Thus, we control the $L^2$ norm of this term with $C\|\partial_r^\varphi P\|_{H^{s,k}}$ by the product estimate (A.1).
This eventually yields a control of $\|\nabla_x^\varphi P^\Lambda\|_{L^2}$ that is uniform in $\mu$, and allows us to conclude:

$$ \int_{S_r} \nabla^\varphi P^\Lambda \cdot U^\Lambda (1 + \epsilon h) \le C (\|V^\Lambda\|_{L^2} + \|V\|_{H^s} + \sqrt{\mu}\|w\|_{H^s} + \|\eta\|_{H^s})^2 $$

(vii) For the term (vii), we use Cauchy-Schwarz inequality, Lemma 3.5 as well as the previous bound (3.9) on $\|w^\Lambda\|_{L^2}$, to get

$$ \int_{S_r} F w^\Lambda \le C (\|V^\Lambda\|_{L^2} + \|V\|_{H^s} + \sqrt{\mu}\|w\|_{H^s} + \|\eta\|_{H^s})^2 $$

Summing these estimates together yields

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

Lemma 3.2 allows us to conclude.

<figure><img src="image_5.png" /></figure>

*Remark 3.7.* Once we apply a differential operator $\Lambda$ of the form $|D|\Lambda^{s-1}$ or $\Lambda^{s-k}\partial_r^k$ for $k \le s$ to the system (E), we get the quasilinear system (3.4). Note however that the terms $\nabla_x V^\Lambda$, $\nabla_x w^\Lambda$, $\nabla_x D|\Lambda^{s-1}\eta$ in (3.4) are distributions, as $V, w, \eta$ are in $H^{s,s}(S)$, but may not be in $H^{s+1,s}(S_r)$ (recall the definition (1.19)). This can be solved by applying a smoothing (in x) operator, for instance $\chi(\delta\Lambda)\Lambda$ where $\chi$ is a smooth bump function and $\delta > 0$, instead of $\Lambda$ in the proof of Proposition 3.4. The same estimates hold, are uniform in $\delta$, and sending $\delta \to 0$ yields the estimate of Proposition 3.4. This is detailed in the context of elliptic equations in [Lan13, Lemma 2.38] and we omit it.

*Remark 3.8.* The proof yields a slightly more precise estimate than the one stated in Proposition 3.4, namely

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

The contributions of the different source terms of (3.4) in this estimate have already been summarized in Remark 3.6. Let us now detail how the additional assumptions in [DLS20] allow the authors to obtain a time of existence of order $O((\epsilon/\sqrt{\mu})^{-1})$.

The terms in $R_2, R_5$ are of order 1. However, assuming $V=0$ and making the strong Boussinesq assumption (that is, the coefficient $1/\partial_\nu$ in front of the pressure terms is a constant), as done in [DLS20], sets the terms of order 1 to zero and therefore $R_1$ and $R_5$ are of order $\epsilon$.

In the proof of Proposition 3.4, we then distinguished two cases. In [DLS20], the authors only use the strategy of the first case in the proof of Proposition 3.4, that is the pressure term is canceled by the use of the incompressibility conditions. Hence, they need the trace of the vertical derivatives of $w$ to vanish at the boundary, which is not true in general, but is the purpose of the good preparation of the initial data that they assume. Furthermore, with the assumptions of the strong Boussinesq approximation and constant Brunt-Väisälä frequency, the commutators in the term $F$ are zero. The upside is that they do not have any term of order 1, and no shear velocity at equilibrium, and thus obtain a time of existence of order $\sqrt{\mu}/\epsilon$.

*Remark 3.9.* Finally, let us state that the distinction of the two cases described above holds true in Eulerian coordinates, and is by no mean specific to the isopycnal coordinates. Conversely, we believe that making the same assumptions as in [DLS20] would yield the same time of existence in isopycnal coordinates. However, the propagation of the good preparation assumption seems to require more technical computations than in Eulerian coordinates. The well-preparation of the initial data would then allow to use the strategy of the Case 1 in the proof of Proposition 3.4 for both the case $\Lambda = |D|\Lambda^{s-1}$ and $\Lambda = \Lambda^{s-k}\partial_r^k$, $k \ge 1$. More precisely, with this additional assumption, the integration by parts in the Case 1, (ii) does not yield boundary terms.

## 4. CONCLUSION

We first finish the proof of Theorem 1.1. Then, we state the result of Theorem 1.1 in Eulerian coordinates (see Corollary 4.8), and compare more precisely the differences between Corollary 4.8 and [DLS20, Theorem 2].
where we used Proposition 2.4 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 (3.3) (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.3) and (B.6). We get:

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

**Estimate for $F$:**

The commutator estimate (A.6) and Proposition 2.4 allow us to bound the terms of $F$:

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

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

□

**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 (3.3). The proof yields a slightly more precise estimate than the one stated in Proposition 3.4, namely

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

This is done by testing the equations (3.3) against $(\varrho(1 + \epsilon h)V^\Lambda, \varrho(1 + \epsilon h)w^\Lambda, \varrho'\lambda\eta)$, with $\Lambda := \Lambda^{s-k}\partial_r^k$ with $s \ge s_0 + \frac{5}{2}$, $k \ge 1$, and estimating the subsequent terms. The terms on the left hand-side of (3.3) 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 $\sqrt{\mu}w$.

For the term $F$ in (3.3), 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 $\varrho$ 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 (3.3) are not multiplied by $w$ anymore, and thus are of order 1. The same
**4.1. Completion of the proof of Theorem 1.1.** Note that $(E^{\text{eul}})$ is a quasilinear initial boundary value problem, and as such the existence of solutions is not obvious, see [BGS06] for a quite complete introduction on these systems. However, (E) does not contain any vertical advection, and can therefore be viewed as family of initial value problems on $\mathbb{R}^d$ parametrized by $r \in [0, 1]$. Still, the operator $\nabla^\varphi$ being non-linear, the linear system (that is, taking $\epsilon = 0$ in (E)) and the non-linear system (E) do not have the same structure, and a standard Picard iteration scheme relying on the energy estimates in Proposition 3.4 cannot be used directly. We thus choose to sketch the proof of Theorem 1.1 in this subsection. We first show the existence and uniqueness of solutions for the system (4.1) in Proposition 4.1. Note that we strongly use the semi-lagrangian structure of the system (4.1), see Remark 4.5. Then, we apply Proposition 4.1 to the Euler equations (E) to prove the existence of solutions on a time interval independent of the parameters $\epsilon$ and $\mu$, relying on the energy estimate of Proposition 3.4.
Let $s_0 > d/2$, $s \ge s_0 + \frac{5}{2}$ an integer. In Proposition 4.1, we aim at constructing a solution in $C^0([0, T], H^s(S))$ of the system

$$ (4.1) \qquad \left\{ \begin{array}{ll} \partial_t \mathbf{U} + V \cdot \nabla_x \mathbf{U} + \frac{1}{\varrho} \nabla^\varphi P + \mathbf{b} = 0, & \\ \partial_t \eta + V \cdot \nabla_x \eta - w = 0, & \text{in } [0, T) \times S_r. \\ \nabla^\varphi \cdot \mathbf{U} = 0, & \end{array} \right. $$

Here, we write $\mathbf{U} := (V, w)^T$, $\varphi(t, x, r) := (t, x, r + \eta(t, x, r))$ and $\nabla^\varphi$ is defined through (1.17). Here, we assume that $\mathbf{b}$ is a source term that may depend on the solution $(\mathbf{U}, \eta)$, satisfying

$$ (H.14) \qquad \left\{ \begin{array}{l} \|\mathbf{b}(\mathbf{U}, \eta)\|_{H^s} \le C(1 + \|\langle \mathbf{U}, \eta \rangle\|_{H^s}), \\ \|\mathbf{b}(\mathbf{U}_1, \eta_1) - \mathbf{b}(\mathbf{U}_2, \eta_2)\|_{H^s} \le C \|(\mathbf{U}_1 - \mathbf{U}_2, \eta_1 - \eta_2)\|_{H^s}, \end{array} \right. $$

with $C > 0$ a constant. We also assume that there exist $c_* \ge c_* > 0$ such that

$$ (H.15) \qquad c_* \le 1 + h \le c^*, $$

where $h := -\partial_r \eta$. We assume that the traces $\eta|_{r=-1}$ and $\eta|_{r=0}$ of $\eta$ at the top and bottom are zero, i.e. that the top and bottom boundaries are flat:

$$ (H.16) \qquad \eta|_{r=1} = \eta|_{r=0} = 0. $$

The factor $\varrho$ only depends on $r$ and satisfies

$$ (H.17) \qquad c_* \le \varrho \le c^*. $$

The system (4.1) is completed with the initial data $(\mathbf{U}_{\text{in}}, \eta_{\text{in}}) \in H^s(S)^{d+2}$ satisfying

$$ (H.18) \qquad \|(\mathbf{U}_{\text{in}}, \eta_{\text{in}})\|_{H^s} \le M_{\text{in}}, $$

for a constant $M_{\text{in}} > 0$.

