# Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates

Théo Fradin

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Théo Fradin. Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates. 2024. hal-04615812v2

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# Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates

Théo Fradin

<figure><img src="image_1.png" /></figure>

## To cite this version:

Théo Fradin. Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates. 2024. hal-04615812v3

HAL Id: hal-04615812

https://hal.science/hal-04615812v3

Preprint submitted on 12 Feb 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
WELL-POSEDNESS OF THE EULER EQUATIONS IN A STABLY STRATIFIED
OCEAN IN ISOPYCNAL COORDINATES

THÉO FRADIN

ABSTRACT. This article is concerned with the well-posedness of the incompressible Euler equations describing a stably stratified ocean, reformulated in isopycnal coordinates. Our motivation for using this reformulation is twofold: first, its quasi-2D structure renders some parts of the analysis easier. Second, it closes a gap between the analysis performed in the paper by Bianchini and Duchêne in 2022 in isopycnal coordinates, with shear velocity but with a regularizing term, and the analysis performed in the paper by Desjardins, Lannes, Saut in 2020 in Eulerian coordinates, without any regularizing term but without shear velocity. Our main result is an energy estimate in Sobolev spaces on the system in isopycnal coordinates, with shear velocity, without any regularizing term. With additional assumptions, it is uniform in the shallow-water parameter. The main difficulty consists in transposing to the isopycnal reformulation the symmetric structure of the system which is more straightforward in Eulerian coordinates.

# 1. INTRODUCTION

1.1. **General setting.** The aim of this paper is to study the equations describing the evolution of an ocean, that is in our context a strip of water in which we consider density variations. We start from the incompressible Euler equations in Eulerian coordinates, on a time interval $[0, T)$ where $T > 0$, on the strip $S_z := \mathbb{R}^d \times (-H, 0)$, with $d \in \{1, 2\}$ being the horizontal dimension and $H > 0$ the height of the strip. This set of equations on the velocity $\mathbf{U}^{\text{eul}} := \begin{pmatrix} V^{\text{eul}} \\ w^{\text{eul}} \end{pmatrix}$ is completed by the mass conservation on the density $\rho : \mathbb{R}^d \times [-H, 0] \to [\rho_*, \rho^*]$ with $0 < \rho_* < \rho^*$ two constants,

$$ (E^{\text{eul}}) \qquad \left\{ \begin{array}{ll} \partial_t \mathbf{U}^{\text{eul}} + (\mathbf{U}^{\text{eul}} \cdot \nabla)\mathbf{U}^{\text{eul}} + \frac{1}{\rho}\nabla P^{\text{eul}} - \mathbf{g} = 0, & \\ \partial_t \rho + \mathbf{U}^{\text{eul}} \cdot \nabla \rho = 0, & \text{in } [0, T) \times S_z. \\ \nabla \cdot \mathbf{U}^{\text{eul}} = 0, & \end{array} \right. $$

Here, $\nabla := (\partial_{x_1} \dots \partial_{x_d} \partial_z)^T$ denotes the gradient, $\mathbf{g} = -g\mathbf{e}_z$ with $g > 0$ is the acceleration of gravity and $\mathbf{e}_z$ is the upward vertical unit vector. We assume a flat bottom and a rigid lid, so that the impermeability conditions yield the boundary conditions

$$ (1.1) \qquad w_{|z=0}^{\text{eul}} = w_{|z=-H}^{\text{eul}} = 0 \quad \text{in } [0, T) \times \mathbb{R}^d. $$

<footnote>Date: June 21, 2024.</footnote>

<footnote n="2010">2010 Mathematics Subject Classification. 35Q35, 76B03, 76B70, 86A05.</footnote>

<footnote>Key words and phrases. Euler equations, stratification, isopycnal coordinates, Alinhac's good unknown.</footnote>
WELL-POSEDNESS OF THE EULER EQUATIONS IN A STABLY STRATIFIED OCEAN IN ISOPYCNAL
COORDINATES

THÉO FRADIN

ABSTRACT. This article is concerned with the well-posedness of the incompressible Euler equations describing a stably stratified ocean, reformulated in isopycnal coordinates. Our motivation for using this reformulation is twofold: first, its quasi-2D structure renders some parts of the analysis easier. Second, it closes a gap between the analysis performed in the paper by Bianchini and Duchêne in 2022 in isopycnal coordinates, with shear velocity but with a regularizing term, and the analysis performed in the paper by Desjardins, Lannes, Saut in 2020 in Eulerian coordinates, without any regularizing term but without shear velocity. Our main result is a local well-posedness result in Sobolev spaces on the system in isopycnal coordinates, with shear velocity, without any regularizing term. The time of existence that we obtain is uniform with respect to the size $\epsilon$ of the perturbation, and boils down to the large time $1/\epsilon$ with the assumptions of the paper by Desjardins, Lannes, Saut in 2020. With additional assumptions, it is also uniform in the shallow-water parameter. The main difficulty consists in transposing to the isopycnal reformulation the symmetric structure of the system which is more straightforward in Eulerian coordinates.

1. INTRODUCTION

1.1. **General setting.** The aim of this paper is to study the equations describing the evolution of an ocean, that is in our context a strip of water in which we consider density variations. We start from the incompressible Euler equations in Eulerian coordinates, on a time interval $[0, T)$ where $T > 0$, on the strip $S_z := \mathbb{R}^d \times (-H, 0)$, with $d \in \{1, 2\}$ being the horizontal dimension and $H > 0$ the height of the strip. This set of equations on the velocity $\mathbf{U}^{\text{eul}} := \begin{pmatrix} V^{\text{eul}} \\ w^{\text{eul}} \end{pmatrix}$ is completed by the mass conservation on the density $\rho : \mathbb{R}^d \times [-H, 0] \to [\rho_*, \rho^*]$ with $0 < \rho_* < \rho^*$ two constants,

$$ (E^{\text{eul}}) \qquad \left\{ \begin{array}{ll} \partial_t \mathbf{U}^{\text{eul}} + (\mathbf{U}^{\text{eul}} \cdot \nabla)\mathbf{U}^{\text{eul}} + \frac{1}{\rho}\nabla P^{\text{eul}} - \mathbf{g} = 0, & \\ \partial_t \rho + \mathbf{U}^{\text{eul}} \cdot \nabla \rho = 0, & \text{in } [0, T) \times S_z. \\ \nabla \cdot \mathbf{U}^{\text{eul}} = 0, & \end{array} \right. $$

Here, $\nabla := (\partial_{x_1}, \dots, \partial_{x_d}, \partial_z)^T$ denotes the gradient, $\mathbf{g} = -g\mathbf{e}_z$ with $g > 0$ is the acceleration of gravity and $\mathbf{e}_z$ is the upward vertical unit vector. We assume a flat bottom and a rigid lid, so that the impermeability conditions yield the boundary conditions

$$ (1.1) \qquad w_{|z=0}^{\text{eul}} = w_{|z=-H}^{\text{eul}} = 0 \quad \text{in } [0, T) \times \mathbb{R}^d. $$

This set of equations is completed by the initial data

$$ (1.2) \qquad \begin{cases} \mathbf{U}_{|t=0}^{\text{eul}} = \mathbf{U}_{\text{in}}^{\text{eul}}, \\ \rho_{|t=0} = \rho_{\text{in}}, \end{cases} \quad \text{in } S_z. $$

The well-posedness of the inhomogeneous (that is, when $\rho$ is not constant) incompressible Euler equations is well known, see for instance [Dan10] for a study in the whole space or [IT99] in a more general domain.

In this study we assume that the ocean is stratified, meaning that we can find a function $\eta : \mathbb{R}^d \times [0, 1] \to [-H, 0]$ such that the density is constant along the level sets of $\eta$ with respect to the third variable, i.e.

$$ (1.3) \qquad \rho(t, x, \eta(t, x, r)) = \varrho(r), \quad \forall (t, x, r) \in [0, T] \times \mathbb{R}^d \times [0, 1], $$

Date: February 12, 2025.

2010 Mathematics Subject Classification. 35Q35, 76B03, 76B70, 86A05.

Key words and phrases. Euler equations, stratification, isopycnal coordinates, Alinhac's good unknown.
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Setting in Eulerian coordinates. Isopycnals are sets where the density is constant.</figcaption></figure>

This set of equations is completed by the initial data

$$ (1.2) \qquad \begin{cases} \mathbf{U}_{|t=0}^{\text{eul}} = \mathbf{U}_{\text{in}}^{\text{eul}} \\ \rho_{|t=0} = \rho_{\text{in}} \end{cases} \quad \text{in } S_z. $$

The well-posedness of the inhomogeneous (that is, when $\rho$ is not constant) incompressible Euler equations is well known, see for instance [Dan10] for a study in the whole space or [IT99] in a more general domain.

In this study we assume that the ocean is stratified, meaning that we can find a function $\eta: \mathbb{R}^d \times [0, 1] \to [-H, 0]$ such that the density is constant along the level sets of $\eta$ with respect to the third variable, i.e.

$$ (1.3) \qquad \rho(t, x, \eta(t, x, r)) = \varrho(r), \quad \forall (t, x, r) \in [0, T] \times \mathbb{R}^d \times [0, 1], $$

where $\varrho: [0, 1] \to [\rho_*, \rho^*]$ is thus the function mapping the level set $r$ to the density of the fluid on this level set, see Figure 1.

Assuming further that $\eta$ is a $C^1$ function and that $-\partial_r \eta \ge h_*$ for a certain $h_* > 0$, this yields a diffeomorphism

$$ \varphi: \begin{cases} [0, T) \times S_r \to [0, T) \times S_z \\ (t, x, r) \mapsto \varphi(t, x, r) := (t, x, \eta(t, x, r)). \end{cases} $$

Here, $S_r$ is the domain $\mathbb{R}^d \times (0, 1)$ of the so-called isopycnal coordinates., see Figure 2.

Note that this implies that the density at the rigid lid (resp. bottom) is constant, that is

$$ \begin{cases} \rho_{|z=0} = \rho_*, \\ \rho_{|z=-H} = \rho^*. \end{cases} $$

Therefore, we assume that this condition holds initially. Note that this condition is propagated by the equations thanks to the transport equation on the density $\rho$ and the fact that the velocity field is tangent at the boundary.

Writing the Euler equations in isopycnal coordinates means applying the change of
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Setting in Eulerian coordinates. Isopycnals are sets where the density is constant.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIGURE 2. Setting in isopycnal coordinates. Isopycnals are flattened.</figcaption></figure>

where $\varrho : [0, 1] \to [\rho_*, \rho^*]$ is thus the function mapping the level set $r$ to the density of the fluid on this level set, see Figure 1.

Assuming further that $\eta$ is a $C^1$ function and that $-\partial_r \eta \ge h_*$ for a certain $h_* > 0$, this yields a diffeomorphism

$$ \varphi : \begin{cases} [0, T) \times S_r \to [0, T) \times S_z \\ (t, x, r) \mapsto \varphi(t, x, r) := (t, x, \eta(t, x, r)). \end{cases} $$

Here, $S_r$ is the domain $\mathbb{R}^d \times (0, 1)$ of the so-called isopycnal coordinates, see Figure 2. Note that this implies that the density at the rigid lid (resp. bottom) is constant, that is

$$ \begin{cases} \rho|_{z=0} = \rho_*, \\ \rho|_{z=-H} = \rho^*. \end{cases} $$

Therefore, we assume that this condition holds initially. Note that this condition is propagated by the equations thanks to the transport equation on the density $\rho$ and the fact that the velocity field is tangent at the boundary. Writing the Euler equations in isopycnal coordinates means applying the change of coordinates $\varphi$ to the system ($E^{\text{eul}}$). To this end, we define the new set of unknowns

$$ \mathbf{U}^{\text{iso}} := \mathbf{U}^{\text{eul}} \circ \varphi, \quad V^{\text{iso}} := V^{\text{eul}} \circ \varphi, $$

$$ w^{\text{iso}} := w^{\text{eul}} \circ \varphi, \quad P^{\text{iso}} := P^{\text{eul}} \circ \varphi, $$
<figure><img src="image_3.png" /><figcaption>FIGURE 2. Setting in isopycnal coordinates. Isopycnals are flattened.</figcaption></figure>

coordinates $\varphi$ to the system ($E^{\text{eul}}$). To this end, we define the new set of unknowns

$$ \mathbf{U}^{\text{iso}} := \mathbf{U}^{\text{eul}} \circ \varphi, \qquad V^{\text{iso}} := V^{\text{eul}} \circ \varphi, $$

$$ w^{\text{iso}} := w^{\text{eul}} \circ \varphi, \qquad P^{\text{iso}} := P^{\text{eul}} \circ \varphi, $$

as well as the operators

$$ \partial_t^\varphi f^{\text{iso}} := \partial_t(f^{\text{eul}}) \circ \varphi, \qquad \nabla_x^\varphi f^{\text{iso}} := \nabla_x(f^{\text{eul}}) \circ \varphi, $$

$$ \partial_r^\varphi f^{\text{iso}} := \partial_r(f^{\text{eul}}) \circ \varphi, \qquad \nabla^\varphi = \begin{pmatrix} \nabla_x^\varphi \\ \partial_r^\varphi \end{pmatrix}, $$

for $f^{\text{eul}}$ an unknown in Eulerian coordinates and $f^{\text{iso}} := f^{\text{eul}} \circ \varphi$ its expression in isopycnal coordinates. We thus obtain the first reformulation of the momentum equation and incompressibility condition in isopycnal coordinates

$$ \left\{ \begin{array}{l} \partial_t^\varphi \mathbf{U}^{\text{iso}} + (\mathbf{U}^{\text{iso}} \cdot \nabla^\varphi) \mathbf{U}^{\text{iso}} + \frac{1}{\rho} \nabla^\varphi P^{\text{iso}} - \mathbf{g} = 0, \\ \nabla^\varphi \cdot \mathbf{U}^{\text{iso}} = 0, \end{array} \right. \quad \text{in } [0, T) \times S_r. $$

This set of equations is completed with the boundary conditions

$$ (1.4) \qquad w_{|r=0}^{\text{iso}} = w_{|r=1}^{\text{iso}} = 0 \quad \text{in } [0, T) \times \mathbb{R}^d, $$

and initial data

$$ \left\{ \begin{array}{l} \mathbf{U}_{|t=0}^{\text{iso}} = \mathbf{U}_{\text{in}}^{\text{iso}}, \\ \eta_{|t=0} = \eta_{\text{in}}, \end{array} \right. \quad \text{in } S_r. $$

Using the chain rule, we can compute

$$ \nabla_{t,x}^\varphi f^{\text{iso}} = \nabla_{t,x} f^{\text{iso}} - \frac{\nabla_{t,x} \eta}{\partial_r \eta} \partial_r f^{\text{iso}}, $$

$$ \partial_r^\varphi f^{\text{iso}} = \frac{1}{\partial_r \eta} \partial_r f^{\text{iso}}, $$

for any function $f^{\text{iso}}$ defined in isopycnal coordinates.

Thus, we need to complete the momentum equation in isopycnal coordinates with an
as well as the operators

$$ \partial_t^\varphi f^{\text{iso}} := \partial_t (f^{\text{eul}}) \circ \varphi, \qquad \nabla_x^\varphi f^{\text{iso}} := \nabla_x (f^{\text{eul}}) \circ \varphi, $$

$$ \partial_r^\varphi f^{\text{iso}} := \partial_r (f^{\text{eul}}) \circ \varphi, \qquad \nabla^\varphi = \begin{pmatrix} \nabla_x^\varphi \\ \partial_r^\varphi \end{pmatrix}, $$

for $f^{\text{eul}}$ an unknown in Eulerian coordinates and $f^{\text{iso}} := f^{\text{eul}} \circ \varphi$ its expression in isopycnal coordinates. We thus obtain the first reformulation of the momentum equation and incompressibility condition in isopycnal coordinates

$$ \left\{ \begin{array}{l} \partial_t^\varphi \mathbf{U}^{\text{iso}} + (\mathbf{U}^{\text{iso}} \cdot \nabla^\varphi) \mathbf{U}^{\text{iso}} + \frac{1}{\rho} \nabla^\varphi P^{\text{iso}} - \mathbf{g} = 0, \\ \nabla^\varphi \cdot \mathbf{U}^{\text{iso}} = 0, \end{array} \right. \quad \text{in } [0, T) \times S_r. $$

This set of equations is completed with the boundary conditions

$$ (1.4) \qquad w_{|r=0}^{\text{iso}} = w_{|r=1}^{\text{iso}} = 0 \quad \text{in } [0, T) \times \mathbb{R}^d, $$

and initial data

$$ \left\{ \begin{array}{l} \mathbf{U}_{|t=0}^{\text{iso}} = \mathbf{U}_{\text{in}}^{\text{iso}}, \\ \eta_{|t=0} = \eta_{\text{in}}, \end{array} \right. \quad \text{in } S_r. $$

Using the chain rule, we can compute

$$ \nabla_{t,x}^\varphi f^{\text{iso}} = \nabla_{t,x} f^{\text{iso}} - \frac{\nabla_{t,x} \eta}{\partial_r \eta} \partial_r f^{\text{iso}}, $$

$$ \partial_r^\varphi f^{\text{iso}} = \frac{1}{\partial_r \eta} \partial_r f^{\text{iso}}, $$

for any function $f^{\text{iso}}$ defined in isopycnal coordinates.

Thus, we need to complete the momentum equation in isopycnal coordinates with an equation on $\eta$. This is done by differentiating (1.3) with respect to $x$ and $t$, namely

$$ \left\{ \begin{array}{l} \nabla_x \rho(t, x, \eta(t, x, r)) + \nabla_x \eta \partial_z \rho(t, x, \eta(t, x, r)) = 0, \\ \partial_t \rho(t, x, \eta(t, x, r)) + \partial_t \eta \partial_z \rho(t, x, \eta(t, x, r)) = 0. \end{array} \right. $$

Using the mass conservation equation in $(E^{\text{eul}})$, this yields

$$ (1.5) \qquad (\partial_t \eta + V \cdot \nabla_x \eta - w) \partial_z \rho = 0. $$

Now restricting to the case of a stable stratification, that is, there exists $c > 0$ such that $-\partial_z \rho \ge c > 0$, the above equation (1.5) yields an evolution equation on $\eta$. Note that differentiating (1.3) with respect to $r$ yields

$$ \partial_r \eta \partial_z \rho = \rho'. $$

Thus, as $-\partial_r \eta \ge h_* > 0$, the stable stratification assumption is equivalent to the existence of $c_* > 0$ such that $\rho' \ge c_*$.

Using (1.5), we can compute

$$ (1.6) \qquad \partial_t^\varphi + \mathbf{U}^{\text{iso}} \cdot \nabla^\varphi = \partial_t + V^{\text{iso}} \cdot \nabla_x. $$

We can now write the system in isopycnal coordinates according to this remark, namely

$$ (E^{\text{iso}}) \qquad \left\{ \begin{array}{l} \partial_t \mathbf{U}^{\text{iso}} + (V^{\text{iso}} \cdot \nabla_x) \mathbf{U}^{\text{iso}} + \frac{1}{\rho} \nabla^\varphi P^{\text{iso}} - \mathbf{g} = 0, \\ \partial_t \eta + V^{\text{iso}} \cdot \nabla_x \eta - w^{\text{iso}} = 0, \quad \text{in } [0, T) \times S_r. \\ \nabla^\varphi \cdot \mathbf{U}^{\text{iso}} = 0, \end{array} \right. $$

It is important to notice that there is no vertical advection in the formulation $(E^{\text{iso}})$ of the system in isopycnal coordinates, as opposed to the formulation in Eulerian coordinates. For this reason, $(E^{\text{iso}})$ is sometimes said to be quasi two-dimensional [Val17] or semi-Lagrangian [BD24]. In other words, the flow is largely along the isopycnals [Val17], hence the study of large-scale flows in the ocean is sometimes performed in the isopycnal coordinates [GBB+00]. From a technical viewpoint, we crucially use this semi-Lagrangian property in this study
equation on $\eta$. This is done by differentiating (1.3) with respect to $x$ and $t$, namely

$$ egin{cases} \nabla_x \rho(t, x, \eta(t, x, r)) + \nabla_x \eta \partial_z \rho(t, x, \eta(t, x, r)) = 0, \\ \partial_t \rho(t, x, \eta(t, x, r)) + \partial_t \eta \partial_z \rho(t, x, \eta(t, x, r)) = 0. \end{cases} $$

Using the mass conservation equation in $(E^{\text{eul}})$, this yields

$$ (1.5) \qquad (\partial_t \eta + V \cdot \nabla_x \eta - w) \partial_z \rho = 0. $$

Now restricting to the case of a stable stratification, that is, there exists $c > 0$ such that $-\partial_z \rho \ge c > 0$, the above equation (1.5) yields an evolution equation on $\eta$. Note that differentiating (1.3) with respect to $r$ yields

$$ \partial_r \eta \partial_z \rho = \varrho'. $$

Thus, as $-\partial_r \eta \ge h_* > 0$, the stable stratification assumption is equivalent to the existence of $c_* > 0$ such that $\varrho' \ge c_*$.

Using (1.5), we can compute

$$ (1.6) \qquad \partial_t^\varphi + \mathbf{U}^{\text{iso}} \cdot \nabla^\varphi = \partial_t + V^{\text{iso}} \cdot \nabla_x $$

We can now write the system in isopycnal coordinates according to this remark, namely

$$ (E^{\text{iso}}) \quad \left\{ \begin{array}{ll} \partial_t \mathbf{U}^{\text{iso}} + (V^{\text{iso}} \cdot \nabla_x) \mathbf{U}^{\text{iso}} + \frac{1}{\varrho} \nabla^\varphi P^{\text{iso}} - \mathbf{g} = 0, & \\ \partial_t \eta + V^{\text{iso}} \cdot \nabla_x \eta - w^{\text{iso}} = 0, & \text{in } [0, T) \times S_r. \\ \nabla^\varphi \cdot \mathbf{U}^{\text{iso}} = 0, & \end{array} \right. $$

It is important to notice that there is no vertical advection in the formulation $(E^{\text{iso}})$ of the system in isopycnal coordinates, as opposed to the formulation in Eulerian coordinates. For this reason, $(E^{\text{iso}})$ is sometimes said to be quasi two-dimensional [Val17] or semi-Lagrangian [DB22]. In other words, the flow is largely along the isopycnals [Val17], hence the study of large-scale flows in the ocean is sometimes performed in the isopycnal coordinates [GBB$^+$00]. From a technical viewpoint, we crucially use this semi-Lagrangian property in this study for the construction of a solution from a regularized system, which is less obvious in Eulerian coordinates, see Remark 4.1.

1.2. **Ansatz.** The system in Eulerian coordinates $(E^{\text{eul}})$ admits equilibria called shear flows, namely of the form

$$ (V^{\text{eul}}, w^{\text{eul}}, \rho)(t, x, z) := (V_{\text{eq}}^{\text{eul}}, 0, \rho_{\text{eq}})(z). $$

The pressure $P_{\text{eq}}^{\text{eul}}$ is then given by the equation on $w$ in $(E^{\text{eul}})$, and is called the hydrostatic pressure

$$ P_{\text{eq}}^{\text{eul}}(z) := \int_z^0 g \rho_{\text{eq}}(z) dz, $$

where we assumed without loss of generality that the pressure at the rigid lid is zero. In Eulerian coordinates, we make the natural ansatz of studying a perturbation of size $\epsilon \in [0, 1]$ of a reference solution that is an equilibrium:

$$ (\underline{V}^{\text{eul}}(z), 0, \underline{P}^{\text{eul}}(z), \underline{\rho}(z)) := (V_{\text{eq}}^{\text{eul}}(z), 0, P_{\text{eq}}^{\text{eul}}(z), \rho_{\text{eq}}(z)). $$

Namely, we look for solutions under the form

$$ (V_{\text{total}}^{\text{eul}}, w_{\text{total}}^{\text{eul}}, P_{\text{total}}^{\text{eul}}, \rho_{\text{total}}) := (\underline{V}^{\text{eul}}, 0, \underline{P}^{\text{eul}}, \underline{\rho}) + \epsilon (\tilde{V}^{\text{eul}}, \tilde{w}^{\text{eul}}, \tilde{P}^{\text{eul}}, \tilde{\rho}). $$
for the construction of a solution from a regularized system, which is less obvious in Eulerian coordinates, see
Remark 4.5.

1.2. **Ansatz.** The system in Eulerian coordinates ($E^{\text{eul}}$) admits equilibria called shear flows, namely of the form

$$
(V^{\text{eul}}, w^{\text{eul}}, \rho)(t, x, z) := (V_{\text{eq}}^{\text{eul}}, 0, \rho_{\text{eq}})(z).
$$

The pressure $P_{\text{eq}}^{\text{eul}}$ is then given by the equation on $w$ in ($E^{\text{eul}}$), and is called the hydrostatic pressure

$$
P_{\text{eq}}^{\text{eul}}(z) := \int_{z}^{0} g \rho_{\text{eq}}(z) dz,
$$

where we assumed without loss of generality that the pressure at equilibrium at the rigid lid is zero. In Eulerian coordinates, we make the natural ansatz of studying a perturbation of size $\epsilon \in [0, 1]$ of a reference solution that is an equilibrium:

$$
(\underline{V}^{\text{eul}}(z), 0, \underline{P}^{\text{eul}}(z), \underline{\rho}(z)) := (V^{\text{eul}}(z), 0, P^{\text{eul}}(z), \rho^{\text{eq}}(z)).
$$

Namely, we look for solutions under the form

$$
(V_{\text{total}}^{\text{eul}}, w_{\text{total}}^{\text{eul}}, P_{\text{total}}^{\text{eul}}, \rho_{\text{total}}) := (\underline{V}^{\text{eul}}, 0, \underline{P}^{\text{eul}}, \underline{\rho}) + \epsilon(\tilde{V}^{\text{eul}}, \tilde{w}^{\text{eul}}, \tilde{P}^{\text{eul}}, \tilde{\rho}).
$$

The system ($E^{\text{eul}}$) then writes, in $[0, T) \times S_z$:

$$
\begin{equation}
\begin{aligned}
& \left\{
  \begin{array}{@{}l@{}}
    \partial_t \tilde{V}^{\text{eul}} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) \cdot \nabla_x \tilde{V}^{\text{eul}} + \tilde{w}^{\text{eul}} \partial_z (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) + \frac{1}{\underline{\rho} + \epsilon \tilde{\rho}} \nabla_x \tilde{P}^{\text{eul}} = 0, \\
    \partial_t \tilde{w}^{\text{eul}} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) \cdot \nabla_x \tilde{w}^{\text{eul}} + \epsilon \tilde{w}^{\text{eul}} \partial_z \tilde{w}^{\text{eul}} + \frac{1}{\underline{\rho} + \epsilon \tilde{\rho}} \partial_z \tilde{P}^{\text{eul}} + \frac{\tilde{\rho}}{\underline{\rho} + \epsilon \tilde{\rho}} = 0, \\
    \partial_t \tilde{\rho} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) \cdot \nabla_x \tilde{\rho} + \tilde{w}^{\text{eul}} \partial_z (\underline{\rho} + \epsilon \tilde{\rho}) = 0, \\
    \nabla_x \cdot \tilde{V}^{\text{eul}} + \partial_z \tilde{w}^{\text{eul}} = 0.
  \end{array}
\right.
\end{aligned}
\tag{1.7}
\end{equation}
$$

In [DLS20], the authors study the system (1.7) in this particular setting, with the assumption of stable stratifica-
tion, i.e. that the density is strictly decreasing with respect to the vertical coordinate. Under several technical
assumptions, one of which being the absence of background shear-flow, they prove the well-posedness of the
system (1.7), in Sobolev spaces of finite regularity, on a time interval $[0, T)$ with $T$ of order $O(1/\epsilon)$ (see [DLS20, The-
orem 1]). Regarding this issue, Theorem 1.1 is an extension of [DLS20, Theorem 1] to the case of a non-zero shear
flow. Note however that Theorem 1.1 yields a shorter time of existence than [DLS20, Theorem 1], see Remark 1.4.

Let us now turn to the reformulation in isopycnal coordinates. The system in isopycnal coordinates ($E^{\text{iso}}$)
also admits equilibrium solutions ($V_{\text{eq}}^{\text{iso}}$, $0$, $P_{\text{eq}}^{\text{iso}}$, $\eta_{\text{eq}}$) that only depend on the vertical variable $r$ and satisfying the
hydrostatic balance

$$
\partial_r^{\varphi_{\text{eq}}} P_{\text{eq}}^{\text{iso}} = -g \rho,
$$

where $\varphi_{\text{eq}}(t, x, r) := (t, x, \eta_{\text{eq}}(r))$. We make the particular choice $\eta_{\text{eq}}(r) = -rH$. Indeed, if $\rho_{\text{eq}}$ is the density at equilibrium, we can write

$$
\forall r \in [0, 1], \rho_{\text{eq}}(\eta_{\text{eq}}(r)) = \rho_{\text{eq}}(-rH) := \rho(r).
$$

so that the change of coordinates $\varphi_{\text{eq}}(t, x, r) = (t, x, \eta_{\text{eq}}(r))$ does verify

$$
h_{\text{eq}}(r) := -\partial_r \eta_{\text{eq}}(r) = H > 0.
$$

We thus have

$$
(V_{\text{eq}}^{\text{iso}}, P_{\text{eq}}^{\text{iso}})(r) = (V_{\text{eq}}^{\text{eul}}, P_{\text{eq}}^{\text{eul}})(-rH).
$$

In [BD24], the authors study a perturbation of such an equilibrium. Let us make a slightly different ansatz, and look for solutions ($V_{\text{total}}^{\text{iso}}$, $w_{\text{total}}^{\text{iso}}$, $P_{\text{total}}^{\text{iso}}$, $\eta_{\text{total}}$) of the form

$$
(V_{\text{total}}^{\text{iso}}, w_{\text{total}}^{\text{iso}}, P_{\text{total}}^{\text{iso}}, \eta_{\text{total}}) := (\underline{V}^{\text{iso}} + \epsilon \tilde{V}^{\text{iso}}, \epsilon \tilde{w}^{\text{iso}}, \underline{P}^{\text{iso}} + \epsilon \tilde{P}^{\text{iso}}, \underline{\eta} + \epsilon \tilde{\eta}),
$$
The system ($E^{\text{eul}}$) then writes, in $[0, T) \times S_z$:

$$ (1.7) \quad \left\{ \begin{aligned} & \partial_t \tilde{V}^{\text{eul}} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}} \cdot \nabla_x) \tilde{V}^{\text{eul}} + \tilde{w}^{\text{eul}} \partial_z (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) + \frac{1}{\underline{\rho} + \epsilon \tilde{\rho}} \nabla_x \tilde{P}^{\text{eul}} = 0, \\ & (\partial_t \tilde{w}^{\text{eul}} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}} \cdot \nabla_x) \tilde{w}^{\text{eul}} + \epsilon \tilde{w}^{\text{eul}} \partial_z \tilde{w}^{\text{eul}}) + \frac{1}{\underline{\rho} + \epsilon \tilde{\rho}} \partial_z \tilde{P}^{\text{eul}} + \frac{\tilde{\rho}}{\underline{\rho} + \epsilon \tilde{\rho}} = 0, \\ & \partial_t \tilde{\rho} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) \cdot \nabla_x \tilde{\rho} + \tilde{w}^{\text{eul}} \partial_z (\underline{\rho} + \epsilon \tilde{\rho}) = 0, \\ & \nabla_x \cdot \tilde{V}^{\text{eul}} + \partial_z \tilde{w}^{\text{eul}} = 0. \end{aligned} \right. $$

In [DLS20], the authors study the system (1.7) in this particular setting, with the assumption of stable stratification, i.e. that the density is strictly decreasing with respect to the vertical coordinate. Under several technical assumptions, one of which being the absence of background shear-flow, they prove the well-posedness of the system (1.7), in Sobolev spaces of finite regularity, on a time interval $[0, T)$ with $T$ of order $O(1/\epsilon)$ (see [DLS20, Theorem 1]). Regarding this issue, Theorem 1.1 is an extension of [DLS20, Theorem 1] to the case of a non-zero shear flow. Note however that Theorem 1.1 yields a shorter time of existence than [DLS20, Theorem 1], namely a time independent of $\epsilon$.

