Stationary solutions of the Navier-Stokes
equations in the whole plane
Julien Guillod
Description of the problem
- Stationary incompressible Navier–Stokes equations $$-\Delta\bu+\bnabla p + \bu\bcdot\bnabla\bu = \bff \qquad \bnabla\bcdot\bu=0$$ in $\mathbb{R}^n$ with $n=2,3$ and for a given forcing term $\bff$.
- Boundary condition at infinity $$\lim_{|\bx|\to\infty}\bu=\bu_\infty$$ for a given $\bu_\infty\in\mathbb{R}^n$.
Fundamental properties
- a priori bound on the enstrophy
↳ topological methods - invariance by a scaling symmetry
↳ perturbative methods
Physical situations
$$\begin{align} \bu_\infty &\neq \bzero\\ \bF &\neq \bzero \end{align}$$
$$\begin{align} \bu_\infty &= \bzero\\ \bF &\neq \bzero \end{align}$$
$$\begin{align} \bu_\infty &= \bzero\\ \bF &= \bzero \end{align}$$Topological methods
A priori bound on the enstrophy
$$\int_{\mathbb{R}^n} \bnabla\bu:\bnabla\bu + \underbrace{\int_{\mathbb{R}^n} (\bu\bcdot\bnabla\bu)\bcdot\bu}_{=\,0} = \int_{\mathbb{R}^n} \bff\bcdot\bu$$ $$\implies \Vert\bnabla\bu\Vert_{L^2}^2 = \langle\bff,\bu\rangle_{L^2}$$
Formal a priori bound
If $\bff$ satisfies $$|\langle\bff,\bu\rangle_{L^2}| \leq C \Vert\bnabla\bu\Vert_{L^2}$$
for some $C>0$, then
$$\Vert\bnabla\bu\Vert_{L^2} \leq C\,.$$
Weak solutions
Natural space
$$\dot{H}^1_{\sigma} = \bigl\{\bu \in \dot{H}^1: \bnabla\bcdot\bu=0 \bigr\}$$
Definition (weak solutions)
A vector field $\bu∶\mathbb{R}^n\to\mathbb{R}^n$ is called a weak solution if
- $\bu\in\dot{H}^1_{\sigma}$
- $$\bigl\langle\bnabla\bu,\bnabla\bphi\bigr\rangle_{L^{2}}+\bigl\langle\bu\bcdot\bnabla\bu,\bphi\bigr\rangle_{L^{2}}=\bigl\langle\bff,\bphi\bigr\rangle_{L^{2}}$$
$$\forall\bphi\in C^\infty_{0,\sigma}=\bigl\{\bphi\in C^{\infty}_0: \bnabla\bcdot\bphi=0 \bigr\}$$
Existence of weak solutions in $\mathbb{R}^3$
Theorem (Leray, 1933)
If there exists $C>0$ such that
$$|\langle\bff,\bphi\rangle_{L^2}| \leq C \Vert\bnabla\bphi\Vert_{L^2} \quad \forall \bphi\in C^\infty_{0,\sigma}$$
then for any $\bu_\infty\in\mathbb{R}^3$ there exists a weak solution $\bu$ in $\mathbb{R}^3$ such that
$$\int_{S^2}|\bu-\bu_\infty|^2 = O\left(\frac{1}{|\bx|}\right).$$
Remark
Similar result in three-dimensional exterior domains with Lipshitz boundary.
Existence of weak solutions in $\mathbb{R}^2$
Theorem (Guillod & Wittwer, 2018)
If there exists $C>0$ such that
$$|\langle\bff,\bphi\rangle_{L^2}| \leq C \Vert\bnabla\bphi\Vert_{L^2} \quad \forall \bphi\in C^\infty_{0,\sigma}$$
then for any $\bmu\in\mathbb{R}^2$ there exists a weak solution $\bu$ in $\mathbb{R}^2$ such that
$$\textstyle\int_{B_1}\bu=\bmu.$$
Remark
In two-dimensional exterior domains $\Omega\neq\mathbb{R}^2$, the existence of weak solutions was obtained by Leray (1933).
In this case the parameterization of the weak solution by some $\bmu\in\mathbb{R}^2$ is open.