**Proposition 4.1.** Let $(\mathbf{U}_{\text{in}}, \eta_{\text{in}}) \in H^s(S)^{d+2}$ satisfy (H.15), (H.18) and the divergence-free condition in (4.1). Assume that the terms $\mathbf{b}$ and $\varrho$ satisfy (H.14) and (H.17) respectively. Assume that the stratification satisfies (H.15) and (H.16). Then there exists $T > 0$ that only depends on $M_{\text{in}}, c_*, h_*, h^*, d, s_0, s$ and a unique solution $(\mathbf{U}, \eta) \in C^0([0, T], H^s(S)^{d+2})$ to (4.1) with initial data $(\mathbf{U}_{\text{in}}, \eta_{\text{in}})$. Moreover, if $T$ is finite, then

$$ (4.2) \qquad \text{either} \quad \|\mathbf{U}(t, \cdot)\|_{H^s} \xrightarrow{t \to T} +\infty \quad \text{or} \quad \inf_{(x,r) \in S_r} (1+h(t,x,r)) \xrightarrow{t \to T} 0. $$

**Proof of Proposition 4.1.** In order to prove Proposition 4.1, we first write a regularized system, for which classical results on ordinary differential equations (ODE) apply. We then show how to define the pressure term for this system, and that it is Lipschitz continuous with respect to the unknowns $(\mathbf{U}, \eta)$, in the Sobolev spaces $H^s$. Then, we use an energy estimate and a compactness argument to extract a solution of the original system in $L^\infty([0, T], H^s(S)^{d+2})$. We then show that this solution is unique, and derive the blow-up criterion (4.2). Finally, we show that the solution is in $C^0([0, T], H^s(S)^{d+2})$ and is divergence-free.

**Step 0: Some technical tools**

Let $\mathcal{J}_\delta$ for $\delta > 0$ the mollification operator defined with the convolution with a smooth cutoff function in the horizontal Fourier space (see [MBO02]), to regularize $\nabla_x$. It satisfies the following properties, see [MBO02, Lemma 3.5]
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 3.7.

*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 (3.3), with $P$ defined according to Proposition 2.4.

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

$$ (3.6) \qquad \begin{aligned} & \int_{S_r} \varrho(1+\epsilon h)\partial_t V^\Lambda \cdot V^\Lambda + \mu \int_{S_r} \varrho(1+\epsilon h)\partial_t w^\Lambda w^\Lambda + \int_{S_r} \varrho'\partial_t \dot{\eta} \dot{\eta} + (i) \\ & = (ii)+(iii)+(iv)+(v)+(vi)+(vii), \end{aligned} $$

where

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

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

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

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

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

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

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

We start by studying the left-hand side of (3.6). 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

$$ (3.7) \qquad \epsilon \partial_t h + \nabla_x \cdot ((1 + \epsilon h)(\underline{V} + \epsilon V)) = 0. $$

Therefore, one can integrate by parts horizontally in the integrals in (i) and use (3.7) to get

$$ (i) = \frac{1}{2} \int_{S_r} \varrho \partial_t (1 + \epsilon h) |V^\Lambda|^2 + \mu \frac{1}{2} \int_{S_r} \varrho \partial_t (1 + \epsilon h) |w^\Lambda|^2 + \frac{1}{2} \int_{S_r} \varrho' |\dot{\eta}|^2. $$
**Lemma 4.2.** Let $s, t \in \mathbb{R}$, $f \in H^t(\mathbb{R}^d)$. For $\delta > 0$, we can write

$$ (4.3) \qquad |\mathcal{J}_{\delta} f|_{H^s(\mathbb{R}^d)} \le C_{\delta,s,t} |f|_{H^t(\mathbb{R}^d)}, $$

for some constant $C_{\delta,s,t}$.
For $\delta' > 0$, we also have

$$ (4.4) \qquad |(\mathcal{J}_{\delta} - \mathcal{J}_{\delta'})f|_{H^{t-1}} \le C|\delta - \delta'| |f|_{H^t}, $$

where $C$ does not depend on $\delta$ nor $\delta'$.

We will also need a Leray projector defined as follows

**Definition 4.3.** We define

$$ (4.5) \qquad \begin{array}{l} \Pi : L^2(S_r) \to L^2(S_r) \\ \mathbf{U} \mapsto \mathbf{U} - \nabla^\varphi \psi \end{array} $$

where $\psi \in \dot{H}^1/\mathbb{R}$ is the solution to

$$ (4.6) \qquad \begin{array}{l} \nabla^\varphi \cdot \nabla^\varphi \psi' = \nabla^\varphi \cdot \mathbf{U}, \\ \partial_r^\varphi \psi = 0 \quad \text{at } r = -1 \text{ and } r = 0. \end{array} $$

We state the main properties of such an operator.

**Lemma 4.4.** Let $\mathbf{U} \in L^2(S_r)$, $s_0 > \frac{d}{2}$.

* The field $\Pi\mathbf{U}$ is divergence-free:

$$ \nabla^\varphi \cdot \Pi\mathbf{U} = 0. $$

* For $s \ge s_0 + \frac{5}{2}$, The operator $\Pi$ is bounded from $H^s(S_r)$ to $H^s(S_r)$, and its norm only depends on an upper-bound on $\|\eta\|_{H^s(S_r)}$.

* If $\mathbf{U} \in H^1(S_r)$, there exists $C > 0$ such that

$$ (4.7) \qquad \|\nabla^\varphi (\Pi\mathbf{U} - \mathbf{U})\|_{L^2} \le C \|\nabla^\varphi \cdot \mathbf{U}\|_{L^2}. $$

*Proof.* The existence of $\psi$ in the definition (4.5) of $\Pi$ as well as the second point are a direct consequence of Proposition 2.6. The first point follows from the definition of $\psi$ in the definition (4.5) of $\Pi$. For the last point, we take $\tilde{\psi}$ such that $\tilde{\psi}(r + \eta(t, x, r)) = \psi(t, x, r)$. By change of variables, (4.7) boils down to

$$ (4.8) \qquad \|\nabla(\nabla\tilde{\psi})\|_{L^2} \le C\|\nabla\cdot\nabla\tilde{\psi}\|_{L^2}. $$

This last inequality is a div-curl estimate (see for instance [ABDG98, Lemma 2.1]), as $\nabla\tilde{\psi}$ is a curl-free vector field, with zero normal trace to the boundary of the strip, and the boundary is flat so its curvature is zero. $\square$

