Outline
- Steady Navier–Stokes equations
↳ existence and uniqueness
↳ asymptotic behavior
- Non-linear stability
↳ of steady Navier–Stokes flows
↳ of periodic Ginzburg–Landau equilibria - Nonuniqueness for Navier–Stokes
- Models with variable density
↳ existence and uniqueness
↳ long-time behavior
↳ numerical simulations
1. Steady
Navier–Stokes
equations
Description of the problem
- Exterior domain: $$\Omega=\mathbb{R}^n\setminus\overline{B} \quad\text{with}\quad B \quad\text{bounded}$$
- Stationary Navier–Stokes equations:$$-\Delta\bu+\bnabla p+\bu\bcdot\bnabla\bu=\bff\,,\quad\bnabla\bcdot\bu=0$$
- Boundary conditions: $$\bu|_{\partial\Omega}=\bzero \quad\text{and}\quad \lim_{|\bx|\to\infty}\bu = \bu_\infty$$
Fundamental properties
- a priori bound on the enstrophy
↳ topological methods - invariance by a scaling symmetry
↳ perturbative methods
$B$
$\bu_\infty$
$\partial\Omega$
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, 2017)
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.
Limit at infinity in $\mathbb{R}^2$
Theorem (Gilbarg & Weinberger, 1978)
If $\bu$ is a weak solution in $\mathbb{R}^2$, then either there exists $\bu_0\in\mathbb{R^2}$ such that
$$\lim_{\left|\bx\right|\to\infty}\int_{S^{1}}\left|\bu-\bu_{0}\right|^{2}=0$$
or
$$\lim_{\left|\bx\right|\to\infty}\int_{S^{1}}\left|\bu\right|^{2}=\infty$$
Consequence
The limit at infinity of the weak solutions
is widely open in two dimensions.
is widely open in two dimensions.
Perturbative solutions
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; Finn & Smith, 1967; Galdi, 1992)
For any $\bu_\infty\neq\bzero$, if $\bff\in L^1 \cap L^2$ is small enough, then there exists a solution $\bu$
behaving at infinity like the Oseen fundamental solution.
Theorem (Guillod & Wittwer, 2017)
In two dimensions, the vorticity $\omega=\bnabla\bwedge\bu$ does not behave like the Oseen fondamental solution.
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)
If $\bff$ is small enough in a suitable space, there exists a unique solution with $\bu_\infty=\bzero$ behaving at infinity like the Landau solution:
$$\bu = \bU_{\!\bF} + O(|\bx|^{-1-\varepsilon}) \,,$$
where $\bF = \int_{\mathbb{R}^3} \bff$ is the net force.
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 $M$:
$$\bU_{\!M} = \frac{M}{4\pi}\frac{\bx^\perp}{|\bx|^2} \,.$$
Theorem (Guillod, 2015c; Guillod & Wittwer, 2017)
If $\bff$ is small in a suitable space with $\bF=\int_{\mathbb{R}^2}\bff=\bzero$
and belongs to a manifold of codimension two,
there exists a unique solution behaving like
$$\bu = \bU_M + O(|\bx|^{-1-\varepsilon})$$
where $M = \int_{\mathbb{R}^2}\bx\bwedge\bff$ is the net torque.
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
$k=2$
$k=3$
$k=4$
Asymptotic expansion in $\mathbb{R}^2$
when $\bu_\infty=\bzero$ and $\bF\neq\bzero$
Theorem (Guillod & Wittwer, 2015)
For any $\bF\neq\bzero$, there exists an explicit solution
$\bU_{\!\bF}$ with some $\bff$ having net force $\bF$.
$$\bU_{\!\bF} \sim |\bx|^{-1/3}$$
$$\bU_{\!\bF} \sim |\bx|^{-2/3}$$
Conjecture in $\mathbb{R}^2$
when $\bu_\infty=\bzero$ and $\bF\neq\bzero$
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
$$\bu = \bU_{\!\bF} + O(|\bx|^{-1}).$$
$|\bx|\,|\bu-\bU_{\!\bF}|$
2. Non-linear stability
Stability of steady flows
Theorem (Guillod, 2017)
Consider the Navier–Stokes equations
$$\partial_t \bu + \bu\bcdot\bnabla\bu = \Delta \bu - \bnabla p \,, \qquad \bnabla\bcdot\bu=0 \,,$$
in a two-dimensionnal exterior domain.
