1. Introduction

Description of the problem

$B$

$\bu^*$

$\partial B$

$\bu_\infty$

Topological method

Leray–Schauder fixed point

$\bu_\infty$

$\bu_1$

$\bu_2$

$\bu_\infty$

$\bu_\infty$

$\bu_3$

$\bu_\infty$

$\bu_4$

$\bu_5$

$\bu_6$

$\bu_7$

$\bu_8$

$B$

$\partial B$

$\bu^*$

Perturbation method

Banach fixed point

• Linearization around $\bu=\bu_\infty$ $$\Delta\bu-\bnabla p-\bu_\infty\bcdot\bnabla\bu=\bff \qquad \bnabla\bcdot\bu=\bzero \qquad \lim_{|\bx|\to\infty}\bu=\bzero$$
• Oseen equations if $\bu_\infty\neq\bzero$ : well-posed system
  

  existence of solutions for small data through a fixed point argument (Finn, 1965; Finn & Smith, 1967)
• Stokes equations if $\bu_\infty=\bzero$ : ill-posed system
  

  a priori no solutions even for small data (Stokes paradox, 1856)
• In the talk: $\bu_\infty=\bzero$ for small data

2. Navier–Stokes equations
with $\bu_\infty=\bzero$

Scaling symmetry

• Scaling symmetry of the Navier–Stokes equations in $\mathbb{R}^n$ $$\bu(\bx) \mapsto \bu_\lambda(\bx) = \e^{\lambda}\bu(\e^{\lambda}\bx) \quad \text{for all} \quad \lambda\in\mathbb{R}$$
• A Banach space $X$ is scale-invariant if $${\Vert\bu_\lambda\Vert}_X = {\Vert\bu\Vert}_X$$
• Examples of scale-invariant spaces $X$ $$\begin{gathered}L^{n}(\mathbb{R}^{n})\\L^{n,\infty}(\mathbb{R}^{n})\end{gathered} \qquad\begin{gathered}\dot{H}^{n/2-1}(\mathbb{R}^{n})\\BMO^{-1}(\mathbb{R}^{n})\end{gathered} \qquad\begin{gathered}B_{p,q}^{-1+n/p}(\mathbb{R}^{n})\\X_{1}(\mathbb{R}^{n})\end{gathered}$$ where $$X_\alpha(\mathbb{R}^n) = \big\{\bu:\sup_{\bx\in\mathbb{R}^{n}}\vert\bx\vert^\alpha\vert\bu\vert<\infty\big\}$$
• Main characterization$$\vert\bx\vert^{-1} \in X \text{ or nearly}$$

Criticality

for the time-dependant problem

• Power counting near $\vert\bx\vert=0$ $$\begin{align} \bu & \sim\left|\bx\right|^{-\alpha} & \bnabla\bu & \sim\left|\bx\right|^{-\alpha-1}\\ \Delta\bu & \sim\left|\bx\right|^{-\alpha-2} & \bu\bcdot\bnabla\bu & \sim\left|\bx\right|^{-2\alpha-1} \end{align}$$
• Subcritical:
  linearity dominates $\quad\Longleftrightarrow\quad \color{green}{\alpha<1}$
• Critical:
  $-\alpha-2 = -2\alpha-1 \quad\Longleftrightarrow\quad \color{yellow}{\alpha=1}$
• Supercritical:
  nonlinearity dominates $\quad\Longleftrightarrow\quad \color{red}{\alpha>1}$

Criticality

for the steady-state problem

• Power counting near $\vert\bx\vert=\infty$ $$\begin{align} \bu & \sim\left|\bx\right|^{-\alpha} & \bnabla\bu & \sim\left|\bx\right|^{-\alpha-1}\\ \Delta\bu & \sim\left|\bx\right|^{-\alpha-2} & \bu\bcdot\bnabla\bu & \sim\left|\bx\right|^{-2\alpha-1} \end{align}$$
• Subcritical:
  linearity dominates $\quad\Longleftrightarrow\quad \color{green}{\alpha>1}$
• Critical:
  $-\alpha-2 = -2\alpha-1 \quad\Longleftrightarrow\quad \color{yellow}{\alpha=1}$
• Supercritical:
  nonlinearity dominates $\quad\Longleftrightarrow\quad \color{red}{\alpha<1}$