**Step 1: Mollification**

Let $\mathcal{J}_\delta$ for $\delta > 0$ the mollification operator defined above. We then introduce the following mollified system

$$ (E_\delta) \qquad \left\{ \begin{array}{l} \partial_t \mathbf{U}_\delta + \mathcal{J}_\delta [\mathcal{J}_\delta V_\delta \cdot \nabla_x \mathcal{J}_\delta \mathbf{U}_\delta] + \frac{1}{\varrho} \nabla^\varphi P_\delta + \mathbf{b} = 0, \\ \partial_t \eta_\delta + \mathcal{J}_\delta [\mathcal{J}_\delta V \cdot \nabla_x \mathcal{J}_\delta \eta_\delta] - w_\delta = 0, \end{array} \right. $$

together with the initial conditions

$$ \left\{ \begin{array}{l} (\mathbf{U}_\delta)_{t=0} = \mathbf{U}_{\text{in}}, \\ (\eta_\delta)_{t=0} = \eta_{\text{in}}. \end{array} \right. $$

We define the pressure $P_\delta$ through the elliptic equation

$$ (4.9) \qquad \nabla^\varphi \cdot \frac{1}{\varrho} \nabla^\varphi P_\delta = - \nabla^\varphi \cdot (\mathbf{U} \cdot \nabla^\varphi \Pi \mathbf{U}) - \nabla^\varphi \cdot \mathbf{b}, $$

completed with the boundary conditions

$$ (4.10) \qquad \partial_r^\varphi P_\delta = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \cdot (\mathbf{U} \cdot \nabla^\varphi \Pi \mathbf{U} + \mathbf{b}) \quad \text{at } r=0 \text{ and } r=1. $$
By the Leibniz rule, the left-hand side of (3.6) 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

$$ (iv) \le \epsilon \|\Lambda^{s-1} \partial_r^l \tilde{b} \|_{L^2} \|\varrho(1+\epsilon h)w^\Lambda\|_{L^2} + \epsilon \|R_1\|_{L^2} \|\varrho(1+\epsilon h)V^\Lambda\|_{L^2} \\ + \epsilon \|R_3\|_{L^2} \|\varrho(1+\epsilon h)w^\Lambda\|_{L^2}. $$

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,

$$ \text{writing } U^{\Lambda} := \begin{pmatrix} V^{\Lambda} \\ w^{\Lambda} \end{pmatrix}: $$

$$ \begin{align*} (ii) &= \int_{S_r} \nabla^\varphi P^\Lambda \cdot U^\Lambda (1 + \epsilon h) = \int_{r=0;1} P^\Lambda w^\Lambda - \int_{S_r} P^\Lambda \nabla^\varphi \cdot U^\Lambda \\ &= -\frac{1}{\epsilon} \int_{S_r} P^\Lambda \nabla_x^\varphi \cdot \underline{V}^\Lambda - \frac{1}{\epsilon} \int_{S_r} P^\Lambda \epsilon R_6 \\ &= \frac{1}{\epsilon} \int_{S_r} \nabla_x^\varphi P^\Lambda \cdot \underline{V}^\Lambda - \int_{S_r} P^\Lambda R_6. \end{align*} $$

For the first term on the right-hand side, we use Cauchy-Schwarz inequality. Recall the estimate of Proposition 2.4:

$$ \|\sqrt{\mu} \nabla_x^\varphi P^\Lambda\|_{L^2} \le C (\|V\|_{H^s} + \|\sqrt{\mu} w\|_{H^s} + \|\eta\|_{H^s}). $$

Recall also the expression

$$ \underline{V}^{\Lambda} = \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \underline{V}^{\prime}. $$
Note that because $\nabla^{\varphi} \cdot \Pi \mathbf{U} = 0$, (4.9) also writes

$$ (4.11) \qquad \nabla^{\varphi} \cdot \frac{1}{\varrho} \nabla^{\varphi} P_{\delta} = -\partial_{i}^{\varphi} \mathbf{U}_{j} \partial_{j}^{\varphi} (\Pi \mathbf{U})_{i} - \nabla^{\varphi} \cdot \mathbf{b}. $$

Using the boundary condition $w = \Pi w = 0$ at the boundaries, the boundary condition (4.10) can be recast as

$$ (4.12) \qquad \partial_{r}^{\varphi} P_{\delta} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \cdot \mathbf{b} \quad \text{at } r=0 \text{ and } r=1. $$

Note the abuse of notation, as here $\Pi w$ denotes the vertical component of $\Pi \mathbf{U}$.

**Step 2: $(\mathbf{U}, \eta) \mapsto \nabla^{\varphi} P$ is well-defined and Lipschitz continuous**

Recall that $s \ge s_0 + \frac{5}{2}$. As for the proof of Proposition 2.1, the elliptic system (4.9) defines uniquely $P$ in $\dot{H}^1/\mathbb{R}$ according to Proposition 2.3. As for the proof of Proposition 2.1, the system (4.11) yields, according to Proposition 2.6:

$$ \|\nabla^{\varphi} P\|_{H^s} \le C \|(\mathbf{U}, \eta)\|_{H^s}, $$

where $C$ only depends on an upper bound on $\|(\mathbf{U}, \eta)\|_{H^s}$, $M_{\text{in}}$, $c_*$, $c^*$.

Let now $(\mathbf{U}_i, \eta_i) \in (H^s)^{d+2}$ for $i \in \{1, 2\}$ be fields whose $H^s$-norms are bounded by a constant $M > 0$. Let $P_1$ be the pressures defined by the system (4.9) with respect to the solution $(\mathbf{U}_i, \eta_i)$. We write $\varphi_i(t, x, r) := (t, x, -r + \eta_i(t, x, r))$ the change of variables associated to $\eta_i$. Applying $|D|\Lambda^{s-1}$ to the system (4.11) satisfied by both $P_1$ and $P_2$ using Alinhac's good unknown as in (2.14), and then taking the difference between the two equations yields:

$$ \nabla_{\mu}^{\varphi_1} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_1} (P_1^{\Lambda} - P_2^{\Lambda}) = \bar{F}_1 - \bar{F}_2 + \left( \nabla_{\mu}^{\varphi_1} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_1} - \nabla_{\mu}^{\varphi_2} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_2} \right) P_2^{\Lambda}, $$

where $\bar{F}_1 - \bar{F}_2$ is the difference of the source terms in (2.14), which is Lipschitz continuous in the unknowns. Therefore, using Proposition 2.6 we get the bound

$$ \|\nabla_{\mu}^{\varphi_1} (P_1^{\Lambda} - P_2^{\Lambda})\|_{L^2} \le C(\|V_1 - V_2\|_{H^{s,k}} + \sqrt{\mu}\|w_1 - w_2\|_{H^{s,k}} + \|\eta_1 - \eta_2\|_{H^{s,k}}). $$

Here, $C > 0$ depends on $M$.

To conclude, we write

$$ (4.13) \qquad \|\Lambda(\nabla_{\mu}^{\varphi_1} P_1 - \nabla_{\mu}^{\varphi_2} P_2)\|_{L^2} \le \|\nabla_{\mu}^{\varphi_1} (P_1^{\Lambda} - P_2^{\Lambda})\|_{L^2} + \| (\nabla_{\mu}^{\varphi_2} - \nabla_{\mu}^{\varphi_1}) P_2^{\Lambda} \|_{L^2} + \|R\|_{L^2}, $$

where $R$ is the difference of the remainder terms appearing in the relation (B.3) on the commutation between the operators $\Lambda$ and $\nabla_{\mu}^{\varphi_i}$. The term $R$ is multi-linear and as such it is easily bounded by a linear function of $\|\eta_1 - \eta_2\|_{H^s}$ and $\|\nabla_{\mu}^{\varphi_1} P_1 - \nabla_{\mu}^{\varphi_2} P_2\|_{H^{s-1}}$.

The first term of (4.13) has already been treated above, and for the second one we write

$$ \| (\nabla_{\mu}^{\varphi_2} - \nabla_{\mu}^{\varphi_1}) P_2^{\Lambda} \|_{L^2} \le \| \eta_1 - \eta_2 \|_{H^{s_0+\frac{3}{2},2}} \| \nabla_{\mu}^{\varphi_2} P_2 \|_{H^s} \le C \| \eta_1 - \eta_2 \|_{H^{s_0+\frac{3}{2},2}} $$

by Proposition 2.1.

