Stationary weak 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 (large data) - invariance by a scaling symmetry
↳ perturbative methods (small data)
Physical situations
$$\begin{align} \bu_\infty &\neq \bzero\\ \bF &\neq \bzero \end{align}$$
$$\begin{align} \bu_\infty &= \bzero\\ \bF &\neq \bzero \end{align}$$
$$\begin{align} \bu_\infty &= \bzero\\ \bF &= \bzero \end{align}$$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
$$\bF=\bzero \quad\text{and}\quad \bu_\infty=\bzero$$
then the constructed weak solution satisfies
$$\lim_{\left|\bx\right|\to\infty}\bu = \bu_\infty$$
uniformly. Moreover, the solution is unique if
$$\Vert\bff\Vert_{H^{-1}(B_2)} \leq \frac{\varepsilon^2 |\bu_\infty|}{(1+|\bu_\infty|^3)\log^{1/2}(2+|\bu_\infty|^{-1})}$$
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 $r_2/r_1\to\infty$.
Summary in $\mathbb{R}^2$
| | | |||
| $$\bu_\infty\neq\bzero$$ | $$\begin{align} \bu_\infty &= \bzero\\ \bF &\neq \bzero \end{align}$$ | $$\begin{align} \bu_\infty &= \bzero\\ \bF &= \bzero \end{align}$$ | |||
| Small data | ∃! | ? | ∃ | ||
| Forcing smaller than $\bu_\infty$ | ∃ | ? | ∃ | ||
| Large data | ? | ? | ∃ |