或三维半空间中的零粘性极限的论文中的第二篇。 在解析初值的假设下，我们构造了与粘性无关的短时Navier-Stokes方程解。Navier-Stokes方程的解是通过一个复合渐近展开式构造的，该展开式包括第一篇论文中构造的Euler方程和Prandtl方程的解，加上一个误差项。这表明，纳维尔-斯托克斯解在边界层外变为欧拉解，在边界层内变为普朗特方程解。误差项被写为一阶欧拉和普朗特校正加上另一个误差项的总和。误差项的方程是弱非线性的，其线性部分是时间相关的斯托克斯方程。 该误差方程通过斯托克斯方程的反演来求解，通过将解表示为规则（类欧拉）部分加上边界层（类普朗特）部分。在这种分析中的主要技术工具是抽象的柯西-科瓦洛夫斯基定理。
This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.