Therefore, the gradient of the pressure is Lipschitz continuous in $H^s$.

**Step 2: Existence and uniqueness**

We now follow the steps of [MBO02] for the existence. As the system $(E_{\delta})$ is a system of ODEs with a Lipschitz continuous source-term according to the regularizing property in Lemma 4.2 and Step 1, Cauchy-Lipschitz theorem yields a solution $(\mathbf{U}, \eta)_{\delta}$ on a time $T_{\delta} > 0$ for any $\delta > 0$, in the Banach space

$$ \mathcal{B} := \{(\mathbf{U}_{\delta}, \eta_{\delta}) \in C^0([0, T_{\delta}], (H^s)^{d+2}), \|\mathbf{U}_{\delta}, \eta_{\delta}\|_{H^s} \le 2M_0, \frac{1}{2}h_* \le 1 + h \le 2h^*\}, $$

We use a continuity argument to show that the times of existence $T_{\delta}$ are bounded from below by some $T > 0$ independent of $\delta$.

We start by writing the following classical energy estimate (see for example [MBO02, Proposition 3.7]), where $\nabla^{\varphi} P + \mathbf{b}$ is treated as a source term:

$$ (4.14) \qquad \frac{d}{dt} \|(\mathbf{U}_{\delta}(t, \cdot), \eta_{\delta}(t, \cdot))\|_{H^s}^2 \le C \|(\mathbf{U}_{\delta}(t, \cdot), \eta_{\delta}(t, \cdot))\|_{H^s}^2, $$
This yields:

$$ \frac{1}{\epsilon} \int_{S_r} \nabla_x^\varphi P^\Lambda \cdot \underline{V}^\Lambda \leq \frac{|\underline{V}'|_{L^\infty}}{\sqrt{\mu}} C (\|V\|_{H^s} + \|\sqrt{\mu}w\|_{H^s} + \|\eta\|_{H^s}). $$

Here, we use the assumption $|\underline{V}'|_{L^\infty} \leq \sqrt{\mu}$.

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 (2.8) derived in the proof of Proposition 2.4:

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

Hence, using the estimate (2.11) to control the first term independently of $\mu$ and Proposition 2.1 to control the second term, this yields

$$ \|P^\Lambda\|_{L^2} \leq C (\|V\|_{H^s} + \|\sqrt{\mu}w\|_{H^s} + \|\eta\|_{H^s}). $$

We eventually get

$$ (ii) \leq C (\|V\|_{H^s}^2 + \mu \|w\|_{H^s}^2 + \|\eta\|_{H^s}^2). $$

(vii) Recall that $F$ is the sum of two commutators of $\Lambda^{s-1}|D|$ 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 $k \geq 1$

In this case, we use the fact that $k \geq 1$ 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

$$ \|\Lambda^{s-k} \partial_r^{k-1} \partial_r w\|_{L^2} \leq \|\Lambda^{s-k} \partial_r^{k-1} ((1 + \epsilon h) \nabla_x^\varphi \cdot V)\|_{L^2} + \|\Lambda^{s-k} \partial_r^{k-1} (\nabla_x \eta \cdot \underline{V}')\|_{L^2} $$

We use product estimates, as $s-1 \geq s_0 + \frac{1}{2}$:

If $k-1 \geq 1$, we use (A.2) in the expression of $\nabla_x^\varphi \cdot V$ (recall the definition (1.13) of $\nabla_x^\varphi$) to get

$$ \|\Lambda^{s-k} \partial_r^{k-1} ((1 + \epsilon h) \nabla_x^\varphi \cdot V)\|_{L^2} \leq c \|\nabla_x \cdot V\|_{H^{s-1,k-1}} + c \|h\|_{H^{s-1,k-1}} \|\nabla_x \cdot V\|_{H^{s-1,k-1}} \\ + c \|\nabla_x \eta\|_{H^{s-1,k-1}} \|\partial_r V\|_{H^{s-1,k-1}}. $$

If $k-1=0$, we use (A.1) to get

$$ \|\Lambda^{s-k} \partial_r^{k-1} ((1 + \epsilon h) \nabla_x^\varphi \cdot V)\|_{L^2} \leq c \|\nabla_x \cdot V\|_{H^{s-1,0}} + c \|h\|_{H^{s-1,0}} \|\nabla_x \cdot V\|_{H^{s_0+\frac{1}{2},1}} \\ + c \|h\|_{H^{s_0+\frac{1}{2},1}} \|\nabla_x \cdot V\|_{H^{s-1,0}} + c \|\nabla_x \eta\|_{H^{s-1,0}} \|\partial_r V\|_{H^{s_0+\frac{1}{2},1}} \\ + c \|\nabla_x \eta\|_{H^{s_0+\frac{1}{2},1}} \|\partial_r V\|_{H^{s-1,0}}. $$

In any case, as $s \geq s_0 + \frac{5}{2} \geq 2$:

$$ \|\Lambda^{s-k} \partial_r^{k-1} ((1 + \epsilon h) \nabla_x^\varphi \cdot V)\|_{L^2} \leq C (\|V\|_{H^s} + \|\eta\|_{H^s}). $$

The second term $\nabla_x \eta \cdot \underline{V}'$ is controlled the same way and we get

$$ \|w^\Lambda\| \leq C (\|V\|_{H^s} + \|\eta\|_{H^s}). $$

We can now bound the terms (ii) and (vii).
where $C > 0$ is a constant depending only on $M_0, h_*, h^*$. Suppose that $T_\delta$ is finite, for a certain $\delta > 0$. Then, Grönwall lemma allows us to bound $\|(\mathbf{U}_\delta(t, \cdot), \eta_\delta(t, \cdot))\|_{H^s}^2$ by $e^{Ct}M_0$. Taking $T < \ln(2M_0)^{1/C}$, we have that:

$$ \|(\mathbf{U}_\delta(t, \cdot), \eta_\delta(t, \cdot))\|_{H^s}^2 < 2M_0. $$

Recall the equation on $h_\delta$, obtained by differentiating the equation on $\eta_\delta$ in $(E_\delta)$:

$$ (4.15) \qquad \partial_t h_\delta = -\partial_r (\mathcal{J}_\delta(\mathcal{J}_\delta V_\delta \cdot \nabla_x \mathcal{J}_\delta \eta_\delta) - w). $$

Integrating this equation in time allows us to bound $1+h_\delta$ from above and from below by a continuous function of $\|(\mathbf{U}_\delta, \eta_\delta)\|_{H^{0+\frac{3}{2}}}$, namely

$$ (4.16) \qquad 1-t \|(\mathbf{U}_\delta, \eta_\delta)\|_{L^\infty([0,t],H^{0+\frac{3}{2}})} \le 1+h_\delta(t,x,r) \le 1+t \|(\mathbf{U}_\delta, \eta_\delta)\|_{L^\infty([0,t],H^{0+\frac{3}{2}})} $$

Therefore, for $T$ small enough, we still have $\frac{1}{2}h_* \le 1+h(t, x, r) \le 2h^*$ for $t < T$ and all $(x, r) \in S_r$. Hence we can use Cauchy-Lipschitz theorem once again to extend the previous solution up to a time $T$, independent of $\delta$.

The $L^2$ energy estimate (4.14) yields that $((\mathbf{U}, \eta)_\delta)_{\delta}$ is a Cauchy sequence with convergence rate $\delta$. Thus, by interpolation (see Lemma A.8), it is also a Cauchy sequence in $H^{s-1}$. The convergence is thus strong in $H^{s_0+\frac{3}{2},2}$, so that we may pass to the limit in every non-linear term in (4.1) and the limit $(\mathbf{U}, \eta)$ is indeed a solution of (4.1). Finally, as the sequence $((V, w, \eta)_\delta)$ is bounded in $H^s$, it has a weak limit by Banach-Alaoglu's theorem, and this weak-limit coincides with the strong limit in $H^{s_0+\frac{3}{2},2}$. We finally get the existence of a solution of (4.1) in $C^0([0, T], H^{s-1}) \cap L^\infty([0, T], H^s)$.