Net force

$B$

$\partial B$

$\partial B(\bzero,r)$

$D=\Omega\cap B(\bzero,r)$

Net force and criticality

• Power counting $$\mathbf{T}=\bnabla\bu+\left(\bnabla\bu\right)^{T}-p\,\boldsymbol{1}-\bu\otimes\bu \sim \left|\bx\right|^{-\min(\alpha+1,2\alpha)}$$
• Net force in dimension $n$ $$\bF = \lim_{r\to\infty}\int_{\partial B(\bzero,r)}\!\!\mathbf{T}\bcdot\bn = \lim_{r\to\infty}r^{n-1} \int_{S^{n-1}}\!\!\mathbf{T}\bcdot\bn$$
• In two dimensions ($n=2$) $$\bF \neq \bzero \quad\text{and}\quad |\bF|<\infty \quad\Longleftrightarrow\quad \color{red}{\alpha=\frac{1}{2}}$$
• In three dimensions ($n=3$) $$\bF \neq \bzero \quad\text{and}\quad |\bF|<\infty \quad\Longleftrightarrow\quad \color{yellow}{\alpha=1}$$

Consequences

3. Asymptotic behavior
when $\bF\neq\bzero$

Dominant term

of the asymptotic expansion

• Net force invariant: homogenous decay like $|\bx|^{-1/2}$
• Euler equations dominate at infinity $$ -\bnabla p - \bu\bcdot\bnabla\bu = \bff$$
• Exact solution with $\,\bff=\bF\delta^2(\bx)$ $$\boldsymbol{U}_\bF=\frac{\bphi_\bF(\hat{\bx})}{|\bx|^{1/2}}$$ where $\bphi_\bF$ is a function depending only on the direction of $\bx$

Idea: wake

• Idea: inhomogeneous decay so new power counting
• Parts of the linearity and of the nonlinearity remain dominant at infinity
• Wake for $p,q>0$ $$u(r,\theta) = \frac{1}{r^q}\varphi(r^p \theta)$$

$$u(r,\theta) \quad\text{for}\quad \varphi(z)=\mathrm{e}^{-z^2}$$

$$r^q u(r,\theta) \quad\text{for}\quad \varphi(z)=\mathrm{e}^{-z^2}$$

Ansatz

• Wake condition (linearity and non-linearity both dominant) $$2p+q=1$$
• Net force finite and nonzero $$p=q=\frac{1}{3}$$
• Ansatz $$\boldsymbol{U}_\bF = \frac{1}{r^{1/3}} \varphi_\bF\left(r^{1/3}\theta\right)\be_r$$
• Remark on the scaling $$\lambda \boldsymbol{U}_\bF(\lambda\bx) = \boldsymbol{U}_{\lambda\bF}(\bx)\quad\text{for all}\quad\lambda>0$$

Dominant term

of the asymptotic expansion

• Ansatz into the Navier–Stokes equations
• A large distances, dominant term given by an ordinary differential equation
• Explicit solution $$\varphi_\bF(z) = -6 a^2 \mathrm{sech}^2\left(a z\right)$$ where $$a=\left(\frac{|\bF|}{48}\right)^{1/3}$$
• Solution of the Navier–Stokes equations with a remainder $O\big(|\bx|^{-6/3}\big)$
• The decay of the remainder can be impoved by calculating the next orders