Let us now turn to the reformulation in isopycnal coordinates. The system in isopycnal coordinates ($E^{\text{iso}}$) also admits equilibrium solutions ($V_{\text{eq}}^{\text{iso}}, 0, P_{\text{eq}}^{\text{iso}}, \eta_{\text{eq}}$) that only depend on the vertical variable $r$ and satisfying the hydrostatic balance

$$ \partial_r^{\varphi_{\text{eq}}} P_{\text{eq}}^{\text{iso}} = -g \varrho, $$

where $\varphi_{\text{eq}}(t, x, r) := (t, x, \eta_{\text{eq}}(r))$. We make the particular choice $\eta_{\text{eq}}(r) = -rH$. Indeed, if $\rho_{\text{eq}}$ is the density at equilibrium, we can write

$$ \forall r \in [0, 1], \rho_{\text{eq}}(\eta_{\text{eq}}(r)) = \rho_{\text{eq}}(-rH) := \varrho(r). $$

so that the change of coordinates $\varphi_{\text{eq}}(t, x, r) = (t, x, \eta_{\text{eq}}(r))$ does verify

$$ h_{\text{eq}}(r) := -\partial_r \eta_{\text{eq}}(r) = H > 0. $$

We thus have

$$ (V_{\text{eq}}^{\text{iso}}, P_{\text{eq}}^{\text{iso}})(r) = (V_{\text{eq}}^{\text{eul}}, P_{\text{eq}}^{\text{eul}})(-rH). $$

In [DB22], the authors study a perturbation of such an equilibrium. Let us make a slightly different ansatz, and look for solutions ($V_{\text{total}}^{\text{iso}}, w_{\text{total}}^{\text{iso}}, P_{\text{total}}^{\text{iso}}, \eta_{\text{total}}$) of the form

$$ (V_{\text{total}}^{\text{iso}}, w_{\text{total}}^{\text{iso}}, P_{\text{total}}^{\text{iso}}, \eta_{\text{total}}) := (\underline{V}^{\text{iso}} + \epsilon \tilde{V}^{\text{iso}}, \epsilon \tilde{w}^{\text{iso}}, \underline{P}^{\text{iso}} + \epsilon \tilde{P}^{\text{iso}}, \underline{\eta} + \epsilon \tilde{\eta}), $$

where $\epsilon \in [0, 1]$ and

$$ \left\{ \begin{aligned} \underline{\eta}(r) &:= \eta_{\text{eq}}(r), \\ \underline{V}(r)^{\text{iso}} &:= V_{\text{eq}}^{\text{iso}}(r), \\ \underline{P}^{\text{iso}}(t, x, r) &:= P_{\text{eq}}^{\text{iso}} \left( -\frac{1}{H} \left( \underline{\eta}(r) + \epsilon \tilde{\eta}(t, x, r) \right) \right). \end{aligned} \right. $$

We can thus write the change of coordinates as

$$ \varphi(t, x, r) := (t, x, \underline{\eta}(r) + \epsilon \tilde{\eta}(t, x, r)). $$

Remark that the ansatz on the pressure is such that

$$ \nabla_x^{\varphi} \underline{P}^{\text{iso}} = (\nabla_x P_{\text{eq}}^{\text{eul}}(\cdot)) \circ (-\underline{\eta}(r) - \epsilon \tilde{\eta}(t, x, r)) = 0, $$
where $\epsilon \in [0, 1]$ and

$$ 
\begin{cases}
\underline{\eta}(r) := \eta_{\text{eq}}(r), \\
\underline{V}(r)^{\text{iso}} := V_{\text{eq}}^{\text{iso}}(r), \\
\underline{P}^{\text{iso}}(t, x, r) := P_{\text{eq}}^{\text{iso}}\left(-\frac{1}{H}\left(\underline{\eta}(r) + \epsilon\tilde{\eta}(t, x, r)\right)\right).
\end{cases}
$$

We can thus write the change of coordinates as

$$ \varphi(t, x, r) := (t, x, \underline{\eta}(r) + \epsilon \tilde{\eta}(t, x, r)). $$

Remark that the ansatz on the pressure is such that

$$ \nabla_x^\varphi \underline{P}^{\text{iso}} = (\nabla_x P_{\text{eq}}^{\text{eul}}(\cdot)) \circ (-\underline{\eta}(r) - \epsilon \tilde{\eta}(t, x, r)) = 0, $$

as $P_{\text{eq}}^{\text{eul}}$ only depends on the vertical variable. This property will simplify the analysis later on. Note that a Taylor expansion on $P_{\text{eq}}^{\text{iso}}$ yields

$$ \underline{P}^{\text{iso}}(t, x, r) = P_{\text{eq}}^{\text{iso}}(r) + O(\epsilon), $$

so that the difference with this ansatz and the one used in [BD24] consisting in studying directly a perturbation of the equilibrium correspond up to terms of order $\epsilon$.

Using the notation

$$ b := \frac{1}{\epsilon} \left( 1 - \frac{\rho_{\text{eq}}(\eta_{\text{eq}} + \epsilon \tilde{\eta})}{\rho_{\text{eq}}(\eta_{\text{eq}})} \right) g $$

for the buoyancy term, this yields the system

$$ (1.8) \qquad \left\{ \begin{array}{ll} \partial_t \tilde{V}^{\text{iso}} + (\underline{V}^{\text{iso}} + \epsilon \tilde{V}^{\text{iso}}) \cdot \nabla_x \tilde{V}^{\text{iso}} + \frac{1}{\rho} \nabla_x^\varphi \tilde{P}^{\text{iso}} = 0, & \\ (\partial_t \tilde{w}^{\text{iso}} + (\underline{V}^{\text{iso}} + \epsilon \tilde{V}^{\text{iso}}) \cdot \nabla_x \tilde{w}^{\text{iso}}) + \frac{1}{\rho} \partial_r^\varphi \tilde{P}^{\text{iso}} + b = 0, & \text{in } [0, T) \times S_r, \\ \partial_t \tilde{\eta} + (\underline{V}^{\text{iso}} + \epsilon \tilde{V}^{\text{iso}}) \cdot \nabla_x \eta - \tilde{w}^{\text{iso}} = 0, & \\ \nabla_x^\varphi \cdot \tilde{V}^{\text{iso}} + \frac{1}{\epsilon} \nabla_x^\varphi \cdot \underline{V}^{\text{iso}} + \partial_r^\varphi \tilde{w}^{\text{iso}} = 0, & \end{array} \right. $$

with the boundary conditions

$$ (1.9) \qquad \tilde{w}_{|r=0}^{\text{iso}} = \tilde{w}_{|r=1}^{\text{iso}} = 0 \quad \text{in } [0, T) \times \mathbb{R}^d, $$

and the initial conditions

$$ (1.10) \qquad \left\{ \begin{array}{ll} \tilde{V}_{|t=0}^{\text{iso}} = \tilde{V}_{\text{in}}^{\text{iso}}, & \\ \tilde{w}_{|t=0}^{\text{iso}} = \tilde{w}_{\text{in}}^{\text{iso}}, & \text{in } S_r. \\ \tilde{\eta}_{|t=0} = \tilde{\eta}_{\text{in}}, & \end{array} \right. $$

Remark that a Taylor expansion of $\underline{\rho}$ around $\underline{\eta}(r)$ yields the approximate expression

$$ b(t, x, r) = \frac{\underline{\rho}'(r)}{\underline{\rho}(r)} \tilde{\eta}(t, x, r) + O(\epsilon). $$

Recall indeed that $\rho_{\text{eq}}(\eta_{\text{eq}}(r)) = \underline{\rho}(r)$ by definition. This indicates that $b$ is of order 1, and not of order $1/\epsilon$. It also makes appear the quantity

$$ (1.11) \qquad N^2(r) := \frac{\underline{\rho}'(r)}{\underline{\rho}(r)} $$

which is the Brunt-Väisälä frequency. This quantity plays an important role in the dynamic, see for instance [Val17], or [DLS20] for its role in the coupling of internal waves. This argument will be made rigorous in Lemma C.1 thanks to composition estimates.

A variation of the system (1.8) has already been studied in [BD24], though with the more natural ansatz of considering directly a perturbation of an equilibrium and in the free-surface setting. More importantly, the authors consider a diffusion term stemming from mesoscale turbulence introduced in [GM90], which regularizes the
as $P_{\text{eq}}^{\text{eul}}$ only depends on the vertical variable. This property will simplify the analysis later on. Note that a Taylor expansion on $P_{\text{eq}}^{\text{iso}}$ yields

$$ \underline{P}^{\text{iso}}(t, x, r) = P_{\text{eq}}^{\text{iso}}(r) + O(\epsilon), $$

so that the difference with this ansatz and the one used in [DB22] consisting in studying directly a perturbation of the equilibrium correspond up to terms of order $\epsilon$. Using the notation

$$ b := \frac{1}{\epsilon} \left( 1 - \frac{\rho_{\text{eq}}(\eta_{\text{eq}} + \epsilon \tilde{\eta})}{\rho_{\text{eq}}(\eta_{\text{eq}})} \right) g $$

for the buoyancy term, this yields the system

$$ (1.8) \qquad \left\{ \begin{array}{ll} \partial_t \tilde{V}^{\text{iso}} + (\underline{V}^{\text{iso}} + \epsilon \tilde{V}^{\text{iso}}) \cdot \nabla_x \tilde{V}^{\text{iso}} + \frac{1}{\rho} \nabla_x^\varphi \tilde{P}^{\text{iso}} = 0, & \\ (\partial_t \tilde{w}^{\text{iso}} + (\underline{V}^{\text{iso}} + \epsilon \tilde{V}^{\text{iso}}) \cdot \nabla_x \tilde{w}^{\text{iso}}) + \frac{1}{\rho} \partial_r^\varphi \tilde{P}^{\text{iso}} + b = 0, & \text{in } [0, T) \times S_r, \\ \partial_t \tilde{\eta} + (\underline{V}^{\text{iso}} + \epsilon \tilde{V}^{\text{iso}}) \cdot \nabla_x \eta - \tilde{w}^{\text{iso}} = 0, & \\ \nabla_x^\varphi \cdot \tilde{V}^{\text{iso}} + \frac{1}{\epsilon} \nabla_x^\varphi \cdot \underline{V}^{\text{iso}} + \partial_r^\varphi \tilde{w}^{\text{iso}} = 0, & \end{array} \right. $$

with the boundary conditions

$$ (1.9) \qquad \tilde{w}_{|r=0}^{\text{iso}} = \tilde{w}_{|r=1}^{\text{iso}} = 0 \quad \text{in } [0, T) \times \mathbb{R}^d, $$

and the initial conditions

$$ (1.10) \qquad \left\{ \begin{array}{ll} \tilde{V}_{|t=0}^{\text{iso}} = \tilde{V}_{\text{in}}^{\text{iso}}, & \\ \tilde{w}_{|t=0}^{\text{iso}} = \tilde{w}_{\text{in}}^{\text{iso}}, & \text{in } S_r. \\ \tilde{\eta}_{|t=0} = \tilde{\eta}_{\text{in}}, & \end{array} \right. $$

Remark that a Taylor expansion of $\underline{\rho}$ around $\underline{\eta}(r)$ yields the approximate expression

$$ b(t, x, r) = \frac{\rho'(r)}{\rho(r)} \tilde{\eta}(t, x, r) + O(\epsilon). $$

Recall indeed that $\rho_{\text{eq}}(\eta_{\text{eq}}(r)) = \rho(r)$ by definition. This indicates that $b$ is of order 1, and not of order $1/\epsilon$. It also makes appear the quantity $N^2(r) := \frac{\rho'(r)}{\rho(r)}$ which is the Brunt-Väisälä frequency. This quantity plays an important role in the dynamic, see for instance [DLS20] for its role in the coupling of internal waves. This argument will be made rigorous in Lemma C.1 thanks to composition estimates.

A variation of the system (1.8) has already been studied in [DB22], though with the more natural ansatz of considering directly a perturbation of an equilibrium and in the free-surface setting. More importantly, the authors consider a diffusion term stemming from mesoscale turbulence introduced in [GM90], which regularizes the system. One striking fact is that they crucially use the regularizing properties of the diffusion term to prove the well-posedness of the system (1.8), whereas the analogous system (1.7) in Eulerian coordinates has been proved to be well-posed without this term in [DLS20].

This discrepancy motivates the present investigation of the well-posedness of this system in isopycnal coordinates without the regularization used in [DB22]. Regarding this issue, Theorem 1.1 is an extension of the existence result in [DB22, Theorem 1.2].
system. One striking fact is that they crucially use the regularizing properties of the diffusion term to prove the well-posedness of the system (1.8), whereas the analogous system (1.7) in Eulerian coordinates has been proved to be well-posed without this term in [DLS20].

This discrepancy motivates the present investigation of the well-posedness of this system in isopycnal coordinates without the regularization used in [BD24]. Regarding this issue, Theorem 1.1 is an extension of the existence result in [BD24, Theorem 1.2].

### 1.3. Non-dimensionalization.
We now turn to the non-dimensionalization of the equations, first in Eulerian coordinates. We write $\tilde{x} = \frac{x}{L}$, $\tilde{t} = \frac{\sqrt{gH}}{L}t$, $\tilde{z} = \frac{z}{H}$, where L and H are characteristic horizontal and vertical scales, respectively. We write $\mu := \frac{H^2}{L^2}$ for the shallowness parameter. In Eulerian coordinates, the system (1.7) becomes

$$ (1.12) \qquad \left\{ \begin{array}{l} \partial_t \tilde{V}^{\text{eul}} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) \cdot \nabla_x \tilde{V}^{\text{eul}} + \tilde{w}^{\text{eul}} \partial_z (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) + \frac{1}{\underline{\rho} + \epsilon \tilde{\rho}} \nabla_x \tilde{P}^{\text{eul}} = 0, \\[1em] \mu (\partial_t \tilde{w}^{\text{eul}} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) \cdot \nabla_x \tilde{w}^{\text{eul}} + \epsilon \tilde{w}^{\text{eul}} \partial_z \tilde{w}^{\text{eul}}) + \frac{1}{\underline{\rho} + \epsilon \tilde{\rho}} \partial_z \tilde{P}^{\text{eul}} + \frac{\tilde{\rho}}{\underline{\rho} + \epsilon \tilde{\rho}} = 0, \\[1em] \partial_t \tilde{\rho} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) \cdot \nabla_x \tilde{\rho} + \tilde{w}^{\text{eul}} \partial_z (\underline{\rho} + \epsilon \tilde{\rho}) = 0, \\[1em] \nabla_x \cdot \tilde{V}^{\text{eul}} + \partial_z \tilde{w}^{\text{eul}} = 0. \end{array} \right. $$

We are interested in the regime $\mu \ll 1$, as for oceanic applications one can have $\mu \sim 10^{-6}$ (see [Val17]). Therefore, it is natural to look for a time of existence at least uniform with respect to $\mu$. The hydrostatic equations (that is, setting $\mu = 0$ in (1.12)) are very singular: in the homogeneous case (that is, $\underline{\rho}$ being constant and $\tilde{\rho}$ being zero in (1.12)), ill-posedness of the initial value problem of the linearized system (that is, setting $\epsilon = 0$ in (1.12)) was proved by Renardy in [Ren09]. However, Masmoudi and Wong showed in [MW12] existence and uniqueness of solutions for the two-dimensional, homogeneous case, under the Rayleigh criterion

$$ \partial_z^2 (\underline{V} + \epsilon \tilde{V}^{\text{eul}}) \geq c_* > 0. $$

In the case of a stable stratification, the Miles and Howard criterion

$$ (1.13) \qquad \forall z \in [-H, 0], g \frac{-\rho'_{\text{eq}}(z)}{\rho_{\text{eq}}(z)|V'_{\text{eq}}(z)|^2} \geq \frac{1}{4} $$

(see [Mil61, How61]) gives a sufficient condition to prevent unstable modes for the linearized equations around an equilibrium shear flow ($\rho_{\text{eq}}(z)$, $V_{\text{eq}}(z)$). However, this is not enough to conclude well-posedness of the hydrostatic equations, which is still an open problem. Nonetheless, in [BZE24], the authors found a stratified shear flow around which the hydrostatic system is ill-posed.

In [DLS20], using the stable stratification assumption, but also restricting to the case of uniform stratification (that is, $\rho_{\text{eq}}$ is an exponential), using the Boussinesq approximation (that is, the density is assumed constant in the momentum equations, except in the buoyancy term $b$), with well-prepared initial data and no shear flow, the authors prove the existence and stability of solutions on a time interval $[0, T)$ with $T$ of order $\frac{\sqrt{\mu}}{\epsilon}$. In Theorem 1.1 below, we state a similar result in the case of a non-zero shear flow, without those additional assumptions. However our analysis only gives rise to a time of existence uniform in $\epsilon$ and $\mu$. A more thorough comparison with their result is provided in Remark 1.4.

In isopycnal coordinates, equations (1.8) become, in non-dimensional form and dropping the accents and the superscripts 'iso' for readability

$$ (E) \qquad \left\{ \begin{array}{l} \partial_t V + (\underline{V} + \epsilon V) \cdot \nabla_x V + \frac{1}{\underline{\rho}} \nabla_x^\varphi P = 0, \\[1em] \mu (\partial_t w + (\underline{V} + \epsilon V) \cdot \nabla_x w) + \frac{1}{\underline{\rho}} \partial_r^\varphi P + b = 0, \qquad \text{in } [0, T) \times S_r. \\[1em] \partial_t \eta + (\underline{V} + \epsilon V) \cdot \nabla_x \eta - w = 0, \\[1em] \nabla_x^\varphi \cdot V + \frac{1}{\epsilon} \nabla_x^\varphi \cdot \underline{V} + \partial_r^\varphi w = 0, \end{array} \right. $$
1.3. **Non-dimensionalization.** We now turn to the non-dimensionalization of the equations, first in Eulerian coordinates. We write $\tilde{x} = \frac{x}{L}$, $\tilde{t} = \frac{\sqrt{gH}}{L}t$, $\tilde{z} = \frac{z}{H}$, where L and H are characteristic horizontal and vertical scales, respectively. We write $\mu := \frac{H^2}{L^2}$ for the shallowness parameter. In Eulerian coordinates, the system (1.7) becomes

$$ (1.11) \qquad \left\{ \begin{aligned} & \partial_t \tilde{V}^{\text{eul}} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}} \cdot \nabla_x) \tilde{V}^{\text{eul}} + \tilde{w}^{\text{eul}} \partial_z (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) + \frac{1}{\underline{\rho} + \epsilon \tilde{\rho}} \nabla_x \tilde{P}^{\text{eul}} = 0, \\ & \mu (\partial_t \tilde{w}^{\text{eul}} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}} \cdot \nabla_x) \tilde{w}^{\text{eul}} + \epsilon \tilde{w}^{\text{eul}} \partial_z \tilde{w}^{\text{eul}}) + \frac{1}{\underline{\rho} + \epsilon \tilde{\rho}} \partial_z \tilde{P}^{\text{eul}} + \frac{\tilde{\rho}}{\underline{\rho} + \epsilon \tilde{\rho}} = 0, \\ & \partial_t \tilde{\rho} + (\underline{V}^{\text{eul}} + \epsilon \tilde{V}^{\text{eul}}) \cdot \nabla_x \tilde{\rho} + \tilde{w}^{\text{eul}} \partial_z (\underline{\rho} + \epsilon \tilde{\rho}) = 0, \\ & \nabla_x \cdot \tilde{V}^{\text{eul}} + \partial_z \tilde{w}^{\text{eul}} = 0. \end{aligned} \right. $$

We are interested in the regime $\mu \ll 1$, as for oceanic applications one can have $\mu \sim 10^{-6}$ (see [Val17]). Therefore, it is natural to look for a time of existence at least uniform with respect to $\mu$. The hydrostatic equations (that is, setting $\mu = 0$ in (1.11)) are very singular: in the homogeneous case (that is, $\underline{\rho}$ being constant and $\tilde{\rho}$ being zero in (1.11)), ill-posedness of the initial value problem of the linearized system (that is, setting $\epsilon = 0$ in (1.11)) was proved by Renardy in [Ren09]. However, Masmoudi and Wong showed in [MW12] existence and uniqueness of solutions for the two-dimensional, homogeneous case, under the Rayleigh criterion

$$ \partial_z^2 (\underline{V} + \epsilon \tilde{V}^{\text{eul}}) \geq c_* > 0. $$

In the case of a stable stratification, the Miles and Howard criterion

$$ (1.12) \qquad \forall z \in [-H, 0], g \frac{-\rho'_{\text{eq}}(z)}{\rho_{\text{eq}}(z) |V'_{\text{eq}}(z)|^2} \geq \frac{1}{4} $$

(see [Mil61, How61]) gives a sufficient condition to prevent unstable modes for the linearized equations around an equilibrium shear flow $(\rho_{\text{eq}}(z), V_{\text{eq}}(z))$. However, this is not enough to conclude well-posedness of the hydrostatic equations, which is still an open problem. Nonetheless, in [BZE24], the authors found a stratified shear flow around which the hydrostatic system is ill-posed.

In [DLS20], using the stable stratification assumption, but also restricting to the case of uniform stratification (that is, $\rho_{\text{eq}}$ is an exponential), using the Boussinesq approximation (that is, the density is assumed constant in the momentum equations, except in the buoyancy term $b$), with well-prepared initial data and no shear flow, the authors prove the existence and stability of solutions on a time interval $[0, T)$ with $T$ of order $\frac{\sqrt{H}}{\epsilon}$. In Theorem 1.1 below, we state a similar result in the case of a non-zero shear flow, without those additional assumptions. However our analysis only gives rise to a time of existence uniform in $\epsilon$ and $\mu$. A more thorough comparison with their result is provided after the statement of Theorem 1.1.
Here, the buoyancy term b reads, in non-dimensional form:

$$ (1.14) \qquad b := \frac{1}{\epsilon} \left( 1 - \frac{\rho_{\text{eq}}(\eta_{\text{eq}} + \epsilon\eta)}{\rho_{\text{eq}}(\eta_{\text{eq}})} \right). $$

Note that the existence result in [BD24, Theorem 1.2] gives a time of existence uniform in the shallow-water parameter $\mu$, though by using a regularizing term.

**1.4. Main result.** We now state our main result, that is the well-posedness of the system (E) in Sobolev spaces under the assumption of stable stratification, with a time of existence uniform in $\epsilon$. Assuming further a medium amplitude asymptotic regime ($\epsilon \le \sqrt{\mu}$) and small shear velocity $\underline{V}'$, it is also uniform in $\mu$.

Let us state the classical assumptions used in Theorem 1.1. Let $s \in \mathbb{N}$. First, let $\underline{M} > 0$. We assume that $\underline{V}, \underline{\rho} \in W^{s+1,\infty}$, with

$$ (H.1) \qquad |\underline{V}|_{W^{s+1,\infty}} + |\underline{\rho}|_{W^{s+1,\infty}} \le \underline{M}, $$

as well as the strict stability assumption, that there exists $c_* > 0$ such that

$$ (H.2) \qquad \underline{\rho}' \ge c_* $$

Second, let $M_{\text{in}} > 0$. If $(V_{\text{in}}, w_{\text{in}}, \eta_{\text{in}}) \in H^s(S_r)^{d+2}$ correspond to initial data, then we assume

$$ (H.3) \qquad \|V_{\text{in}}\|_{H^s}^2 + \mu \|w_{\text{in}}\|_{H^s}^2 + \|\eta_{\text{in}}\|_{H^s}^2 \le M_{\text{in}}. $$

Moreover, the initial data satisfy the boundary condition and incompressibility condition

$$ (H.4) \qquad w_{\text{in}}|_{r=0} = w_{\text{in}}|_{r=1} = 0, $$

$$ \nabla_x^\varphi \cdot (\underline{V} + \epsilon V_{\text{in}}) + \epsilon \partial_r^\varphi w_{\text{in}} = 0, $$

with the notation

$$ \nabla_x^\varphi := \nabla_x + \epsilon \frac{\nabla_x \eta}{1 + \epsilon h} \partial_r, $$

$$ \partial_r^\varphi := \frac{-1}{1 + \epsilon h} \partial_r, $$

where $h := -\partial_r \eta$. Note the slight abuse of notation, since the operators $\nabla_x^\varphi$ and $\partial_r^\varphi$ in (H.4) are defined relatively to $\eta_{\text{in}}$ and $h_{\text{in}}$, rather that $\eta$ and $h$.

Define

$$ h_{\text{in}} := -\partial_r \eta_{\text{in}}. $$

We assume that there exist $h^* \ge h_* > 0$ such that

$$ (H.5) \qquad h_* \le 1 + \epsilon h_{\text{in}} \le h^*. $$

**Theorem 1.1.** Let $d \in \mathbb{N}^*$, $s_0 > \frac{d}{2}$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{5}{2}$. Let $\epsilon \in [0, 1]$, $\mu \in [0, 1]$. Let $\underline{V}, \underline{\rho} \in W^{s+1,\infty}((0, 1))$, $(V_{\text{in}}, w_{\text{in}}, \eta_{\text{in}}) \in H^s(S_r)^{d+2}$ such that assumptions (H.1), (H.2), (H.3), (H.4), (H.5) are satisfied. Assume further that the shear flow is small with respect to $\mu$ and that we are in the weakly non-linear regime

$$ |\underline{V}'|_{L^\infty} \le \sqrt{\mu}, $$

$$ \epsilon \le \sqrt{\mu}. $$

Then there exists $T > 0$ independent of $\epsilon, \mu$ such that there exists a unique solution $(V, w, \eta) \in C^0([0, T], H^s(S_r)^{d+2})$ to (E) with the initial data $(V_{\text{in}}, w_{\text{in}}, \eta_{\text{in}})$.

**Remark 1.2.** More precisely, the time of existence $T$ in Theorem 1.1 admits a lower bound that depends only on $M_{\text{in}}, \underline{M}, c_*, h_*, h^*, d, s_0, s$.

**Remark 1.3.** This result is to be compared with [BD24, Theorem 1.2], where the authors prove in particular the well-posedness of a system similar to (E), but with a regularizing term and a slightly different ansatz. Here the use of Alinhac's good unknown, presented in Appendix B, allows us to study the system (E) without this regularizing term. The ansatz that we use here allows us to keep the symmetry of the system (1.7) (see the term (iii) in the proof of Proposition 3.4).

This result is in accordance with [DLS20, Theorem 2], where the well-posedness is established in Eulerian coordinates without any regularizing term, in the case of no background current $\underline{V}$.
In isopycnal coordinates, equations (1.8) become, in non-dimensional form and dropping the accents and the superscripts 'iso' for readability

$$ (E) \qquad \left\{ \begin{array}{l} \partial_t V + (\underline{V} + \epsilon V) \cdot \nabla_x V + \frac{1}{\rho} \nabla_x^\varphi P = 0, \\[1em] \mu (\partial_t w + (\underline{V} + \epsilon V) \cdot \nabla_x w) + \frac{1}{\rho} \partial_r^\varphi P + b = 0, \quad \text{in } [0, T) \times S_r. \\[1em] \partial_t \eta + (\underline{V} + \epsilon V) \cdot \nabla_x \eta - w = 0, \\[1em] \nabla_x^\varphi \cdot V + \frac{1}{\epsilon} \nabla_x^\varphi \cdot \underline{V} + \partial_r^\varphi w = 0, \end{array} \right. $$

Here, the buoyancy term b reads, in non-dimensional form:

$$ b := \frac{1}{\epsilon} \left( 1 - \frac{\rho_{\text{eq}}(\eta_{\text{eq}} + \epsilon\eta)}{\rho_{\text{eq}}(\eta_{\text{eq}})} \right). $$

Note that the existence result in [DB22, Theorem 1.2] gives a time of existence uniform in the shallow-water parameter $\mu$, though by using a regularizing term.

1.4. Main result. We now state our main result, that is the well-posedness of the system (E) in Sobolev spaces under the assumption of stable stratification, with a time of existence uniform in $\epsilon$. Assuming further a medium amplitude asymptotic regime ($\epsilon \le \sqrt{\mu}$) and small shear velocity $\underline{V}'$, it is also uniform in $\mu$.