Uniqueness follows from the $L^2$ energy estimate (4.14) performed on (4.1) directly (or, equivalently, on $(E_\delta)$ with $\delta = 0$).

### Step 3: Continuity in time of the solution

Following very closely [MBO02, Theorem 3.5], we can prove the following

$$ \|\mathbf{U}(t, \cdot) - \mathbf{U}(t_0, \cdot)\|_{H^s} \underset{t \to t_0}{\longrightarrow} 0, $$

for any $t_0 \in [0, T)$. The proof does not need any adaptation and we omit it.

### Step 4: Blow-up criterion

Let us assume that $(\mathbf{U}, \eta)$ is a bounded solution of (4.1) in $C^0([0, T], H^s)$ for some $T > 0$, satisfying the assumption (H.15). Then by using the solution of (4.1) constructed in Step 2 starting from $(\mathbf{U}(T), \eta(T))$ at initial time $T$, as well as its uniqueness, we may extend $(\mathbf{U}, \eta)$ on $[0, T]$ for some $T' > T$. Therefore, it $T^*$ is the maximal time of existence of this solution, then either (4.2) holds or

$$ (4.17) \qquad \|\eta(t, \cdot)\|_{H^s} \underset{t \to T^*}{\longrightarrow} +\infty. $$

We now show that the case (4.2) happens in any case. Standard energy estimates on the equation on $\eta$ in (4.1) yield

$$ (4.18) \qquad \frac{d}{dt} \|\eta\|_{H^s} \le C (\|\eta\|_{H^s} + \|\mathbf{U}\|_{H^s}), $$

where $C$ only depends on an upperbound on $\|\mathbf{U}\|_{H^s}$. Therefore, by Gronwall Lemma, if (4.17) holds true, then so does the first case in (4.2).

### Step 5: Propagation of the divergence-free condition

Using the equation on $\eta$ in (4.1), we can write the equation on $\mathbf{U}$ in (4.1) as

$$ \partial_t^\varphi \mathbf{U} + \mathbf{U} \cdot \nabla^\varphi \mathbf{U} + \nabla^\varphi P + \mathbf{b} = 0. $$

Applying $\nabla^\varphi \cdot$ to this equation and using the equation (4.11) satisfied by the pressure, we get

$$ (4.19) \qquad \partial_t^\varphi \nabla^\varphi \cdot \mathbf{U} + \partial_i^\varphi \mathbf{U}_j \partial_j^\varphi (\Pi \mathbf{U} - \mathbf{U})_i + \mathbf{U} \cdot \nabla^\varphi (\nabla^\varphi \cdot \mathbf{U}) = 0. $$

Testing this equation against $(1+h)\nabla^\varphi \cdot \mathbf{U}$, we get

$$ (4.20) \qquad \frac{d}{dt} \|\nabla^\varphi \cdot \mathbf{U}\|_{L^2} \le C (\|\nabla^\varphi \cdot \mathbf{U}\|_{L^2} + \|\nabla^\varphi (\Pi \mathbf{U} - \mathbf{U})\|_{L^2}), $$

where $C$ depends on $M_{in}, h_*, h^*$. Using the estimate (4.7) and Gronwall Lemma, we get

$$ \|\nabla^\varphi \cdot \mathbf{U}(t, \cdot)\|_{L^2} \le \|\nabla^\varphi \cdot \mathbf{U}(0, \cdot)\|_{L^2} e^{Ct}. $$

Hence the divergence-free condition in (4.1) is propagated by the equation if it holds initially. $\square$
(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 2.4 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.2) together with (B.3) to write

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

The last two terms are controlled with equations 2.3 and (2.11) respectively. For the first one, remark that

$$ \Lambda \nabla_x^\varphi P = \Lambda^{s-k} \partial_r^{k-1} \partial_r \nabla_x^\varphi P = \Lambda^{s-k} \partial_r^{k-1} (1 + \epsilon h) \partial_r^\varphi \nabla_x^\varphi P = \Lambda^{s-k} \partial_r^{k-1} (1 + \epsilon h) \nabla_x^\varphi \partial_r^\varphi P. $$

Thus, we control the $L^2$ norm of this term with $C\|\partial_r^\varphi P\|_{H^{s,k}}$ by product estimates (A.1).

This eventually yields a control of $\|\nabla_x^\varphi P^\Lambda\|_{L^2}$ that is uniform in $\mu$, and allows us to conclude:

$$ \int_{S_r} \nabla^\varphi P^\Lambda \cdot U^\Lambda (1 + \epsilon h) \le C (\|V^\Lambda\|_{L^2} + \|V\|_{H^s} + \sqrt{\mu} \|w\|_{H^s} + \|\eta\|_{H^s})^2 $$

(vii) For the term (vii), we use Cauchy-Schwarz inequality, Lemma 3.5 as well as the previous bound on $\|w^\Lambda\|_{L^2}$, to get

$$ \int_{S_r} F w^\Lambda \le C (\|V^\Lambda\|_{L^2} + \|V\|_{H^s} + \sqrt{\mu} \|w\|_{H^s} + \|\eta\|_{H^s})^2 $$

Summing these estimates together yields

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

Lemma 3.2 allows us to conclude.

□

**Remark 3.7.** The proof yields a slightly more precise estimate than the one stated in Proposition 3.4, namely

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

The contributions of the different source terms of (3.3) in this estimate have already been summarized in Remark 3.6. Let us now detail how the additional assumptions in [DLS20] allow the authors to obtain a time of existence of order $O((\epsilon/\sqrt{\mu})^{-1})$.

The terms in $R_2, R_5$ are of order 1. However, assuming $V = 0$ and making the strong Boussinesq assumption (that is, the coefficient $1/\partial$ in front of the pressure terms is a constant), as done in [DLS20], sets the terms of order 1 to zero and therefore these terms are of order $\epsilon$.

In the proof of Proposition 3.4, we then distinguished two cases. In [DLS20], the authors only use the strategy of the first case in the proof of Proposition 3.4, that is the pressure term is canceled by the use of the incompressibility conditions. Hence, they need the trace of the vertical derivatives of $w$ to vanish at the boundary, which is not true in general, but is the purpose of the good preparation of the initial data that they
**Remark 4.5.** In Step 1, we crucially used that the only unbounded operator in (E) is $\nabla_x$, which we can easily mollify using Fourier transform as $x \in \mathbb{R}^d$. In Eulerian coordinates, note that there is a term $w\partial_z V$, see for instance $(E^{\text{eul}})$. Thus, as we do not have any boundary condition on $V$, such a regularization is not possible. Therefore, the proof in Eulerian coordinates would require other ingredients.

**Remark 4.6.** In Step 5 of the proof of Proposition 4.1, we used the div-curl estimate (4.7). With a non-flat bottom or rigid lid, this estimate would contain boundary terms on the right-hand side involving the trace of **U**, and therefore the bootstrap argument in Step 5 would not hold true. Hence, the proof in the case of non-flat boundaries would require adaptations.

**Remark 4.7.** This strategy can be adapted to the free-surface case, however we leave this adaptation for a future work.

To conclude the proof of Theorem 1.1, we apply Proposition 4.1 with

$$ 
\begin{aligned}
\mathbf{U} &= \left( \frac{V + \epsilon V}{\sqrt{\mu} w} \right), && \eta = \epsilon \eta, \\
\mathbf{b} &= (0, b)^T,
\end{aligned}
$$

where $b$ is defined through (1.14). Therefore, there exists a unique solution of (E). The blow-up criterion (4.2) together with the estimate of Proposition 3.4 yields a time of existence independent of $\epsilon$ and $\mu$.