If $\bar{\bu}$ is a steady-state such that
$$\bigl(\bv\bcdot\bnabla\bar{\bu}, \bv\bigr) \leq \delta\Vert\bnabla\bv\Vert^2 \quad\text{for all}\quad \bv \in \dot{H}^1_{0,\sigma}$$
for some $\delta\in[0,1)$, then $\bar{\bu}$ is asymptotically stable in $L^2$.
In particular, this is the case when $$|\bar{\bu}| \leq \frac{\delta}{2|\bx|\log|\bx|} \,.$$
In particular, this is the case when $$|\bar{\bu}| \leq \frac{\delta}{2|\bx|\log|\bx|} \,.$$
Stability for Ginzburg–Landau
Theorem (Guillod, Schneider, Wittwer & Zimmermann, 2018)
Consider the real Ginzburg–Landau equation
$$\partial_{t}A=\partial_{x}^{2}A+A\bigl(1-|A|^{2}\bigr)$$
on the real line. The periodic equilibria
$$A_q(x) = \sqrt{1-q^2} \e^{i q x}$$
is asymptotically stable under small perturbation in $L^1 \cap L^\infty$
if and only if $q^2\leq \frac{1}{3}$.
The long-time behavior is governed by the linear diffusion when $q^2<\frac{1}{3}$, but is non-linear when $q^2=\frac{1}{3}$.
The long-time behavior is governed by the linear diffusion when $q^2<\frac{1}{3}$, but is non-linear when $q^2=\frac{1}{3}$.
3. Nonuniqueness
for Navier–Stokes
Navier–Stokes equations
$$\begin{cases}
\partial_{t}\bu+\bu\bcdot\bnabla\bu=\Delta\bu-\bnabla p\\
\bnabla\bcdot\bu=0\\
\bu(0,\cdot)=\bu_{0}
\end{cases}
\quad \text{in} \quad \mathbb{R}^3\tag{NS}$$
Main question
Well-posedness of the Cauchy problem (existence, uniqueness, regularity)
Main properties
- energy conservation ⇒ compactness
- scaling invariance ⇒ Banach fixed point
Weak solutions
- A priori bound on the energy:$$\left\Vert \bu(t)\right\Vert _{L^{2}}^{2}+2\int_{0}^{t}\left\Vert \bnabla\bu(\tau)\right\Vert _{L^{2}}^{2}\rd\tau\leq\left\Vert \bu_0\right\Vert _{L^{2}}^{2}$$
- Leray–Hopf solution: $$\bu \in L^{\infty}(0,T;L^{2})\cap L^{2}(0,T;\dot{H}^{1})$$ satisfying (NS) as a distribution and the local energy inequality.
Theorem (Leray, 1934)
For any $\bu_0\in L^2$, existence of a global Leray–Hopf weak solution ($T=\infty$). The regularity and uniqueness of these weak solutions are open.
Scaling invariance
- Scaling symmetry: $$\begin{aligned}\bu_{\lambda}(t,\bx) & =\e^\lambda\bu(\e^{2\lambda}t,\e^\lambda\bx)\\ \bu_{0\lambda}(\bx) & =\e^\lambda\bu_{0}(\e^\lambda\bx)\end{aligned}$$
- Mild solution: $$\bu(t)=\e^{\Delta t}\bu_0+\int_{0}^{t}\e^{\Delta(t-\tau)}\mathbb{P}\bigl(\bu(\tau)\bcdot\bnabla\bu(\tau)\bigr)\,\rd\tau$$
- Critical space $X$ of functions $(0,\infty)\times\mathbb{R}^3\to\mathbb{R}^3$: $$\Vert\bu_\lambda\Vert_X = \Vert\bu\Vert_X$$
- Critical spaces are a priori the largest ones where a fixed point argument is expected to close.
Perturbative solutions
Theorem (Kato, 1984)
For $\bu_0\in L^3$, existence of a local mild solution $\bu\in L^\infty(0,T;L^3)$, which is global if $\bu_0$ is small enough.
Theorem (Escauriaza, Seregin & Šverák, 2003)
If $\bu\in L^{\infty}(0,T;L^{3})$ is a Leray–Hopf solution, then $\bu$ is smooth and unique on $(0,T)$.
Theorem (Koch & Tataru, 2001)
For $\bu_0\in BMO^{-1}$, existence of a unique global solution provided $\bu_0$ is small enough in $BMO^{-1}$.
Conclusions
- Global existence for large data in $L^{2}$ but no regularity nor uniqueness.
