Stability of a gravity-driven fluid

Julien Guillod

joint work with Anne-Laure Dalibard and Antoine Leblond

1. Introduction

Problem

Question

Stability of stratified flow and long-time behavior?

Motivation from oceanography

Lock exchange test case

Variables and domain

Horizontal variable $x$ and vertical variable $z$
Velocity field $\bu$ and density $\rho$
Periodic channel $\Omega=\mathbb{T}\times(0,1)$

Description of the model

Stokes approximation (Boussinesq and Reynold zero) $$-\Delta u+\bnabla p=-\rho\be_z$$
Incompressible fluid $$\bnabla\bcdot\bu =0$$
Pure transport of the density $$\partial_{t}\rho+\bu\bcdot\bnabla\rho=0$$
Dirichlet boudnary conditions $$\bu|_{\partial\Omega}=\bzero$$

A priori estimates

Potential energy $$E(\rho)=\int_{\Omega}z\rho$$
From transport equation $$\frac{\mathrm{d}}{\mathrm{d}t}E(\rho)=\int_{\Omega}z\partial_{t}\rho=-\int_{\Omega}z\bnabla\bcdot(\rho\bu)=\int_{\Omega}\rho\bu\bcdot\be_{z}$$
From velocity equation $$\|\bnabla\bu\|_{L^2}^2=\int_{\Omega}\rho\bu\bcdot\be_{z}$$
Energy balance $$\frac{\mathrm{d}E(\rho)}{\mathrm{d}t}=-\|\bnabla\bu\|_{L^2}^2$$

Steady states

Proposition

Steady states are precisely stratified density profiles, i.e density profiles depending only on the vertical variable $z$.

Sketch of the proof

From energy estimate $\|\bnabla\bu\|_{L^2}=0$, hence $\bu=\bzero$.
Hence velocity equation reduces to $$\bnabla p=-\rho\be_{z}$$ so the pressure is independent of $x$, hence $\rho$ too.

Difficulty

Continuum of steady states, hence asymptotic stability is not expected to hold → slightly modified stratified profile.

Global well-posedness

Theorem (Leblond, 2021, 2023)

Let $q\in(2,\infty]$. For any $\rho_{0}\in L^q$, (ST) has a unique solution $$\rho\in L^{\infty}(\mathbb{R},L^{q}), \qquad \bu\in L^{\infty}(\mathbb{R},W^{1,\infty}).$$

Previous results

For patches: Antontsev, Meirmanov & Yurinsky (2000)
In $\mathbb{R}^3$: Höfer (2018), Mecherbet (2018), Höfer & Schubert (2021), Mecherbet & Sueur (2022), Inversi (2023)

2. Stability of
continuous
density profiles

Stability of $\Theta(z)=1-z$

Theorem (Dalibard, Guillod & Leblond, 2024)

There exists $\varepsilon_0 >0$ small such that for any $\rho_0 \in H^6$ satisfying $\|\rho_0-\Theta\|_{H^6} \leq \epsilon_0$ and $\rho_0-\Theta\in H^2_0$, the solution of (ST) satisfies: $$\begin{align*} \|\rho-\rho_\infty\|_{L^2(\Omega)} &\lesssim \frac{\varepsilon_0}{1+t},& \|\rho-\rho_\infty\|_{H^4(\Omega)} \lesssim \varepsilon_0, \end{align*}$$ for some $\rho_\infty$ depending only on $z$.

Remarks

The perturbation need to vanish near the boundary.
Can be generalized to other profiles $\Theta(z)$.

Identification of the limit

Corollary (Dalibard, Guillod & Leblond, 2024)

The long-time asymptotic $\rho_\infty$ is given by the decreasing vertical rearrangement of $\rho_0$: $$\rho_\infty(z) = \rho_0^* := \int_0^\infty \boldsymbol{1}_{0 \leq z \leq |\{\rho_0>\lambda\}|}\mathrm{d}\lambda.$$

Sketch of the proof

$\rho_\infty$ depends only on $z$
$\rho_\infty$ is strictly decreasing as a perturbation of $\Theta$
levels sets of $\rho$ are preserved by the time evolution, so $\rho_\infty$ is a rearrangement of $\rho_0$
unique decreasing vertical rearrangement