Let us state the classical assumptions used in Theorem 1.1. Let $s \in \mathbb{N}$. First, let $\underline{M} > 0$. We assume that $\underline{V}, \rho \in W^{s+1,\infty}$, with

$$ (H.1) \qquad |\underline{V}|_{W^{s+1,\infty}} + |\rho|_{W^{s+1,\infty}} \le \underline{M}, $$

as well as the strict stability assumption, that is there exists $c_* > 0$ such that

$$ (H.2) \qquad \rho' \ge c_* $$

Second, let $M_{\text{in}} > 0$. If $(V_{\text{in}}, w_{\text{in}}, \eta_{\text{in}}) \in H^s(S_r)^{d+2}$ correspond to initial data, then we assume

$$ (H.3) \qquad \|V_{\text{in}}\|_{H^s}^2 + \mu \|w_{\text{in}}\|_{H^s}^2 + \|\eta_{\text{in}}\|_{H^s}^2 \le M_{\text{in}}. $$

Moreover, the initial data satisfy the boundary condition and incompressibility condition

$$ (H.4) \qquad w_{\text{in}}|_{r=0} = w_{\text{in}}|_{r=1} = 0, $$

$$ \nabla_x^\varphi \cdot (\underline{V} + \epsilon V_{\text{in}}) + \epsilon \partial_r^\varphi w_{\text{in}} = 0, $$

where we recall the notation

$$ \nabla_x^\varphi := \nabla_x + \epsilon \frac{\nabla_x \eta}{1 + \epsilon h} \partial_r, $$

$$ \partial_r^\varphi := \frac{-1}{1 + \epsilon h} \partial_r, $$

where $h := -\partial_r \eta$. Note the slight abuse of notation, since the operators $\nabla_x^\varphi$ and $\partial_r^\varphi$ in (H.4) are defined relatively to $\eta_{\text{in}}$ and $h_{\text{in}}$, rather that $\eta$ and $h$.

Define

$$ h_{\text{in}} := -\partial_r \eta_{\text{in}}. $$

We assume that there exist $h^* \ge h_* > 0$ such that

$$ (H.5) \qquad h_* \le 1 + \epsilon h_{\text{in}} \le h^*. $$
Remark 1.4. In order to compare our result with [DLS20, Theorem 2], let us assume the Boussinesq approximation (that is, $\rho$ is constant in (E) except in the buoyancy term b) and assume that the initial data is well-prepared in the sense of [DLS20, Theorem 2]. Then in this case our analysis (or a slight variation from it, see Remark 3.9) yields a more precise time of existence of the solution than the one stated in Theorem 1.1, namely

$$ (1.15) \qquad T = O\left( \left( \left| (N^2)' \right|_{W^{s-1,\infty}([0,1])} + \frac{\epsilon}{\sqrt{\mu}} + \frac{|V'|_{L^\infty}}{\sqrt{\mu}} \right)^{-1} \right), $$

where $N^2$ is the Brunt-Väisälä frequency, defined through (1.11) and the prime denotes the derivative. In [DLS20, Theorem 2], the authors also assume that $N^2$ is constant and $V=0$. In this case, our time of existence boils down to $T = O(\frac{\sqrt{\mu}}{\epsilon})$, which is precisely the time of existence of [DLS20]. Such a result is qualified as large time well-posedness, as the time of existence grows as $\epsilon$ goes to 0.

In the case of $N^2$ non-constant and $V \neq 0$, the time of existence in (1.15) does not grow as $\epsilon \to 0$, and such a result is qualified as short-time well-posedness. Let us mention that such a short-time well-posedness is not uncommon among the study of perturbations of non-constant equilibria of non-linear systems, or more generally, of non-linear systems with non-constant coefficients. For example, the non-linear shallow water equations with non-flat bottom suffer from the same difficulty, although it has been resolved in [BM10]. Another example is the water-waves system, for which the large time well-posedness is known for the flat bottom case, but is still an open problem for the non-flat bottom case (see [Lan13],[MG17],[Duc22, Sections 2.5, 8.7]).

In the spirit of weakening the assumptions made in [DLS20], our analysis does not need the Boussinesq approximation nor the well-preparation of the initial data stated above. Doing so, other contributions to the time of existence in (1.15) appear, and we do not keep track of them here, but rather state that $T$ is independent of $\epsilon$ and $\mu$, under the assumptions of Theorem 1.1. A more detailed comparison is provided in Remark 3.8.

Remark 1.5. Let us discuss the smallness assumption on the shear velocity now. First notice that this implies that the Miles and Howard criterion (1.13) is verified for $\mu$ small enough. Then, notice that the total vorticity is defined as

$$ \omega_{\text{total}} := \left( \partial_r V_{\text{total}}^{\perp} - \nabla_x^{\perp} w_{\text{total}} - \nabla_x^{\perp} \cdot V_{\text{total}} \right). $$

Here, $\nabla_x^{\perp} := \left( -\partial_{x_2} \atop \partial_{x_1} \right)$.

Thus, the assumption of small shear velocity $|V'|_{L^\infty} \le \sqrt{\mu}$ yields that the vorticity at equilibrium is of order $\sqrt{\mu}$. Together with the assumption $\epsilon \le \sqrt{\mu}$, this implies that the total vorticity is of order $\sqrt{\mu}$. This is in accordance with the result in [CL15], where the authors study the well-posedness of the water waves equations with vorticity under this assumption.

Remark 1.6. Let us finally comment on the regularity $s \ge s_0 + \frac{5}{2}$ needed here. Note that this is equivalent to $s > \frac{d+1}{2} + 2$, where $d+1$ is the dimension of the strip $S_r$. The classical critical regularity for similar results on the euler equation is $\frac{d+1}{2} + 1$, where $d+1$ denotes the dimension of the domain occupied by the fluid. This discrepancy is due to the non-linearity introduced by the operators $\nabla^\varphi$, namely recall the expression of the term $\nabla_x^\varphi P$ in (E)

$$ \nabla_x^\varphi P := \nabla_x P + \epsilon \frac{\nabla_x \eta}{1 + \epsilon h} \partial_r P, $$

where $h := -\partial_r \eta$. On a technical viewpoint, this is the sharp regularity needed on $\eta$ so that $(\nabla_x \eta, h) \in W^{1,\infty}$ by Sobolev embeddings.

1.5. **Perspectives.** Let us mention a few perspectives.

As shown in this study, the energy method can be applied to the reformulation (E) of the Euler equations in isopycnal coordinates, without the regularization used in [BD24]. This offers the possibility of using the reformulation in isopycnal coordinates for the study of stratified flows. For instance, one advantage of the isopycnal coordinates, besides the quasi-2D structure, is the rectification of the isopycnals. In the free surface case, assuming that the surface is initially an isopycnal, this change of coordinates also rectifies the surface (see for instance [BD24]): in this setting the free-surface case is quite close to the rigid lid case, and we expect that our study can
**Theorem 1.1.** Let $d \in \mathbb{N}^*$, $s_0 > \frac{d}{2}$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{5}{2}$. Let $\epsilon \in [0, 1]$, $\mu \in [0, 1]$. Let $\underline{V}$, $\underline{\rho} \in W^{s+1,\infty}((0,1))$, $(V_{\text{in}}, w_{\text{in}}, \eta_{\text{in}}) \in H^s(S_r)^{d+2}$ such that assumptions (H.1), (H.2), (H.3), (H.4), (H.5) are satisfied. Assume further that the shear flow is small with respect to $\mu$ and that we are in the weakly non-linear regime

$$|\underline{V}'|_{L^\infty} \le \sqrt{\mu}, \\ \epsilon \le \sqrt{\mu}.$$

Then there exists $T > 0$ depending only on $M$, $M_{\text{in}}$ such that the following holds. There exists a unique solution $(V, w, \eta) \in L^\infty([0, T], H^s(S_r)^{d+2})$ to (E) with initial data $(V_{\text{in}}, w_{\text{in}}, \eta_{\text{in}})$.

**Remark 1.2.** More precisely, with the notations of Theorem 1.1, our analysis yields a time of existence of the solution

$$T = O\left(\left(1 + \frac{\epsilon}{\sqrt{\mu}} + \frac{|\underline{V}'|_{L^\infty}}{\sqrt{\mu}}\right)^{-1}\right).$$

**Remark 1.3.** This result is to be compared with [DB22, Theorem 1.2], where the authors prove in particular the well-posedness of a system similar to (E), but with a regularizing term and a slightly different ansatz. Here the use of Alinhac's good unknown, presented in Appendix B, allows us to study the system (E) without this regularizing term. The ansatz that we use here allows us to keep the symmetry of the system (1.7) (see the term (iii) in the proof of Proposition 3.4).

This result is in accordance with [DLS20, Theorem 2], where the well-posedness is established in Eulerian coordinates without any regularizing term, in the case of no background current $\underline{V}$. However, note that the assumptions in [DLS20, Theorem 2] are more restrictive than the ones in Theorem 1.1. Namely, in the present study we do not assume that the Brunt-Väisälä frequency is constant, the strong Boussinesq assumption (i.e. that the density is assumed constant in the momentum equations, except in the buoyancy term $b$), nor the good preparation of the initial data, and we allow a background current. The downside is that the time of existence in [DLS20, Theorem 2] is of order $\sqrt{\mu}/\epsilon$ whereas in Theorem 1.1, it is only uniform with respect to $\epsilon/\sqrt{\mu}$ (in the case of no background current). A more detailed comparison is provided in Remark 3.7.

**Remark 1.4.** Let us discuss the smallness assumption on the shear velocity now. First notice that this implies that the Miles and Howard criterion (1.12) is verified for $\mu$ small enough. Then, notice that the total vorticity is defined as

$$\omega_{\text{total}} := \left( \frac{\partial_r V_{\text{total}}^{\perp} - \nabla_x^{\perp} w_{\text{total}}}{-\nabla_x^{\perp} \cdot V_{\text{total}}} \right).$$

Here, $\nabla_x^{\perp} := \begin{pmatrix} -\partial_{x_2} \\ \partial_{x_1} \end{pmatrix}$.

Thus, the assumption of small shear velocity $|\underline{V}'|_{L^\infty} \le \sqrt{\mu}$ yields that the vorticity at equilibrium is of order $\sqrt{\mu}$. Together with the assumption $\epsilon \le \sqrt{\mu}$, this implies that the total vorticity is of order $\sqrt{\mu}$. This is in accordance with the result in [CL14], where the authors study the well-posedness of the water waves equations with vorticity under this assumption.

**Remark 1.5.** Let us finally comment on the regularity $s \ge s_0 + \frac{5}{2}$ needed here. Note that this is equivalent to $s > \frac{d+1}{2} + 2$, where $d+1$ is the dimension of the strip $S_r$. The
be extended to the free-surface case with minor adaptations, see Remarks 4.5 and 4.7.

Recall now that under the assumptions of no shear flow, uniform stratification and strong Boussinesq assumption, Theorem 1.1 yields a time of existence of order 1 with respect to $\epsilon$. However, with the additional assumption of good preparation on the initial data, [DLS20, Theorem 2] yields an existence time of order $\frac{1}{\epsilon}$ with respect to $\epsilon$. Therefore, it remains to study the phenomenon arising in the long-time dynamics when no assumption of good preparation is made. In [KL23], the authors perform a similar study, though in a different regime, and show the existence of a boundary layer at the bottom.

Eventually, it would be of utmost interest to study a specific stratification that is ubiquitous in oceanography, namely two layers of (almost) constant density, separated by a pycnocline, where the density varies sharply but in a continuous manner. In [DLS20], the authors suggested to rely on the modal decomposition they introduced to obtain the two-layer shallow water equations as a limit when the thickness of the pycnocline and the shallow water parameter $\mu$ both tend to zero. The isopycnal coordinates seem a useful setting, as the isopycnals are straightened, therefore so is the pycnocline - which can be viewed as a blurred interface between the two layers. Relying on the regularization used in [BD24], the authors of [ABD24] proved the convergence of this continuous stratification with thin pycnocline setting towards the two-layer setting, in the hydrostatic case.

**1.6. Structure of the proof and plan of the paper.** In Section 2, we show how the pressure is defined from the other unknowns through an elliptic equation. We state in Subsection 2.1 two estimates on the pressure term. In Subsection 2.2, we prove these estimates.

In Section 3, we derive the energy estimate on the unknowns $V, w, \eta$, that we want uniform in $\epsilon$ and $\mu$. The approach in Subsection 3.2 yields a loss of derivatives, but is used in Subsection 3.3 together with Alinhac's good unknown to derive energy estimates without any loss of derivatives.

In Section 4, we finish the proof of Theorem 1.1. In particular, we show in Proposition 4.1 how the quasi-2D structure of (E) and the elliptic estimates for the pressure in Proposition 2.1 allow us to use the classical energy method to prove the existence of solutions. We also briefly show how Theorem 1.1 yields existence and uniqueness of solutions in Eulerian coordinates as well.

Appendix A is concerned with product, commutator and composition estimates, and Appendix B gathers definitions and results about Alinhac's good unknown. Appendix C states an estimate on the non-linear term and the buoyancy term in (E). In this appendix we also show how to use the incompressibility condition to get a control on $\partial_r w$ that is uniform in $\mu$.

**1.7. Notations.**

* Bold letters denote vectors of dimension $d+1$, such as $\mathbf{g}$.

* If $A$ is a rectangular matrix, $A^T$ denotes its transpose.

* If $(a, b) \in \mathbb{R}^2$, then $a \vee b$ denotes the maximum between $a$ and $b$, and $a \wedge b$ denotes the minimum.

* Let $f$ a function, $s_0, s \in \mathbb{R}$. We define

$$ \langle f \rangle_{s \ge s_0}(s) = \begin{cases} 0 & \text{if } s < s_0 \\ f(s) & \text{if } s \ge s_0. \end{cases} $$

* Throughout the paper, the letters with superscript .eul (as $V^{\text{eul}}, w^{\text{eul}}$) denote quantities in Eulerian coordinates, and letters with superscript .iso (as $V^{\text{iso}}, w^{\text{iso}}$) denote quantities in Eulerian coordinates. Where there is no possible confusion, we drop the superscript .iso for readability.

* The letter $\Lambda$ generically represents a differential operator. We denote by $\Lambda$ the Fourier multiplier on $\mathbb{R}^d$ of symbol $(1+|\xi|^2)^{\frac{1}{2}}$ and $|D|$ the Fourier multiplier on $\mathbb{R}^d$ of symbol $|\xi|$.

* We denote by

$$ S_z := \mathbb{R}^d \times (-H, 0) $$

the strip occupied by the fluid written in Eulerian coordinates, and by

$$ S_r := \mathbb{R}^d \times (0, 1) $$

the strip occupied by the fluid written in isopycnal coordinates.
classical critical regularity for similar results on the euler equation is $\frac{N}{2} + 1$, where N denotes the dimension of the domain occupied by the fluid. This discrepancy is due to the non-linearity introduced by the operators $\nabla^\varphi$, namely recall the expression of the term $\nabla_x^\varphi P$ in (E)

$$ \nabla_x^\varphi P := \nabla_x P + \epsilon \frac{\nabla_x \eta}{1 + \epsilon h} \partial_r P, $$

where $h := -\partial_r \eta$. On a technical viewpoint, this is the sharp regularity needed on $\eta$ so that $(\nabla_x \eta, h) \in W^{1,\infty}$ by Sobolev embeddings.

### 1.5. Perspectives.
Let us mention a few perspectives.

As shown in this study, the energy method can be applied to the reformulation (E) of the Euler equations in isopycnal coordinates, without the regularization used in [DB22]. This offers the possibility of using the reformulation in isopycnal coordinates for the study of stratified flows. For instance, one advantage of the isopycnal coordinates, besides the quasi-2D structure, is the rectification of the isopycnals. In the free surface case, assuming that the surface is initially an isopycnal, this change of coordinates also rectifies the surface (see for instance [DB22]): in this setting the free-surface case is quite close to the rigid lid case, and we expect that our study can be extended to the free-surface case with minor adaptations.

Recall now that under the assumptions of no shear flow, uniform stratification and strong Boussinesq assumption, Theorem 1.1 yields a time of existence of order 1 with respect to $\epsilon$. However, with the additional assumption of good preparation on the initial data, [DLS20, Theorem 2] yields an existence time of order $\frac{1}{\epsilon}$ with respect to $\epsilon$. Therefore, it remains to study the phenomenon arising in the long-time dynamics when no assumption of good preparation is made. In [KL23], the authors perform a similar study, and show the existence of a boundary layer at the bottom.

Eventually, it would be of utmost interest to study a specific stratification that is ubiquitous in oceanography, namely two layers of (almost) constant density, separated by a pycnocline, where the density varies sharply but in a continuous manner. In [DLS20], the authors suggested to rely on the modal decomposition they introduced to obtain the two-layer shallow water equations as a limit when the thickness of the pycnocline and the shallow water parameter $\mu$ both tend to zero. The isopycnal coordinates seem a useful setting, as the isopycnals are straightened, therefore so is the pycnocline - which can be viewed as a blurred interface between the two layers. Relying on the regularization used in [DB22], the authors of [ABD24] proved the convergence of this continuous stratification with thin pycnocline setting towards the two-layer setting, in the hydrostatic case.

### 1.6. Structure of the proof and plan of the paper.
In Section 2, we show how the pressure is defined from the other unknowns through an elliptic equation. We derive in Subsection 2.1 a first set of elliptic estimates on the gradient of the pressure, showing how the standard strategy yields a loss of derivatives in the estimates. These estimates are then used in Subsection 2.2 together with Alinhac's good unknown to derive estimates without any loss of derivatives.
• We denote by $\rho(r)$ the density of the layer $r$ for $r \in [0, 1]$. The stable stratification assumption yields that $\rho$ is strictly increasing, as $r = 0$ denotes the surface, and we assume that $0 < \rho(0) < \rho(1)$. The precise dependence on these two constants $\rho_0 := \rho(0)$ and $\rho_1 := \rho(1)$ is not specified in the rest of the study.

• We use an isopycnal change of coordinates

$$ \varphi(t, x, r) = (t, x, -r + \epsilon\eta(t, x, r)), $$

where the function $(t, x, r) \mapsto \eta(t, x, r)$ is a $C^2$ function, such that for $(t, x) \in [0, T] \times \mathbb{R}^d$, the application $r \mapsto -r + \epsilon\eta(t, x, r)$ is bijective, and

$$ (\underline{\rho} + \epsilon\underline{\rho})(t, x, -r + \epsilon\eta(t, x, r)) = \underline{\rho}(r). $$

We write

$$ h := -\partial_r \eta. $$

The Jacobian of $\varphi$ is thus given by

$$ (1.16) \qquad \partial \varphi = \begin{pmatrix} 1 & 0 & \epsilon \partial_t \eta \\ 0 & I_d & \epsilon \nabla_x \eta \\ 0 & 0 & -1 - \epsilon h \end{pmatrix} $$

and its determinant is $-(1 + \epsilon h)$.

The gradient in Eulerian coordinates expressed in isopycnal coordinates is defined as

$$ \nabla^{\varphi} f := (\nabla(f \circ \varphi^{-1})) \circ \varphi. $$

The chain rule yields

$$ (1.17) \qquad \begin{aligned} \partial_t^{\varphi} f &= \partial_t f + \epsilon \frac{\partial_t \eta}{1 + \epsilon h} \partial_r f, & \partial_r^{\varphi} f &= -\frac{1}{1 + \epsilon h} \partial_r f, \\ \nabla_x^{\varphi} f &= \nabla_x f + \epsilon \frac{\nabla_x \eta}{1 + \epsilon h} \partial_r f. \end{aligned} $$

We also write $\nabla^{\varphi} := (\nabla_x^{\varphi}, \partial_r^{\varphi})^T$.

• For $\mu \in (0, 1]$, we define

$$ (1.18) \qquad \nabla_{\mu}^{\varphi} := \left( \frac{\sqrt{\mu} \nabla_{x}^{\varphi}}{\partial_{r}^{\varphi}} \right). $$

• For an unknown $f$ in isopycnal coordinates, we usually denote by $f^{\text{eul}}$ the corresponding unknown in Eulerian coordinates, namely $f^{\text{eul}} \circ \varphi = f$.

• For $p \in [1, \infty]$, we use the standard $L^p$-based Sobolev spaces

$$ (1.19) \qquad W^{s,p}(\mathbb{R}^d) := \{\phi \in L^p(\mathbb{R}^d), |\phi|_{W^{s,p}} < +\infty\}, $$

with the norm

$$ |\phi|_{W^{s,p}} := |\Lambda^s \phi|_{L^p}. $$

We denote with simple bars $| \cdot |$ norms in dimensions 1 or $d$. We write

$$ H^s := W^{s,2}. $$

We use the functional spaces, for $s \in \mathbb{R}_+$, $0 \le k \le s$, $p \in [1, \infty]$:

$$ W^{s,k,p}(S_r) := \{\phi \in L^p(S_r), \|\phi\|_{W^{s,k,p}} < \infty\}, $$

where we define the norm

$$ \|\phi\|_{W^{s,k,p}} := \sum_{l=0}^{k} \|\Lambda^{s-l} \partial_r^l \phi\|_{L^p}, $$

and write $H^{s,k} := W^{s,k,2}$. In the case $s = k$, we write $H^s(S_r) := H^{s,k}(S_r)$. We only use $L^2$ and $L^\infty$ based Sobolev spaces.
In Section 3, we derive the energy estimate on the unknowns $V, w, \eta$, that we want uniform in $\epsilon$ and $\mu$. Once again, the approach in Subsection 3.2 yields a loss of derivatives, but is used in Subsection 3.3 together with Alinhac's good unknown to derive energy estimates without any loss of derivatives.

In Section 4, we finish the proof of Theorem 1.1. In particular, we show how the quasi-2D structure of (E) and the elliptic estimates for the pressure in Proposition 2.4 allow us to use the classical energy method to prove the existence of solutions. We also briefly show how Theorem 1.1 yields existence and uniqueness of solutions in Eulerian coordinates as well.

Appendix A is concerned with product, commutator and composition estimates, and Appendix B gathers definitions and results about Alinhac's good unknown. Appendix C states an estimate on the non-linear term and the buoyancy term in (E). In this appendix we also show how to use the incompressibility condition to get a control on $\partial_r w$ that is uniform in $\mu$.

## 1.7. Notations.

* Uppercase letters usually denote vectors of dimension $d$, such as $V$. Bold letters usually denote vectors of dimension $d+1$, such as $g$.

* If $A$ is a rectangular matrix, $A^T$ denotes its transpose.

* If $(a, b) \in \mathbb{R}^2$, then $a \vee b$ denotes the maximum between $a$ and $b$, and $a \wedge b$ denotes the minimum.

* Let $f$ a function, $s_0, s \in \mathbb{R}$. We define

$$ \langle f \rangle_{s \ge s_0}(s) = \begin{cases} 0 & \text{if } s < s_0 \\ f(s) & \text{if } s \ge s_0 \end{cases}. $$

* Throughout the paper, the letters with superscript $^{.eul}$ (as $V^{\text{eul}}, w^{\text{eul}}$) denote quantities in Eulerian coordinates, and letters with superscript $^{.iso}$ (as $V^{\text{iso}}, w^{\text{iso}}$) denote quantities in Eulerian coordinates. Where there is no possible confusion, we drop the superscript $^{.iso}$ for readability.

* The letter $\Lambda$ generically represents a differential operator. We denote by $\Lambda$ the Fourier multiplier on $\mathbb{R}^d$ of symbol $(1+|\xi|^2)^{\frac{1}{2}}$ and $|D|$ the Fourier multiplier on $\mathbb{R}^d$ of symbol $|\xi|$.

* We denote by

$$ S_z := \mathbb{R}^d \times (-H, 0) $$

the strip occupied by the fluid written in Eulerian coordinates, and by

$$ S_r := \mathbb{R}^d \times (0, 1) $$

the strip occupied by the fluid written in isopycnal coordinates.

* We denote by $\rho(r)$ the density of the layer $r$ for $r \in [0, 1]$. The stable stratification assumption yields that $\rho$ is strictly increasing, as $r = 0$ denotes the surface, and we assume that $0 < \rho(0) < \rho(1)$. The precise dependence on these two constants $\rho_0 := \rho(0)$ and $\rho_1 := \rho(1)$ is not specified in the rest of the study.

* We use an isopycnal change of coordinates

$$ \varphi(t, x, r) = (t, x, -r + \epsilon\eta(t, x, r)), $$
* We introduce the space of locally square-integrable functions $L_{\text{loc}}^2(S_r)$ as the functions that are square integrable on any compact subset of $S_r$. We then define the homogeneous Sobolev spaces, for $s \in \mathbb{N}^*$

$$ \dot{H}^s(S_r) := \{\phi \in L_{\text{loc}}^2(S_r), \nabla \phi \in H^{s-1}(S_r)\}, $$

with the convention $H^0 := L^2$.

* Let $I \subset \mathbb{R}$ an interval, $X$ a Banach space. The space $C^k(I, X)$ denotes the space of $k$ times differentiable functions from $I$ to $X$, with continuous $k^{\text{th}}$-differential.

* Let $\Lambda$ be a differential operator, and $f, g$ two functions. We define the commutator

$$ [\Lambda, f]g := \Lambda(fg) - f\Lambda(g), $$

as well as the symmetric commutator

$$ [\Lambda; f, g] := \Lambda(fg) - \Lambda(f)g - f\Lambda(g). $$

## 2. PRESSURE ESTIMATES

In this section we show how the pressure is defined from the other unknowns, using the incompressibility condition. We also obtain an estimate on the pressure term thanks to the elliptic equation satisfied by the pressure. In Subsection 2.1, we state the two main results of this section: Proposition 2.1 states that the pressure is well defined and as regular as the unknowns $(\eta, V, w)$. However, the estimate (2.4) involves $\nabla_{\mu}^{\varphi} P$ (defined through (1.18)). Proposition 2.2 states a similar result involving $\nabla^{\varphi} P$, however with a loss of derivatives, i.e. $\nabla^{\varphi} P$ is less regular than $(\eta, V, w)$.

In subsection 2.2, we first state two elliptic estimates, namely in Propositions 2.3 and 2.6. Proposition 2.6 is classical, and we only prove Proposition 2.6. We use these two propositions to prove Propositions 2.1 and 2.2.

### 2.1. Statement of the results.
To get the elliptic problem (2.1) satisfied by the pressure, we mimic the method in Eulerian coordinates: we multiply the equation on the horizontal momentum by $\mu$, and want to apply $\nabla_{\mu}^{\varphi}$. On this equation, $\partial_{r}^{\varphi}$ on the equation on the vertical momentum, then sum the two expressions. However, note that $\partial_{t}$ and $\nabla^{\varphi}$ do not commute. To bypass this difficulty, note that the system can still be written without the property of quasi-2D advection (1.6), and reads

$$ (E') \qquad \left\{ \begin{array}{l} \partial_t^\varphi (\underline{V} + \epsilon V) + (\underline{U} + \epsilon \mathbf{U}) \cdot \nabla^\varphi (\underline{V} + \epsilon V) + \frac{\epsilon}{\rho} \nabla_x^\varphi P = 0, \\ \mu (\partial_t^\varphi w + (\underline{U} + \epsilon \mathbf{U}) \cdot \nabla^\varphi w) + \frac{1}{\rho} \partial_r^\varphi P + b = 0, \\ \partial_t \eta + (\underline{V} + \epsilon V) \cdot \nabla_x \eta - w = 0, \\ \nabla_x^\varphi (\underline{V} + \epsilon V) + \epsilon \partial_r^\varphi w = 0. \end{array} \right. $$

Here, $\underline{U} := (\underline{V}, 0)^T$ and recall the expression (1.14) of b. The operators $\partial_t^\varphi$ and $\nabla^\varphi$ commute, as can be seen by change of coordinates. Therefore, we apply $\frac{\mu}{\epsilon} \nabla_x^\varphi$ to the first equation, $\partial_r^\varphi$ to the second one. Summing these two expressions yields

$$ (2.1) \qquad \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho(r)} \nabla_{\mu}^{\varphi} P = -\frac{\mu}{\epsilon} \nabla^{\varphi} \cdot ((\underline{U} + \epsilon \mathbf{U}) \cdot \nabla^{\varphi} (\underline{U} + \epsilon \mathbf{U})) - \partial_{r}^{\varphi} b. $$

This is completed with Neumann conditions at the bottom and rigid lid, by taking the trace of the equation on the vertical momentum in (E')

$$ (2.2) \qquad \frac{1}{\rho(r)} \partial_r^\varphi P \Big|_{r=0;r=1} = -b \Big|_{r=0;r=1}. $$

We also wrote

$$ \nabla_{\mu}^{\varphi} := \left( \frac{\sqrt{\mu} \nabla_{x}^{\varphi}}{\partial_{r}^{\varphi}} \right), \qquad \nabla_{\mu} := \left( \frac{\sqrt{\mu} \nabla_{x}}{\partial_{r}} \right). $$
where the function $(t, x, r) \mapsto \eta(t, x, r)$ is a $C^2$ function, such that for $(t, x) \in [0, T] \times \mathbb{R}^d$, the application $r \mapsto -r + \epsilon\eta(t, x, r)$ is bijective, and

$$ (\underline{\rho} + \epsilon\underline{\rho})(t, x, -r + \epsilon\eta(t, x, r)) = \underline{\rho}(r). $$

We write

$$ h := -\partial_r \eta. $$

The Jacobian of $\varphi$ is thus given by

$$ \partial_t \varphi = \begin{pmatrix} 1 & 0 & \epsilon \partial_t \eta \\ 0 & I_d & \epsilon \nabla_x \eta \\ 0 & 0 & -1 - \epsilon h \end{pmatrix} $$

and its determinant is $-(1 + \epsilon h)$.

The gradient in Eulerian coordinates expressed in isopycnal coordinates is defined as

$$ \nabla^{\varphi} f := (\nabla(f \circ \varphi^{-1})) \circ \varphi. $$

The chain rule yields

$$ \begin{aligned} \partial_t^\varphi f &= \partial_t f + \epsilon \frac{\partial_t \eta}{1 + \epsilon h} \partial_r f, & \partial_r^\varphi f &= -\frac{1}{1 + \epsilon h} \partial_r f, \\ \nabla_x^\varphi f &= \nabla_x f + \epsilon \frac{\nabla_x \eta}{1 + \epsilon h} \partial_r f. \end{aligned} \tag{1.13} $$

We also write $\nabla^\varphi := (\nabla_x^\varphi, \partial_r^\varphi)^T$.