- Local well-posedess for large data in $L^{3}$ but not in $L^{3,\infty}$.
- Fundamental difference between the two spaces: $$\begin{align*}\frac{1}{|\bx|} & \notin L^{3} \,, & \frac{1}{|\bx|} & \in L^{3,\infty} \subset BMO^{-1} \end{align*}$$
- Hidden smallness condition: $$\bu_{0\lambda}\big|_{B_R}\to0 \quad\text{as}\quad \lambda\to -\infty \quad\text{for any}\quad \bu_0\in L^{3}$$
Question on the nonperturbative regime
What is happening for large data in $L^{3,\infty}$?
Scale-invariant solutions
- Scale-invariant initial data: $$\bu_{0\lambda}=\bu_0 \quad\textit{i.e.}\quad \bu_0(\bx)=\frac{\varphi(\hat{\bx})}{|\bx|}$$
- Scale-invariant solutions: $$\bu_{\lambda}=\bu \quad\textit{i.e.}\quad \bu(t,\bx)=\frac{1}{\sqrt{2\kappa t}}\bU\Bigl(\frac{\bx}{\sqrt{2\kappa t}}\Bigr)$$
- Forward self-similar if $\kappa>0$, backward self-similar if $\kappa < 0$.
- Ansatz into (NS): $$\begin{cases} \Delta\bU+\kappa\bigl(\bx\bcdot\bnabla\bU+\bU\bigr)-\bU\bcdot\bnabla\bU-\bnabla P=\bzero\\ \bnabla\bcdot\bU=0\\ \bU(\bx)=\bu_{0}(\bx)+o(\left|\bx\right|^{-1})\quad\text{as}\quad\left|\bx\right|\to\infty \end{cases}\tag{NS-self}$$
Numerical simulations
$\sigma=\,$0
Nonuniqueness of Navier–Stokes
Numerical result (Guillod & Šverák, 2017)
Two forward self-similar solutions with the same scale-invariant initial data $\bu_0 \in C^\infty(\mathbb{R}^3\setminus\{\bzero\})$ can be constructed numerically.
Intuition
The two solutions are obtained through a pitchfork bifurcation happening when the size of the initial data $\bu_0$ is increased.
Corollary (adapted from Jia & Šverák, 2015)
Under a spectral assumption verified numerically, there exists two different Leray–Hopf solutions
with the same compactly supported initial data $\bu_{0}\in C^{\infty}(\mathbb{R}^{3}\setminus\{\bzero\}) \cap L^{3,\infty}(\mathbb{R}^{3})$.
Ill-posedness in $L^{p}(0,T;L^{q})$
Summary
- The Navier–Stokes equations are locally ill-posed for $$\bu_0\in L^2 \cap L^{3,\infty}$$
- The Leray–Hopf solutions are not unique in $$\bu \in L^{p}(0,T;L^{q})$$ with $$\tfrac{2}{p}+\tfrac{3}{q}>1$$ even with the energy equality.
4. Models with
variable density
Models with variable density
- Pure transport of the density: $$\partial_t \rho + \bu\bcdot\bnabla\rho = 0$$
- Coupled to an equation for the fluid motion, e.g.: $$\partial_t \bu + \bu\bcdot\bnabla\bu - \Delta \bu + \bnabla p = -\rho \be_z \qquad \bnabla\bcdot\bu=0$$
- Different options: free-surface, hydrostatic approximation, ...
Questions
- Well-posedness for smooth initial densities or patches
- Long-time behavior and stability
Aims
Validation and improvement of the numerical schemes
Stokes-transport equations
Theorem (Antoine Leblond, 2020)
Consider the Stokes-transport system:
$$\partial_t \rho + \bu\bcdot\bnabla\rho = 0 \,, \qquad \rho|_{t=0} = \rho_0 \,,$$
$$-\Delta \bu +\bnabla p = -\rho \be_z \,, \qquad \bnabla\bcdot\bu = 0 \,,$$
in either a bounded domain or in a two-dimensional strip.
For initial data $\rho_0\in L^\infty$, this problem is well-posed in $L^\infty(0,\infty;L^\infty)$.
The velocity field is uniformly Lipshitz in time, so density patches remain patches forever.
Questions
- Does graphs remain graphs?
- What is the long-time behavior?
- Are monotonous steady-states stable?
Open problems
Open problems
- What about the hydrostatic approximation?
- What about Navier–Stokes-Transport?
- What about free surface?