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.

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 (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 $$\bu = \bU_{\!\bF} + O(|\bx|^{-1}).$$

$|\bx|\,|\bu-\bU_{\!\bF}|$

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.

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 $\bu_{0}\in C^{\infty}(\mathbb{R}^{3}\setminus\{\bzero\}) \cap L^{3,\infty}(\mathbb{R}^{3})$.

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 $$|\bar{\bu}| \leq \frac{\delta}{2|\bx|\log|\bx|} \,.$$

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 $q^2=\frac{1}{3}$.

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 $z$.

Theorem (Di Martino, El Hassanieh, Godlewski, Guillod & Sainte-Marie, 2024)

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.