二维Navier-Stokes方程垂直粘性消失时的边界层与初始-边界角渐近性
Boundary Layers and Initial-Boundary Corner Asymptotics for the 2D Navier-Stokes Equations with Vanishing Vertical Viscosity
- Zhuhai NO.1 High School（珠海市第一中学）
- MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究二维Navier-Stokes方程垂直粘性消失极限，构造边界层剖面并给出误差估计，刻画初始-边界角渐近行为，分离固定与短时间尺度。
AI中文摘要：
受地球物理流体动力学中各向异性粘性的启发，其中垂直动量扩散通常远弱于水平扩散，在内部可能可忽略但在固体壁面附近仍然至关重要，我们研究了上半平面中二维不可压缩Navier-Stokes方程在垂直粘性消失极限下的行为，其中水平粘性固定为1，垂直粘性为\\(\varepsilon^2\\)。对于任意无散度的\\(H^4\\)无滑移初始数据，且无需时间微分相容性条件，我们在每个给定的有限区间\\([0,T]\\)上构造了极限水平粘性流和边界层剖面。首阶修正近似在\\(L^\infty\\)中具有\\(O(\varepsilon)\\)误差，而完整的有限阶展开将此误差降至\\(O(\varepsilon^{3/2})\\)，且一致地直到\\(t=0\\)。我们进一步描述了从初始时在壁面处消失的数据发展出边界层的过程。在初始-边界角处，我们确定了前两个自相似系数，并通过精确剖面尾部刻画匹配，从而将固定时间尺度\\(\varepsilon\\)与短时间尺度\\(\varepsilon\sqrt t\\)分离开来。当初始壁面加速度非零时，首阶修正率和第一个归一化角余项都是尖锐的。最后，针对固定数据的连续性估计给出了精确解的小时间和小粘性联合极限，以及有限\\(L^p\\)渐近性。
英文摘要：
Motivated by anisotropic viscosity in geophysical fluid dynamics, where vertical momentum diffusion is often much weaker than horizontal diffusion and may be negligible in the interior yet remains essential near a solid wall, we study the vanishing vertical viscosity limit for the two-dimensional incompressible Navier--Stokes equations in the upper half-plane, with horizontal viscosity fixed at one and vertical viscosity \(\varepsilon^2\). For arbitrary divergence-free \(H^4\) no-slip initial data, with no time-differentiated compatibility conditions required, we construct the limiting horizontally viscous flow and the boundary-layer profiles on every prescribed finite interval \([0,T]\). The leading-order corrected approximation has \(O(\varepsilon)\) error in \(L^\infty\), and the full finite-order expansion reduces this error to \(O(\varepsilon^{3/2})\), uniformly down to \(t=0\). We further describe the development of the layer from data that vanish initially at the wall. At the initial--boundary corner, we determine the first two self-similar coefficients and characterize matching through exact profile tails, thereby separating the fixed-time scale \(\varepsilon\) from the short-time scale \(\varepsilon\sqrt t\). When the initial wall acceleration is nonzero, both the leading correction rate and the first normalized corner remainder are sharp. Finally, continuity estimates for fixed data yield joint small-time and small-viscosity limits for the exact solution, as well as finite-\(L^p\) asymptotics.