**4.2. Well-posedness in Eulerian coordinates.** We now state the result of Theorem 1.1 in Eulerian coordinates, for the sake of completeness. Let us first detail the assumptions necessary for Corollary 4.8 to hold. Let $s \in \mathbb{N}$. First, let $\underline{M} > 0$. We assume that $\underline{V}, \underline{\rho} \in W^{s+1,\infty}$ and

$$ (H.19) \qquad |\underline{V}^{\text{eul}}|_{W^{s+1,\infty}} + |\underline{\rho}|_{W^{s+1,\infty}} \le \underline{M}. $$

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

$$ (H.20) \qquad \|V_{\text{in}}^{\text{eul}}\|_{H^s}^2 + \mu \|w_{\text{in}}^{\text{eul}}\|_{H^s}^2 + \|\rho_{\text{in}}\|_{H^s} \le M_{\text{in}}. $$

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

$$ 
\begin{aligned}
(H.21) \qquad & w_{\text{in}}^{\text{eul}}|_{|r=0} = w_{\text{in}}^{\text{eul}}|_{|r=1} = 0, \\
& \nabla_x \cdot V_{\text{in}}^{\text{eul}} + \partial_z w_{\text{in}}^{\text{eul}} = 0.
\end{aligned}
$$

The stratification is stable, i.e. there exists $c_* > 0$ such that:

$$ (H.22) \qquad \underline{\rho}' \ge c_* $$

We now state our main result in Eulerian coordinates.

**Corollary 4.8.** Let $d \in \mathbb{N}^*$, $s_0 > \frac{d}{2}$, $s \in \mathbb{N}$ with $s \ge s_0+\frac{5}{2}$. Let $\underline{V}^{\text{eul}}, \underline{\rho} \in W^{s+1,\infty}$ satisfying (H.19), and $(V_{\text{in}}^{\text{eul}}, w_{\text{in}}^{\text{eul}}, \rho_{\text{in}}) \in H^s(S_z)^{d+2}$ satisfying (H.20), (H.21), (H.22). We also make the assumptions

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

Then there exists $T > 0$ depending only on $\underline{M}, M_{\text{in}}$ such that the following holds. There exists a unique solution $(V^{\text{eul}}, w^{\text{eul}}, \eta) \in L^\infty([0, T], H^s(S_z)^{d+2})$ to (1.7) with initial data $(V_{\text{in}}^{\text{eul}}, w_{\text{in}}^{\text{eul}}, \rho_{\text{in}})$.

**Proof.** Let us define the quantities involved in Theorem 1.1. Let $(t, x, r) \in S_r$ and

$$ 
\begin{aligned}
\underline{\eta}(r) &:= -r, \\
\underline{V} &:= \underline{V}^{\text{eul}}(\underline{\eta}(r)), \\
V_{\text{in}}(x, r) &:= V_{\text{in}}^{\text{eul}}(\underline{\eta}(r) + \epsilon \eta_{\text{in}}(x, r)), \\
w_{\text{in}}(x, r) &:= w_{\text{in}}^{\text{eul}}(\underline{\eta}(r) + \epsilon \eta_{\text{in}}(x, r)).
\end{aligned}
$$
assume. Furthermore, with the assumptions of the strong Boussinesq approximation and constant Brunt-Väisälä frequency, the commutators in the term $F$ are zero. The upside is that they do not have any term of order 1, and no shear velocity at equilibrium, and thus obtain a time of existence of order $\frac{\sqrt{\mu}}{\epsilon}$.

*Remark 3.8.* Finally, let us state that the distinction of the two cases described above holds true in Eulerian coordinates, and is by no mean specific to the isopycnal coordinates. Conversely, we believe that making the same assumptions as in [DLS20] would yield the same time of existence in isopycnal coordinates. However, the propagation of the good preparation assumption seems to require more technical computations than in Eulerian coordinates.

## 4. CONCLUSION

We first finish the proof of Theorem 1.1. Then, we state the result of Theorem 1.1 in Eulerian coordinates (see Corollary 4.2), and compare more precisely the differences between Corollary 4.2 and [DLS20, Theorem 2].

### 4.1. Completion of the proof of Theorem 1.1.
We now sketch the proof of Theorem 1.1, which heavily relies on the energy estimate of Proposition 3.4.

*Proof of Theorem 1.1.* In order to prove Theorem 1.1, we first briefly show how the pressure term is Lipschitz continuous with respect to the unknowns $(V, w, \eta)$, in the Sobolev spaces $H^s$. Then, we use the energy estimate of Proposition 3.4 to prove the uniqueness of solutions, and we turn to the existence of solutions. We first mention how to mollify the system. Finally, we use Cauchy-Lipschitz theorem to solve the mollified system, and use a compactness argument to extract a solution of the original system. Let $s_0 > \frac{d}{2}$.

**Step 1:** $(V, w, \eta) \to \nabla^\varphi P$ is Lipschitz continuous in $H^s(S_r)$ for $s \ge s_0 + \frac{5}{2}$

Let $(V_i, w_i, \eta_i) \in (H^s)^{d+2}$ for $i \in \{1, 2\}$ be two solutions of $(E)$ with the same initial data, and assume that their $H^s$-norms are bounded by a constant $M > 0$. Let $P_i$ be the pressures defined by Proposition 2.4 with respect to the solution $(V_i, w_i, \eta_i)$. We write $\phi_i(t, x, r) := (t, x, -r + \epsilon\eta_i(t, x, r))$ the change of variables associated to $\eta_i$. Then we can write an elliptic problem satisfied by the difference $P := P_1 - P_2$:

$$ \nabla_{\mu}^{\varphi_1} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_1} P = F_1 - F_2 + \left( \nabla_{\mu}^{\varphi_1} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_1} - \nabla_{\mu}^{\varphi_2} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_2} \right) P_2, $$

where $F_i$ is the right-hand side of the elliptic problem (2.1) satisfied by the pressure $P_i$. Note that the two terms on the right-hand side can be bounded by linear quantities in $\eta_1 - \eta_2, V_1 - V_2$ and $\sqrt{\mu}(w_1 - w_2)$. Reasoning as for Proposition 2.1, we thus get that $\|\nabla_{\mu}^{\varphi_1} P\|_{H^s}$ is bounded by the $H^{s+1}$-norm of the difference $\eta_1 - \eta_2$, as well as the $H^s$-norm of $V_1 - V_2$ and $\sqrt{\mu}(w_1 - w_2)$.

To obtain a bound without any loss of derivatives, we must use Alinhac's good unknowns, as in Proposition 2.4. Indeed, applying $\Lambda := |D|\Lambda^{s-1}$ or $\Lambda^{s-l}\partial_r^l$ for $1 \le l \le s$ to the elliptic problems satisfied by $P_1$ and $P_2$, we get:

$$ \nabla_{\mu}^{\varphi_1} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_1} (P_1^{\Lambda} - P_2^{\Lambda}) = \tilde{F}_1 - \tilde{F}_2 + \left( \nabla_{\mu}^{\varphi_1} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_1} - \nabla_{\mu}^{\varphi_2} \cdot \frac{1}{\varrho} \nabla_{\mu}^{\varphi_2} \right) P_2^{\Lambda}, $$

where $\tilde{F}_1 - \tilde{F}_2$ is the difference of the source terms in (2.7), which is Lipschitz-continuous in the unknowns. Therefore, reasoning as in the proof of Proposition 2.4, we get the
Note the second term in $V_{\text{in}}^{\text{eff}}$ that, in spite of the factor $\frac{1}{\epsilon}$, is of order 1 (by Taylor expansion, or composition estimates). Because of this term, we can write

$$ (\underline{V} + \epsilon V_{\text{in}}^{\text{eff}})(x, r) = (\underline{V}^{\text{eul}} + \epsilon V_{\text{in}}^{\text{eul}})(t, x, \underline{\eta}(r) + \epsilon \eta_{\text{in}}(x, r)), $$

and $\underline{V}$ is independent of $x$.