• For $\mu \in (0, 1]$, we define

$$ (1.14) \qquad \nabla_{\mu}^{\varphi} := \left( \frac{\sqrt{\mu} \nabla_{x}^{\varphi}}{\partial_{r}^{\varphi}} \right). $$

• For an unknown $f$ in isopycnal coordinates, we usually denote by $f^{\text{eul}}$ the corresponding unknown in Eulerian coordinates, namely $f^{\text{eul}} \circ \varphi = f$.

• For $p \in [1, \infty]$, we use the standard $L^p$-based Sobolev spaces

$$ W^{s,p}(\mathbb{R}^d) := \{\phi \in L^p(\mathbb{R}^d), |\phi|_{W^{s,p}} < +\infty\}, $$

with the norm

$$ |\phi|_{W^{s,p}} := |\Lambda^s \phi|_{L^p}. $$

We denote with simple bars $|\cdot|$ norms in dimensions 1 or $d$.

We write

$$ H^s := W^{s,2}. $$

We use the functional spaces, for $s \in \mathbb{R}_+$, $0 \le k \le s$, $p \in [1, \infty]$:

$$ W^{s,k,p}(S_r) := \{\phi \in L^p(S_r), \|\phi\|_{W^{s,k,p}} < \infty\}, $$

where we define the norm

$$ \|\phi\|_{W^{s,k,p}} := \sum_{l=0}^{k} \|\Lambda^{s-l} \partial_{r}^{l} \phi\|_{L^{p}}, $$

and write $H^{s,k} := W^{s,k,2}$.

In the case $s = k$, we write $H^s(S_r) := H^{s,k}(S_r)$.

We only use $L^2$ and $L^\infty$ based Sobolev spaces.
Note that (2.1) is valid at any given time $t$, it is not an evolution equation. Therefore, we omit the dependence in time in this section, to simplify the notations.
Using the divergence-free condition in (E'), we can write

$$ (2.3) \qquad \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho(r)} \nabla_{\mu}^{\varphi} P = -\frac{\mu}{\epsilon} \partial_{i}^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_{j} \partial_{j}^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_{i} - \partial_{r}^{\varphi} b, $$

with the convention of summing over repeated indices. The equation (2.3) is completed with the boundary conditions (2.2). Let $s > 0, 0 \le k \le s$, and $\Lambda := \Lambda^{s-k} \partial_r^k$. Let $\underline{M}, M > 0$, as well as $s_0 > d/2, s \ge s_0 + \frac{5}{2}, 2 \le k \le s$. Throughout this section, we assume

$$ (H.6) \qquad \|\eta\|_{H^{s,k}} + \|V\|_{H^{s,k}} + \sqrt{\mu} \|w\|_{H^{s,k}} \le M. $$

Recall that $\nabla_{\mu}^{\varphi}$ is defined through (1.18). We also need to assume the particular asymptotic regime

$$ (H.7) \qquad \epsilon \le \sqrt{\mu}, $$

for the following estimates to be independent of $\mu$. We now state the main results of this section.

**Proposition 2.1.** Let $s_0 > d/2, s \in \mathbb{R}$ with $s \ge s_0 + \frac{5}{2}, 3 \le k \le s$ with $k \in \mathbb{N}$. We further assume (H.1), (H.5), (H.6), and (H.7) for some constants $\underline{M}, M$. Then the pressure $P$ defined through (2.1) is well-defined in $\dot{H}^1/\mathbb{R}$ and satisfies the following bound, with $C > 0$ a constant depending only on $M, \underline{M}$

$$ (2.4) \qquad \|\nabla_{\mu}^{\varphi} P\|_{H^{s,k}} \le C (\|\eta\|_{H^{s,k}} + \|V\|_{H^{s,k}} + \sqrt{\mu} \|w\|_{H^{s,k}}). $$

We now state a similar result, where the bound is uniform in $\mu$ but there is a loss of derivatives due to the use of the hydrostatic pressure in the proof. Recall indeed that the estimate in Proposition 2.1 involves $\nabla_{\mu}^{\varphi} P := (\sqrt{\mu}\nabla_{x}^{\varphi} P, \partial_{r}^{\varphi} P)^T$, whereas the following estimate is on $\nabla_{\varphi} P := (\nabla_{x}^{\varphi} P, \partial_{r}^{\varphi} P)^T$.

**Proposition 2.2.** With the same assumptions and notations as in Proposition 2.1, we have the bound

$$ (2.5) \qquad \|\nabla_{\varphi} P\|_{H^{s,k}} \le C (\|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s,k}} + \sqrt{\mu} \|w\|_{H^{s,k}}). $$

**2.2. Proofs of the results.** In order to prove Propositions 2.1 and 2.2, we first state Proposition 2.3, which is a classical estimate on the elliptic equation (2.6). However, this result involves a loss of derivatives in $\eta$. We show how to deal with this issue in Proposition 2.6, which does not involve any loss of derivatives. Finally, we show how these two propositions imply Propositions 2.1 and 2.2.

**2.2.1. A first elliptic system.** We solve the more general elliptic system

$$ (2.6) \qquad \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho(r)} \nabla_{\mu}^{\varphi} P = \nabla_{\mu}^{\varphi} \cdot \mathbf{G}. $$

This is completed with Neumann conditions

$$ (2.7) \qquad \frac{1}{\rho(r)} \mathbf{n}_i \cdot \nabla_{\mu}^{\varphi} P \Big|_{r=i} = \mathbf{n}_i \cdot \mathbf{G} \Big|_{r=i}, $$

for $i \in \{0, 1\}$. Here, $\mathbf{G}$ is a source term, and

$$ (2.8) \qquad \varphi(t, x, r) = (t, x, -r + \epsilon \eta(t, x, r)). $$

Note that it is not assumed that $\eta$ vanishes at the boundaries $r = 0$ and $r = 1$, and thus $\mathbf{n}_i$ denotes the unitary outward normal at the boundary $r = i$, namely

$$ (2.9) \qquad \begin{aligned} \mathbf{n}_0 &:= \frac{1}{\sqrt{1 + \epsilon^2 |\nabla_x \eta|_{r=0}^2}} \begin{pmatrix} -\epsilon \nabla_x \eta|_{r=0} \\ 1 \end{pmatrix}, \\ \mathbf{n}_1 &:= \frac{1}{\sqrt{1 + \epsilon^2 |\nabla_x \eta|_{r=1}^2}} \begin{pmatrix} \epsilon \nabla_x \eta|_{r=1} \\ -1 \end{pmatrix}. \end{aligned} $$

This would allow us to consider a non-flat bottom. Then, $\nabla_{\varphi}, \nabla_{\mu}^{\varphi}$ are defined from $\varphi$ through (1.17) and (1.18) respectively.
* We introduce the space of locally square-integrable functions $L_{\text{loc}}^2(S_r)$ as the functions that are square integrable on any compact subset of $S_r$. We then define the homogeneous Sobolev spaces, for $s \in \mathbb{N}^*$

$$ \dot{H}^s(S_r) := \{\phi \in L_{\text{loc}}^2(S_r), \nabla \phi \in H^{s-1}(S_r)\}, $$

with the convention $H^0 := L^2$.

* Let $n \in \mathbb{N}^*$, $k \in \mathbb{N}$, $\Omega$ an open set in $\mathbb{R}^n$, $X$ a Banach space. The space $C^k(\Omega, X)$ denotes the space of $k$ times differentiable functions from $\Omega$ to $X$, with continuous $k^{\text{th}}$-differential.

* Let $\Lambda$ be a differential operator, and $f, g$ two functions. We define the commutator

$$ [\Lambda, f]g := \Lambda(fg) - f\Lambda(g), $$

as well as the symmetric commutator

$$ [\Lambda; f, g] := \Lambda(fg) - \Lambda(f)g - f\Lambda(g). $$

## 2. PRESSURE ESTIMATES

In this section we show how the pressure is defined from the other unknowns, using the incompressibility condition. We also obtain an estimate on the pressure term thanks to the elliptic equation satisfied by the pressure. In Subsection 2.1, we prove a first set of elliptic estimates. However, they involve a loss of derivative in $\eta$ (see Proposition 2.1). We still use this result in Subsection 2.2, together with Alinhac's good unknown, to get the second set of elliptic estimates, without loss of derivatives (see Proposition 2.4).

To get the elliptic problem (2.1) satisfied by the pressure, we mimic the method in Eulerian coordinates: we multiply the equation on the horizontal momentum by $\mu$, and want to apply $\nabla_x^\varphi$ on this equation, $\partial_t^\varphi$ on the equation on the vertical momentum, then sum the two expressions. However, note that $\partial_t$ and $\nabla^\varphi$ do not commute. To bypass this difficulty, note that the system can still be written without the property of quasi-2D advection (1.6), and reads

$$ (E') \qquad \left\{ \begin{array}{l} \partial_t^\varphi (\underline{V} + \epsilon V) + (\underline{U} + \epsilon \underline{U}) \cdot \nabla^\varphi (\underline{V} + \epsilon V) + \frac{\epsilon}{\rho} \nabla_x^\varphi P = 0, \\ \mu (\partial_t^\varphi w + (\underline{U} + \epsilon \underline{U}) \cdot \nabla^\varphi w) + \frac{1}{\rho} \partial_r^\varphi P + b = 0, \\ \partial_t \eta + (\underline{V} + \epsilon V) \cdot \nabla_x \eta - w = 0, \\ \nabla_x^\varphi \cdot (\underline{V} + \epsilon V) + \epsilon \partial_r^\varphi w = 0. \end{array} \right. $$

Here, $\underline{U} := (\underline{V}, 0)^T$ and recall the expression of $b$

$$ b := \frac{1}{\epsilon} \left( 1 - \frac{\rho_{\text{eq}}(\eta_{\text{eq}} + \epsilon \eta)}{\rho_{\text{eq}}(\eta_{\text{eq}})} \right). $$

The operators $\partial_t^\varphi$ and $\nabla^\varphi$ commute, as can be seen by change of coordinates. Therefore, we apply $\frac{\mu}{\epsilon} \nabla_x^\varphi$ to the first equation, $\partial_t^\varphi$ to the second one. Summing these two expressions yields

$$ (2.1) \qquad \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho(r)} \nabla_{\mu}^{\varphi} P = -\frac{\mu}{\epsilon} \nabla^{\varphi} \cdot ((\underline{U} + \epsilon \underline{U}) \cdot \nabla^{\varphi} (\underline{U} + \epsilon \underline{U})) - \partial_{r}^{\varphi} b. $$
**Proposition 2.3.** Let $s_0 > d/2, 2 \le k \le s_0 + \frac{3}{2}$ with $k \in \mathbb{N}$. We assume, (H.1), (H.5), (H.6) with $s \ge s_0 + \frac{5}{2}$. Then there exists a unique solution $P \in \dot{H}^1(S_r)/\mathbb{R}$ solution to (2.6) and there exists a constant $C > 0$ depending only on $M, \underline{M}$ such that

$$ \|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{3}{2},k}} \le C(1 + \|\eta\|_{H^{s_0+\frac{5}{2},k+1}}) \|G\|_{H^{s_0+\frac{3}{2},k}}. $$

The proof is a standard result on elliptic equations and we omit it. This is done thoroughly in [Lan13, Lemma 2.38], [BD24, Lemma 3.1] or [DLS20, Lemma 4].

**Remark 2.4.** Note that in Proposition 2.3, the factor $1 + \|\eta\|_{H^{s_0+\frac{5}{2},k+1}}$ could be included in the constant $C$, as $C$ depends on $\|\eta\|_{H^s}$ with $s \ge s_0 + \frac{5}{2}$ according to (H.6). We keep this factor in Proposition 2.3 to emphasize that the gradient of the pressure $\nabla_{\mu}^{\varphi} P$ is less regular than $\eta$. This result is not optimal, and it will be improved in Proposition 2.6 thanks to Alinhac's good unknown.

Another difference with Proposition 2.6 is that the source term in (2.6) solved in Proposition 2.3 needs to be in divergence form in order to have existence of a solution to (2.6) with the Neumann boundary conditions (2.7). In the equation (2.10) solved in Proposition 2.6, the source term is not in divergence form, and we need to assume the existence of a solution, as well as the control of the low regularity part, namely (H.9).

**Remark 2.5.** One can replace the boundary condition (2.7) with homogeneous Dirichlet conditions, and the proof requires only very minor adaptation, see Remark 2.7 below.

### 2.2.2. A second elliptic system.
We provide estimates on the quantity $P$ solving the following elliptic system

$$ (2.10) \qquad \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho(r)} \nabla_{\mu}^{\varphi} P = \sqrt{\mu} G^{(1)} + \nabla_{\mu}^{\varphi} \cdot G^{(2)}, $$

where $G^{(1)}, G^{(2)}$ are source terms. Once again, $\varphi$ is defined through (2.8). Note that we do not assume that $\eta$ vanishes at the boundaries $r=0$ or $r=1$ for Proposition 2.6. This would allow us to consider a non-flat bottom. We denote by

$$ (2.11) \qquad \eta_i := \eta|_{r=i}, $$

for $i \in \{0,1\}$, the traces of $\eta$, for which we assume

$$ (H.8) \qquad |\nabla_x \eta_i|_{H^{s-\frac{1}{2}}} \le M, $$

for some $s \ge s_0 + \frac{5}{2}$. The system (2.10) is completed with Neumann conditions

$$ (2.12) \qquad \frac{1}{\varrho(r)} \mathbf{n}_i \cdot \nabla_{\mu}^{\varphi} P \Big|_{r=i} = \sqrt{\mu} G_i + \mathbf{n}_i \cdot G^{(2)}|_{r=i}, $$

for $i \in \{0,1\}$. Here, $G_0$ and $G_1$ are source terms, $G^{(2)}|_{r=i}$ denotes the trace of $G^{(2)}$ at $r=i$, and $\mathbf{n}_i$ is the unitary outward normal to the boundary defined through (2.9). We assume that $P$ is in $\dot{H}^1/\mathbb{R}$ and moreover that

$$ (H.9) \qquad \|\nabla^{\varphi} P\|_{H^{s_0+\frac{3}{2},k}} \le M. $$

**Proposition 2.6.** Let $s_0 > d/2, s \ge s_0 + \frac{5}{2}, 3 \le k \le s$ with $k \in \mathbb{N}$. We assume that there exists $\underline{M}, M$ such that (H.1), (H.5), (H.6), (H.7), (H.8) hold.

Assume that there exists a solution $P \in \dot{H}^1(S_r)/\mathbb{R}$ to (2.6) satisfying (H.9). Then there exists a constant $C > 0$ depending only on $M, \underline{M}$ such that

$$ (2.13) \qquad \|\nabla_{\mu}^{\varphi} P\|_{H^{s,k}} \le C \left( \|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{3}{2},2}} + \|(G_0, G_1)\|_{H^{s-\frac{1}{2}}} + \|G^{(1)}\|_{H^{s-1,k-1}} + \|G^{(2)}\|_{H^{s,k}} \right). $$

The main difference with the proof of the elliptic estimates performed in [BD24, Lemma 3.1] is that we use the relation (B.3) and Alinhac's good unknown to commute the differential operators $\Lambda$ (defined in the proof) and $\nabla^{\varphi}$ without loss of derivatives.

**Remark 2.7.** One can replace the boundary condition (2.7) with homogeneous Dirichlet conditions at the top and/or bottom, and the proof requires only very minor adaptations. Namely, the boundary terms in the integration by parts performed to derived (2.16) vanish.
This is completed with Neumann conditions at the bottom and rigid lid, by taking the trace of the equation on the vertical momentum in (E')

$$ (2.2) \qquad \frac{1}{\rho(r)} \partial_r^\varphi P \bigg|_{r=0, r=1} = -b \bigg|_{r=0, r=1}. $$

We also wrote

$$ \nabla_{\mu}^{\varphi} := \left( \frac{\sqrt{\mu} \nabla_{x}^{\varphi}}{\partial_{r}^{\varphi}} \right), \qquad \nabla_{\mu} := \left( \frac{\sqrt{\mu} \nabla_{x}}{\partial_{r}} \right). $$

Note that (2.1) is valid at any given time $t$, it is not an evolution equation. Therefore, we omit the dependence on time in this section, to simplify the notations.

**2.1. Estimates with loss of regularity.** The main result of this subsection is the estimates of Proposition 2.1. Recall that $\nabla_x^\varphi$ and $\partial_r^\varphi$ are defined through (1.13). Roughly speaking, we have a control on

$$ \| (\sqrt{\mu} \nabla_x^\varphi P, \partial_r^\varphi P) \|_{H^{s_0+\frac{3}{2}}} $$

by a quantity that depends (among others) on $\|\eta\|_{H^{s_0+\frac{5}{2}}}$, hence we qualify this estimate as an estimate with loss of derivative. Note that it could be extended to a control in $H^s$ for $s > 0$, but it is not useful in our study and would burden the statement with heavier notations.

Note that the above control on the gradient of the pressure degenerates as $\mu \to 0$. In Proposition 2.2, we show how to obtain a control on

$$ \| (\nabla_x^\varphi P, \partial_r^\varphi P) \|_{H^{s_0+\frac{3}{2}}} $$

Let $M, M > 0$, as well as $s_0 > d/2$ and $2 \le k \le s_0 + \frac{3}{2}$. Throughout this subsection, we assume

$$ (H.6) \qquad |\rho|_{W^{s_0+\frac{3}{2}, \infty}} + |V|_{W^{k+1, \infty}} \le \underline{M}, $$

as well as

$$ (H.7) \qquad \|\eta\|_{H^{s_0+\frac{5}{2}, k+1}} + \|V\|_{H^{s_0+\frac{3}{2}, k}} + \sqrt{\mu} \|w\|_{H^{s_0+\frac{3}{2}, k}} \le M. $$

Recall that $\nabla_\mu^\varphi$ is defined through (1.14). We now state the main result of this subsection.

**Proposition 2.1.** Let $s_0 > d/2$, $2 \le k \le s_0 + \frac{3}{2}$ with $k \in \mathbb{N}$. We assume (H.5), (H.6), (H.7) and $\epsilon \le \sqrt{\mu}$.

Then there exists a unique solution $P \in \dot{H}^1(S_r)/\mathbb{R}$ solution to (2.1) and there exists a constant $C > 0$ depending only on $M, \underline{M}$ such that

$$ \|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{3}{2}, k}} \le C \left( \|\eta\|_{H^{s_0+\frac{5}{2}, k+1}} + \|V\|_{H^{s_0+\frac{3}{2}, k}} + \sqrt{\mu} \|w\|_{H^{s_0+\frac{3}{2}, k}} \right). $$

The proof is a standard result on elliptic equations and we omit it. This is done thoroughly in [Lan13, Lemma 2.38], [DB22, Lemma 3.1] or [DLS20, Lemma 4].

We now state a similar result. Note that the following bound is uniform in $\mu$, with the smallness assumption $\epsilon \le \sqrt{\mu}$, but involves a loss of derivatives, due to the use of the hydrostatic pressure in the proof.
### Proof of Proposition 2.6. Step 1: The elliptic equation verified by Alinhac's good unknown

Let $\Lambda$ be an operator of the form $|D|\Lambda^{s-1}$ for $s \ge 1$ or $\Lambda^{s-k}\partial_r^k$. We apply $\Lambda$ to the elliptic problem (2.10) satisfied by the pressure, assuming that $P$ is smooth enough so that the computations needed to derive the estimate in 2.1 make sense. To close the argument, one should use a smoothing operator $|D|\Lambda^{s-1}$ instead of $|D|\Lambda^{s-1}$ in Step 2 below, and check that the estimates are uniform with respect to $\iota > 0$. Then, taking the limit $\iota \to 0$ yields the result. This is detailed in [Lan13, Chapter 2] so that we directly apply $|D|\Lambda^{s-1}$ and assume the needed regularity. We apply $\Lambda$ to $\nabla_\mu^\varphi \cdot \frac{1}{\rho}\nabla_\mu^\varphi P$. This yields, using (B.6):

$$ 
\begin{aligned}
\Lambda\nabla_\mu^\varphi \cdot \frac{1}{\rho}\nabla_\mu^\varphi P &= \nabla_\mu^\varphi \cdot \left(\frac{1}{\rho}\nabla_\mu^\varphi P\right)^\Lambda + \epsilon(\Lambda\eta\partial_r^\varphi)\left(\nabla_\mu^\varphi \cdot \frac{1}{\rho}\nabla_\mu^\varphi P\right) + \epsilon R_2^\Lambda(\nabla_\mu \frac{1}{\rho}\nabla_\mu^\varphi P) \\
&= \nabla_\mu^\varphi \cdot \left(\frac{1}{\rho}\Lambda(\nabla_\mu^\varphi P) + \frac{\epsilon\Lambda\eta}{1+\epsilon h}\partial_r\left(\frac{1}{\rho}\nabla_\mu^\varphi P\right)\right) + \epsilon(\Lambda\eta\partial_r^\varphi)\left(\nabla_\mu^\varphi \cdot \frac{1}{\rho}\nabla_\mu^\varphi P\right) \\
&\quad + \epsilon R_2^\Lambda(\nabla_\mu \frac{1}{\rho}\nabla_\mu^\varphi P) + \nabla_\mu^\varphi \cdot \left(\Lambda, \frac{1}{\rho}\right]\nabla_\mu^\varphi P.
\end{aligned}
$$

Using once again (B.3), the right-hand side of the previous expression becomes

$$ 
\begin{aligned}
\nabla_\mu^\varphi \cdot \left( \frac{1}{\rho} \nabla_\mu^\varphi P^\Lambda + \epsilon \frac{1}{\rho} (\Lambda \eta \partial_r^\varphi) \nabla_\mu^\varphi P + \epsilon R_1^\Lambda (\nabla_\mu P) + \frac{\epsilon \Lambda \eta}{1+\epsilon h} \partial_r \left( \frac{1}{\rho} \nabla_\mu^\varphi P \right) \right) \\
+ \epsilon (\Lambda \eta \partial_r^\varphi) \left( \nabla_\mu^\varphi \cdot \frac{1}{\rho} \nabla_\mu^\varphi P \right) + \epsilon R_2^\Lambda (\nabla_\mu \frac{1}{\rho} \nabla_\mu^\varphi P) + \nabla_\mu^\varphi \cdot \left( \Lambda, \frac{1}{\rho} \right) \nabla_\mu^\varphi P.
\end{aligned}
$$

According to the elliptic equation (2.10) and multiplying by $1 + \epsilon h$, we thus get the equation:

$$ 
\begin{aligned}
-(1 + \epsilon h)\nabla_\mu^\varphi \cdot \frac{1}{\rho}\nabla_\mu^\varphi P^\Lambda &= (1 + \epsilon h)\nabla_\mu^\varphi \cdot \left(\frac{1}{\rho}(\epsilon\Lambda\eta\partial_r^\varphi)\nabla_\mu^\varphi P\right) \\
&\quad + \epsilon(1 + \epsilon h)\nabla_\mu^\varphi \cdot R_1^\Lambda(\nabla_\mu P) + (1 + \epsilon h)\nabla_\mu^\varphi \cdot \left(\frac{\epsilon\Lambda\eta}{1 + \epsilon h}\partial_r\left(\frac{1}{\rho}\nabla_\mu^\varphi P\right)\right) \\
&\quad + \epsilon(1 + \epsilon h)(\Lambda\eta\partial_r^\varphi)\left(\nabla_\mu^\varphi \cdot \frac{1}{\rho}\nabla_\mu^\varphi P\right) + \epsilon(1 + \epsilon h)R_2^\Lambda(\nabla_\mu \frac{1}{\rho}\nabla_\mu^\varphi P) \\
&\quad + (1 + \epsilon h)\nabla_\mu^\varphi \cdot \left(\left[\Lambda, \frac{1}{\rho}\right]\nabla_\mu^\varphi P\right) + \sqrt{\mu}|D|\left[(1 + \epsilon h)\Lambda^{s-1}G^{(1)}\right] + \sqrt{\mu}[|D|, 1 + \epsilon h]\Lambda^{s-1}G^{(1)} \\
&\quad + (1 + \epsilon h)\nabla_\mu^\varphi \cdot \mathbf{G}^{(2)\Lambda} + \epsilon(1 + \epsilon h)\Lambda\eta\partial_r^\varphi\nabla_\mu^\varphi \cdot \mathbf{G}^{(2)} + \epsilon(1 + \epsilon h)R_2^\Lambda(\nabla_\mu \mathbf{G}^{(2)}).
\end{aligned}
\quad (2.14) $$

Note that we used (B.6) on the source term $\nabla_\mu^\varphi \cdot \mathbf{G}^{(2)}$. In the case $\Lambda = |D|\Lambda^{s-1}$ (that is, we only differentiate (2.10) in the tangential direction with respect to the boundary of the strip $S_r$), we can apply $\Lambda$ to the boundary conditions (2.12) and the system (2.14) is completed with the boundary conditions, for $i$ in $\{0, 1\}$

$$ 
\begin{aligned}
& \left( \frac{1}{\rho} \mathbf{n}_i \cdot \nabla_\mu^\varphi P^\Lambda + \epsilon \frac{1}{\rho} \mathbf{n}_i \cdot \Lambda \eta \partial_r^\varphi \nabla_\mu^\varphi P + \epsilon \frac{1}{\rho} \mathbf{n}_i \cdot R_1^\Lambda (\nabla_\mu P) + [\Lambda, \mathbf{n}_i] \cdot \nabla_\mu^\varphi P \right) \Big|_{r=i} \\
& = \left( \Lambda G_i + \mathbf{n}_i \cdot \mathbf{G}^{(2)\Lambda} - \epsilon \Lambda \eta_i \mathbf{n}_i \cdot \partial_r^\varphi \mathbf{G}^{(2)} + [\Lambda, \mathbf{n}_i] \cdot \mathbf{G}^{(2)} \right) \Big|_{r=i}.
\end{aligned}
\quad (2.15) $$

### Step 2: Estimate on $||\nabla^\varphi P^\Lambda||_{L^2}$ for $\Lambda = |D|\Lambda^{s-1}$

We use (2.14) with $\Lambda := |D|\Lambda^{s-1}$. Note that in this case $[\Lambda, \frac{1}{\rho}] = 0$ as $\rho$ only depends on $r$.

We use an induction on $s$. The base case $s = s_0 + \frac{3}{2}$ in (2.13) is automatically satisfied since $||\nabla_\mu^\varphi P||_{H^{s_0+\frac{3}{2},2}}$ appears
**Proposition 2.2.** With the same assumptions and notations as the previous lemma we have the following bound:

$$ (2.3) \qquad \|\nabla^\varphi P\|_{H^{s_0+\frac{3}{2},k}} \le C \left(1 + \frac{\epsilon}{\sqrt{\mu}}\right) \left(\|\eta\|_{H^{s_0+\frac{5}{2},k+1}} + \|V\|_{H^{s_0+\frac{3}{2},k}} + \sqrt{\mu}\|w\|_{H^{s_0+\frac{3}{2},k}}\right). $$

This yields a uniform estimates with respect to the parameters $\epsilon$ and $\mu$ in the medium amplitude regime $\epsilon \le \sqrt{\mu}$.

*Proof.* We start by defining a pressure, reminiscent of the hydrostatic pressure

$$ P_h(t, x, r) := \int_0^r \varrho(r')b(t, x, r')dr', $$

so that

$$ \frac{1}{\varrho} \partial_r^\varphi P_h + b = \epsilon \frac{h}{1 + \epsilon h} b. $$

By the product estimate (A.3) we readily get

$$ \|\nabla^\varphi P_h\|_{H^{s,k}} \le C \|\eta\|_{H^{s+1,k+1}}. $$

Notice that there is a loss of derivative, but the estimate is uniform in $\mu$.

We now define $P_{nh} := P - P_h$, and look for an estimate on $\|\nabla P_{nh}\|_{H^{s,k}}$ uniform in $\mu$. Together with the estimate on $P_h$, it then shows the desired estimate.