Strategy of proof
Method of exhaustion
- Find a large enough Hilbert subspace $\mathcal{H}\subset\dot{H}^1_\sigma$ such that $$\mathcal{H}\subset H^1_{\mathrm{loc}} \Subset L^4_{\mathrm{loc}}$$ ↳ Hardy inequality
- Construct a weak solution $\bu_k=\bu_{\infty,1} + \bv_k$ in the ball $B_k$ such that
the sequence $(\bv_k)_{k\geq1}$ is uniformly bounded in $\mathcal{H}$
↳ a priori bound in bounded domains
↳ compactness of the nonlinearity in bounded domains
↳ Leray–Schauder fixed point theorem - Pass to the limit in the weak formulation
↳ local compactness property of $\mathcal{H}$
Hardy inequality in $\mathbb{R}^3$
Hardy inequality (Hardy, 1934)
For any $\bu\in \dot{H}^1_\sigma$, there exists a unique $\bu_\infty\in\mathbb{R}^3$ such that
$$\left\Vert \frac{\bu-\bu_\infty}{|\bx|} \right\Vert_{L^2} \leq 2 \Vert \bnabla\bu \Vert_{L^2}$$
and moreover $\bu-\bu_\infty\in L^6$ and
$$\int_{S^2}|\bu-\bu_\infty|^2 = O\left(\frac{1}{|\bx|}\right).$$
Hilbert subspace
The following subspace of $\dot{H}^1_\sigma$ has a nondegenerate norm:
$$\mathcal{H} = \bigl\{\bu\in\dot{H}^1_\sigma : \bu_\infty=\bzero\bigr\}
= \dot{H}^1_\sigma \cap L^6
\subset H^1_{\mathrm{loc}} \Subset L^4_{\mathrm{loc}}$$
Hardy inequality in $\mathbb{R}^2$
Hardy inequality (Guillod & Wittwer, 2018)
There exists a constant $C>$ such that for any $\bu\in \dot{H}^1_\sigma$,
$$\left\Vert \frac{\bu-\bmu}{\langle\bx\rangle\langle\log\langle\bx\rangle\rangle} \right\Vert_{L^2} \leq C \Vert \bnabla\bu \Vert_{L^2}$$
where $\bmu\in\mathbb{R}^2$ is the mean of $\bu$ on the ball of radius one
$$\bmu = \frac{1}{|B_1|}\int_{B_1} \bu \,.$$
Hilbert subspace
The following subspace of $\dot{H}^1_\sigma$ has a nondegenerate norm:
$$\mathcal{H} = \bigl\{\bu\in\dot{H}^1_\sigma : \bmu=\bzero\bigr\}
\subset H^1_{\mathrm{loc}} \Subset L^4_{\mathrm{loc}}$$
Remark on the hypothesis on $\bff$
Corollary in $\mathbb{R}^3$
In $\mathbb{R}^3$ if $|\bx|\bff \in L^2$, then there exists $C>0$ such that
$$|\langle\bff,\bphi\rangle_{L^2}| \leq C \Vert\bnabla\bphi\Vert_{L^2} \quad \forall \bphi\in C^\infty_{0,\sigma}$$
Corollary in $\mathbb{R}^2$
In $\mathbb{R}^2$ if $\langle\bx\rangle\langle\log\langle\bx\rangle\rangle\bff \in L^2$ and
$$\int_{\mathbb{R}^2}\bff=\bzero$$
then there exists $C>0$ such that
$$|\langle\bff,\bphi\rangle_{L^2}| \leq C \Vert\bnabla\bphi\Vert_{L^2} \quad \forall \bphi\in C^\infty_{0,\sigma}$$
Difficulties and problems
Difficulties with $\mathbb{R}^2$ compared to $\mathbb{R}^3$
- Hypothesis on $\bff$ implies a vanishing net force: $$\bF = \int_{\mathbb{R}^2} \bff = \bzero $$ ↳ otherwise the a priori bound in $\dot{H}^1$ is not valid.
