• Exterior domain: $$\Omega=\mathbb{R}^2\setminus\overline{B} \quad\text{with}\quad B \quad\text{bounded}$$
• Stationary Navier-Stokes equations:$$\Delta\bu-\bnabla p=\bu\bcdot\bnabla\bu\,,\quad\quad\bnabla\bcdot\bu=0$$
• Boundary conditions: $$\bu|_{\partial B}=\bu^* \quad\text{and}\quad \lim_{|\bx|\to\infty}\bu = \bu_\infty$$

Theorem (Babenko, 1970)

If $\bu$ is a physically reasonable solution, then $$\bu(\bx)=\bu_{\infty}-\left(\sqrt{8\pi}F_{1}r^{-1/2}+O(r^{-1})\right)\e^{-r\left(1-\cos\theta\right)}$$ $$\quad\quad{}+2F_1\frac{\be_r}{r}+2F_2\frac{\be_\theta}{r}+O(r^{-1-\varepsilon})$$ where $\bF\in\mathbb{R}^{2}$ is the net force acting on the body.

Theorem (Babenko, 1970)

If $\bu$ is a physically reasonable solution, then $$\omega(\bx)=-\sqrt{8\pi}F_{1}\sin\theta\,r^{-1/2}\e^{-r\left(1-\cos\theta\right)}$$ $$\quad\quad\quad{}+O\bigl(r^{-3/2}\e^{-\mu r\left(1-\cos\theta\right)}\bigr)$$ for any $\mu\in(0,1)$.

Theorem (Guillod & Wittwer, 2016)

If $\bu$ is a physically reasonable solution, then $$\omega(\bx)=r^{F_1(1-\cos\theta)+F_2\sin\theta}r^{-1/2}\left[\mu(\theta)+O(r^{-\varepsilon})\right]\e^{-r\left(1-\cos\theta\right)}$$ where $\mu$ is a $2\pi$-periodic Hölder continuous function.

Idea of the proof

The correct linearization for this problem is not around $\bu=\bu_\infty$ but around the harmonic function $$\bu_\infty+F_1\frac{\be_r}{r}+F_2\frac{\be_\theta}{r}\,.$$ The proof use an explicit change of variable to analyze the corresponding linear system.