Plugging the previous relation into (2.1) yields

$$ (1 + \epsilon h) \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho(r)} \nabla_{\mu}^{\varphi} P_{nh} = -\frac{\mu}{\epsilon}(1 + \epsilon h) \nabla^{\varphi} \cdot ((\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})) \\ - \epsilon \partial_r \left( b \frac{h}{1 + \epsilon h} \right) - \mu \nabla_x^{\varphi} \cdot \frac{1}{\varrho} \nabla_x^{\varphi} P_h, $$

with the boundary condition

$$ \frac{1}{\varrho(r)} \partial_r^\varphi P_{nh} \Big|_{r=0; r=1} = -\epsilon b \frac{h}{1 + \epsilon h} \Big|_{r=0; r=1}. $$

From standard results on elliptic equations (see [Lan13, Lemma 2.38], [DB22, Lemma 3.1] and [DLS20, Lemma 4]), we get

$$ \|\nabla_{\mu}^{\varphi} P_{nh}\|_{H^{s,k}} \le (\sqrt{\mu} + \epsilon)C \left( \|\eta\|_{H^{s_0+\frac{5}{2},k+1}} + \|V\|_{H^{s_0+\frac{3}{2},k+1}} + \sqrt{\mu}\|w\|_{H^{s_0+\frac{3}{2},k+1}} \right). $$

Recalling that $\nabla_{\mu}^{\varphi} := (\sqrt{\mu} \nabla_x^{\varphi}) / \partial_r^{\varphi}$ yields the desired estimate, after dividing by $\sqrt{\mu}$. $\square$

**Remark 2.3.** The estimate in Proposition 2.1 involves a loss of derivative due to the terms $\nabla\eta$ in the operators $\nabla_{\mu}^{\varphi}$ and $\partial_r^{\varphi}$ in (2.1). On the other hand, the estimate in Proposition 2.2 involves another loss of derivative, due to the definition of the hydrostatic pressure, as well as the one due to the terms $\nabla\eta$ in the operators $\nabla_{\mu}^{\varphi}$ and $\partial_r^{\varphi}$. However, the estimate in Proposition 2.2 yields a control on $(\nabla_x P, \partial_r P)$ whereas the estimate in Proposition 2.1 only states a control on $(\sqrt{\mu} \nabla_x P, \partial_r P)$. Therefore, Proposition 2.2 yields a better estimate than the one in 2.1. We still choose to keep both propositions, as an effort to show where these distinct difficulties arise, and to ease the comparison with the next subsection.
on the right-hand side. We thus take $s \ge s_0 + \frac{5}{2}$. We set

$$
\begin{equation} \tag{2.16}
\begin{aligned}
\epsilon f_0 &:= \epsilon (\Lambda \eta \partial_r^\varphi) (\nabla_\mu^\varphi \cdot \frac{1}{\varrho} \nabla_\mu^\varphi P) + \epsilon R_2^\Lambda (\nabla_\mu \frac{1}{\varrho} \nabla_\mu^\varphi P) + \sqrt{\mu} [|D|, 1 + \epsilon h] \Lambda^{s-1} G^{(1)} \\
&\quad + \epsilon (1 + \epsilon h) \Lambda \eta \partial_r^\varphi \nabla_\mu^\varphi \cdot \mathbf{G}^{(2)} + \epsilon (1 + \epsilon h) R_2^\Lambda (\nabla_\mu \mathbf{G}^{(2)}), \\
\mathbf{F}_1 &= \epsilon \frac{1}{\varrho} (\Lambda \eta \partial_r^\varphi) \nabla_\mu^\varphi P + \epsilon R_1^\Lambda (\nabla_\mu P) + \frac{\epsilon \Lambda \eta}{1 + \epsilon h} \partial_r (\frac{1}{\varrho} \nabla_\mu^\varphi P) + \mathbf{G}^{(2)\Lambda}, \\
f_2 &= (1 + \epsilon h) \Lambda^{s-1} G^{(1)}, \\
(\epsilon + \sqrt{\mu}) f_{3,i} &= \left( \sqrt{\mu} \Lambda G_i - \epsilon \Lambda \eta_i \mathbf{n}_i \cdot \nabla_r^\varphi \mathbf{G}^{(2)} + [\Lambda, \mathbf{n}_i] \cdot \mathbf{G}^{(2)} + \epsilon \frac{1}{\varrho} \mathbf{n}_i \cdot \Lambda \eta \partial_r^\varphi \nabla_\mu^\varphi P \right. \\
&\qquad \left. + \epsilon \frac{1}{\varrho} \mathbf{n}_i \cdot R_1^\Lambda (\nabla_\mu P) + [\Lambda, \mathbf{n}_i] \cdot \nabla_\mu^\varphi P \right) \Big|_{r=i}.
\end{aligned}
\end{equation}
$$

We now test (2.14) against $P^\Lambda$. Integrating by parts and using the boundary conditions (2.15) yields

$$
\begin{equation}
\begin{aligned}
\| \sqrt{1 + \epsilon h} \nabla_{\mu}^{\varphi} P^{\Lambda} \|_{L^2}^2 &\le \epsilon \|(1 + \epsilon h) f_0 \|_{L^2} \| P^{\Lambda} \|_{L^2} + \| \mathbf{F}_1 \|_{L^2} \| \nabla_{\mu}^{\varphi} P^{\Lambda} \|_{L^2} \\
&\quad + \| f_2 \|_{L^2} \| \sqrt{\mu} |D| P^{\Lambda} \|_{L^2} + (\epsilon + \sqrt{\mu}) \int_{r=i} f_{3,i} P^{\Lambda} \big|_{r=i} .
\end{aligned}
\tag{2.16}
\end{equation}
$$

Note that the last term is implicitly the sum of the term for $i=0$ (that is, the boundary term at the rigid lid) and for $i=1$ (that is, the boundary term at the bottom). For these boundary terms, we use the $H^{-\frac{1}{2}}-H^{\frac{1}{2}}$ estimate and the trace estimate of Lemma A.2 to write

$$
\left| \int_{r=i} f_{3,i} P^{\Lambda} \right|_{r=i} \le C |f_{3,i}|_{H^{-\frac{1}{2}}} |P^{\Lambda}|_{H^{\frac{1}{2}}} \le C |f_{3,i}|_{H^{-\frac{1}{2}}} (\|P^{\Lambda}\|_{L^2} + \|\nabla P^{\Lambda}\|_{L^2}).
$$

Here, $C$ denotes a constant depending only on $M$ and $\underline{M}$. We use Young's inequality for the first term of (2.16), and introduce a parameter $0 < \delta \le 1$ to be fixed later (see (2.24)), independent of $\epsilon$ and $\mu$. This yields

$$
(2.17) \quad \| \sqrt{1 + \epsilon h} \nabla_{\mu}^{\varphi} P^{\Lambda} \|_{L^2} \le \delta \|(1 + \epsilon h) f_0 \|_{L^2} + (\epsilon + \sqrt{\mu}) \frac{1}{\delta} \| P^{\Lambda} \|_{L^2} + C |f_{3,i}|_{H^{-\frac{1}{2}}} + \|(\mathbf{F}_1, f_2)\|_{L^2}.
$$

We now need to estimate each of those terms. We first need a control on $\|P^\Lambda\|_{L^2}$. This is provided by induction, namely we have:

$$
P^{\Lambda} := \Lambda P + \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \partial_{r} P = \Lambda P - \epsilon \Lambda \eta \partial_{r}^{\varphi} P.
$$

Thus we have a control on $\|P^\Lambda\|_{L^2}$ by induction on $s$ (recall that $\Lambda := |D|\Lambda^{s-1}$ and the product estimate (A.1)):

$$
(2.18) \qquad \| P^\Lambda \|_{L^2} \le \frac{C}{\sqrt{\mu}} \| \nabla_\mu P \|_{H^{s-1,0}} + \epsilon C \| \eta \|_{H^{s,0}} \| \nabla_\mu^\varphi P \|_{H^{s_0+\frac{1}{2},1}}.
$$

Note that there is a singularity in $\mu$, however it is compensated by the factor $\epsilon + \sqrt{\mu}$ in (2.17), thanks to (H.7).
For the first term of $f_0$, we use the product estimate (A.1) to get

$$
\| (\epsilon \Lambda \eta \partial_r^\varphi) (\nabla_\mu^\varphi \cdot \frac{1}{\varrho} \nabla_\mu^\varphi P) \|_{L^2} \le C \epsilon \|\eta\|_{H^{s,0}} \|\nabla_\mu^\varphi \cdot \frac{1}{\varrho} \nabla_\mu^\varphi P\|_{H^{s_0+\frac{3}{2},2}}.
$$

We use the equation (2.13) to replace $\nabla_\mu^\varphi \cdot \frac{1}{\varrho} \nabla_\mu^\varphi P$ by the source term. The product estimate (A.1) then yields

$$
\| (\epsilon \Lambda \eta \partial_r^\varphi) (\nabla_\mu^\varphi \cdot \frac{1}{\varrho} \nabla_\mu^\varphi P) \|_{L^2} \le C \epsilon \|\eta\|_{H^{s,0}} \|\sqrt{\mu} G^{(1)} + \nabla_\mu^\varphi \cdot \mathbf{G}^{(2)} \|_{H^{s_0+\frac{3}{2},2}}.
$$

For the second term of $f_0$, we use the estimate (B.12) to get

$$
\| R_2^\Lambda (\nabla_\mu \frac{1}{\varrho} \nabla_\mu^\varphi P) \|_{L^2} \le C \| \nabla \eta \|_{H^{s-1,0}} \| \nabla_\mu^\varphi \frac{1}{\varrho} \nabla_\mu^\varphi P \|_{H^{s-1,0}},
$$

For the third term in $f_0$, we use the commutator estimate (A.5) to get

$$
\|[D, 1 + \epsilon h] \Lambda^{s-1} G^{(1)}\|_{L^2} \le C \epsilon \|\eta\|_{H^{s_0+\frac{5}{2},3}} \|G^{(1)}\|_{H^{s-1,2}}.
$$
2.2. **Estimates without loss of regularity.** In this subsection we improve the estimates of Proposition 2.1, in the sense that the estimates in Proposition 2.4 provide a control on

$$ \| (\sqrt{\mu} \nabla_x^\varphi P, \partial_r^\varphi P) \|_{H^s} $$

by a quantity that depends (among others) on $\|\eta\|_{H^s}$. Therefore, we speak of estimates without any loss of derivative, in opposition to the estimate of Proposition 2.1, where the same quantity is controlled by terms depending on $\|\eta\|_{H^{s+1}}$. However, note the assumption $s \ge s_0 + \frac{5}{2}$ in Proposition 2.4, so that we still use Proposition 2.1 in the low regularity case $s = s_0 + \frac{3}{2}$.

Let $s > 0, 0 \le k \le s$, and $\Lambda := \Lambda^{s-k} \partial_r^k$. Recall that $\nabla^\varphi$ is defined through (1.13). The loss of derivatives in Proposition 2.1 is due to the fact that the operators $\Lambda$ and $\nabla^\varphi$ do not commute, and the resulting commutator is bounded, among other factors, by $\|\eta\|_{H^{s+1,k+1}}$. We can however use the following identity, showing how to commute these operators properly:

$$ \Lambda \nabla^\varphi f = \nabla^\varphi f^\Lambda + (\epsilon \Lambda \eta \partial_r^\varphi) \nabla^\varphi f + \epsilon R_1^\Lambda (\nabla f). $$

Here, $f \in H^{s,k}$ is a function and Alinhac's good unknown $f^\Lambda$ is defined as

$$ f^\Lambda := \Lambda f + \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \partial_r f. $$

See [Ali89], as well as [Lan13] for its use in the study of the Water Waves equation. The remainder term $R_1^\Lambda(\nabla f)$ is bounded without any loss of derivatives on $\eta$, see Lemma B.2. This identity and other results on Alinhac's good unknown are presented in Appendix B.

The purpose of this subsection is to prove the following elliptic estimates.

**Proposition 2.4.** Let $s_0 > d/2$, $s \in \mathbb{R}$ with $s \ge s_0 + \frac{5}{2}$, $3 \le k \le s$ with $k \in \mathbb{N}$. We further assume (H.1), (H.5), $\epsilon \le \sqrt{\mu}$, and

$$ \|\eta\|_{H^{s,k}} + \|V\|_{H^{s,k}} + \sqrt{\mu} \|w\|_{H^{s,k}} \le M, $$

for $M > 0$ a constant. Then the pressure $P$ defined through (2.1) satisfies the following bound, with $C > 0$ a constant depending only on $M$, $\underline{M}$

$$ \| \nabla_{\mu}^{\varphi} P \|_{H^{s,k}} \le C \left( \| \eta \|_{H^{s,k}} + \| V \|_{H^{s,k}} + \sqrt{\mu} \| w \|_{H^{s,k}} \right). $$

In order to prove Proposition 2.4, we first state the following elliptic estimates. Due to the source term $f_0$ on the right-hand side of (2.4), we assume a priori the existence of a solution to the elliptic equation (2.4) in $H^1$, and only prove the estimate.

**Lemma 2.5.** Let $s_0 > \frac{d}{2}$ and assume $\eta \in W^{1,\infty}$, as well as (H.5).

Let $f_0 \in L^2(S_r)$, $F_1 \in L^2(S_r)^{d+1}$, $F_2 \in L^2(S_r)^{d+1}$, $f_3 \in L^2(S_r)$.

Suppose that $Q$ is a $H^1$-function that satisfies, in the variational sense,

$$ (2.4) \qquad (1 + \epsilon h) \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} Q = f_0 + (1 + \epsilon h) \nabla_{\mu}^{\varphi} \cdot \mathbf{F}_1 + \nabla_{\mu} \cdot \mathbf{F}_2 + \sqrt{\mu} |D| f_3, $$

completed with Neumann boundary conditions

$$ (2.5) \qquad \frac{1}{\varrho} e_3 \cdot \nabla_{\mu}^{\varphi} Q \Big|_{r=0;r=1} = e_3 \cdot (\mathbf{F}_1 + \mathbf{F}_2)|_{r=0;r=1}. $$

Then there exists $C > 0$ depending only on $\|\eta\|_{W^{2,\infty}}$ such that

$$ \| \nabla_{\mu} Q \|_{L^2} \le C \left( \sqrt{\| f_0 \|_{L^2} \| Q \|_{L^2}} + \| \mathbf{F}_1 \|_{L^2} + \| \mathbf{F}_2 \|_{L^2} + \| f_3 \|_{L^2} \right). $$
The other terms of $f_0$ are treated the same way and we omit them. We can eventually write

$$ \|f_0\|_{L^2} \le C(\sqrt{\mu}\|G^{(1)}\|_{H^{s-1,2}} + \|G^{(2)}\|_{H^{s,3}} + \|\nabla_{\mu}^{\varphi}P\|_{H^{s,1}}). $$

For the first term of $\mathbf{F}_1$, we use the product estimate (A.1) to get

$$ \| \frac{1}{\varrho} (\epsilon \Lambda \eta \partial_r^\varphi) \nabla_\mu^\varphi P \| \le \epsilon C \|\eta\|_{H^{s,0}} \|\nabla_\mu^\varphi P\|_{H^{s_0+\frac{3}{2},2}}. $$

For the second term of $\mathbf{F}_1$, we use estimate (B.12)

$$ \|R_1^{\Lambda}(\nabla_{\mu}^{\varphi} P)\|_{L^2} \le C \|\nabla \eta\|_{H^{s-1,0}} \|\nabla_{\mu}^{\varphi} P\|_{H^{s-1,0}}. $$

For the third term of $\mathbf{F}_1$, we use the product estimate (A.1):

$$ \| \frac{\Lambda \eta}{1 + \epsilon h} \partial_{\varrho} \left( \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P \right) \| \le C \| \eta \|_{H^{s,0}} \| \nabla_{\mu}^{\varphi} P \|_{H^{s_0+\frac{3}{2},2}}. $$

The bound on the last term of $\mathbf{F}_1$ comes directly from the definition (B.5) and the product estimate (A.2), and we can thus write

$$ \|\mathbf{F}_1\|_{L^2} \le \epsilon C (\|\nabla^{\varphi} P\|_{H^{s-1,0}} + \|G^{(2)}\|_{H^{s,2}}). $$

The term in $f_2$ is directly controlled by $\|G^{(1)}\|_{H^{s-1,0}}$. For the terms of $f_{3,i}$, we only treat the third term, as the other terms are either treated the same way or have already been treated above. From the commutator estimate (A.4), the definition (2.9) of $\mathbf{n}_i$ together with the composition estimate (A.11) and the trace estimate in Lemma A.2, we get

$$ \| [\Lambda, \mathbf{n}_i] \cdot \mathbf{G}^{(2)} \|_{H^{-\frac{1}{2}}} \le C \epsilon \|\mathbf{G}^{(2)}\|_{H^{s,2}}. $$

We eventually get

$$ |f_{3,i}|_{H^{-\frac{1}{2}}} \le C (|G_i|_{H^{s-\frac{1}{2}}} + \|G^{(2)}\|_{H^{s,2}} + \|\nabla_{\mu}^{\varphi} P\|_{H^{s-1,1}}). $$

From the control on $f_0 \mathbf{F}_1$, $f_2$, $f_{3,i}$ above and (2.17), we deduce

$$ \begin{aligned} \| \sqrt{1 + \epsilon h} \nabla_{\mu}^{\varphi} P^{\Lambda} \|_{L^2} &\le C \delta (\sqrt{\mu} \|G^{(1)}\|_{H^{s-1}} + \epsilon \|\nabla_{\mu}^{\varphi} P\|_{H^{s,1}}) \\ &\quad + C(\epsilon + \sqrt{\mu}) \frac{1}{\delta} \left( \frac{1}{\sqrt{\mu}} \|\nabla_{\mu} P\|_{H^{s-1,0}} + \epsilon \|\eta\|_{H^{s,0}} \|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{1}{2},1}} \right) \\ &\quad + C(|G_i|_{H^{s-\frac{1}{2}}} + \|G^{(2)}\|_{H^{s,2}} + \|\nabla_{\mu}^{\varphi} P\|_{H^{s-1,1}}) \\ &\quad + \epsilon C \|\eta\|_{H^{s,0}} \|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{1}{2},1}} + \epsilon C \|\nabla^{\varphi} P\|_{H^{s-1,0}}. \end{aligned} \quad (2.19) $$

We now use (B.17), (H.5) and (H.7) to simplify the above expression to get

$$ \begin{aligned} \| \nabla_{\mu}^{\varphi} P \|_{H^{s,0}} &\le C \left( \|G^{(1)}\|_{H^{s-1,1}} + \|G^{(2)}\|_{H^{s,2}} + \frac{1}{\delta} \| \nabla_{\mu}^{\varphi} P \|_{H^{s_0+\frac{1}{2},1}} + |G_i|_{H^{s-\frac{1}{2}}} \right) \\ &\quad + C \epsilon \delta \| \nabla_{\mu}^{\varphi} P \|_{H^{s,1}} + C \frac{1}{\delta} \| \nabla_{\mu}^{\varphi} P \|_{H^{s-1,1}}. \end{aligned} \quad (2.20) $$

### Step 3: Control of higher vertical derivatives

We now want an estimate on $\|\nabla_{\mu} P\|_{H^{s,k}}$ for $k \ge 1$. To this end, we start by noticing

$$ \|\nabla_{\mu}^{\varphi} P\|_{H^{s,k}} \le c \|\nabla_{\mu}^{\varphi} P\|_{H^{s,k-1}} + c \|\Lambda^{s-k} \partial_r^{k-1} \partial_r^{\varphi} P\|_{L^2}, \quad (2.21) $$

where $c$ only depends on $s$ and $k$. We aim to use the equation (2.10) to control the vertical derivatives. We use an induction on $s$ and $k$. More precisely, the base case $k=0$ and any $s \ge 0$ is given by the previous step, although the term $\|\nabla_{\mu}^{\varphi} P\|_{H^{s,1}}$ in (2.20) needs to be taken care of; this is done at the end of this step. The base case $s \le s_0 + \frac{3}{2}$, $k \le s$ is given by Assumption (H.9). Therefore, we assume from now on that $\|\nabla_{\mu}^{\varphi} P\|_{H^{s',k'}}$ is controlled for either $s' < s$, $k' \le s'$ and for $s' = s$, $k' < k$.

The first term in (2.21) is controlled by induction. For the second term, we use the equation (2.10): Let $\Lambda :=$
*Proof.* Testing (2.4) against a function $\phi \in H^{1,1}$ and performing integrations by parts (thanks to Lemma A.1) yields the variational formulation (2.6)

$$ \forall \phi \in H^{1,1}, \int (1+\epsilon h) \nabla_{\mu}^{\varphi} \phi \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} Q = \int f_0 \phi + \int (1+\epsilon h) \mathbf{F}_1 \cdot \nabla_{\mu}^{\varphi} \phi + \int \mathbf{F}_2 \nabla_{\mu} \cdot \phi + \int f_3 |D| \phi. $$

Note that here $Q$ is already assumed to be in $H^{1,1}$, and we are not concerned about existence. We test equation (2.6) against $Q$:

$$ \|\nabla_{\mu}^{\varphi} Q\|_{L^2}^2 \le C (\|f_0\|_{L^2} \|Q\|_{L^2} + \|\mathbf{F}_1\|_{L^2} \|\nabla_{\mu} Q\|_{L^2} + \|\mathbf{F}_2\|_{L^2} \|\nabla_{\mu} Q\|_{L^2} + \|f_3\|_{L^2} \|\nabla_{\mu} Q\|_{L^2}). $$

Recalling (B.14):

$$ \|\nabla_{\mu} Q\|_{L^2} \le C \|\nabla_{\mu}^{\varphi} Q\|_{L^2}, $$

this yields the desired estimate.

<figure><img src="image_4.png" /></figure>

We now prove Proposition 2.4. The main difference with the proof of the elliptic estimates performed in [DB22, Lemma 3.1] is that we use the relation (B.2) and Alinhac's good unknown to commute the differential operators $\Lambda$ and $\nabla^{\varphi}$ without loss of derivatives.

**Proof of Proposition 2.4. Step 1: The elliptic equation verified by Alinhac's good unknown**

We apply $|D|\Lambda^{s-1}$ to the elliptic problem (2.1) verified by the pressure, assuming that $P$ is smooth enough so that the computations needed to derive the estimate in 2.4 make sense. To close the argument, one should use a smoothing operator $|D|\Lambda_{\delta}^{s-1}$ instead of $|D|\Lambda^{s-1}$, and check that the previous estimate is uniform with respect to $\delta > 0$. Then, taking the limit $\delta \to 0$ yields the result. This is detailed in [Lan13, Chapter 2] so that we directly apply $|D|\Lambda^{s-1}$ and assume the needed regularity.

Let $\Lambda$ be an operator of the form $|D|\Lambda^{s-1}$ for $s \ge 1$ or $\Lambda^{s-k}\partial_r^k$.

Applying $\Lambda$ to $\nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P$ and using identity (B.2) yields

$$ \begin{aligned} \Lambda \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P &= \nabla_{\mu}^{\varphi} \cdot \left( \frac{1}{\rho} \nabla_{\mu}^{\varphi} P \right)^{\Lambda} + \epsilon (\Lambda \eta \partial_{r}^{\varphi}) \left( \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P \right) + \epsilon R_{2}^{\Lambda} (\nabla_{\mu} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P) \\ &= \nabla_{\mu}^{\varphi} \cdot \left( \frac{1}{\rho} \Lambda (\nabla_{\mu}^{\varphi} P) + \frac{\epsilon \Lambda \eta}{1+\epsilon h} \partial_{r} \left( \frac{1}{\rho} \nabla_{\mu}^{\varphi} P \right) \right) + \epsilon (\Lambda \eta \partial_{r}^{\varphi}) \left( \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P \right) \\ &\quad + \epsilon R_{2}^{\Lambda} (\nabla_{\mu} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P) + \nabla_{\mu}^{\varphi} \cdot \left( [\Lambda, \frac{1}{\rho}] \nabla_{\mu}^{\varphi} P \right). \end{aligned} $$

Using once again (B.2), the right-hand side of the previous expression becomes

$$ \begin{aligned} & \nabla_{\mu}^{\varphi} \cdot \left( \frac{1}{\rho} \nabla_{\mu}^{\varphi} P^{\Lambda} + \epsilon \frac{1}{\rho} (\Lambda \eta \partial_{r}^{\varphi}) \nabla_{\mu}^{\varphi} P + \epsilon R_{1}^{\Lambda} (\nabla_{\mu} P) + \frac{\epsilon \Lambda \eta}{1+\epsilon h} \partial_{r} \left( \frac{1}{\rho} \nabla_{\mu}^{\varphi} P \right) \right) \\ & + \epsilon (\Lambda \eta \partial_{r}^{\varphi}) \left( \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P \right) + \epsilon R_{2}^{\Lambda} (\nabla_{\mu} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P) + \nabla_{\mu}^{\varphi} \cdot \left( [\Lambda, \frac{1}{\rho}] \nabla_{\mu}^{\varphi} P \right). \end{aligned} $$
$\Lambda^{s-k} \partial_r^{k-1}$, we apply $\Lambda$ to (2.10). Recall (2.14) derived in Step 1. We write $R$ for the right-hand side, multiply by $1 + \epsilon h$ and expand the left-hand side to get:

$$ (2.22) \qquad \left( \frac{1 + \mu |\nabla_x \eta|^2}{\rho(1 + \epsilon h)^2} \right) \partial_r^2 P^{\Lambda} = -\mu \nabla_x \cdot \left( \frac{(1 + \epsilon h)}{\rho} \nabla_x P^{\Lambda} + \epsilon \frac{\nabla_x \eta}{\rho} \partial_r P^{\Lambda} \right) + \mu \partial_r \left( \epsilon \frac{\nabla_x \eta \cdot \nabla_x P}{\rho} \right) \\ \qquad \qquad + \partial_r \left( \frac{1 + \mu \epsilon^2 |\nabla_x \eta|^2}{\rho(1 + \epsilon h)} \right) \partial_r P^{\Lambda} + (1 + \epsilon h) R. $$

We take the $L^2$-norm of the previous expression. Recall that $1 + \epsilon h \le h^*$. We use the product estimate (A.1), and the composition estimate (A.11) for $\frac{1}{1+\epsilon h} = 1 - \frac{\epsilon h}{1+\epsilon h}$. We get:

$$ (2.23) \qquad \| \partial_r^2 P^{\Lambda} \|_{L^2} \le C (\|R\|_{L^2} + \| \nabla_{\mu}^{\varphi} P^{\Lambda} \|_{H^{1,0}}), $$

by using that $\frac{h^*}{\rho_0} \frac{1}{\rho(1+\epsilon h)} \ge c_* h_* > 0$.

The estimation of the terms in $R$ is very similar to the previous step, except for two terms. The first one is $\nabla_{\mu}^{\varphi} \cdot R_1^{\Lambda}(\nabla_{\mu} P)$. Recall that, up to a harmless factor, $R_1^{\Lambda}(\nabla_{\mu} P)$ is the symmetric commutator $[\Lambda; (\partial \varphi)^T, \nabla_{\mu}^{\varphi} P]$ so that we write:

$$ \nabla_{\mu}^{\varphi} \cdot [\Lambda; (\partial \varphi)^T, \nabla_{\mu}^{\varphi} P] = [\nabla_{\mu}^{\varphi} \Lambda; (\partial \varphi)^T, \nabla_{\mu}^{\varphi} P] + [\nabla_{\mu}^{\varphi}, \nabla_{\mu}^{\varphi} P] \Lambda (\partial \varphi)^T + [\nabla_{\mu}^{\varphi}, (\partial \varphi)^T] \Lambda \nabla_{\mu}^{\varphi} P, $$

and the commutator estimates (A.8) (in the case $k \ge 2$) and (A.10) (in the case $k = 1$) yield:

$$ \| \nabla_{\mu}^{\varphi} \cdot R_{1}^{\Lambda} (\nabla_{\mu} P) \|_{L^2} \le C (\| \nabla_{\mu}^{\varphi} P \|_{H^{5/2+1/2}} + \| \nabla_{\mu}^{\varphi} P \|_{H^{s-1, k\Lambda(s-1)}}). $$

The second term is $\nabla_{\mu}^{\varphi} \cdot [\Lambda, \frac{1}{\rho}] \nabla_{\mu}^{\varphi} P$. Note however that, as $\rho$ only depends on $r$, we get:

$$ [\Lambda, \frac{1}{\rho}] = [\partial_r^{k-1}, \frac{1}{\rho}] \Lambda^{s-k}, $$

and the commutator estimate (A.7) allows us to conclude:

$$ \| \nabla_{\mu}^{\varphi} \cdot [\Lambda, \frac{1}{\rho}] \nabla_{\mu}^{\varphi} P \|_{L^2} \le C \| \nabla_{\mu}^{\varphi} P \|_{H^{s-1, k-1}}. $$

We now turn to the control of $\| \nabla_{\mu}^{\varphi} P^{\Lambda} \|_{H^{1,0}}$ in (2.23). We apply $\Lambda$ to (B.3) to get:

$$ \| \nabla_{\mu}^{\varphi} P^{\Lambda} \|_{H^{1,0}} \le C \| \nabla_{\mu}^{\varphi} P \|_{H^{s,k-1}} + C \epsilon \| \eta \|_{H^{s,k-1}} \| \nabla_{\mu}^{\varphi} P \|_{H^{5/2+1/2}} \\ + C \epsilon \| \eta \|_{H^{sv(s_0+3/2),3v(k\Lambda(s-1))}} \| \nabla_{\mu}^{\varphi} P \|_{H^{(s-1)v(s_0+5/2), (k-1)\Lambda(s-2)}}, $$

where we used (B.4) (for $k \ge 2$) or (B.12) (for $k = 1$) to control the term $R_1^{\Lambda}(\nabla_{\mu} P)$ from (B.3). Therefore, by induction, we have a bound for $\| \nabla_{\mu}^{\varphi} P^{\Lambda} \|_{H^{1,0}}$. That is to say, we have a bound for $\| \partial_r \partial_r^{\varphi} P^{\Lambda} \|_{L^2}$. Now recall:

$$ P^{\Lambda} := \Lambda^{s-k} \partial_r^{k-1} P + \frac{\Lambda^{s-k} \partial_r^{k-1} \eta}{1 + \epsilon h} \partial_r P, $$

so that, by applying $\partial_r$ and using triangular inequality:

$$ \| \partial_r \Lambda^{s-k} \partial_r^{k-1} \partial_r^{\varphi} P \|_{L^2} \le \| \partial_r \partial_r^{\varphi} P^{\Lambda} \|_{L^2} + C \| \eta \|_{H^{s-1,2}} \| \nabla_{\mu}^{\varphi} P \|_{H^{5/2+1/2}}. $$

Finally, note that:

$$ \| \partial_r \partial_r^{\varphi} P^{\Lambda} \|_{L^2} = \| \partial_r \left( \frac{1}{1 + \epsilon h} \partial_r P^{\Lambda} \right) \|_{L^2} \le C \| \partial_r P^{\Lambda} \|_{L^2} + C \| \partial_r^2 P^{\Lambda} \|_{L^2}, $$

as the factor that appears are controlled by $\| \eta \|_{H^{5/2+1/2}}$.