- Subspace $\mathcal{H}$ with $\bu_1=\bzero$ not suitable to study the asymptotic behavior
↳ functions are not even bounded. - Use of $\bu_\infty$ on the artificial boundary (as in $\mathbb{R}^3$)
↳ hard to get local control
↳ hope to obtain something on the asymptotic behavior
Existence of weak solutions (version 2)
Theorem (Guillod, Korobkov & Ren, 2022)
If $\mathrm{supp} \bff \subset B_1$ and $\bff \in H^{-1}(B_2)$, then the solution $\bu_k$ in $B_k$ with $\bu_k|_{\partial B_k}=\bu_\infty$ satisfies:
$$\Vert\bnabla\bu_k\Vert^2_{L^2} \leq C A \left(A+|\bu_\infty|+A^{1/3}\right)$$
$$\sup_{1\leq r\leq k} \bar{\bu}_k(r) \leq C \left(A+|\bu_\infty|+A^{1/3}\right)$$
where $\bar{\bu}_k(r)$ denotes the average of $\bu_k$ on $\partial B_r$ and $A=\Vert\bff\Vert_{H^{-1}(B_2)}$.
Corollary
This proves the existence of a weak solution $\bu$ in $\mathbb{R}^2$. We note that the net force might be nonzero.
Limit at infinity in $\mathbb{R}^2$
Theorem (Guillod, Korobkov & Ren, 2022)
Given $\bff$ and $\bu_\infty$, there exists a constant $\varepsilon>0$ such that if:
$$\Vert\bff\Vert_{H^{-1}(B_2)} \leq \frac{\varepsilon^2 |\bu_\infty|}{\log^{1/2}(2+|\bu_\infty|^{-1})}$$
or
$$\bF=\int_{B_1}\bff=\bzero \quad\text{and}\quad \bu_\infty=\bzero$$
then the constructed weak solution satisfies
$$\lim_{\left|\bx\right|\to\infty}\bu = \bu_\infty$$
uniformly.
Ideas of the proof
Remark
In case $\bu_\infty = \bzero$ and $\bF\neq\bzero$ the limit at infinity in open.
This might be related to the Stokes paradox.
This might be related to the Stokes paradox.
Ideas of the proof
- For any weak solution in an annulus $A_{r_1,r_2}$, estimate $|\bar{\bu}(r_1)-\bar{\bu}(r_2)|$ in terms of $\Vert\bnabla\bu\Vert_{L^2(A_{r_1,r_2})}$.
- Rescale the solution only in space from $B_k$ to $B_1$ in order to obtain the Euler equation in the limit.
- Establish properties on the pressure of this Euler system.
- Keep the essence of the first estimate in the limit $r_1\to\infty$ with $r_2/r_1\to\infty$.
Perturbative methods
Scaling symmetry
Scaling symmetry
$$\begin{align}
\bu(\bx) &\mapsto \bu_\lambda(\bx) = \lambda \bu(\lambda\bx)\\
p(\bx) &\mapsto p_\lambda(\bx) = \lambda^2 p(\lambda\bx)\\
\bff(\bx) &\mapsto \bff_\lambda(\bx) = \lambda^3 \bff(\lambda\bx)\\
\end{align}$$
Remark
The scaling symmetry describes the link
between the linearity and the nonlinearity.
between the linearity and the nonlinearity.
Criticality
Criticality
- subcritical space
↳ linearity dominant - critical space
↳ linearity and nonlinearity balanced - supercritical space
↳ nonlinearity dominant
Criticality of a Banach space $X$
$$\Vert \bu_\lambda \Vert_X = \lambda^{1-\alpha} \Vert\bu\Vert_X$$
- $\alpha<1$ : subcritical locally and supercritical at infinity
- $\alpha=1$ : critical both locally and at infinity
- $\alpha>1$ : supercritical locally and subcritical at infinity
Examples of function spaces
Function space for weak solutions
$\dot{H}^1_\sigma$ has $\alpha=\frac{n}{2}-1$, so for $n=2,3$ it is
- subcritical locally (linearity locally dominant)
↳ no regularity problem - supercritical at infinity (nonlinearity dominant at infinity)
↳ a priori no boostrap from weak solutions
Function space for perturbative solutions
$$E_\alpha = \bigl\{ \bu\in L^1_{\mathrm{loc}}: {\textstyle\sup_{\bx\in\mathbb R^n}} (1+|\bx|^{\alpha}) |\bu(\bx)| <\infty \bigr\}$$
is subcritical locally and
- subcritical at infinity if $\alpha>1$
- supercritical at infinity if $\alpha<1$
Perturbation method
when $\bu_\infty\neq\bzero$
Linearization around $\bu_\infty\neq\bzero$ (Oseen equations)
$$-\Delta\bu+\bnabla p + \bu_\infty\bcdot\bnabla\bu = \bff \qquad \bnabla\bcdot\bu=0$$
Theorem (Finn, 1965; Babenko, 1973; Galdi, 1992)
If $\bu$ is a weak solution in $\mathbb{R}^3$ with $\bu_\infty\neq\bzero$, then
$\bu$ behaves at infinity like the Oseen fundamental solution.