Behavior of the asymptotic expansion

up to second order

The second order $\boldsymbol{U}_\bF^{(2)}$ is explicit and decays like $|\bx|^{-2/3}$ at infinity

$$|\bx|^{1/3} \left| \boldsymbol{U}_\bF(\bx) + \boldsymbol{U}_\bF^{(2)}(\bx) \right|$$

$$\bu \sim |\bx|^{-1/3}$$

$$\bu \sim |\bx|^{-1}$$

Numerical validation

• Exterior of a disk with $\bu^*=(\lambda,0)$
• Domain truncated by a ball of radius $R=10^5$
• Boundary condition on $\partial B(\bzero,R)$ : $\bu=\bzero$

$|\bx|^{1/3}\,|\bu|$

$|\bx|\,\big|\bu-\boldsymbol{U}_\bF-\boldsymbol{U}_\bF^{(2)}\big|$

Conjecture

4. Asymptotic behavior
when $\bF=\bzero$

Solutions of the linearized problem

• Asymptotic expansion of the Stokes solutions: $$\bu = \mathcal{M} \bu_0 + \mathcal{A} \bu_a + \mathcal{B} \bu_b + O\big(\color{green}{|\bx|^{-2}}\big) \quad\text{with}\quad \mathcal{M},\mathcal{A},\mathcal{B}\in\mathbb{R}$$

$|\bx|\,|\bu_0|$

$|\bx|\,|\bu_a|$

$|\bx|\,|\bu_b|$

Dominant term

of the asymptotic expansion

• Critical case with a decay like $|\bx|^{-1}$
• By analogy with the three-dimensional case: scale-invariant solution
• Scale-invariant solutions $$e^\lambda\bu(e^\lambda\bx) = \bu(\bx)\quad\text{for all}\quad \lambda\in\mathbb{R}$$
• The scale-invariant solutions are (Šverák, 2011) $$\bu_0 = \frac{\mu}{r}\be_\theta \quad\quad \text{or} \quad\quad \bu_k = \frac{\varphi_k(\theta-\theta_0)}{r}\be_r$$ for $\mu\in\mathbb{R}$, $\theta_0\in\mathbb{R}$, and $k\in\mathbb{N}$ with $k\geq 2$

Dominant term

second attempt

Existence and uniqueness

of generalized scale-invariant solutions

Generalized scale-invariant solutions

for $\eta=\,$ 0.0

$k=2$

$k=3$

$k=4$

Small solutions

and asymptotic behavior

• Only two small solutions $$k=0 \;\;\&\;\; |\mu|\ll 1 \quad\quad\text{and}\quad\quad k=2 \;\;\&\;\; \theta_0\in\mathbb{R} \;\;\&\;\; |\eta|\ll 1$$
• Three parameters but only two solutions (no superposition)

Numerical simulations

for $\boldsymbol{\mathcal{M}}=\,$ 0.0$\pi$ and $\boldsymbol{\mathcal{A}}=\,$ 0.0$\pi$

$\bu\sim|\bx|^-$^0.00

$\mathcal{A}$

$\mathcal{M}$

$|\bx|$^0.00$\quad\;|\bu|$

5. Conclusions

Conclusions

• Formal arguments and numerical simulations indicate the existence of stationary solutions to the Navier–Stokes equations with a zero velocity at infinity.
• Conjectures on the asymptotic behaviors with a velocity field decaying like $|\bx|^{-1/3}$.
• Linearization around the asymptote: supercritical perturbation of the Stokes equations, so difficult problem.
• Interplay between the linearity and the nonlinearity might be crucial in other open problems.

References

• Guillod J. & Wittwer P. 2015a
  Asymptotic behaviour of solutions to the stationary Navier–Stokes equations in two-dimensional exterior domains with zero velocity at infinity.
  Math. Mod. Meth. Appl. S.

• Guillod J. & Wittwer P. 2015b
  Generalized scale-invariant solutions to the two-dimensional stationary Navier–Stokes equations. SIAM J. Math. Anal.

• Guillod J. 2015
  Steady solutions of the Navier–Stokes equations in the plane.
  arXiv:1511.03938