We eventually get a control on $\| \Lambda^{s-k} \partial_r^{k} \partial_r^{\varphi} P \|_{L^2}$ so that by (2.21) and induction, we get, for any $s \ge s_0 + \frac{5}{2}$ and $0 \le k \le s$:

$$ (2.24) \qquad \| \nabla_{\mu}^{\varphi} P \|_{H^{s,k}} \le C \left( \| G^{(1)} \|_{H^{s-1,k-1}} + \| G^{(2)} \|_{H^{s,k}} + \frac{1}{\delta} \| \nabla_{\mu}^{\varphi} P \|_{H^{5/2+1/2}} + |G_i|_{H^{s-1/2}} \right) \\ \qquad \qquad + C \epsilon \delta \| \nabla_{\mu}^{\varphi} P \|_{H^{s,1}} + C \frac{1}{\delta} \| \nabla_{\mu}^{\varphi} P \|_{H^{s-1,1}}. $$

We choose $\delta \le \frac{1}{2C}$ so that, as $\epsilon \le 1$, we have $C \epsilon \delta \le \frac{1}{2}$ and the last term in (2.24) can be absorbed on the left-hand side when $k \ge 1$. Note that $\delta$ is indeed independent of $\epsilon$ and $\mu$, in particular $\frac{1}{\delta}$ remains bounded. This yields (2.13). $\square$
According to the elliptic equation (2.1) satisfied by the pressure, we thus get the equation:

$$-\nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P^{\wedge} = \frac{\mu}{\epsilon} \Lambda(\partial_{i}^{\varphi}(\mathbf{U} + \epsilon \mathbf{U})_{j} \partial_{j}^{\varphi}(\mathbf{U} + \epsilon \mathbf{U})_{i}) + \nabla_{\mu}^{\varphi} \cdot \left( \frac{1}{\varrho} (\epsilon \Lambda \eta \partial_{r}^{\varphi}) \nabla_{\mu}^{\varphi} P \right) \\ + \epsilon \nabla_{\mu}^{\varphi} \cdot R_{1}^{\wedge}(\nabla_{\mu} P) + \nabla_{\mu}^{\varphi} \cdot \left( \frac{\epsilon \Lambda \eta}{1 + \epsilon h} \partial_{r} \left( \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P \right) \right) \\ + \epsilon (\Lambda \eta \partial_{r}^{\varphi}) \left( \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P \right) + \epsilon R_{2}^{\wedge}(\nabla_{\mu} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P) \\ + \Lambda \partial_{r}^{\varphi} b + \nabla_{\mu}^{\varphi} \cdot \left[ \Lambda, \frac{1}{\varrho} \right] \nabla_{\mu}^{\varphi} P.$$

We use once again Alinhac's good unknown on the buoyancy term. We multiply these equations by $(1 + \epsilon h)$ to get:

$$\begin{aligned} -(1 + \epsilon h) \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P^{\wedge} = & \frac{\mu}{\epsilon} (1 + \epsilon h) \Lambda (\partial_{i}^{\varphi} (\mathbf{U} + \epsilon \mathbf{U})_{j} \partial_{j}^{\varphi} (\mathbf{U} + \epsilon \mathbf{U})_{i}) \\ & + (1 + \epsilon h) \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} (\epsilon \Lambda \eta \partial_{r}^{\varphi}) \nabla_{\mu}^{\varphi} P \\ & + \epsilon (1 + \epsilon h) \nabla_{\mu}^{\varphi} \cdot R_{1}^{\wedge} (\nabla_{\mu} P) + \nabla_{\mu}^{\varphi} \cdot [\Lambda, \frac{1}{\varrho}] \nabla_{\mu}^{\varphi} P \\ & + (1 + \epsilon h) \nabla_{\mu}^{\varphi} \cdot \frac{\epsilon \Lambda \eta}{1 + \epsilon h} \partial_{r} \left( \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P \right) \\ & + (1 + \epsilon h) (\epsilon \Lambda \eta \partial_{r}^{\varphi}) \left( \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P \right) \\ & + \epsilon (1 + \epsilon h) R_{2}^{\wedge} (\nabla_{\mu} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P) + \partial_{r} b^{\wedge} + \epsilon \Lambda \eta \partial_{r}^{\varphi} \partial_{r}^{\varphi} b + \epsilon R_{1}^{\wedge} (\nabla b), \end{aligned}$$

In the case $\Lambda = |D|\Lambda^{s-1}$ (that is, we only differentiate (2.1) in the tangential direction with respect to the boundary of the strip $S_r$), we can apply $\Lambda$ to the boundary conditions of equation (2.1) and the system (2.7) is completed with the boundary conditions

$$\left( \frac{1}{\varrho} \partial_r^{\varphi} P^{\wedge} + \epsilon \frac{1}{\varrho} \Lambda \eta \partial_r^{\varphi} \partial_r^{\varphi} P + \epsilon R_1^{\wedge} (\nabla_{\mu} P) \right) \Big|_{r=0;1} = - \Lambda b |_{r=0;1}.$$

**Step 2: Estimate on $\|\nabla^{\varphi} P^{\wedge}\|_{L^2}$ for $s \ge s_0 + \frac{3}{2}$**

We use (2.7) with $\Lambda := |D|\Lambda^{s-1}$. Note that in this case $[\Lambda, \frac{1}{\varrho}] = 0$ as $\varrho$ only depends on $r$. We use an induction on $s$, with the base case $s \le s_0 + \frac{3}{2}$ given in Proposition 2.1. We would like to apply Lemma 2.5, and therefore we need a control on $\|P^{\wedge}\|_{L^2}$. This is provided by induction, namely we have:

$$P^{\wedge} := \Lambda P + \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \partial_r P = \Lambda P - \epsilon \Lambda \eta \partial_r^{\varphi} P.$$

Thus we have a control on $\|P^{\wedge}\|_{L^2}$ by induction on $s$ (recall that $\Lambda := |D|\Lambda^{s-1}$ and the product estimate (A.1)):

$$\|P^{\wedge}\|_{L^2} \le \frac{1}{\sqrt{\mu}} \|\nabla_{\mu} P\|_{H^{s-1,0}} + \epsilon \|\eta\|_{H^{s,0}} \|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{1}{2},1}}.$$

Note that there is a singularity in $\mu$, however it will be compensated by a factor $\epsilon$ later on.
2.2.3. *Proof of Proposition 2.1.* First, we apply Proposition 2.3 to the system (2.1), with

$$G := \left( \frac{\sqrt{\mu}}{\epsilon}(\mathbf{U} + \epsilon\mathbf{U}) \cdot \nabla^{\varphi}(\underline{V} + \epsilon V), \frac{\mu}{\epsilon}(\mathbf{U} + \epsilon\mathbf{U}) \cdot \nabla^{\varphi} w + b \right)^T,$$

and $\eta_{|r=0} = \eta_{|r=1} = 0$. Then, Lemma C.1 yields

$$\|G\|_{H^{s,k}} \le C (\|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s+1,k+1}} + \sqrt{\mu}\|w\|_{H^{s+1,k+1}}).$$

We thus get the existence and uniqueness of the pressure $P$, as well as the bound

$$\|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{3}{2},k}} \le C (\|\eta\|_{H^{s_0+\frac{3}{2},k+1}} + \|V\|_{H^{s_0+\frac{3}{2},k+1}} + \sqrt{\mu}\|w\|_{H^{s_0+\frac{3}{2},k+1}}),$$

for $0 \le k \le s_0 + \frac{3}{2}$.

Because $\nabla^{\varphi} \cdot (\mathbf{U} + \epsilon\mathbf{U}) = 0$, the pressure $P$ also satisfies (2.3), with the boundary condition (2.2). We use Proposition 2.6 with

$$\left\{
\begin{array}{l}
G^{(1)} := \partial_i^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_j \partial_j^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_i, \\
G^{(2)} := (0, b)^T, \\
G_0 := 0, \\
G_1 := 0.
\end{array}
\right.
$$

Using also Lemma C.1, we directly get (2.4), for $s \ge s_0 + \frac{5}{2}$, $3 \le k \le s$.

2.2.4. *Proof of Proposition 2.2.* We start by defining a pressure, reminiscent of the hydrostatic pressure

$$P_h(t, x, r) := \int_0^r \rho(r')b(t, x, r')dr',$$

so that

$$\frac{1}{\rho} \partial_r^{\varphi} P_h + b = \epsilon \frac{h}{1 + \epsilon h} b.$$

By the product estimate (A.3) and the estimate on $b$ in C.1, we readily get

$$\|\nabla^{\varphi} P_h\|_{H^{s,k}} \le C \|\eta\|_{H^{s+1,k+1}}.$$

Notice that there is a loss of derivative, but the estimate is uniform in $\mu$.

We now define $P_{nh} := P - P_h$, and look for an estimate on $\|\nabla^{\varphi} P_{nh}\|_{H^{s,k}}$ uniform in $\mu$. Together with the estimate on $P_h$, it then shows the desired estimate.

Plugging the previous relation into (2.1) yields

$$(1 + \epsilon h) \nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho(r)} \nabla_{\mu}^{\varphi} P_{nh} = -\frac{\mu}{\epsilon}(1 + \epsilon h) \partial_{i}^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_{j} \partial_{j}^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_{i} \\ - \epsilon \partial_{r} (b \frac{h}{1 + \epsilon h}) - \mu \nabla_{x}^{\varphi} \cdot \frac{1}{\rho} \nabla_{x}^{\varphi} P_{h},$$

with the boundary condition

$$\frac{1}{\rho(r)} \partial_r^{\varphi} P_{nh}|_{r=0;r=1} = -\epsilon b \frac{h}{1 + \epsilon h} \Big|_{r=0;r=1}.$$

Using Proposition 2.6, we get

$$\|\nabla_{\mu}^{\varphi} P_{nh}\|_{H^{s,k}} \le (\sqrt{\mu} + \epsilon)C (\|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s,k}} + \sqrt{\mu}\|w\|_{H^{s,k}}).$$

Recalling that $\nabla_{\mu}^{\varphi} := (\sqrt{\mu}\nabla_{x}^{\varphi} \quad \partial_{r}^{\varphi})$ yields the desired estimate, after dividing by $\sqrt{\mu}$.
We now apply Lemma 2.5 with

$$
\begin{cases}
Q = P^{\Lambda}, \\
f_0 = \epsilon (\Lambda \eta \partial_r^{\varphi}) (\nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P) + \epsilon R_2^{\Lambda} (\nabla_{\mu} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P) + \epsilon \Lambda \eta \partial_r^{\varphi} \partial_r^{\varphi} b + \epsilon R_1^{\Lambda} (\nabla b), \\
\mathbf{F}_1 = \epsilon \frac{1}{\varrho} (\Lambda \eta \partial_r^{\varphi}) \nabla_{\mu}^{\varphi} P + \epsilon R_1^{\Lambda} (\nabla_{\mu} P) + \frac{\epsilon \Lambda \eta}{1 + \epsilon h} \partial_r (\frac{1}{\varrho} \nabla_{\mu}^{\varphi} P), \\
\mathbf{F}_2 = b^{\Lambda}, \\
f_3 = - \frac{\sqrt{\mu}}{\epsilon} \Lambda^{s-1} (\partial_i^{\varphi} (\mathbf{U} + \epsilon \mathbf{U})_j) \partial_j^{\varphi} (\mathbf{U} + \epsilon \mathbf{U})_i.
\end{cases}
$$

Note that the boundary conditions of (2.7) and the fact that $\Lambda\eta$ is zero on the top and bottom ensures the boundary conditions required in order to use Lemma 2.5.
We now need to estimate each of those terms. In what follows, $C_0$ denotes a constant depending only on $\|\eta\|_{H^{s_0+\frac{3}{2},3}}$, $\|V\|_{H^{s_0+\frac{3}{2},2}}$, $\sqrt{\mu}w\|_{H^{s_0+\frac{3}{2},2}}$.

(0) For the first term of $f_0$, we use the product estimate (A.1) to get

$$
\| (\epsilon \Lambda \eta \partial_r^{\varphi}) \left( \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P \right) \|_{L^2} \le C_0 \epsilon \|\eta\|_{H^{s,0}} \| \nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P \|_{H^{s_0+\frac{3}{2},2}}
$$

We use the equation (2.1) to replace $\nabla_{\mu}^{\varphi} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi} P$ by the non-linear and buoyancy
terms. Lemma C.1 together with the product estimate (A.1) then yields

$$
\| (\epsilon \Lambda \eta \partial_r^{\varphi}) (\nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P) \|_{L^2} \le C \epsilon \|\eta\|_{H^{s,0}} (\|\eta\|_{H^{s_0+\frac{3}{2},2}} + \|V\|_{H^{s_0+\frac{3}{2},2}} + \sqrt{\mu} \|w\|_{H^{s_0+\frac{3}{2},2}}).
$$

For the second term of $f_0$, we use the estimate (B.6):

$$
\begin{align*}
\| R_2^{\Lambda} (\nabla_{\mu} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P) \|_{L^2} &\le C_0 \|\eta\|_{H^{sV(s_0+\frac{5}{2}),3}} \|\nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P\|_{H^{(s-1)V(s_0+\frac{3}{2}),2}} \\
&\quad + C_0 \|\eta\|_{H^{sV(s_0+\frac{5}{2}),3}}^2 \|\nabla_{\mu}^{\varphi} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{1}{2},1}}.
\end{align*}
$$

We use equation (2.1) together with Lemma C.1 to control the first term. We use Proposition 2.1 to control the second term, so that

$$
\| R_2^{\Lambda} (\nabla_{\mu} \cdot \frac{1}{\rho} \nabla_{\mu}^{\varphi} P) \|_{L^2} \le C ( \| \eta \|_{H^{sV(s_0+\frac{5}{2}),3}} + \| V \|_{H^{sV(s_0+\frac{5}{2}),3}} + \sqrt{\mu} \| w \|_{H^{sV(s_0+\frac{5}{2}),3}} + \| \eta \|_{H^{sV(s_0+\frac{5}{2}),3}} ).
$$

For the third term in $f_0$, we use product estimates (A.1) together with the esti-
mate on the buoyancy term in Lemma C.1:

$$
\| \Lambda \eta \partial_r^\varphi \partial_r^\varphi b \|_{L^2} \le C ( \| \eta \|_{H^{s,0}} + \| b \|_{H^{s_0+\frac{5}{2},3}} ) \le C \| \eta \|_{H^{s,0}}.
$$

For the last term in $f_0$, we use (B.12):

$$
R_1^{\Lambda}(\partial_r^{\varphi} b) \le C \|\eta\|_{H^{sV(s_0+\frac{5}{2}),3}} \|b\|_{H^{sV(s_0+\frac{5}{2}),1}},
$$

and conclude with Lemma C.1. We can eventually write

$$
\| f_0 \|_{L^2} \le \epsilon C ( \| \eta \|_{H^{sV(s_0+\frac{5}{2}),3}} + \| V \|_{H^{sV(s_0+\frac{5}{2}),3}} + \sqrt{\mu} \| w \|_{H^{sV(s_0+\frac{5}{2}),3}} ).
$$

(1) For the first term of $\mathbf{F}_1$, we use the product estimate (A.1) to get

$$
\| \frac{1}{\rho} ( \epsilon \Lambda \eta \partial_r^{\varphi} ) \nabla_{\mu}^{\varphi} P \| \le \epsilon C \| \eta \|_{H^{s,0}} \| \nabla_{\mu}^{\varphi} P \|_{H^{s_0+\frac{3}{2},2}}.
$$
### 3. A PRIORI ENERGY ESTIMATES

The goal of this section is to derive energy estimates for smooth solutions of (E). In Subsection 3.1, we define the energy functional that we will use. One main concern is that, roughly speaking, we only control $\sqrt{\mu}w$ in the energy. Therefore, any source term containing a factor $w$ has to either cancel out (with a symmetry argument, see the proof of Proposition 3.4), or be multiplied by a factor $\sqrt{\mu}$.

In Subsection 3.2, we derive a first energy estimate, with loss of derivative. In Subsection 3.3, we use Alinhac's good unknown to improve this result into an energy estimate without loss of derivatives. Note that this improved estimate relies on the one derived in Subsection 3.2, for the low regularity case $s \le s_0 + \frac{3}{2}$.

**3.1. The energy functional.** We define the energy that we will use in the following definition. Note that there is a low regularity part and a high regularity part.

**Definition 3.1.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{3}{2}$. Let $(V, w, \eta) \in L^\infty([0, T], (H^s)^{d+2})$ for a certain $T > 0$. Let

$$ (3.1) \qquad \mathcal{D} := \{\Lambda^{s-l} \partial_r^l, 1 \le l \le s\} \cup \{|D|\Lambda^{s-1}\} $$

be the set of operators that we will use in order to define the energy functional. We define, for $0 \le t < T$:

$$ \mathcal{E}_0(t) := \|V(t, \cdot)\|_{H^{s_0+\frac{3}{2},2}}^2 + \mu \|w(t, \cdot)\|_{H^{s_0+\frac{3}{2},2}}^2 + \|\eta(t, \cdot)\|_{H^{s_0+\frac{3}{2},2}}^2, $$

and

$$ (3.2) \qquad \mathcal{E}(t) := \mathcal{E}_0 + \sum_{\Lambda \in \mathcal{D}} \|\sqrt{\rho(1 + \epsilon h)} V^{\Lambda}(t, \cdot)\|_{L^2}^2 + \mu \|\sqrt{\rho(1 + \epsilon h)} w^{\Lambda}(t, \cdot)\|_{L^2}^2 + \|\sqrt{\rho'} \Lambda \eta(t, \cdot)\|_{L^2}^2, $$

where $V^\Lambda$ and $w^\Lambda$ are Alinhac's good unknowns associated with the operator $\Lambda$, namely

$$ V^{\Lambda} := \Lambda V + \frac{\Lambda \eta}{1 + \epsilon h} \partial_r V, \qquad w^{\Lambda} := \Lambda w + \frac{\Lambda \eta}{1 + \epsilon h} \partial_r w. $$

In the next lemma we prove that this energy is equivalent to the $H^s$-norm of the unknowns $(V, w, \eta)$.

**Lemma 3.2.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{3}{2}$. Let $(V, w, \eta) \in L^\infty([0, T], (H^s)^{d+2})$ for a certain $T > 0$. Recall the definition (3.1) of $\mathcal{D}$. We make the assumptions (H.2), (H.5) as well as

$$ \|\eta\|_{H^s} \le M, $$

for $M > 0$ a constant. Then there exists a constant $C > 0$ depending only on $M$ such that

$$ \frac{1}{C} (\|V\|_{H^s}^2 + \mu \|w\|_{H^s}^2 + \|\eta\|_{H^s}^2) \le \mathcal{E} \le C (\|V\|_{H^s}^2 + \mu \|w\|_{H^s}^2 + \|\eta\|_{H^s}^2). $$

*Proof.* Recall that $H^s(S_r) = H^{s,k}(S_r)$ when $k = s \in \mathbb{N}$. The assumptions (H.2), (H.5) yield the existence of a constant $c > 0$ such that $\frac{1}{c} \le \rho(1 + \epsilon h) \le c$ and $\frac{1}{c} \le \rho' \le c$. Using this, as well as (B.15) on $V$ and $\sqrt{\mu}w$ yields the result. $\square$

**3.2. Estimates with loss of regularity.** The main result of this subsection is the energy estimate with loss of derivatives in Proposition 3.3. Note that in the estimate of Proposition 3.3, the functional $\mathcal{E}_0$ only controls $H^{s_0+\frac{3}{2}}$-norms of the unknowns, whereas it is bounded by a constant depending on the $H^{s_0+\frac{5}{2}}$-norms of the unknowns.

Let $\Lambda$ is a differential operator of the form $\Lambda^{s-l} \partial_r^l$ for $0 \le l \le s$. Applying $\Lambda$ to (E) formally yields

$$ (3.3) \qquad \left\{ \begin{array}{ll} \partial_t \dot{V} = R_1, & (1) \\ \mu \partial_t \dot{w} = R_2, & (2) \\ \partial_t \dot{\eta} = R_3, & (3) \end{array} \right. $$
For the second term of $\mathbf{F}_1$, we use estimate (B.12)

$$ \|R_1^{\mathbf{A}}(\nabla_{\mu}^{\varphi} P)\|_{L^2} \le C \|\eta\|_{H^{s-1,0}} + C \|\nabla_{\mu}^{\varphi} P\|_{H^{s-1,0}} . $$

For the third term of $\mathbf{F}_1$, we use the product estimate (A.1):

$$ \| \frac{\partial \eta}{1 + \epsilon h} \partial_{\rho} \left( \frac{1}{\rho} \nabla_{\mu}^{\varphi} P \right) \| \le C \| \eta \|_{H^{s,0}} \| \nabla_{\mu}^{\varphi} P \|_{H^{s_0 + \frac{3}{2}, 2}}. $$

We can thus write, using the estimate of Proposition 2.1:

$$ \|\mathbf{F}_1\|_{L^2} \le \epsilon C (\|\eta\|_{H^{s_0+\frac{5}{2},3}} + \|V\|_{H^{s_0+\frac{3}{2},2}} + \sqrt{\mu}\|w\|_{H^{s_0+\frac{3}{2},2}} + \|\nabla^{\varphi} P\|_{H^{s-1,0}}). $$

(2) For the term of $\mathbf{F}_2$, equation (B.4) together with the estimate of the buoyancy term in Lemma C.1 yields

$$ \|\mathbf{F}_2\|_{L^2} \le C \|\eta\|_{H^{s,2}}. $$

(3) For the term of $f_3$, Lemma C.1 directly yields:

$$ \|f_3\|_{L^2} \le \sqrt{\mu} C (\|\eta\|_{H^{s,2}} + \|V\|_{H^{s,2}} + \sqrt{\mu}\|w\|_{H^{s,2}}). $$

Thanks to Lemma 2.5, we can eventually write:

$$ \|\nabla_{\mu} P^{\mathbf{A}}\|_{L^2} \le C_0 (\sqrt{\|f_0\|_{L^2} \|P^{\mathbf{A}}\|_{L^2}} + \|\mathbf{F}_1\|_{L^2} + \|\mathbf{F}_2\|_{L^2} + \|f_3\|_{L^2}). $$

From the control on $f_0$ above, we deduce

$$ \begin{aligned} \|\nabla_{\mu} P^{\mathbf{A}}\|_{L^2} &\le \frac{\epsilon}{\sqrt{\mu}} C \sqrt{ (\|\eta\|_{H^{sV(s_0+\frac{5}{2}),3}} + \|V\|_{H^{sV(s_0+\frac{3}{2}),3}} + \sqrt{\mu}\|w\|_{H^{sV(s_0+\frac{5}{2}),3}} ) \|\nabla_{\mu} P\|_{H^{s-1,0}} } \\ &\quad + \epsilon^2 C \|\eta\|_{H^{s,0}} \|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{3}{2},1}} + C \|\eta\|_{H^{s,2}} \\ &\quad + \epsilon C ( \|\eta\|_{H^{s_0+\frac{5}{2},3}} + \|V\|_{H^{s_0+\frac{3}{2},2}} + \sqrt{\mu}\|w\|_{H^{s_0+\frac{3}{2},2}} + \|\nabla_{\mu}^{\varphi} P\|_{H^{s-1,0}} ) \\ &\quad + \sqrt{\mu} C ( \|\eta\|_{H^{s,2}} + \|V\|_{H^{s,2}} + \sqrt{\mu}\|w\|_{H^{s,2}} ). \end{aligned} $$

We now use (B.17) and simplify the above expression to get

$$ \begin{aligned} \|\nabla_{\mu}^{\varphi} P\|_{H^{s,0}} &\le C \frac{\epsilon}{\sqrt{\mu}} \sqrt{ (\|\eta\|_{H^{sV(s_0+\frac{5}{2}),3}} + \|V\|_{H^{sV(s_0+\frac{3}{2}),3}} + \sqrt{\mu}\|w\|_{H^{sV(s_0+\frac{5}{2}),3}} ) \|\nabla_{\mu} P\|_{H^{s-1,0}} } \\ &\quad + C ( \|\eta\|_{H^{sV(s_0+\frac{5}{2}),3}} + \|V\|_{H^{sV(s_0+\frac{3}{2}),3}} + \sqrt{\mu}\|w\|_{H^{sV(s_0+\frac{5}{2}),3}} ), \end{aligned} $$

where we used Proposition 2.1 to estimate $\|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{3}{2},1}}$.

By induction on $s$, as the base case $\|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{3}{2},2}}$ is controlled thanks to Proposition 2.1:

$$ \|\nabla_{\mu}^{\varphi} P\|_{H^{s,0}} \le C (\|\eta\|_{H^{sV(s_0+\frac{5}{2}),3}} + \|V\|_{H^{sV(s_0+\frac{3}{2}),3}} + \sqrt{\mu}\|w\|_{H^{sV(s_0+\frac{5}{2}),3}}), $$

where $C$ depends only on $M, \mathcal{M}, h_k, h^*$.

### Step 3: Control of higher vertical derivatives

We now want an estimate on $\|\nabla_{\mu} P\|_{H^{s,k}}$ for $k \ge 1$. To this end, we start by noticing

$$ (2.9) \qquad \|\nabla_{\mu}^{\varphi} P\|_{H^{s,k}} \le c \|\nabla_{\mu}^{\varphi} P\|_{H^{s,k-1}} + c \|\Lambda^{s-k} \partial_r^{k-1} \partial_r^{\varphi} P\|_{L^2}, $$

where $c$ only depends on $s$ and $k$. As for the low regularity case, we aim to use the equation (2.1) to control the vertical derivatives. We use an induction on $s$ and $k$. More precisely, the base case $k=0$ and any $s \ge 0$ is given by the previous step (and Proposition 2.1 for the low regularity case), and the base case $s \le s_0 + \frac{3}{2}$, $k \le s$ is given by Proposition 2.1. Therefore, we assume from now on that $\|\nabla_{\mu}^{\varphi} P\|_{H^{s',k'}}$ is controlled
where $(\dot{V}, \dot{w}, \dot{\eta}) := \mathbb{A}(V, w, \eta)$ and

$$R_1 := -\mathbb{A}((\underline{V} + \epsilon V) \cdot \nabla_x V + \frac{1}{\underline{\rho}} \nabla_x^\varphi P), \quad R_3 := -\mathbb{A}((\underline{V} + \epsilon V) \cdot \nabla_x \eta + w),$$

$$R_2 := -\mu \mathbb{A}((\underline{V} + \epsilon V) \cdot \nabla_x w) - \mathbb{A}\left(\frac{1}{\underline{\rho}} \partial_r^\varphi P + b\right).$$

We make the assumptions that the unknowns $(V, w, \eta)$ satisfy

$$(H.10) \qquad \|\eta(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} + \|V(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} + \sqrt{\mu}\|w(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} \le M_0,$$

with $M_0 > 0$ a constant.

We also make the assumption that the stratification $\underline{\rho}$ and the shear flow $\underline{V}$ satisfy

$$(H.11) \qquad |\underline{\rho}|_{W_r^{s_0+\frac{5}{2}, \infty}} + |\underline{V}|_{W_r^{s_0+\frac{5}{2}, \infty}} \le \underline{M},$$

with $\underline{M} > 0$ a constant. We now state the energy estimates with a loss of derivatives.

**Proposition 3.3.** Let $s_0 > d/2$. We assume that there exists $T > 0$ and $(V, w, \eta) \in L^\infty([0, T], (H^{s_0+\frac{5}{2}})^{d+2})$ satisfying (E) as well as (H.2), (H.5), (H.10), (H.11). Then there exists $C > 0$ a constant depending only on $M_0$, $\underline{M}$ such that:

$$\frac{d}{dt} \mathcal{E}_0(t) \le C \left( \|\eta(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} + \|V(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} + \sqrt{\mu} \|w(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} \right)^2.$$

The proof consists in estimating the terms $R_1, R_2, R_3$ with product, commutators and composition estimates, and Proposition 2.1. This is a standard strategy and we refer to [DLS20, Theorem 1] and [BD24, Lemma 3.6] where very similar systems are thoroughly studied.

**3.3. Estimates without loss of regularity.** The main result of this subsection is the energy estimates without loss of derivatives provided in Proposition 3.4. As opposed to the estimate in Proposition 3.3, the following energy estimate is without loss of derivatives. However, it requires the additional assumption $|\underline{V}'|_{L^\infty} \le \sqrt{\mu}$, see Remark 3.8 for a more detailed comparison. The proof relies on the use of Alinhac's good unknowns. Before proving the proposition, we derive the quasilinearized system satisfied by Alinhac's good unknowns in Lemma 3.5.

**Proposition 3.4.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{5}{2}$. We make the assumption $\epsilon \le \sqrt{\mu}$ as well as

$$|\underline{V}'|_{L^\infty} \le \sqrt{\mu}.$$

Assume that there exists $T > 0$ and $(V, w, \eta) \in L^\infty([0, T], (H^s)^{d+2})$ satisfying (E) as well as (H.2), (H.5), (H.12), (H.13).

Then there exists $C > 0$ a constant depending only on $\underline{M}, M$ such that

$$\frac{d}{dt} \mathcal{E}(t) \le C \mathcal{E}(t).$$

Before proving Proposition 3.4, let us first exhibit the quasilinear structure of the system. Let $s \ge s_0 + \frac{5}{2}$. Let $\mathbb{A}$ be a differential operator of the form $\mathbb{A} := \Lambda^{s-k} \partial_r^k$ for $k \ge 1$ or $|D|\Lambda^{s-1}$. Applying $\mathbb{A}$ to (E) formally yields

$$(3.4) \qquad \left\{ \begin{array}{l} \partial_t V^\Lambda + (\underline{V} + \epsilon V) \cdot \nabla_x V^\Lambda + \frac{1}{\underline{\rho}} \nabla_x^\varphi P^\Lambda = \epsilon R_1 + R_2, \quad (1) \\ \mu(\partial_t w^\Lambda + (\underline{V} + \epsilon V) \cdot \nabla_x w^\Lambda) + \frac{1}{\underline{\rho}} \partial_r^\varphi P^\Lambda + \frac{\underline{\rho}'}{\underline{\rho}} \dot{\eta} = F - \epsilon \mathbb{A} \bar{b} + \epsilon R_3 + \mu R_4, \quad (2) \\ \partial_t \dot{\eta} + (\underline{V} + \epsilon V) \cdot \nabla_x \dot{\eta} - w^\Lambda = R_5, \quad (3) \\ \nabla_x^\varphi \cdot V^\Lambda + \partial_r^\varphi w^\Lambda = \frac{\nabla_x \eta}{1 + \epsilon h} \cdot \partial_r \underline{V}^\Lambda + \epsilon R_6, \quad (4) \end{array} \right.$$
for either $s' < s$, $k' \le s'$ and for $s' = s$, $k' < k$.