Theorem (Finn & Smith, 1967; Galdi, 1992)
Let $q\in(1,6/5]$ and $\bu_\infty\neq\bzero$. If $\bff\in L^q$ is small enough, then there exists a solution $\bu$
behaving at infinity like the Oseen fundamental solution.
Net force and decay
when $\bu_\infty=\bzero$
- Navier–Stokes equivalent to $\bnabla\bcdot\mathbf{T}=\bff$ where $$\mathbf{T}=\bnabla\bu+\left(\bnabla\bu\right)^{T}-p\,\boldsymbol{1}-\bu\otimes\bu$$
- Invariant quantity $$\int_{B_r}\bff = \int_{\partial B_r}\!\mathbf{T}\bcdot\bn$$
- Net force $$\bF = \int_{\mathbb{R}^n}\bff = \lim_{r\to\infty}\int_{\partial B_r}\!\mathbf{T}\bcdot\bn = \lim_{|\bx|\to\infty}|\bx|^{n-1}\int_{S^{n-1}}\!\mathbf{T}\bcdot\bn$$
Consequence
If $\bu\in E_\alpha$ with $\alpha>\frac{n-1}{2}$, then $\bF=\bzero$.
Perturbative solutions in $\mathbb{R}^3$
when $\bu_\infty=\bzero$
Landau solution
For $\bF\in\mathbb{R}^3$, the Landau solution $\bU_{\!\bF}$ is an explicit scale-invariant solution of the Navier–Stokes equations with
$\bff = \bF \delta(\bx)$.
Theorem (Korolev & Šverák, 2011)
Let $\varepsilon\in(0,1)$. If $\bff\in E_{3+\varepsilon}$ is small enough, then there exists a unique solution $\bu\in E_1$ such that
$$\bu = \bU_{\!\bF} + O(|\bx|^{-1-\varepsilon})$$
where $\bF = \int_{\mathbb{R}^3}\bff$. Moreover this solution is unique among weak solutions having $\bu_\infty=\bzero$.
Perturbative solutions in $\mathbb{R}^2$
when $\bu_\infty=\bzero$
Proposition (Guillod & Hillairet, 2015)
For any $\bF\neq\bzero$, there exists an exact solution
$$\bU_{\!\bF} = \frac{\bphi(\hat\bx)}{|\bx|^{1/2}} \in E_{1/2}$$
of the Euler equation
$$\bnabla p + \bu\bcdot\bnabla\bu = \bff \qquad \bnabla\bcdot\bu=0$$
for $\bff = \bF \delta(\bx)$.
Problem
In Navier–Stokes, the next order correction has necessarily a flux greater than $3\pi$.
Perturbative solutions in $\mathbb{R}^2$
when $\bu_\infty=\bzero$ and $\bF=\bzero$
Tork solution
For $M\in\mathbb{R}$, the tork solution $\bU_{\!M}$ is the scale-invariant solution of the Navier–Stokes equations with
tork $M$:
$$\bU_{\!M} = \tfrac{M}{4\pi} \bx^\perp |\bx|^{-2} \,.$$
Theorem (Guillod, 2015; Guillod & Wittwer, 2017)
Let $\varepsilon\in(0,1)$. If $\bff\in E_{3+\varepsilon}$ is small enough with $\bF=\bzero$
and belongs to a manifold of codimension two,
then there exists a unique solution $\bu\in E_1$ such that
$$\bu = \bU_{\!M} + O(|\bx|^{-1-\varepsilon})$$
where $M = \int_{\mathbb{R}^2}\bx\bwedge\bff$.
If $M=0$, this solution is unique among the
weak solutions having the same $\bmu$.
