Asymptotic behavior of
viscous flows past an obstacle

velocity field

Julien Guillod

Sorbonne Université

23 janvier 2019

vorticity

1. Introduction

Viscous flow past a body

  • Exterior domain: $$\Omega=\mathbb{R}^2\setminus\overline{B} \quad\text{with}\quad B \quad\text{bounded}$$
  • Stationary Navier–Stokes equations:$$\Delta\bu-\bnabla p-\bu\bcdot\bnabla\bu=\bff\,,\quad\bnabla\bcdot\bu=0$$
  • Boundary conditions: $$\bu|_{\partial B}=\bzero \quad\text{and}\quad \lim_{|\bx|\to\infty}\bu = \bu_\infty$$
  • Without lost of generality: $\bu_\infty = 2\be_1$
schema
$B$
$\bu_\infty$

Existence results

Weak solution (Leray, 1933)
Existence of a weak solution $\bu\in\dot{H}^1_0(\Omega)$ even for large data. However, the convergence of $\bu$ to $\bu_\infty$ at large distances is open.
Physically reasonable solution (Finn & Smith, 1967)
Existence of a physically reasonable solution $$\bu=\bu_\infty+\bigO\left(|\bx|^{-1/4-\varepsilon}\right)$$ with $\varepsilon>0$ for small data. The existence of physically reasonable solutions is open for large data.

Aim and outline

Main question
What is the asymptotic behavior of $\bu$ and $\omega = \bnabla\bwedge\bu$ of the physically reasonable solutions?
Outline
  1. Introduction
  2. Asymptotic behavior of $\bu$
  3. Asymptotic behavior of $\omega = \bnabla\bwedge\bu$

2. Asymptotic behavior
of the velocity

Linearization

  • Linearization around $\bu_\infty=2\be_1$ in $\mathbb{R}^2$ for simplicity: $$\Delta\bv-2\partial_1\bv-\bnabla q=\bff\,,\quad\bnabla\bcdot\bv=0\,,\quad \lim_{|\bx|\to\infty}\bv = \bzero$$
  • Green function: $$\bv(\bx) = \int_{\mathbb{R}^2} \mathbf{E}(\bx-\bx_0)\,\bff(\bx_0)\,\rd \bx_0$$ where $$\mathbf{E}=\begin{pmatrix}\partial_{1}\psi-G & \partial_{2}\psi\\ \partial_{2}\psi & -\partial_{1}\psi \end{pmatrix}, \quad \psi=\frac{H+G}{2}$$ with $$\Delta H=\delta_{0}\,, \quad \Delta G-2\partial_{1}G=-\delta_{0}$$ or explicity $$H=\frac{1}{2\pi}\log r\,, \quad G =\frac{1}{2\pi}\e^{r\cos\theta}K_{0}(r) \sim \frac{1}{r^{1/2}}\e^{-r(1-\cos\theta)}$$

Asymptotic behavior

of the linearization

  • Assuming that $\bff$ has compact support: $$\begin{align} \bv(\bx) &= \mathbf{E}(\bx)\bcdot \bF + \bigO\left(D^{|1|}\mathbf{E}(\bx)\right)\\ &= -\sqrt{2\pi}F_1\frac{\be_1}{r^{1/2}}\e^{-r(1-\cos\theta)} + 2F_1 \frac{\be_r}{r} - 2F_2 \frac{\be_\theta}{r}\\ & \quad\quad\quad\quad + \bigO\left(\left(\frac{\left|\theta\right|^{2}}{r^{1/2}}+\frac{1}{r^{3/2}}\right)\e^{-r\left(1-\cos\theta\right)}\right) \end{align}$$ where $$\bF = \int_{\mathbb{R}^2}\bff(\bx)\,\rd \bx$$
  • Remark: $$|\theta|\e^{-r(1-\cos\theta)} \lesssim (1-\cos\theta)^{1/2}\e^{-r(1-\cos\theta)} \lesssim \frac{1}{r^{1/2}}\e^{-r(1-\cos\theta)/2}$$

