Julien Guillod
en collaboration avec Mmes Dalibard, Deshayes, El Hassanieh, Godlewski & Gorce et MM. Albritton, Di Martino, Leblond, Korobkov, Meunier, Ren, Sainte-Marie, Schneider, Šverák, Wittwer & Zimmermann
Ma vie
et
mon œuvre
PRÉFACE DE MME. CÉCILE HUNEAU
Édition temporaire, pas revue et ni corrigée
DMA, PARIS
1. Steady
Navier–Stokes
equations
$$\left\{ \begin{aligned}
-\Delta\bu+\bnabla p+\bu\bcdot\bnabla\bu &= \bff\\
\bnabla\bcdot\bu &= 0\\
\lim_{|\bx|\to\infty}\bu &= \bu_{\infty}
\end{aligned}
\right.
\quad \text{in} \quad \mathbb{R}^2$$Existence of weak solutions in $\mathbb{R}^2$
Theorem (Guillod & Wittwer, 2018)
If $\bff \in H^{-1}(\mathbb{R}^2)$, then for any $\bmu\in\mathbb{R}^2$ there exists a weak solution $\bu$ in $\mathbb{R}^2$ such that
$$\textstyle\int_{B_1}\bu=\bmu$$
Note that the net force $\int\!\bff$ is zero.
Theorem (Guillod, Korobkov & Ren, 2022)
If $\mathrm{supp} \bff \subset B_1$ and $\bff \in H^{-1}(B_2)$, then there exists a weak solution $\bu$ in $\mathbb{R}^2$.
Note that the net force $\int\!\bff$ might be non-zero.
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
$$\int\!\bff=\bzero \quad\text{and}\quad \bu_\infty=\bzero$$
then the constructed weak solution satisfies
$$\lim_{\left|\bx\right|\to\infty}\bu = \bu_\infty$$
uniformly.
Conjecture in $\mathbb{R}^2$
when $\bu_\infty=\bzero$ and $\int\!\bff\neq\bzero$
Conjecture (Guillod & Wittwer, 2015)
For a large class of small $\bff$ with $\int\!\bff\neq\bzero$,
there exists a solutions to the Navier–Stokes equations such that
$$\bu = \bU_{\!\bF} + O(|\bx|^{-1}).$$
$|\bx|\,|\bu-\bU_{\!\bF}|$
2. Nonuniqueness
for Navier–Stokes
$$\left\{ \begin{aligned}
\partial_{t}\bu+\bu\bcdot\bnabla\bu &= \Delta\bu-\bnabla p\\
\bnabla\bcdot\bu &= 0\\
\bu(0,\cdot) &= \bu_{0}
\end{aligned} \right.
\quad \text{in} \quad \mathbb{R}^{2,3}$$Nonuniqueness of Navier–Stokes in 3d
Existence theorem (Jia & Šverák, 2013)
For any scale-invariant initial data $\bu_0 \in C^\infty(\mathbb{R}^3\setminus\{\bzero\})$,
there exists a scale-invariant solution which is smooth for $t>0$.
there exists a scale-invariant solution which is smooth for $t>0$.
Numerical result (Guillod & Šverák, 2023)
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.
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 $\bu_{0}\in C^{\infty}(\mathbb{R}^{3}\setminus\{\bzero\}) \cap L^{3,\infty}(\mathbb{R}^{3})$.
Nonuniqueness of Navier–Stokes in 2d
Existence theorem (Albritton, Guillod, Korobkov & Ren, 2026)
For any scale-invariant initial data $\bu_0 \in C^\infty(\mathbb{R}^2\setminus\{\bzero\})$,
there exists a scale-invariant solution which is smooth for $t>0$.
there exists a scale-invariant solution which is smooth for $t>0$.
Numerical result (Albritton, Guillod, Korobkov & Ren, 2026)
Two forward self-similar solutions with the same scale-invariant initial data $\bu_0 \in C^\infty(\mathbb{R}^2\setminus\{\bzero\})$ can be constructed numerically.
Wrong Corollary!
There exists two different Leray–Hopf solutions in $\mathbb{R}^2$ with the same compactly supported initial data.
3. 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 $q^2=\frac{1}{3}$.
Stability for Stokes-transport
Theorem (Dalibard, Guillod & Leblond, 2024)
Consider the Stokes-transport
$$\left\{\begin{aligned}
-\Delta u+\bnabla p &=-\rho\be_z\\
\bnabla\bcdot\bu &=0\\
\partial_{t}\rho+\bu\bcdot\bnabla\rho &=0\\
\bu|_{\partial\Omega}&=\bzero
\end{aligned}\right. \quad \text{in} \quad \Omega=\mathbb{T}\times(0,1)$$
The stratified density profile $(1-z)$ is stable under small perturbation in $H^6 \cap H^2_0$:
$$\begin{align*}
\|\rho-\rho_\infty\|_{L^2} &\lesssim \frac{\varepsilon_0}{1+t}
\end{align*}$$
for some $\rho_\infty$ depending only on $z$.
4. Hyperbolicity of hydrostatic Euler equations
$$\left\{ \begin{aligned} \frac{\partial\bu}{\partial t}+\bu\cdot\nabla_{\bx}\bu+w\partial_{z}\bu+g\nabla_{\bx}\eta & =0 & & \text{ in }(0,T)\times\Omega_{t}\\ \nabla_{\bx}\cdot\bu+\partial_{z}w & =0 & & \text{ in }(0,T)\times\Omega_{t}\\ \frac{\partial\eta}{\partial t}+\bu_{s}\cdot\nabla_{\bx}\eta-w_{s} & =0 & & \text{ in }(0,T)\times\mathbb{R}\\ \bu_{b}\cdot\nabla_{\bx}z_{b}-w_{b} & =0 & & \text{ in }(0,T)\times\mathbb{R}\\ \end{aligned} \right.$$Generalized hyperbolicity condition
Theorem (Di Martino, El Hassanieh, Godlewski, Guillod & Sainte-Marie, 2025)
By a semi-Lagrangian change of coordinates, the hydrostatic free-surface Euler system
can be rewritten as:
$$\partial_{t}\bU+A_{0}(\bU)\partial_{x}\bU=0 \quad \text{in} \quad \mathbb{R}\times(0,1)$$
where
$$A_{0}(\bU)=\begin{pmatrix}U_1 & U_2\\
g\int_{0}^{1}\cdot\,\mathrm{d}\lambda & U_1
\end{pmatrix}.$$
We can know study the spectrum of the operator $A_{0}(\bU)$ to see when this is somehow a generalized hyperbolic system.
5. Oceanography applications
Generation of equilibrium ocean states
Generative models on simplified ocean equations
(Deshayes, Gorce & Meunier)
(Deshayes, Gorce & Meunier)