Some possible asymptotic behaviors
when $\bu_\infty=\bzero$ and $\bF=\bzero$
Idea (Guillod & Wittwer, 2015)
Combine the scaling and rotational symmetries to generalize the concept of scale-invariant solutions:
$$e^\lambda\bu(e^\lambda\bx) = \mathbf{R}^{-1}_{\eta\lambda}\bu(\mathbf{R}_{\eta\lambda}\bx)$$
i.e.
$$\bu(r,\theta) = \frac{\bphi(\theta + \eta \log r)}{r}$$
Generalized scale-invariant solutions
for $\eta=\,$ 0.0
$k=2$
$k=3$
$k=4$
Conjectures on the
existence of solutions
in $\mathbb{R}^2$ with $\bu_\infty=\bzero$
Open problems
Open problems when $\bF\neq\bzero$
- existence of weak solutions with $\bu_\infty=\bzero$
- existence of perturbative solutions with $\bu_\infty=\bzero$
Open problems when $\bF=\bzero$
- existence of perturbative solutions if $\bff$ do not satisfies the two compatibility conditions
Question
What is happening without any compatibility conditions on $\bff$, even under smallness assumptions on $\bff$?
Asymptotic expansion when $\bF\neq\bzero$
Theorem (Guillod & Wittwer, 2015)
For any $\bF\neq\bzero$, there exists a solution
$\bU_{\!\bF}\in E_{1/3}$ with some $\bff\in E_{7/3}$ having net force $\bF$.
$$\bU_{\!\bF} \sim |\bx|^{-1/3}$$
$$\bU_{\!\bF} \sim |\bx|^{-2/3}$$
Numerical simulations when $\bF\neq\bzero$
- Take some $\bff$ having compact support and $\bF\neq\bzero$
- Truncate the domain to a ball of radius $r=10^5$
- Add zero boundary condition on $\partial B_r$: $\bu|_{\partial B_r}=\bzero$
$|\bx|^{1/3}\,|\bu|$
$|\bx|\,|\bu-\bU_{\!\bF}|$
Conjecture when $\bF\neq\bzero$
Conjecture (Guillod & Wittwer, 2015)
For a large class of small $\bff$ with $\bF\neq\bzero$,
there exists a solutions to the Navier–Stokes equations such that
$$\bu = \bU_{\!\bF} + O(|\bx|^{-1}).$$
Main problem
How to invert the linear system
$$-\Delta\bu+\bnabla p+\bU_{\!\bF}\bcdot\bnabla\bu+\bu\bcdot\bnabla\bU_{\!\bF}=\bff \qquad \bnabla\bcdot\bu=\bzero$$
since $\bU_{\!\bF}$ decays only like $|\bx|^{-1/3}$?
Numerical simulations when $\bF=\bzero$
Solution of the linear equations
Up to a rotation around the origin, the asymptotic expansion of the Stokes solution is:
$$ \bu = \bU_{\! M} + \bU_{\! A,0} + O(|\bx|^{-2})$$
where
$$M = \int_{\mathbb{R}^2} x_1 f_2 - x_2 f_1 \quad\text{and}\quad A = \int_{\mathbb{R}^2} x_1 f_1 - x_2 f_2$$
Choice of forcing
Take the simplest forcing $\bff$ generating $M$ and $A$:
$$\bff = \frac{M}{\pi} (-x_2,x_1) \, \e^{-|\bx|^2} + \frac{A}{\pi} (x_1,-x_2) \, \e^{-|\bx|^2}$$
Numerical simulations when $\bF=\bzero$
for $M=\,$ 0.0$\pi$ and $A=\,$ 0.0$\pi$
$A$
$M$
$\bu\sim|\bx|^-$0.00
$|\bx|$0.00$\quad|\bu|$
Summary for $\bu_{\infty}=\bzero$
| $n=2$ | $n=3$ | |||||
| $\bF\neq\bzero$ | $\bF=\bzero$ | $\bF\neq\bzero$ | $\bF=\bzero$ | |||
| Stokes solutions | $\log|\bx|$ | $|\bx|^{-1}$ | $|\bx|^{-1}$ | $|\bx|^{-2}$ | ||
| Navier–Stokes solutions | $\color{red}{|\bx|^{-1/3}}$ | $\color{red}{|\bx|^{-1/3}}$ | $\color{yellow}{|\bx|^{-1}}$ | $\color{green}{|\bx|^{-2}}$ | ||
| Asymptotic behavior | single wake? | double wake, spirals,...? | Landau solution | Stokes solution | ||
Conclusion
In $\mathbb{R}^2$ the nonlinearity “helps” for the existence!