Downstream asymptotics in exterior domains: from stationary wakes to time periodic flows

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 $x^{-{3/2}+\epsilon}$, 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 $x^{-{9/8}+\epsilon}$. 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.