Strategy of proof

Rewriting & leading terms

Perturbation $\theta$: $$\rho = \Theta + \theta$$
Decomposition into horizontal average and complement: $$\theta = \bar\theta + \theta' \qquad \bar\theta = \frac{1}{2\pi}\int_0^{2\pi}\theta \, \mathrm{d}x$$
As $t\to\infty$, one expect that $\bar\theta\to\rho_\infty-\Theta$ and $\theta'\to0$
Leading part (using vorticity formulation): $$\left\{ \begin{aligned} \partial_t\theta' &= (1-\partial_z\bar\theta)\partial_{x}\psi & \theta'|_{t=0} &= \theta'_0 \\ \Delta^2\psi &= \partial_x\theta' & \psi|_{\partial\Omega} &= \partial_\bn\psi|_{\partial\Omega} = 0 \end{aligned} \right.$$

Strategy of proof

Bounds & Bootstrap

$$\left\{ \begin{aligned} \partial_t\theta' &= (1-\partial_z\bar\theta)\partial_{x}\psi & \theta'|_{t=0} &= \theta'_0 \\ \Delta^2\psi &= \partial_x\theta' & \psi|_{\partial\Omega} &= \partial_\bn\psi|_{\partial\Omega} = 0 \end{aligned} \right.$$

Main steps

Analysis when $\partial_z\bar\theta$ is small and given:
1. Uniform bounds of $\|\Delta^{s/2}\theta'\|_{L^2}$ for $s=0,4$ only
  ↳ because only the following traces are zero: $$\psi|_{\partial\Omega} \quad \partial_z\psi|_{\partial\Omega} \quad \Delta^2\psi|_{\partial\Omega} \quad \partial_z\Delta^2\psi|_{\partial\Omega}$$
2. Decay of $\|\theta'\|_{L^2}$
  ↳ using Gagliardo–Nirenberg interpolation
Bootstrap argument

Optimality

Theorem (Leblond, 2023), adapted from Kiselev-Yao (2023)

Any stationary solution is unstable under a small perturbation in $H^{2-}$, but stability holds in $H^{6}$.

Theorem (Dalibard, Guillod & Leblond, 2024)

The leading order term is given by a boundary layer $\rho_\BL$ and $\|\rho_\BL\|_{L^2} \sim (1+t)^{-9/8}$, so $(1+t)^{-1}$ is nearly optimal.

What is happening in general?

Proposition (Leblond, 2023)

For $\rho_0\in L^\infty$, let denote by $\rho_0^*$ the decreasing vertical rearrangement of $\rho_0$, and by $\rho$ the solution of (ST).

The potential energy $E(t)$ converges to some $E_\infty\geq E(\rho_0^*)$.
The $\omega$-limit set $$\omega_{H^{-1}} := \bigcap_{t\geq0} \overline{\{\rho(\tau):\tau\geq t\}}^{H^{-1}}$$ is non-empty.
Any $\rho_\infty\in\omega_{H^{-1}}$ is a stratified rearrangment of $\rho_0$ and $E(\rho_\infty) = E_\infty$.
If $E_\infty=E(\rho_0^*)$, then $\omega_{H^{-1}} = \{\rho_0^*\}$ and $\rho(t)\to\rho_0^*$ in $H^{-1}$.

What can go wrong?

3. Stability of the interface problem

Stability of the constant interface

Theorem (Gancedo, Granero-Belinchón & Salguero, 2022)

For $3/2 < m < 2$, there exists $\varepsilon_0>0$ small such that for any $\zeta_0\in H^3(\mathbb{T})$ with $\|\zeta_0\|_{H^3}\leq\varepsilon_0$, the solution of (ST) in $\mathbb{T}\times\mathbb{R}$ with $\rho_0(x,z)=\mathbf{1}_{\{z<\zeta_0(x)\}}$ is given by $\rho(t,x,z)=\mathbf{1}_{\{z<\zeta(t,x)\}}$ where: $$\|\zeta-\bar\zeta_0\|_{L^2} \lesssim \frac{\varepsilon_0}{1+t^m}, \qquad \|\zeta-\bar\zeta_0\|_{H^3} \lesssim \varepsilon_0.$$