The first term in the previous expression is controlled by induction. For the second term, we use the equation (2.1): Let $\Lambda := \Lambda^{s-k} \partial_r^{k-1}$, we apply $\Lambda$ to (2.1). Recall (2.7) derived in Step 1. We write $R$ for the right-hand side, multiply by $1 + \epsilon h$ and expand the left-hand side to get:

$$ (2.10) \qquad \left( \frac{1 + \mu |\nabla_x \eta|^2}{\rho(1 + \epsilon h)^2} \right) \partial_r^2 P^\Lambda = -\mu \nabla_x \cdot \left( \frac{(1 + \epsilon h)}{\rho} \nabla_x P^\Lambda + \epsilon \frac{\nabla_x \eta}{\rho} \partial_r P^\Lambda \right) + \mu \partial_r \left( \epsilon \frac{\nabla_x \eta \cdot \nabla_x P}{\rho} \right) \\ \qquad \qquad + \partial_r \left( \frac{1 + \mu \epsilon^2 |\nabla_x \eta|^2}{\rho(1 + \epsilon h)} \right) \partial_r P^\Lambda + (1 + \epsilon h)R. $$

We take the $L^2$-norm of the previous expression. Recall that $1 + \epsilon h \le h^*$. We use the product estimate (A.1), and the composition estimate (A.10) for $\frac{1}{1+\epsilon h} = 1 - \frac{\epsilon h}{1+\epsilon h}$. We get:

$$ \|\partial_r^2 P^\Lambda\|_{L^2} \le C(\|R\|_{L^2} + \|\nabla_\mu^\varphi P^\Lambda\|_{H^{1,0}}) $$

by using that $\frac{h^*}{\rho_0} \frac{1}{\rho(1+\epsilon h)} \ge c_* h_* > 0$. The estimation of the terms in $R$ is very similar to the previous step, except for two terms. The first one is $\nabla_\mu^\varphi \cdot R_1^\Lambda(\nabla_\mu P)$. Recall that, up to a harmless factor, $R_1^\Lambda(\nabla_\mu P)$ is the symmetric commutator $[\Lambda; (\partial_\varphi)^T, \nabla_\mu^\varphi P]$ so that we write:

$$ \nabla_\mu^\varphi \cdot [\Lambda; (\partial_\varphi)^T, \nabla_\mu^\varphi P] = [\nabla_\mu^\varphi \Lambda; (\partial_\varphi)^T, \nabla_\mu^\varphi P] + [\nabla_\mu^\varphi, \nabla_\mu^\varphi P] \Lambda (\partial_\varphi)^T + [\nabla_\mu^\varphi, (\partial_\varphi)^T] \Lambda \nabla_\mu^\varphi P, $$

and commutator estimates (A.7) (in the case $k \ge 2$) and (A.9) (in the case $k = 1$) yield:

$$ \|\nabla_{\mu}^{\varphi} \cdot R_{1}^{\Lambda}(\nabla_{\mu} P)\|_{L^{2}} \leq C(\|\nabla_{\mu}^{\varphi} P\|_{H^{s_{0}+\frac{3}{2}}} + \|\nabla_{\mu}^{\varphi} P\|_{H^{s-1,k\Lambda(s-1)}}). $$

The second term is $\nabla_\mu^\varphi \cdot [\Lambda, \frac{1}{\rho}] \nabla_\mu^\varphi P$. Note however that, as $\rho$ only depends on $r$, we get

$$ [\Lambda, \frac{1}{\rho}] = [\partial_r^{k-1}, \frac{1}{\rho}] \Lambda^{s-k}, $$

and the commutator estimate (A.6) allows us to conclude

$$ \|\nabla_{\mu}^{\varphi} \cdot [\Lambda, \frac{1}{\rho}] \nabla_{\mu}^{\varphi} P\|_{L^2} \le C \|\nabla_{\mu}^{\varphi} P\|_{H^{s-1, k-1}}. $$

We now turn to the control of $\|\nabla_\mu^\varphi P^\Lambda\|_{H^{1,0}}$. We apply $\Lambda$ to (B.2) to get

$$ \|\nabla_\mu^\varphi P^\Lambda\|_{H^{1,0}} \le C_0 \|\nabla_\mu^\varphi P\|_{H^{s,k-1}} + C_0 \epsilon \|\eta\|_{H^{s,k-1}} \|\nabla_\mu^\varphi P\|_{H^{s_0+\frac{3}{2},2}} \\ + C_0 \epsilon \|\eta\|_{H^{s+(s_0+\frac{3}{2}),3\nu(k\Lambda(s-1))}} \|\nabla_\mu^\varphi P\|_{H^{(s-1)[s_0+\frac{3}{2}],(k-1)\Lambda(s-2)}} $$

where we used (B.3) (for $k \ge 2$) or (B.12) (for $k = 1$) to control the term $R_1^\Lambda(\nabla_\mu P)$ from (B.2). Therefore, by induction, we have a bound for $\|\nabla_\mu^\varphi P^\Lambda\|_{H^{1,0}}$. That is to say, we have a bound for $\|\partial_r \partial_r^\varphi P^\Lambda\|_{L^2}$.

Now recall

$$ P^\Lambda := \Lambda^{s-k} \partial_r^{k-1} P + \frac{\Lambda^{s-k} \partial_r^{k-1} \eta}{1 + \epsilon h} \partial_r P, $$

so that, by applying $\partial_r$ and using triangular inequality:

$$ \|\partial_r \Lambda^{s-k} \partial_r^{k-1} \partial_r^\varphi P\|_{L^2} \le \|\partial_r \partial_r^\varphi P^\Lambda\|_{L^2} + C \|\eta\|_{H^{s-1,2}} \|\nabla_\mu P\|_{H^{s_0+\frac{3}{2},1}}. $$

Finally, note that:

$$ \|\partial_r \partial_r^\varphi P^\Lambda\|_{L^2} \le C \|\partial_r^2 P^\Lambda\|_{L^2}, $$
where $V^{\Lambda}$, $w^{\Lambda}$, $\underline{V}^{\Lambda}$, $P^{\Lambda}$ are Alinhac's good unknowns defined through (B.2), $\dot{\eta} := \Lambda\eta$ and

$$
\begin{equation} \tag{3.5}
\left\{
\begin{aligned}
\tilde{b} &:= \frac{1}{\epsilon} \left( b - \frac{\varrho'}{\varrho} \eta \right), \\
R_1 &:= \Lambda \eta \partial_r^\varphi \partial_t^\varphi V + R_1^\Lambda (\partial_t V) - (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot (\Lambda \eta \partial_r^\varphi \nabla^\varphi V + R_1^\Lambda (\nabla V)), \\
R_2 &:= [\Lambda, \underline{\mathbf{U}} + \epsilon \mathbf{U}] \cdot \nabla^\varphi V + [\Lambda, \frac{1}{\varrho}] \nabla_x^\varphi P + \epsilon \frac{1}{\varrho} (\Lambda \eta \partial_r^\varphi \nabla_x^\varphi P + R_1^\Lambda (\nabla P)), \\
R_3 &:= \Lambda \eta \partial_r^\varphi \partial_t^\varphi w + R_1^\Lambda (\partial_t w) - (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot (\Lambda \eta \partial_r^\varphi \nabla^\varphi w + R_1^\Lambda (\nabla w)), \\
&\quad + \frac{1}{\varrho} (\Lambda \eta \partial_r^\varphi \partial_t^\varphi P + R_1^\Lambda (\nabla P)), \\
R_4 &:= [\Lambda, \underline{\mathbf{U}} + \epsilon \mathbf{U}] \cdot \nabla^\varphi w, \\
R_5 &:= [\Lambda, \underline{V} + \epsilon V] \cdot \nabla_x \eta + \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \partial_r w, \\
R_6 &:= \Lambda \eta \partial_r^\varphi (\nabla^\varphi \cdot \mathbf{U}) + R_2^\Lambda (\nabla \mathbf{U}) + \Lambda \eta \partial_r^\varphi (\nabla^\varphi \cdot \underline{V}) + R_2^\Lambda (\nabla \underline{V}), \\
F &:= [\Lambda, \frac{1}{\varrho}] \partial_r^\varphi P + [\Lambda, \frac{\varrho'}{\varrho}] \eta.
\end{aligned}
\right.
\end{equation}
$$

In order to justify this quasilinearized system in the following lemma, we first make the assumptions that the unknowns $(V, w, \eta)$ satisfy

$$
(H.12) \qquad \|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu} \|w(t, \cdot)\|_{H^s} \le M,
$$

with $M > 0$ a constant.

We also make the assumption that the stratification $\varrho$ and the shear flow $\underline{V}$ satisfy

$$
(H.13) \qquad |\varrho|_{W_r^{s+1,\infty}} + |\underline{V}|_{W_r^{s+1,\infty}} \le \underline{M},
$$

with $\underline{M} > 0$ a constant.

**Lemma 3.5.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s > s_0 + \frac{5}{2}$. We make the assumption $\epsilon \le \sqrt{\mu}$. Assume that there exists $T > 0$ and $(V, w, \eta) \in L^\infty([0, T], (H^s)^{d+2})$ satisfying (E) as well as (H.2), (H.5), (H.12), (H.13).

Then there exists $C > 0$ a constant depending only on $\underline{M}$, $M$ such that

$$
\left\{
\begin{aligned}
\|\tilde{b}\|_{H^s} &\le C (\|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu}\|w(t, \cdot)\|_{H^s}), \\
\|(R_1, R_2, R_3, R_4, R_5, R_6)\|_{L^2} &\le C (\|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu}\|w(t, \cdot)\|_{H^s}), \\
\|F\|_{L^2} &\le C (\|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu}\|w(t, \cdot)\|_{H^s}).
\end{aligned}
\right.
$$

Moreover, if $\Lambda = |D|\Lambda^{s-1}$ then $F = 0$.

*Proof.* First note that the gradient of pressure $\nabla^\varphi P$ is well defined and has the same regularity as $V, w$, and $\eta$
according to Proposition 2.1.

To obtain estimates on the unknowns, we apply $\Lambda := \Lambda^{s-k} \partial_r^k$ or $|D|\Lambda^{s-1}$ to (E'). Recall the definition (B.2) of Alinhac's good unknowns $(V^{\Lambda}, w^{\Lambda}, P^{\Lambda})$. we now apply $\Lambda$ to (E'); using (B.3) yields

$$
(3.6) \qquad \left\{
\begin{array}{l}
\displaystyle \partial_t^\varphi V^\Lambda + (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V^\Lambda + \frac{1}{\varrho} \nabla_x^\varphi P^\Lambda = \epsilon R_1 + R_2 \qquad (1) \\[2ex]
\displaystyle \mu (\partial_t^\varphi w^\Lambda + (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \nabla^\varphi w^\Lambda) + \frac{1}{\varrho} \partial_r^\varphi P^\Lambda + \frac{\varrho'}{\varrho} \dot{\eta} = F - \epsilon \Lambda \tilde{b} + \epsilon R_3 + \mu R_4, \quad (2) \\[2ex]
\displaystyle \partial_t \dot{\eta} + (\underline{V} + \epsilon V) \cdot \nabla_x \dot{\eta} - w^\Lambda = R_5 \qquad (3) \\[2ex]
\displaystyle \nabla_x^\varphi V^\Lambda + \partial_r^\varphi w^\Lambda = \frac{\dot{\eta}}{1 + \epsilon h} \underline{V}' + \epsilon R_6, \qquad (4)
\end{array}
\right.
$$

$$
\partial_t \dot{\eta} + (\underline{V} + \epsilon V) \cdot \nabla_x \dot{\eta} - w^\Lambda = R_5 \quad (3)
$$

$$
\nabla_x^\varphi V^\Lambda + \partial_r^\varphi w^\Lambda = \frac{\dot{\eta}}{1 + \epsilon h} \underline{V}' + \epsilon R_6, \quad (4)
$$

with the notations defined in (3.5). Note that using the semi-Lagrangian property (1.6) of the isopycnal coordi-
nates, we get exactly (3.4).

In what follows, $c$ denotes a constant that only depends on $s_0$, $s$. For simplicity we write

$$
\bar{M} := \|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu} \|w(t, \cdot)\|_{H^s}
$$
as the factor that appears is controlled by $||\eta||_{H^{s_0+\frac{3}{2},2}}$.

We eventually get a control on $||\Lambda^{s-k} \partial_r^k \partial_r^\varphi P||_{L^2}$ so that by (2.9), we have the desired control on $||\nabla_\mu^\varphi P||_{H^{s,k}}$.

□

We now state a similar result, where the bound is uniform in $\mu$ but there is a loss of derivatives due to the use of the hydrostatic pressure in the proof. Recall indeed that the estimate in Proposition 2.4 concerns $\nabla_\mu^\varphi P := (\sqrt{\mu}\nabla_x^\varphi P, \partial_r^\varphi P)^T$, whereas the following estimate is on $\nabla^\varphi P := (\nabla_x^\varphi P, \partial_r^\varphi P)^T$.

**Proposition 2.6.** With the same assumptions and notations as in the previous lemma, we have the bound

$$ (2.11) \qquad \|\nabla^\varphi P\|_{H^{s,k}} \le C \left( 1 + \frac{\epsilon}{\sqrt{\mu}} \right) \left( \|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s+1,k+1}} + \sqrt{\mu} \|w\|_{H^{s+1,k+1}} \right). $$

The proof is very similar to the proof of Proposition 2.2 and we omit it.

### 3. A PRIORI ENERGY ESTIMATES

The goal of this section is to derive energy estimates for smooth solutions of (E). In Subsection 3.1, we define the energy functional that we will use. One main concern is that, roughly speaking, we only control $\sqrt{\mu}w$ in the functional. Therefore, any source term containing a factor $w$ has to be either set to zero (with a symmetry argument, see the proof of Proposition 3.4), or multiplied by a factor $\sqrt{\mu}$.

In Subsection 3.2, we derive a first energy estimate, with loss of derivative. In Subsection 3.3, we use Alinhac's good unknown to improve this result into an energy estimate without loss of derivatives. Note however that this improved estimate relies on the one derived in Subsection 3.2, for the low regularity case $s \le s_0 + \frac{3}{2}$.

**3.1. The energy functional.** We define the energy that we will use in the following definition. Note that there is a low regularity part and a high regularity part.

**Definition 3.1.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{3}{2}$. Let $(V, w, \eta) \in L^\infty([0, T), (H^s)^{d+2})$ for a certain $T > 0$.

Let

$$ \mathcal{D} := \{\Lambda^{s-l} \partial_r^l, 1 \le l \le s\} \cup \{|D| \Lambda^{s-1}\} $$

be the set of operators that we will use in order to define the energy functional. We define, for $0 \le t < T$:

$$ \mathcal{E}_0(t) := \|V(t, \cdot)\|_{H^{s_0+\frac{3}{2},2}}^2 + \mu \|w(t, \cdot)\|_{H^{s_0+\frac{3}{2},2}}^2 + \|\eta(t, \cdot)\|_{H^{s_0+\frac{3}{2},2}}^2, $$

and

$$ (3.1) \qquad \mathcal{E}(t) := \mathcal{E}_0 + \sum_{\Lambda \in \mathcal{D}} \| \sqrt{\varrho(1+\epsilon h)} V^{\Lambda}(t, \cdot) \|_{L^2}^2 + \mu \| \sqrt{\varrho(1+\epsilon h)} w^{\Lambda}(t, \cdot) \|_{L^2}^2 + \| \sqrt{\varrho'} \Lambda \eta(t, \cdot) \|_{L^2}^2, $$

where $V^\Lambda$ and $w^\Lambda$ are Alinhac's good unknowns associated with the operator $\Lambda$, namely

$$ V^{\Lambda} := \Lambda V + \frac{\Lambda \eta}{1 + \epsilon h} \partial_r V, \quad w^{\Lambda} := \Lambda w + \frac{\Lambda \eta}{1 + \epsilon h} \partial_r w. $$
so that we now bound the $H^s$-norm of $\tilde{b}$ and the $L^2$-norm of the other terms from above by $C\tilde{M}$.

**Estimate for $\tilde{b}$:**

Let us write

$$F(r, y) := \frac{1}{\epsilon^2} \left( 1 - \frac{\varrho(r+y)}{\varrho(r)} - \frac{\varrho'(r)}{\varrho(r)} y \right),$$

so that $\tilde{b} = F(r, \epsilon\eta)$. Now note that the assumed regularity on $\varrho$ implies that $F$ is at least $W^{3,\infty}$.
As $F(0) = \partial_y F = 0$, we can write, by Taylor-Laplace formula:

$$F(r, y) = \frac{1}{\epsilon^2} \frac{1}{2} y^2 \int_0^1 \partial_y^2 F(r, \alpha y) \alpha d\alpha.$$

Thus the composition estimate (A.11) yields

$$\|\tilde{b}\|_{H^s} \le C \|\eta\|_{H^s} \le C\tilde{M}.$$

Note in particular that there is no singularity in $\epsilon$.

**Estimate for $R_1$ and $R_3$:**

The terms in $R_1$ and $R_3$ have the same form and we only treat $R_1$. For the first and third terms, we write

$$\Lambda\eta\partial_t^\varphi\partial_t^\varphi V + (\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot \Lambda\eta\partial_r^\varphi\nabla^\varphi V = \Lambda\eta\partial_r^\varphi(\partial_t^\varphi V + (\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot \nabla^\varphi V) - \Lambda\eta\partial_r^\varphi(\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot \nabla^\varphi V.$$

Taking the $L^2$-norm and using the product estimate (A.1), we can bound the right-hand side of the previous expression from above by

$$c \|\eta\|_{H^{s,k}} \left( \|\partial_r^\varphi (\partial_t^\varphi V + (\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot \nabla^\varphi V)\|_{H^{s+\frac{1}{2},1}} + \|\partial_r^\varphi (\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot \nabla^\varphi V\|_{H^{s+\frac{1}{2},1}} \right).$$

For the first term, we use the equation (E') together with the product estimate (A.2):

$$\|\partial_r^\varphi (\partial_t^\varphi V + (\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot \nabla^\varphi V)\|_{H^{s+\frac{1}{2},1}} \le C \left\| \frac{1}{\varrho} \nabla_x^\varphi P \right\|_{H^{s+\frac{3}{2},2}} \le C\tilde{M}.$$

We also used the estimate on the pressure given by (2.5).

The second term is controlled by product estimates (A.2) and (A.3), namely

$$\|\partial_r^\varphi (\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot \nabla^\varphi V\|_{H^{s+\frac{1}{2},1}} \le C (|\underline{V}|_{W^{2,\infty}} + \epsilon \|\mathbf{U}\|_{H^{s+\frac{3}{2},2}}) \|V\|_{H^{s+\frac{3}{2},2}} \le C\tilde{M}.$$

This eventually yields a control on the first and third terms of $R_1$.

For the second term, estimate (B.4) yields:

$$\|R_1^\Lambda (\partial_t V)\|_{L^2} \le C \|\nabla_{t,x,r} \eta\|_{H^{s-1}} \|\nabla_{t,x,r}^\varphi V\|_{H^{s-1}}$$

Note that we crucially used $k=s$ in (B.4) in order to gain one derivative. We now use the equations (E') in order to trade the time derivatives in the right-hand side for space derivatives. Namely, taking the $H^{s-1}$ norm of the first and third equations in (E') yields

$$\left\{ \begin{array}{l} \|\partial_t^\varphi V\|_{H^{s-1}} \le \|(\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot \nabla^\varphi V + \frac{1}{\varrho} \nabla_x^\varphi P\|_{H^{s-1}}, \\ \|\partial_t \eta\|_{H^{s-1}} \le \|(\underline{V} + \epsilon V) \cdot \nabla_x \eta - w\|_{H^{s-1}}. \end{array} \right.$$

We now use triangular inequality, product estimates (A.2) and (A.3) as well as the control on the pressure provided in Proposition 2.1, to write

$$\left\{ \begin{array}{l} \|\partial_t^\varphi V\|_{H^{s-1}} \le C\tilde{M}, \\ \|\partial_t \eta\|_{H^{s-1}} \le C(\|\eta\|_{H^s} + \|w\|_{H^{s-1}}). \end{array} \right.$$

Eventually, we use the incompressibility constraint to estimate $\|w\|_{H^{s-1}}$ uniformly in $\mu$, by loosing one derivative, as detailed in Lemma C.2, to get

$$\|\partial_t \eta\|_{H^{s-1}} \le C\tilde{M}.$$

We thus write

$$\|R_1^\Lambda (\partial_t V)\|_{L^2} \le C\epsilon\|\eta\|_{H^s}\|V\|_{H^s} + C\epsilon\|\eta\|_{H^s}\|V\|_{H^s}.$$

The term $R_1^\Lambda(\nabla V)$ is directly controlled using (B.4).
In the next lemma we prove that this energy is equivalent to the $H^s$-norm of the unknowns, which we will use to derive the energy estimates.

**Lemma 3.2.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{3}{2}$. Let $(V, w, \eta) \in L^\infty([0, T], (H^s)^{d+2})$ for a certain $T > 0$.

Let

$$ \mathcal{D} := \{\Lambda^{s-l} \partial_r^l, 1 \le l \le s\} \cup \{|D|\Lambda^{s-1}\}. $$

We make the assumptions (H.2), (H.5) as well as

$$ \|\eta\|_{H^s} \le M, $$

for $M > 0$ a constant. Then there exists a constant $C' > 0$ depending only on $M$ such that

$$ \frac{1}{C'} (\|V\|_{H^s}^2 + \mu \|w\|_{H^s}^2 + \|\eta\|_{H^s}^2) \le \mathcal{E} \le C' (\|V\|_{H^s}^2 + \mu \|w\|_{H^s}^2 + \|\eta\|_{H^s}^2). $$

*Proof.* Recall that $H^s(S_r) = H^{s,k}(S_r)$ when $k = s \in \mathbb{N}$. The assumptions (H.2), (H.5) yield the existence of a constant $c > 0$ such that $\frac{1}{c} \le \rho(1 + \epsilon h) \le c$ and $\frac{1}{c} \le \rho' \le c$. Using this, as well as (B.15) on $V$ and $\sqrt{\mu}w$ yields the result. $\square$

3.2. **Estimates with loss of regularity.** The main result of this subsection is the energy estimate with loss of derivatives in Proposition 3.3. Note that in the estimate of Proposition 3.3, the functional $\mathcal{E}_0$ only controls $H^{s_0+\frac{3}{2}}$-norms of the unknowns, whereas it is bounded by a constant depending on the $H^{s_0+\frac{5}{2}}$-norms of the unknowns. Therefore, a bootstrap argument cannot be used.

Let $\Lambda$ is a differential operator of the form $\Lambda^{s-l} \partial_r^l$ for $0 \le l \le s$. Applying $\Lambda$ to (E) formally yields

$$ (3.2) \qquad \begin{cases} \partial_t \dot{V} = R_1, & (1) \\ \mu \partial_t \dot{w} = R_2, & (2) \\ \partial_t \dot{\eta} = R_3, & (3) \end{cases} $$

where $(\dot{V}, \dot{w}, \dot{\eta}) := \Lambda(V, w, \eta)$ and

$$ \begin{align*} R_1 &:= -\Lambda\left(\left(\underline{V} + \epsilon V\right) \cdot \nabla_x V + \frac{1}{\underline{\rho}} \nabla_x^\varphi P\right), & R_3 &:= -\Lambda\left(\left(\underline{V} + \epsilon V\right) \cdot \nabla_x \eta + w\right), \\ R_2 &:= -\mu\Lambda\left(\left(\underline{V} + \epsilon V\right) \cdot \nabla_x w\right) - \Lambda\left(\frac{1}{\underline{\rho}} \partial_r^\varphi P + b\right). \end{align*} $$

We make the assumptions that the unknowns $(V, w, \eta)$ satisfy

$$ (H.8) \qquad \|\eta(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} + \|V(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} + \sqrt{\mu} \|w(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} \le M_0, $$

with $M_0 > 0$ a constant.

We also make the assumption that the stratification $\rho$ and the shear flow $\underline{V}$ satisfy

$$ (H.9) \qquad |\underline{\rho}|_{W_r^{s_0+\frac{3}{2}, \infty}} + |\underline{V}|_{W_r^{s_0+\frac{5}{2}, \infty}} \le \underline{M}, $$

with $\underline{M} > 0$ a constant. We now state the energy estimates with a loss of derivatives.
For the two terms remaining, note that they are the product of $\underline{V} + \epsilon V$ with terms coming from Alinhac's good unknown. Therefore, we use product estimates (A.1) and (A.3) to bound the product with the $L^2$-norm of $\Lambda\eta$ and the $H^{s_0+\frac{1}{2},1}$-norm of the factors depending on $\underline{V}$, $V$, $P$. As $s \ge s_0 + \frac{1}{2}$, we get the result

$$ \|R_1\|_{L^2} \le C\tilde{M}. $$

**Estimate for $R_2$ and $R_4$:**

For the first term of $R_2$, we apply the commutator estimates (A.5) and (A.7). We distinguish the case of $\underline{V}$ and $V$ as one is controlled in $W^{s,\infty}$ and the other in $H^s$:

$$ \begin{aligned} \|\underline{[A, \underline{V} + \epsilon V]} \cdot \nabla_x V\|_{L^2} &\le c\|\underline{[A, \underline{V}]\nabla_x V}\|_{L^2} + \epsilon c\|\underline{[A, V]\nabla_x V}\|_{L^2} \\ &\le c\|\underline{V}\|_{W^{k,\infty}}\|\nabla_x V\|_{H^{s-1,k-1}} + \epsilon c\|V\|_{H^{s,k}}\|\nabla_x V\|_{H^{s-1,k-1}} \\ &\le C\tilde{M}. \end{aligned} $$

We control the second term of $R_2$ the exact same way, by $\|\nabla_x^\varphi P\|_{H^{s-1}}$. Thus, by (2.5), we get

$$ \|\nabla_x^\varphi P\|_{H^{s-1,k-1}} \le C\tilde{M}. $$

Note that the bound is uniform in $\mu$.

For the last two terms of $R_2$, we use the product estimate (A.2) and the bound (B.4) to get

$$ \epsilon A \eta \partial_r^\varphi \nabla_x^\varphi P + \epsilon R_1^\lambda (\nabla^\varphi P) \le \epsilon C \|\nabla_x^\varphi P\|_{H^{s,k}} \le \frac{\epsilon}{\sqrt{\mu}} C \tilde{M}, $$

where we used Proposition 2.1 to estimate the pressure. Assuming $\epsilon \le \sqrt{\mu}$ allows us to conclude. The term of $R_4$ is also of the same form as the first term of $R_2$, therefore

$$ \|R_2\|_{L^2} \le C\tilde{M}, $$

$$ \|R_3\|_{L^2} \le C\tilde{M}. $$

Note that the last two terms of (3.4) (2) are treated the same way. However, these terms are zero when $k=0$ and thus will be carefully looked at in the next section.

**Estimate for $R_5$:**

The first term of $R_5$ is of the same form as the first term of $R_2$ and is treated the same way. The second term is controlled by the product estimate (A.1). We get

$$ \|R_5\|_{L^2} \le C\tilde{M}. $$

**Estimate for $R_6$:**

The first and third terms of $R_6$ are bounded by the product estimate (A.1). The second and fourth terms are controlled thanks to estimate (B.4) and (B.7). We get:

$$ \|R_6\|_{L^2} \le C\tilde{M}. $$

**Estimate for $F$:**

The commutator estimate (A.7) and Proposition 2.1 allow us to bound the terms of $F$:

$$ \begin{aligned} \|\underline{A}, \frac{1}{\underline{\varrho}}\partial_r^\varphi P\|_{L^2} + \|\underline{[A, \frac{\varrho'}{\underline{\varrho}}]\eta}\|_{L^2} &\le C(\|\partial_r^\varphi P\|_{H^{s-1,k-1}} + \|\eta\|_{H^{s-1,k-1}}) \\ &\le C\tilde{M}. \end{aligned} $$

Note that as $\varrho$ and $\varrho'$ only depend on $r$, if $k=0$ then the commutators in $F$ are zero. $\square$

**Remark 3.6.** Before turning to the proof of the energy estimates, let us give an insight on the contributions of the different terms in the quasilinearized system (3.4). The proof yields a slightly more precise estimate than the one stated in Proposition 3.4, namely

$$ \frac{d}{dt} \mathcal{E}(t) \le C \left( 1 + \frac{\epsilon}{\sqrt{\mu}} + \frac{|\underline{V}'|_{L^\infty}}{\sqrt{\mu}} \right) \mathcal{E}(t). $$

This is done by testing the equations (3.4) against $(\varrho(1 + \epsilon h)V^A, \varrho(1 + \epsilon h)w^A, \varrho'(A\eta)$, with $A := \Lambda^{s-k}\partial_r^k$ with $s \ge s_0 + \frac{5}{2}$, $k \ge 1$ or $A = |D|\Lambda^{s-1}$, and estimating the subsequent terms. The terms on the left hand-side of (3.4) cancel each other out. The terms in $R_2$, $R_5$ are of order 1. The terms $\tilde{b}$ and $R_3$ account for the factor $\frac{\epsilon}{\sqrt{\mu}}$ above, as they are multiplied by $w$ in the energy estimate, and the energy functional only controls $\sqrt{\mu}w$.
**Proposition 3.3.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s \le s_0 + \frac{3}{2}$. We assume that there exists $T > 0$, $(V, w, \eta) \in L^\infty([0, T], (H^{s_0+\frac{5}{2}})^{d+2})$ satisfying (E) as well as (H.2), (H.5), (H.8), (H.9). Then there exists $C > 0$ a constant depending only on $M_0$, $\underline{M}$ such that:

$$ \frac{d}{dt} \mathcal{E}_0(t) \le C \left( \|\eta(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} + \|V(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} + \sqrt{\mu} \|w(t, \cdot)\|_{H^{s_0+\frac{5}{2}}} \right)^2. $$

The proof consists in estimating the terms $R_1, R_2, R_3$ with product, commutators and composition estimates, and Proposition 2.1. This is a standard strategy and we refer to [DLS20, Theorem 1] and [DB22, Lemma 3.6] where very similar systems are thoroughly studied.

**3.3. Estimates without loss of regularity.** The main result of this subsection is the energy estimates without loss of derivatives provided in Proposition 3.4. As opposed to the estimate in Proposition 3.3, the following energy estimate is without loss of derivatives. However, it requires the additional assumption $|V'|_{L^\infty} \le \sqrt{\mu}$, see Remark 3.7 for a more detailed comparison. The proof relies on the use of Alinhac's good unknowns. Before proving the proposition, we derive the quasilinearized system satisfied by Alinhac's good unknowns in Lemma 3.5.

