外部区域的下游渐近性：从定常尾流到时间周期流
Downstream asymptotics in exterior domains: from stationary wakes to time periodic flows
摘要：
本文考虑半空间中的含时Navier-Stokes方程，边界条件为$(x,y)=(x_0,y)$线上的时间周期（或定常），在无穷远处有固定的渐近速度${\bf u}_{\infty}=(1,0)$.我们证明了存在（局部）唯一的解决方案，所有的数据满足相容性条件，在一定的功能类。此外，我们证明，渐近涡分解本身在一个占主导地位的固定部分的抛物线尺度$y\sim\sqrt{x}$和修正的顺序$x^{-{3/2}+ε}$，而速度场分解本身在一个占主导地位的固定部分的形式，一个明确的多尺度扩展的尺度$y\sim\sqrt{x}$和$y\sim x$和修正衰减至少像$x^{-{9/8}+ε}$。渐近场由泛函数的线性组合组成，其系数轻度依赖于边界数据。平行于${\bf u}_{\infty}$的分量的渐近展开式包含振幅为$\ln(x)x^{-1}$和$x^{-1}$的抛物线尺度中的“非平凡”项。对于一阶，我们的结果还意味着时间周期尾流的行为类似于$x\to\infty$的平稳尾流。所使用的函数类是“自然的”，因为“物理上合理的”（在Finn和Smith的意义上）N.- S.如果半空间选择在下游足够远的地方，则障碍物周围的方程被覆盖。出现在渐近中的系数然后可以与作用在障碍物上的净力线性相关。据我们所知，这是第一次，估计揭露的$\ln(x)x^{-1}$校正证明在此设置。
英文摘要：
We consider the time-dependent Navier-Stokes equations in a half-space with boundary data on the line $(x,y)=(x_0,y)$ assumed to be time-periodic (or stationary) with a fixed asymptotic velocity ${\bf u}_{\infty}=(1,0)$ at infinity. We show that there exist (locally) unique solutions for all data satisfying a compatibility condition in a certain class of fuctions. Furthermore, we prove that asymptotically the vorticity decompose itself in a dominant stationary part on the parabolic scale $y\sim\sqrt{x}$ and corrections of order $x^{-{3/2}+ε}$, while the velocity field decompose itself in a dominant stationary part in form of an explicit multiscale expansion on the scales $y\sim\sqrt{x}$ and $y\sim x$ and corrections decaying at least like $x^{-{9/8}+ε}$. The asymptotic fields are made of linear combinations of universal functions with coefficients depending mildly on the boundary data. The asymptotic expansion for the component parallel to ${\bf u}_{\infty}$ contains `non-trivial' terms in the parabolic scale with amplitude $\ln(x)x^{-1}$ and $x^{-1}$. To first order, our results also imply that time-periodic wakes behave like stationary ones as $x\to\infty$. The class of functions used is `natural' in the sense that `Physically Reasonable' (in the sense of Finn & Smith) stationary solutions of the N.-S. equations around an obstacle are covered if the half-space is choosen sufficiently far downstream. The coefficients appearing in the asymptotics may then be linearly related to the net force acting on the obstacle. To our knowledge, it is the first time that estimates uncovering the $\ln(x)x^{-1}$ correction are proved in this setting.
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