Theorem 1.1 then claims that there exists a unique solution $(V^{\text{eff}}, w, \eta)$ to (E), related to the quantities at equilibrium and initial conditions above. We can then define

$$ \left\{ \begin{aligned} \epsilon \rho(t, x, z) &:= (\underline{\eta} + \epsilon \underline{\eta})^{-1}(t, x, z) - \underline{\rho}(z), \\ V^{\text{eul}}(t, x, z) &:= V^{\text{eff}}(t, x, \underline{\rho}(z) + \epsilon \rho(t, x, z)) - \frac{1}{\epsilon} (\underline{V}^{\text{eul}}(\underline{\eta}(r) + \epsilon \eta(t, x, r)) - \underline{V}(r)), \\ w^{\text{eul}}(t, x, z) &:= w(t, x, \underline{\rho}(z) + \epsilon \rho(t, x, z)). \end{aligned} \right. $$

Note that the definition of $V^{\text{eff}}$ is such that $\underline{V}^{\text{eul}} + \epsilon V^{\text{eul}} = (\underline{V} + \epsilon V^{\text{eff}})(t, x, \underline{\rho} + \epsilon \rho)$.

Changing variables in (E') shows that $(V^{\text{eul}}, w^{\text{eul}}, \underline{\rho})$ is indeed a solution to $(E^{\text{eul}})$ with the desired boundary conditions and initial data.

For the uniqueness, note that two solutions of $(E^{\text{eul}})$ with the same initial data yield, via change of variables, two solutions of (E). Thus, according to Theorem 1.1, the solutions are the same. □

# APPENDIX A. PRODUCT, COMMUTATOR AND COMPOSITION ESTIMATES

In this appendix we gather technical results in Sobolev spaces, such as product, commutator and composition estimates. Most of these results has already been derived, see for instance [Lan13], [BD24].

**Lemma A.1 (Integration by parts in isopycnal coordinates).** If $f$ and $g$ are $H^1(S_r)$, then

$$ \int (1 + \epsilon h) g \nabla^\varphi \cdot f dx dr = - \int_{r=0} f \cdot \mathbf{e}_3 g dx + \int_{r=1} f \cdot \mathbf{e}_3 g dx - \int (1 + \epsilon h) f \nabla^\varphi g dx dr, $$

where $\mathbf{e}_3$ is the vertical upward vector, i.e. the normal vector to the bottom and rigid lid, pointing upward.

*Proof.* The proof consists in going back in Eulerian coordinates to use the standard integration by parts formula, and then doing the isopycnal change of coordinates again. Note that $(1 + \epsilon h)$ is the Jacobian of the isopycnal change of coordinates. □

**Lemma A.2 (A continuous embedding for anisotropic spaces).** Let $s \in \mathbb{R}$, and $f \in H^{s+\frac{1}{2},1}$, we can write

$$ \sup_{0 \le r \le 1} |f(\cdot, r)|_{H^s} \lesssim \|f\|_{H^{s+\frac{1}{2},1}}. $$

*Proof.* See [BD24, Lemma A.1] or [Lan13, Proposition 2.12]

**Lemma A.3 (Standard product estimates in Sobolev spaces $H^s(\mathbb{R}^d)$).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$, $s \ge -s_0$. There exists a constant $C > 0$ such that for any $f \in H^s(\mathbb{R}^d) \cap H^{s_0}(\mathbb{R}^d)$, $g \in H^s(\mathbb{R}^d)$, we can write

$$ |f g|_{H^s(\mathbb{R}^d)} \le C |f|_{H^{s_0}} |g|_{H^s} + C \langle |f|_{H^s} |g|_{H^{s_0}} \rangle_{s>s_0}. $$

**Lemma A.4 (Product estimates in anisotropic spaces, [BD24]).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$, $s \in \mathbb{R}$ and $k \in \mathbb{N}$ with $0 \le k \le s$.

* There exists $C > 0$ such that for any $f \in L^2(S_r)$ and $g \in H^{s_0+\frac{1}{2},1}(S_r)$, we can write

$$ \|fg\|_{L^2} \le C \|f\|_{L^2} \|g\|_{H^{s_0+1/2,1}}. \quad (A.1) $$

* If $s \ge s_0 + \frac{1}{2}$, $k \ge 1$, then there exists $C > 0$ such that for any $g \in H^{s,k}$, $f \in H^{s,k}$

$$ \|fg\|_{H^{s,k}} \le C \|f\|_{H^{s,k}} \|g\|_{H^{s,k}}. \quad (A.2) $$

* If $f \in W_r^{k,\infty}$ is independent of $x$ and $g \in H^{s,k}$, then

$$ \|fg\|_{H^{s,k}} \le C|f|_{W^{k,\infty}} \|g\|_{H^{s,k}}. \quad (A.3) $$
bound

$$ \|\nabla_{\mu}^{\varphi_1}(P_1^{\Lambda} - P_2^{\Lambda})\|_{L^2} \le C(\|V_1 - V_2\|_{H^{s,k}} + \sqrt{\mu}\|w_1 - w_2\|_{H^{s,k}} + \|\eta_1 - \eta_2\|_{H^{s,k}}). $$

Here, $C > 0$ depends on $M$.

To conclude, we write

$$ (4.1) \qquad \|\Lambda(\nabla_{\mu}^{\varphi_1} P_1 - \nabla_{\mu}^{\varphi_2} P_2)\|_{L^2} \le \|\nabla_{\mu}^{\varphi_1} (P_1^{\Lambda} - P_2^{\Lambda})\|_{L^2} + \|(\nabla_{\mu}^{\varphi_2} - \nabla_{\mu}^{\varphi_1}) P_2^{\Lambda}\|_{L^2} + \|R\|_{L^2}, $$

where $R$ is the difference of the remainder terms appearing in the relation (B.2) on the commutation between the operators $\Lambda$ and $\nabla_{\mu}^{\varphi_1}$. The term $R$ is multi-linear and as such it is easily bounded by a linear function of $\|\eta_1 - \eta_2\|_{H^s}$ and $\|\nabla_{\mu}^{\varphi_1} P_1 - \nabla_{\mu}^{\varphi_2} P_2\|_{H^{s-1}}$. The first term of (4.1) has already been treated above, and for the second one we write

$$ \| (\nabla_{\mu}^{\varphi_2} - \nabla_{\mu}^{\varphi_1}) P_2^{\Lambda} \|_{L^2} \le \| \eta_1 - \eta_2 \|_{H^{s_0+\frac{3}{2},2}} \| \nabla_{\mu}^{\varphi_2} P_2 \|_{H^s} \le C \| \eta_1 - \eta_2 \|_{H^{s_0+\frac{3}{2},2}} $$

by Proposition 2.4.

Therefore, the gradient of the pressure is Lipschitz continuous in $H^s$.

### Step 2: Uniqueness

For the uniqueness, let $(V_1, w_1, \eta_1)$ and $(V_2, w_2, \eta_2)$ be two solutions of (E) with the same initial data. We write the system verified by the difference $(V, w, \eta) := (V_1, w_1, \eta_1) - (V_2, w_2, \eta_2)$. We treat the pressure as a Lipschitz-continuous source-term and proceed as in Proposition 3.3 to derive the $L^2$ energy estimate on $(V, w, \eta)$. Grönwall lemma then shows that as the solutions coincide at $t = 0$, their difference remains 0 for $t > 0$.

We now follow the steps of [MBO02] for the existence. Note that the gradient of the pressure is a Lipschitz-continuous operator, and we treat it as a source term. Thus, the only unbounded operator in $H^{s,k}$ here is $\nabla_x$. Here, we crucially use the quasi-2D structure of the isopycnal coordinates.