Asymptotic behavior

of the velocity field

velocity field
$\bu\sim r^{-1/2} + O(r^{-1})$
$\bu\sim r^{-1} + O(r^{-2})$

Asymptotic behavior

of the velocity field

Theorem (Babenko, 1970)
If $\bu$ is a physically reasonable solution, then $$\begin{align} \bu &= \bu_\infty - \sqrt{2\pi}F_1\frac{\be_1}{r^{1/2}}\e^{-r(1-\cos\theta)}+2F_1\frac{\be_r}{r}-2F_2\frac{\be_\theta}{r}\\ & \quad\quad\,\, + \bigO\left(\left(\frac{\left|\theta\right|\log r}{r^{1/2}}+\frac{1}{r}\right)\e^{-r(1-\cos\theta)}+\frac{1}{r^{1+\varepsilon}}\right) \end{align}$$ where $\bF\in\mathbb{R}^{2}$ is the net force acting on the obstacle.
Idea of the proof
The asymptotic behavior is given by the linearization around $\bu_\infty$, so this can be obtained by bootstrapping from the decay of the physically reasonable solution.

3. Asymptotic behavior
of the vorticity

Linearization

  • Linearization around $\bu_\infty = 2\be_1$ in $\mathbb{R}^2$ for simplicity: $$\Delta\omega-2\partial_1\omega=\bnabla\bwedge\bff$$
  • Green function: $$\begin{align} \omega(\bx) &= -\int_{\mathbb{R}^2} G(\bx-\bx_0)\,(\bnabla\bwedge\bff)(\bx_0)\,\rd \bx_0\\ &= \int_{\mathbb{R}^2} \bnabla G(\bx-\bx_0)\, \bff^\perp(\bx_0)\,\rd \bx_0 \end{align}$$
  • Naïve asymptotic expansion: $$\begin{align} \omega(\bx) &= \bnabla G(\bx) \, \bF^\perp + \bigO\left(D^{|2|}G(\bx)\right)\\ &= \left( -\sqrt{8\pi} F_1 \sin\theta + \bigO\left( |\theta|^2 + r^{-1} \right)\right) \frac{\e^{-r(1-\cos\theta)}}{r^{1/2}} \end{align}$$

Asymptotic behavior

of the linearization around $\bu_\infty$

  • Asymptotic expansion of the kernel: $$\bnabla G(\bx-\bx_0) = \bnabla G(\bx) \, \e^{r_0(\cos(\theta-\theta_0)-\cos(\theta_0))} + \bigO\left(\frac{\e^{-r(1-\cos\theta)}}{r^{3/2}}\right)$$
  • If $\bff$ has compact support: $$\begin{align} \omega(\bx) &= \bnabla G(\bx) \, Z(\theta) + \bigO\left(\frac{\e^{-r(1-\cos\theta)}}{r^{3/2}}\right)\\ &= \left(\mu(\theta) + \bigO(r^{-1}) \right) \frac{\e^{-r(1-\cos\theta)}}{r^{1/2}} \end{align}$$ where $$Z(\theta) = \int_{\mathbb{R}^2} \e^{r_0(\cos(\theta-\theta_0)-\cos(\theta_0))}\,\bff(\bx_0)\,\rd\bx_0\,,\quad \mu(\theta) \propto Z(\theta)$$

Main idea

Problem
Bootstrap cannot be used, because we cannot optain exponential decays from polynomial decays.
Main idea
View the vorticity equation $$\Delta\omega - 2\partial_1\omega = \bv\bcdot\bnabla\omega + \bnabla\bwedge\bff \tag{$*$}$$ as a linear equation in $\mathbb{R}^2\setminus B_R$ with $\bv$ and $\omega|_{\partial B_R}$ fixed.
Steps
  1. For $R$ large enough, construct a solution $\omega$ of $(*)$ decaying exponentially.
  2. Prove that the solution of $(*)$ is unique in $L^4\cap\dot{H}^1$.

Intuition

  • Power counting: $$\begin{align} (\Delta - 2\partial_1) (\e^{-r(1-\cos\theta)}) &\sim \frac{\e^{-r(1-\cos\theta)}}{r}\\ \bnabla (\e^{-r(1-\cos\theta)}) &\sim \e^{-r(1-\cos\theta)} \end{align}$$ so at least the terms in $r^{-1}$ in $\bv$ are critical.
  • Remainder: $$\bv = \bu - \bu_\infty = \bu_h + \bar{\bu}$$ where $$\bu_h = 2F_1 \frac{\be_r}{r} - 2F_2 \frac{\be_\theta}{r} \,,\quad |\bar{\bu}| = \bigO\left(\frac{\e^{-r(1-\cos\theta)}}{r^{1/2}}+\frac{1}{r^{3/2}}\right)$$
  • Linear operator: $$L\omega ≔ \Delta\omega - 2\partial_1\omega - \bu_h\bcdot\bnabla\omega = \underbrace{\bar{\bu}\bcdot\bnabla\omega}_{\text{sub-critical}} + \bnabla\bwedge\bff$$