**Proposition 3.4.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{5}{2}$. We make the assumption $\epsilon \le \sqrt{\mu}$ as well as

$$ |V'|_{L^\infty} \le \sqrt{\mu}. $$

Assume that there exists $T > 0$ and $(V, w, \eta) \in L^\infty([0, T], (H^s)^{d+2})$ satisfying (E) as well as (H.2), (H.5), (H.10), (H.11). Then there exists $C > 0$ a constant depending only on $\underline{M}, M$ such that

$$ \frac{d}{dt} \mathcal{E}(t) \le C \mathcal{E}(t). $$

Before proving Proposition 3.4, let us first exhibit the quasilinear structure of the system. Let $s \ge s_0 + \frac{5}{2}$. Let $\Lambda$ be a differential operator of the form $\Lambda := \Lambda^{s-k} \partial_r^k$ for $k \ge 1$ or $|D|\Lambda^{s-1}$. Applying $\Lambda$ to (E) formally yields

$$ (3.3) \quad \left\{ \begin{array}{l} \partial_t V^\Lambda + (\underline{V} + \epsilon V) \cdot \nabla_x V^\Lambda + \frac{1}{\rho} \nabla_x^\varphi P^\Lambda = \epsilon R_1 + R_2, \quad (1) \\[2ex] \mu (\partial_t w^\Lambda + (\underline{V} + \epsilon V) \cdot \nabla_x w^\Lambda) + \frac{1}{\rho} \partial_r^\varphi P^\Lambda + \frac{\rho'}{\rho} \dot{\eta} = F - \epsilon \Lambda \tilde{b} + \epsilon R_3 + \mu R_4, \quad (2) \\[2ex] \partial_t \dot{\eta} + (\underline{V} + \epsilon V) \cdot \nabla_x \dot{\eta} - w^\Lambda = R_5, \quad (3) \\[2ex] \nabla_x^\varphi V^\Lambda + \partial_r^\varphi w^\Lambda = \frac{\nabla_x \eta}{1 + \epsilon h} \cdot \partial_r \underline{V}^\Lambda + \epsilon R_6, \quad (4) \end{array} \right. $$
For the term $F$ in (3.4), we then distinguish two cases. First, if we only differentiate the system with respect to the horizontal variable $x$ (see case 1 in the proof of Proposition 3.4, with $\Lambda = |D|\Lambda^{s-1}$), then $F = 0$, as $\rho$ only depends on $r$. The pressure term is canceled by the use of the incompressibility condition, as well as the boundary condition on $w$.

Second, if we also differentiate the system with respect to the vertical variable $r$ (see the case 2 in the proof, with $\Lambda = \lambda^{s-k} \partial_r^k$), then we only need to control (derivatives of) $\partial_r w$, and not $w$ itself. Moreover, the incompressibility condition allows us to express $\partial_r w$ with respect to the other unknowns. Therefore, the commutators in the second equation of (3.4) are not multiplied by $w$ anymore, and thus are of order 1. The same discussion holds for the pressure term and yields the contribution of size $\frac{|V'|_L^\infty}{\sqrt{\mu}}$. A comparison with the result in [DLS20] is provided in remark 3.8.

**Proof of Proposition 3.4.** Let $0 \le k \le s$ and $\Lambda$ be either $|D|\Lambda^{s-1}$ or $\Lambda^{s-k} \partial_r^k$ (for $k \ge 1$). Recall that $V^\Lambda$ is Alinhac's good unknown associated with the operator $\Lambda$.
According to Lemma 3.5, $(V^\Lambda, w^\Lambda, \dot{\eta})$ (with $\dot{\eta} := \Lambda\eta$) satisfy (3.4), with $P$ defined according to Proposition 2.1.

Testing the first equation against $\rho(1+\epsilon h)V^\Lambda$, the second one against $\rho(1+\epsilon h)w^\Lambda$, the third one against $\rho'\dot{\eta}$ and summing the corresponding expressions, we get:

$$ (3.7) \qquad \begin{aligned} & \int_{S_r} \rho(1+\epsilon h) \partial_t V^\Lambda \cdot V^\Lambda + \mu \int_{S_r} \rho(1+\epsilon h) \partial_t w^\Lambda w^\Lambda + \int_{S_r} \rho' \partial_t \dot{\eta} \dot{\eta} + (i) \\ & = (ii) + (iii) + (iv) + (v) + (vi) + (vii), \end{aligned} $$

where

$$ (i) := \int_{S_r} (\underline{V} + \epsilon V) \cdot \nabla_x V^\Lambda \cdot V^\Lambda \rho(1 + \epsilon h) + \mu \int_{S_r} (\underline{V} + \epsilon V) \cdot \nabla_x w^\Lambda w^\Lambda \rho(1 + \epsilon h) \\ \qquad + \int_{S_r} \rho' (\underline{V} + \epsilon V) \cdot \nabla_x \dot{\eta} \dot{\eta}, $$

$$ (ii) := \int_{S_r} \nabla_x^\varphi P^\Lambda \cdot V^\Lambda (1 + \epsilon h) + \int_{S_r} \partial_r^\varphi P^\Lambda \cdot w^\Lambda (1 + \epsilon h), $$

$$ (iii) := \int_{S_r} \frac{\rho'}{\rho} \dot{\eta} w^\Lambda \rho(1 + \epsilon h) - \int_{S_r} w^\Lambda \dot{\eta} \rho', $$

$$ (iv) := \epsilon \int_{S_r} (\Lambda \tilde{b} w^\Lambda \rho(1 + \epsilon h) + R_1 V^\Lambda \rho(1 + \epsilon h) + R_3 w^\Lambda \rho(1 + \epsilon h)), $$

$$ (v) := \mu \int_{S_r} R_4 w^\Lambda \rho(1 + \epsilon h), $$

$$ (vi) := \int_{S_r} (R_2 V^\Lambda \rho(1 + \epsilon h) + R_5 \cdot \eta \rho'), $$

$$ (vii) := \int_{S_r} F w^\Lambda \rho(1 + \epsilon h). $$

We start by studying the left-hand side of (3.7). Note that as $\eta$ is in $H^{s_0+\frac{5}{2},3}$, we can differentiate equation (E)[(3)] on the evolution of $\eta$ with respect to $r$ to obtain an equation on $h$:

$$ \partial_t h + (\underline{V} + \epsilon V) \cdot \nabla_x h + \nabla_x \eta \cdot \partial_r (\underline{V} + \epsilon V) - \partial_r w = 0. $$

By the incompressibility condition, the two last terms are equal to $(1+\epsilon h)\nabla_x \cdot (\underline{V} + \epsilon V)$ so that we can write
$$ (3.8) \qquad \epsilon \partial_t h + \nabla_x \cdot ((1+\epsilon h)(\underline{V} + \epsilon V)) = 0. $$

Therefore, one can integrate by parts horizontally in the integrals in (i) and use (3.8) to get

$$ (i) = \frac{1}{2} \int_{S_r} \rho \partial_t (1 + \epsilon h) |V^\Lambda|^2 + \mu \frac{1}{2} \int_{S_r} \rho \partial_t (1 + \epsilon h) |w^\Lambda|^2 + \frac{1}{2} \int_{S_r} \rho' |\dot{\eta}|^2. $$
where $V^\Lambda$, $w^\Lambda$, $V^\Lambda$, $P^\Lambda$ are Alinhac's good unknowns defined through (B.2), $\dot{\eta} := \Lambda\eta$ and

$$ (3.4) \quad \left\{ \begin{aligned} \tilde{b} &:= \frac{1}{\epsilon} \left(b - \frac{\varrho'}{\varrho}\eta\right), \\ R_1 &:= \Lambda\eta\partial_r^\varphi \partial_t^\varphi V + R_1^\Lambda(\partial_t V) - (\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot (\Lambda\eta\partial_r^\varphi \nabla^\varphi V + R_1^\Lambda(\nabla V)), \\ R_2 &:= [\Lambda, \underline{\mathbf{U}} + \epsilon\mathbf{U}] \cdot \nabla^\varphi V + [\Lambda, \frac{1}{\varrho}] \nabla_x^\varphi P + \epsilon \frac{1}{\varrho} (\Lambda\eta\partial_r^\varphi \nabla_x^\varphi P + R_1^\Lambda(\nabla P)), \\ R_3 &:= \Lambda\eta\partial_r^\varphi \partial_t^\varphi w + R_1^\Lambda(\partial_t w) - (\underline{\mathbf{U}} + \epsilon\mathbf{U}) \cdot (\Lambda\eta\partial_r^\varphi \nabla^\varphi w + R_1^\Lambda(\nabla w)), \\ &\quad + \frac{1}{\varrho} (\Lambda\eta\partial_r^\varphi \partial_r^\varphi P + R_1^\Lambda(\nabla P)), \\ R_4 &:= [\Lambda, \underline{\mathbf{U}} + \epsilon\mathbf{U}] \cdot \nabla^\varphi w, \\ R_5 &:= [\Lambda, \underline{V} + \epsilon V] \cdot \nabla_x \eta + \epsilon \frac{\Lambda\eta}{1 + \epsilon h} \partial_r w, \\ R_6 &:= \Lambda\eta\partial_r^\varphi (\nabla^\varphi \cdot \mathbf{U}) + R_2^\Lambda(\nabla \cdot \mathbf{U}) + \Lambda\eta\partial_r^\varphi (\nabla^\varphi \cdot \underline{V}) + R_2^\Lambda(\nabla \cdot \underline{V}), \\ F &:= [\Lambda, \frac{1}{\varrho}] \partial_r^\varphi P + [\Lambda, \frac{\varrho'}{\varrho}] \eta. \end{aligned} \right. $$

In order to justify this quasilinearized system in the following lemma, we first make the assumptions that the unknowns $(V, w, \eta)$ satisfy

$$ (H.10) \qquad \|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu}\|w(t, \cdot)\|_{H^s} \le M, $$

with $M > 0$ a constant.

We also make the assumption that the stratification $\varrho$ and the shear flow $\underline{V}$ satisfy

$$ (H.11) \qquad |\varrho|_{W_r^{s+1,\infty}} + |\underline{V}|_{W_r^{s+1,\infty}} \le \underline{M}, $$

with $\underline{M} > 0$ a constant.

**Lemma 3.5.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s > s_0 + \frac{5}{2}$. We make the assumption $\epsilon \le \sqrt{\mu}$. Assume that there exists $T > 0$ and $(V, w, \eta) \in L^\infty([0, T), (H^s)^{d+2})$ satisfying (E) as well as (H.2), (H.5), (H.10), (H.11).

Then there exists $C > 0$ a constant depending only on $\underline{M}, M$ such that

$$ \left\{ \begin{aligned} \|\tilde{b}\|_{H^s} &\le C (\|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu}\|w(t, \cdot)\|_{H^s}), \\ \|(R_1, R_2, R_3, R_4, R_5, R_6)\|_{L^2} &\le C (\|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu}\|w(t, \cdot)\|_{H^s}), \\ \|F\|_{L^2} &\le C (\|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu}\|w(t, \cdot)\|_{H^s}). \end{aligned} \right. $$

Moreover, if $\Lambda = |D|\Lambda^{s-1}$ then $F = 0$.

*Proof.* First note that the gradient of pressure $\nabla^\varphi P$ is well defined and has the same regularity as $V, w$, and $\eta$ according to Proposition 2.4.

To obtain estimates on the unknowns, we apply $\Lambda := \Lambda^{s-k} \partial_r^k$ or $|D|\Lambda^{s-1}$ to $(E')$. To this end, we need once again Alinhac's good unknowns on $V$ and $w$. Thanks to equation (B.2), we can write

$$ \left\{ \begin{aligned} \Lambda \partial_t^\varphi V &= \partial_t^\varphi V^\Lambda + \epsilon \Lambda \eta \partial_r^\varphi \partial_t^\varphi V + \epsilon R_1^\Lambda (\partial_t V), \\ \Lambda \nabla^\varphi V &= \nabla^\varphi V^\Lambda + \epsilon \Lambda \eta \partial_r^\varphi \nabla^\varphi V + \epsilon R_1^\Lambda (\nabla V), \end{aligned} \right. $$
By the Leibniz rule, the left-hand side of (3.7) becomes

$$ \frac{1}{2} \frac{d}{dt} \left( \int_{S_r} \varrho(1 + \epsilon h) |V^{\Lambda}|^2 + \mu \frac{1}{2} \int_{S_r} \varrho(1 + \epsilon h) |w^{\Lambda}|^2 + \frac{1}{2} \int_{S_r} \varrho' |\dot{\eta}|^2 \right). $$

Therefore, the left-hand side is a summand of $\mathcal{E}$.

We now bound the terms on the right-hand side.

(iii) Note that the leading order terms in $\epsilon$ of (iii) cancel each other. Therefore, we can bound the remaining term by Cauchy-Schwarz inequality by

$$ \epsilon C \|w^{\Lambda}\|_{L^2} \|\dot{\eta}\|_{L^2}. $$

(iv) We use Cauchy-Schwarz inequality on each term of (iv), to get

$$ (iv) \le \epsilon \|\Lambda^{s-1} \partial_r^l \bar{b}\|_{L^2} \|\varrho(1+\epsilon h)w^{\Lambda}\|_{L^2} + \epsilon \|R_1\|_{L^2} \|\varrho(1+\epsilon h)V^{\Lambda}\|_{L^2} \\ + \epsilon \|R_3\|_{L^2} \|\varrho(1+\epsilon h)w^{\Lambda}\|_{L^2}. $$

Using Lemma 3.5 and $\epsilon \le \sqrt{\mu}$, we can bound the terms in (iv) by $C\mathcal{E}$.

(v) As for the terms in (iv), we get

$$ \mu \int_{S_r} R_4 w^{\Lambda} \varrho(1 + \epsilon h) \le C \mathcal{E}. $$

(vi) As for the terms in (iv), we get

$$ \int_{S_r} (R_2 V^{\Lambda} \varrho(1 + \epsilon h) + R_5 \cdot \eta \varrho') \le C \mathcal{E}. $$

We now turn to the terms in (ii) and (vii). For these terms, we need to distinguish two cases :

**Case 1:** $\Lambda = |D|\Lambda^{s-1}$

In this case, we use an integration by parts to get rid of the pressure term. Note indeed that the impermeability condition yields : $|D|\Lambda^{s-1} w|_{r=0;1} = 0$ and $\eta$ is constant on the boundaries so $\Lambda^{s-1}|D|\eta|_{r=0;1} = 0$. Thus, $w^{\Lambda}$ has zero trace on the top and bottom.

(ii) We integrate by parts in order to use the incompressibility condition. Namely, writing $U^{\Lambda} := \begin{pmatrix} V^{\Lambda} \\ w^{\Lambda} \end{pmatrix}$:

$$ \begin{aligned} (ii) &= \int_{S_r} \nabla^\varphi P^{\Lambda} \cdot U^{\Lambda}(1 + \epsilon h) = \int_{r=0;1} P^{\Lambda} w^{\Lambda} - \int_{S_r} P^{\Lambda} \nabla^\varphi \cdot U^{\Lambda} \\ &= -\frac{1}{\epsilon} \int_{S_r} P^{\Lambda} \nabla_x^\varphi \cdot \underline{V}^{\Lambda} - \frac{1}{\epsilon} \int_{S_r} P^{\Lambda} \epsilon R_6 \\ &= \frac{1}{\epsilon} \int_{S_r} \nabla_x^\varphi P^{\Lambda} \cdot \underline{V}^{\Lambda} - \int_{S_r} P^{\Lambda} R_6. \end{aligned} $$

For the first term on the right-hand side, we use Cauchy-Schwarz inequality. Recall the estimate of Proposition 2.1:

$$ \|\sqrt{\mu} \nabla_x^\varphi P^{\Lambda}\|_{L^2} \le C (\|V\|_{H^s} + \|\sqrt{\mu} w\|_{H^s} + \|\eta\|_{H^s}). $$

Recall also the expression

$$ \underline{V}^{\Lambda} = \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \underline{V}'. $$

This yields:

$$ \frac{1}{\epsilon} \int_{S_r} \nabla_x^\varphi P^{\Lambda} \cdot \underline{V}^{\Lambda} \le \frac{|\underline{V}'|_{L^\infty}}{\sqrt{\mu}} C (\|V\|_{H^s} + \|\sqrt{\mu} w\|_{H^s} + \|\eta\|_{H^s}). $$

Here, we use the assumption $|\underline{V}'|_{L^\infty} \le \sqrt{\mu}$.

For the second term, we use Cauchy-Schwarz inequality. Recall the bound on $R_6$ provided in Lemma 3.5.

For the pressure, recall the expression (2.18) derived in the proof of Proposition 2.1:

$$ \|P^{\Lambda}\|_{L^2} \le \|\nabla P\|_{H^{s-1,0}} + \epsilon \|\eta\|_{H^{s,0}} \|\nabla_{\mu}^{\varphi} P\|_{H^{s_0+\frac{1}{2},1}}. $$
and same expressions hold for $w$ and $P$, with Alinhac's good unknowns defined through (B.2). We now apply $\Lambda := \Lambda^{s-k} \partial_r^k$ or $|D|\Lambda^{s-1}$ to (E'):

$$ (3.5) \qquad \partial_t^\varphi V^\Lambda + (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V^\Lambda + \frac{1}{\varrho} \nabla_x^\varphi P^\Lambda = \epsilon R_1 + R_2 \quad (1) $$

$$ \mu(\partial_t^\varphi w^\Lambda + (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \nabla^\varphi w^\Lambda) + \frac{1}{\varrho} \partial_r^\varphi P^\Lambda + \frac{\varrho'}{\varrho} \dot{\eta} = F - \epsilon \Lambda \tilde{b} + \epsilon R_3 + \mu R_4, \quad (2) $$

$$ \partial_t \dot{\eta} + (\underline{V} + \epsilon V) \cdot \nabla_x \dot{\eta} - w^\Lambda = R_5 \quad (3) $$

$$ \nabla_x^\varphi \cdot V^\Lambda + \partial_r^\varphi w^\Lambda = \frac{\dot{\eta}}{1 + \epsilon h} V' + \epsilon R_6, \quad (4) $$

with the notation defined in (3.4). Note that using the semi-Lagrangian property (1.6) of the isopycnal coordinates, we get exactly (3.3).

In what follows, $c$ denotes a constant that only depends on $s_0, s$. For simplicity we write

$$ \tilde{M} := \|\eta(t, \cdot)\|_{H^s} + \|V(t, \cdot)\|_{H^s} + \sqrt{\mu} \|w(t, \cdot)\|_{H^s} $$

so that we now bound the $H^s$-norm of $\tilde{b}$ and the $L^2$-norm of the other terms from above by $C\tilde{M}$.

**Estimate for $\tilde{b}$:**

Let us write

$$ F(r, y) := \frac{1}{\epsilon^2} \left( 1 - \frac{\varrho(r+y)}{\varrho(r)} - \frac{\varrho'(r)}{\varrho(r)} y \right), $$

so that $\tilde{b} = F(r, \epsilon\eta)$. Now note that the assumed regularity on $\varrho$ implies that $F$ is at least $W^{3,\infty}$.

As $F(0) = \partial_y F = 0$, we can write, by Taylor-Laplace formula:

$$ F(r, y) = \frac{1}{\epsilon^2} \frac{1}{2} y^2 \int_0^1 \partial_y^2 F(r, \alpha y) \alpha \, da. $$

Thus the composition estimate (A.10) yields

$$ \|\tilde{b}\|_{H^s} \le C \|\eta\|_{H^s} \le C \tilde{M}. $$

Note in particular that there is no singularity in $\epsilon$.

**Estimate for $R_1$ and $R_3$:**

The terms in $R_1$ and $R_3$ have the same form and we only treat $R_1$. For the first and third terms, we write

$$ \Lambda \eta \partial_r^\varphi \partial_t^\varphi V + (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \Lambda \eta \partial_r^\varphi \nabla^\varphi V = \Lambda \eta \partial_r^\varphi (\partial_t^\varphi V + (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V) - \Lambda \eta \partial_r^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V. $$

Taking the $L^2$-norm and using the product estimate (A.1), we can bound the right-hand side of the previous expression from above by

$$ c \|\eta\|_{H^{s,k}} \left( \|\partial_r^\varphi (\partial_t^\varphi V + (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V)\|_{H^{s+\frac{1}{2},1}} + \|\partial_r^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V\|_{H^{s+\frac{1}{2},1}} \right). $$

For the first term, we use the equation (E') together with the product estimate (A.2):

$$ \|\partial_r^\varphi (\partial_t^\varphi V + (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V)\|_{H^{s+\frac{1}{2},1}} \le C \left\| \frac{1}{\varrho} \nabla_x^\varphi P \right\|_{H^{s+\frac{3}{2},2}} \le C \tilde{M}. $$

We also used the estimate on the pressure given by (2.3).

The second term is controlled by product estimates (A.2) and (A.3), namely

$$ \|\partial_r^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V\|_{H^{s+\frac{1}{2},1}} \le C (|V|_{W^{2,\infty}} + \epsilon \|\mathbf{U}\|_{H^{s+\frac{3}{2},2}}) \|V\|_{H^{s+\frac{3}{2},2}} \le C \tilde{M}. $$
Hence, using the estimate (2.5) to control the first term independently of $\mu$ and the estimate (2.4) to control the second term, this yields

$$ \|P^{\Lambda}\|_{L^2} \le C (\|V\|_{H^s} + \|\sqrt{\mu}w\|_{H^s} + \|\eta\|_{H^s}). $$

We eventually get

$$ (ii) \le C (\|V\|_{H^s}^2 + +\mu\|w\|_{H^s}^2 + \|\eta\|_{H^s}^2). $$

(vii) Recall that $F$ is the sum of two commutators of $|D|\Lambda^{s-1}$ with functions that do not depend on $x$, and thus is zero in this case.

**Case 2:** $\Lambda = \Lambda^{s-k} \partial_r^k$ for $k \ge 1$

In this case, we use the fact that $k \ge 1$ to substitute $w^{\Lambda}$ by terms that depend only on $V$ and $\eta$, thanks to the incompressibility constraint. To do so, recall the definition

$$ w^{\Lambda} := \Lambda^{s-k} \partial_r^{l-k} \partial_r w + \frac{\Lambda^{s-k} \partial_r^k \eta}{1 + \epsilon h} \partial_r w. $$

The incompressibility condition yields

$$ \partial_r w = (1 + \epsilon h) \nabla_x^{\varphi} \cdot V + \nabla_x \eta \cdot \underline{V}. $$

Thus, applying $\Lambda^{s-k} \partial_r^{k-1}$ and taking the $L^2$-norm yields

$$ \|\Lambda^{s-k} \partial_r^{k-1} \partial_r w\|_{L^2} \le \|\Lambda^{s-k} \partial_r^{k-1}((1 + \epsilon h)\nabla_x^{\varphi} \cdot V)\|_{L^2} + \|\Lambda^{s-k} \partial_r^{k-1}(\nabla_x \eta \cdot \underline{V}')\|_{L^2} $$

We use product estimates, as $s-1 \ge s_0 + \frac{1}{2}$:

If $k-1 \ge 1$, we use (A.2) in the expression of $\nabla_x^{\varphi} \cdot V$ (recall the definition (1.17) of $\nabla_x^{\varphi}$) to get

$$ \|\Lambda^{s-k} \partial_r^{k-1}((1 + \epsilon h)\nabla_x^{\varphi} \cdot V)\|_{L^2} \le c \|\nabla_x \cdot V\|_{H^{s-1,k-1}} + c \|h\|_{H^{s-1,k-1}} \|\nabla_x \cdot V\|_{H^{s-1,k-1}} \\ + c \|\nabla_x \eta\|_{H^{s-1,k-1}} \|\partial_r V\|_{H^{s-1,k-1}}. $$

If $k-1=0$, we use (A.1) to get

$$ \|\Lambda^{s-k} \partial_r^{k-1}((1 + \epsilon h)\nabla_x^{\varphi} \cdot V)\|_{L^2} \le c \|\nabla_x \cdot V\|_{H^{s-1,0}} + c \|h\|_{H^{s-1,0}} \|\nabla_x \cdot V\|_{H^{s_0+\frac{1}{2},1}} \\ + c \|h\|_{H^{s_0+\frac{1}{2},1}} \|\nabla_x \cdot V\|_{H^{s-1,0}} + c \|\nabla_x \eta\|_{H^{s-1,0}} \|\partial_r V\|_{H^{s_0+\frac{1}{2},1}} \\ + c \|\nabla_x \eta\|_{H^{s_0+\frac{1}{2},1}} \|\partial_r V\|_{H^{s-1,0}}. $$

In any case, as $s \ge s_0 + \frac{5}{2} \ge 2$:

$$ \|\Lambda^{s-k} \partial_r^{k-1}((1 + \epsilon h)\nabla_x^{\varphi} \cdot V)\|_{L^2} \le C (\|V\|_{H^s} + \|\eta\|_{H^s}). $$

The second term $\nabla_x \eta \cdot \underline{V}'$ is controlled the same way and we get

$$ (3.9) \qquad \|w^{\Lambda}\| \le C (\|V\|_{H^s} + \|\eta\|_{H^s}). $$

We can now bound the terms $(ii)$ and $(vii)$.

(ii) Writing $U^{\Lambda} := \begin{pmatrix} V^{\Lambda} \\ w^{\Lambda} \end{pmatrix}$, we bound each term using Cauchy-Schwarz inequality and the previous bound on $\|w^{\Lambda}\|_{L^2}$, namely

$$ \int_{S_r} \nabla^{\varphi} P^{\Lambda} \cdot U^{\Lambda} (1 + \epsilon h) \le \|\nabla^{\varphi} P^{\Lambda}\|_{L^2} (\|V^{\Lambda}\|_{L^2} + \|w^{\Lambda}\|_{L^2}). $$

We use Proposition 2.1 to control $\|\partial_r^{\varphi} P^{\Lambda}\|_{L^2}$. However, in order to have a control on $\|\nabla_x^{\varphi} P^{\Lambda}\|_{L^2}$ that is uniform in $\mu$, we rather use (B.3) together with (B.4) to write

$$ \|\nabla_x^{\varphi} P^{\Lambda}\|_{L^2} \le C \|\Lambda \nabla_x^{\varphi} P\|_{L^2} + C \|\nabla_x^{\varphi} P\|_{H^{s_0+\frac{1}{2},1}} + C \|\nabla_x^{\varphi} P\|_{H^{s-1}}. $$

The last two terms are controlled with Equation (2.5). For the first one, remark that

$$ \begin{aligned} \Lambda \nabla_x^{\varphi} P &= \Lambda^{s-k} \partial_r^{k-1} \partial_r \nabla_x^{\varphi} P \\ &= \Lambda^{s-k} \partial_r^{k-1} (1 + \epsilon h) \partial_r^{\varphi} \nabla_x^{\varphi} P = \Lambda^{s-k} \partial_r^{k-1} (1 + \epsilon h) \nabla_x^{\varphi} \partial_r^{\varphi} P. \end{aligned} $$
This eventually yields a control on the first and third terms of $R_1$. For the second term, estimate (B.3) yields:

$$ \|R_1^A(\partial_t V)\|_{L^2} \le C \|\nabla_{t,x,r} \eta\|_{H^{s-1}} \|\nabla_{t,x,r}^\varphi V\|_{H^{s-1}} $$

Note that we crucially used $k = s$ in (B.3) in order to gain one derivative here. We now use the equations $(E')$ in order to trade the time derivatives in the right-hand side for space derivatives. Namely, taking the $H^{s-1}$ norm of the first and third equations in $(E')$ yields

$$ \left\{ \begin{array}{l} \|\partial_t^\varphi V\|_{H^{s-1}} \le \|(\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi V + \frac{1}{\varrho} \nabla_x^\varphi P\|_{H^{s-1}}, \\ \|\partial_t \eta\|_{H^{s-1}} \le \|(\underline{V} + \epsilon V) \cdot \nabla_x \eta - w\|_{H^{s-1}}. \end{array} \right. $$

We now use triangular inequality, product estimates (A.2) and (A.3) as well as the control on the pressure provided in Proposition 2.4, to write

$$ \left\{ \begin{array}{l} \|\partial_t^\varphi V\|_{H^{s-1}} \le C\tilde{M}, \\ \|\partial_t \eta\|_{H^{s-1}} \le C(\|\eta\|_{H^s} + \|w\|_{H^{s-1}}). \end{array} \right. $$

Eventually, we use the incompressibility constraint to estimate $\|w\|_{H^{s-1}}$ uniformly in $\mu$, by loosening one derivative, as detailed in Lemma C.2, to get

$$ \|\partial_t \eta\|_{H^{s-1}} \le C\tilde{M}. $$

We thus write

$$ \|R_1^A(\partial_t V)\|_{L^2} \le C\epsilon\|\eta\|_{H^s}\|V\|_{H^s} + C\epsilon\|\eta\|_{H^s}\|V\|_{H^s}. $$

The term $R_1^A(\nabla V)$ is directly controlled using (B.3).

For the two terms remaining, note that they are the product of $\underline{V} + \epsilon V$ with terms coming from Alinhac's good unknown. Therefore, we use product estimates (A.1) and (A.3) to bound the product with the $L^2$-norm of $\Lambda\eta$ and the $H^{s_0+\frac{1}{2},1}$-norm of the factors depending on $\underline{V}$, $V$, $P$. As $s \ge s_0 + \frac{1}{2}$, we get the result

$$ \|R_1\|_{L^2} \le C\tilde{M}. $$

**Estimate for $R_2$ and $R_4$:**

For the first term of $R_2$, we apply the commutator estimates (A.4) and (A.6). We distinguish the case of $\underline{V}$ and $V$ as one is controlled in $W^{s,\infty}$ and the other in $H^s$:

$$ \begin{aligned} \|[\Lambda, \underline{V} + \epsilon V] \cdot \nabla_x V\|_{L^2} &\le c \|[\Lambda, \underline{V}] \nabla_x V\|_{L^2} + \epsilon c \|[\Lambda, V] \nabla_x V\|_{L^2} \\ &\le c \|\underline{V}\|_{W^{k,\infty}} \|\nabla_x V\|_{H^{s-1,k-1}} + \epsilon c \|V\|_{H^{s,k}} \|\nabla_x V\|_{H^{s-1,k-1}} \\ &\le C\tilde{M}. \end{aligned} $$

We control the second term of $R_2$ the exact same way, by $\|\nabla_x^\varphi P\|_{H^{s-1}}$. Thus, by (2.11), we get

$$ \|\nabla_x^\varphi P\|_{H^{s-1,k-1}} \le C\tilde{M}. $$

Note that the bound is uniform in $\mu$.

For the last two terms of $R_2$, we use the product estimate (A.2) and the bound (B.3) to get

$$ \epsilon \Lambda \eta \partial_t^\varphi \nabla_x^\varphi P + \epsilon R_1^A (\nabla^\varphi P) \le \epsilon C \|\nabla_x^\varphi P\|_{H^{s,k}} \le \frac{\epsilon}{\sqrt{\mu}} C \tilde{M}, $$