Second main idea

Problem
How to invert $L$ with precise bounds on the asymptote?
Change of variables for harmonic transport
Formally the equation $$\Delta \omega - 2\bnabla\phi\bcdot\bnabla\omega = g$$ where $$\bnabla\phi = \bnabla\bwedge\psi \,,\quad\quad \Delta\phi=\Delta\psi=0$$ is transformed into $$\Delta a - a = h$$ with $$\omega = a(\phi(\bx),\psi(\bx)) \, \e^{\phi(\bx)}, \quad\quad g=h(\bx)\,\e^{\phi(\bx)}\,|\bnabla\phi|^2$$

Asymptotic decay

of the solution of $L\omega=g$

  • In our case: $$\phi = x_1 + F_1 \log r - F_2\theta \,,\quad \psi = -x_2 - F_1\theta - F_2\log r$$ so as $r\to\infty$: $$\sqrt{\phi^2+\psi^2} = r+ F_1\cos\theta\log r +F_2\sin\theta\log r + o(1)$$
  • Formally: $$\begin{align} a(\phi,\psi) &\sim K_0(\sqrt{\phi^2+\psi^2}) \sim \frac{1}{(\phi^2+\psi^2)^{1/4}} \e^{-\sqrt{\phi^2+\psi^2}}\\ &\sim \frac{1}{r^{1/2}} \e^{-r} r^{-F_1 \cos\theta - F_2 \sin\theta} \end{align}$$ so we expect that $$\omega \sim a \e^\phi \sim \frac{1}{r^{1/2}} r^{F_1 (1-\cos\theta) - F_2 \sin\theta} \e^{-r(1-\cos\theta)} \sim r^{F_1 (1-\cos\theta) - F_2 \sin\theta} K_0(r)$$

Asymptotic behavior

of the solution of $L\omega=g$

  • Defining $$\omega(\bx) = r^{F_1 (1-\cos\theta) - F_2 \sin\theta} \e^{r\cos\theta} b(\bx)$$ the equation $(*)$ is transformed into $$\Delta b - b = \bar{\bv}\bcdot(\bnabla b + b \be_r) + kb + R ≔ S$$ where $\bar{\bv},k,R$ are explicit in terms of $\bar{\bu},\bff,F_1,F_2$.
  • If $S$ was of compact support: $$b(\bx) = \int_{\mathbb{R}^2} K_0(|\bx-\bx_0|)\,S(\bx_0)\,\rd\bx_0 = \left( Z(\theta) + \bigO(r^{-1}) \right) \frac{\e^{-r}}{r^{1/2}}$$ where $$Z(\theta) = \int_{\mathbb{R}^2} \e^{r_0\cos(\theta-\theta_0)}\,S(\bx_0)\,\rd\bx_0$$

Asymptotic behavior of the vorticity

Theorem (Guillod & Wittwer, 2017)
If $\bu$ is a physically reasonable solution, then $$\omega=r^{F_1(1-\cos\theta)-F_2\sin\theta}\left(\mu(\theta)+O(r^{-\varepsilon})\right)\frac{\e^{-r\left(1-\cos\theta\right)}}{r^{1/2}}$$ where $\mu\in C^{2\varepsilon}(S^1)$ is a $2\pi$-periodic function depending on the data, and $\bF$ is the net force acting on the obstacle.
Idea of the proof
  1. Show that $S$ is subcritical.
  2. Make $S$ small enough by increasing $R$ in order to close a fixed point argument.
  3. Make bounds to get the asymptotic expansion.

Conclusion

Remarks
  1. Optimal asymptotic behavior: the remainder decays faster than the asymptote in any region.
  2. Asymptotic behavior not given by the linearization around $\bu_\infty$.
  3. Non-universal asymptotic behavior: the polynomial power of decay in front of the exponential factor depends on the data through the net force $\bF$.
  4. Asymptotic behavior for large data, even if uniqueness it not expected.
  5. Nonlinear and nonuniversal nature of the asymptote of the vorticity specific to the two-dimensional case. In three dimensions, everything is easier and known.