### Step 3: Mollification

Let $\mathcal{J}_{\delta}$ for $\delta > 0$ the mollification operator defined with the convolution with a smooth cutoff function in the horizontal Fourier space (see [MBO02]), to regularize $\nabla_x$. We then introduce the following mollified system

$$ (E_{\delta}) \quad \left\{ \begin{array}{l} \partial_t V_{\delta} + \mathcal{J}_{\delta}[\mathcal{J}_{\delta}(\underline{V} + \epsilon V_{\delta}) \cdot \nabla_x \mathcal{J}_{\delta} V_{\delta}] + \frac{1}{\varrho} \nabla_x^{\varphi} P_{\delta} = 0, \\ \mu(\partial_t w_{\delta} + \mathcal{J}_{\delta}[\mathcal{J}_{\delta}(\underline{V} + \epsilon V_{\delta}) \cdot \nabla_x \mathcal{J}_{\delta} w_{\delta}]) + \frac{1}{\varrho} \partial_r^{\varphi} P_{\delta} + b_{\delta} = 0, \\ \partial_t \eta_{\delta} + \mathcal{J}_{\delta}[\mathcal{J}_{\delta}(\underline{V} + \epsilon V) \cdot \nabla_x \mathcal{J}_{\delta} \eta_{\delta}] - w_{\delta} = 0, \end{array} \right. $$

together with the boundary conditions

$$ (w_{\delta})_{|r=0} = (w_{\delta})_{|r=1} = 0, $$

as well as the initial conditions

$$ \begin{cases} (\mathbf{U}_{\delta})_{|t=0} = \mathbf{U}_{\text{in}}, \\ (\eta_{\delta})_{|t=0} = \eta_{\text{in}}. \end{cases} $$

This mollified system $(E_{\delta})$ has the same structure as (E), and one can check that the energy estimate of Proposition 3.4 is thus valid for the regularized system. Cauchy-Lipschitz theorem yields a solution $(V, w, \eta)_{\delta}$ on a time $T_{\delta} > 0$ for any $\delta > 0$, in the
Proof. For the first estimate, we use the standard Sobolev inequality

$$|f|_{L^{\infty}} \leq c|f|_{H^s},$$

for $s \geq s_0 > d/2$ and where $c$ depends on $s_0$. We can then write

$$\|fg\|_{L^2} = \|fg\|_{L_x^2 L_t^2} \leq c \|f\|_{L_x^2} \|g\|_{H^{s_0}} \|f\|_{L_x^2 L_t^2} \sup_{r \in [0,1]} |g|_{H^{s_0}} \leq c \|f\|_{L^2} \|g\|_{H^{s_0+\frac{1}{2},1}}$$

by the embedding of Lemma A.2.

The second inequality is proved by using Leibniz rule for the vertical derivatives and then using the previous estimate. This is thoroughly done in [BD24, Lemma A.3].

For the last one it suffices to estimate terms of the form $\partial_r^l f \partial_r^{l_2} g$ for $l_1 + l_2 = l \leq k$ in $H^{s-l}$. This is done by using $L^\infty - L^2$ Hölder estimate. $\square$

We now state commutator estimates in $H^{s,k}$ spaces. Note the use of $H^{s-1,k\wedge(s-1)}$-norms, so that in the case $k=s$, we gain one derivative.

**Lemma A.5 (Commutator estimates).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$ and $s > 0$.

We start with standard product estimates in $\mathbb{R}^d$. Assume that $s \geq s_0 + 1$, and let $r \in [-\frac{1}{2}, 0]$. If $f \in H^s(\mathbb{R}^d)$ and $g \in H^{s-1}(\mathbb{R}^d)$, then there exists $C > 0$ such that

$$\| [\Lambda^s, f] g \|_{H^r} \leq C |\nabla f|_{H^{s-1}} |g|_{H^s} \quad (A.4)$$

We now turn to commutator estimates in the strip $S_r$. Let $k \in \mathbb{N}$ with $0 \leq k \leq s$.

* If $l \in \mathbb{N}$ with $l \leq k$, then there exists $C > 0$ such that for any $f \in H^{s\vee(s_0+\frac{3}{2}),k\vee 2}$ and $g \in H^{(s-1)\vee(s_0+\frac{1}{2}),2\vee(k\wedge(s-1))}$, we can write

$$\| [\Lambda^{s-l} \partial_r^l, f] g \|_{L^2} \leq C \|f\|_{H^{s\vee(s_0+\frac{3}{2}),k\vee 2}} \|g\|_{H^{(s-1)\vee(s_0+\frac{1}{2}),2\vee(k\wedge(s-1))}} \quad (A.5)$$

* If $s \geq 1$, then there exists $C > 0$ such that for any $f \in H^{s,0} \cap H^{s_0+\frac{3}{2},1}$, $g \in H^{s-1,0}$, we can write

$$\| [D|\Lambda^{s-1}, f] g \|_{L^2} \leq C \| \nabla_x f \|_{H^{s_0+\frac{1}{2},1}} \| g \|_{H^{(s-1)\vee(s_0+\frac{1}{2}),0}} + C \langle \| \nabla_x f \|_{H^{(s-1)\vee(s_0+\frac{1}{2}),0}} \| g \|_{H^{s_0+\frac{1}{2},1}} \rangle_{s>s_0+1} \quad (A.6)$$

* Let $k \geq 1$. There exists $C > 0$ such that if $f \in W_r^{k,\infty}$ and $g \in H^{s,k}$, then

$$\| [\Lambda^{s-k} \partial_r^k, f] g \|_{L^2} \leq C |f|_{W_{k,\infty}} \|g\|_{H^{s-1,k-1}} \quad (A.7)$$

Proof. Estimate (A.4) is a special case of [Lan13, Proposition B.8 (2)].

Estimate (A.5) is detailed in [BD24, Lemma A.8].

Estimate (A.6) is done in [Lan13, Corollary B.16], but we sketch the proof for the sake of completeness. We start by writing the standard the commutator estimate in $H^s(\mathbb{R}^d)$ (see for instance [Lan06, Theorem 6, Remark 38] and references therein), namely

$$\| [D|\Lambda^{s-1}, f] g \|_{L^2} \leq C |\nabla_x f|_{H^{s_0}} |g|_{H^{s-1}} + \langle |\nabla_x f|_{H^{s-1}} |g|_{H^{s_0}} \rangle_{s>s_0+1}$$

Thus, taking the $L^2$-norm in the vertical dimension and using the continuous embedding of Lemma A.2 yields the result

$$\| [D|\Lambda^{s-1}, f] g \|_{L^2} \leq C \| \nabla_x f \|_{H^{s_0+\frac{1}{2},1}} \| g \|_{H^{s-1,0}} + C \langle \| \nabla_x f \|_{H^{s-1,0}} \| g \|_{H^{s_0+\frac{1}{2},1}} \rangle_{s>s_0+1}$$

For inequality (A.7), we start by noticing that as $f$ only depends on $r$, it commutes with $\Lambda^{s-k}$. Then the standard the commutator estimate in $L_r^2$ yields the result. $\square$

**Lemma A.6 (Symmetric commutator estimates).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$, $s \in \mathbb{R}$ with $s \geq s_0 + \frac{5}{2}$ and $k \in \mathbb{N}$ with $2 \leq k \leq s$.

* If $l \in \mathbb{N}$ with $0 \leq l \leq k$. Then there exists $C > 0$ such that for any $f \in H^{s,k}$, $g \in H^{s-1,k\wedge(s-1)}$, we can write

$$\| [\Lambda^{s-l} \partial_r^l; f, g] \|_{L^2} \leq C \|f\|_{H^{s-1,k\wedge(s-1)}} \|g\|_{H^{s-1,k\wedge(s-1)}} \quad (A.8)$$

* For $s \in \mathbb{R}$ and $s \geq 1$, $f \in H^{s-1,0} \cap H^{s_0+\frac{3}{2},1}$, $g \in H^{s-1,0} \cap H^{s_0+\frac{3}{2},1}$ we can write

$$\| [D|\Lambda^{s-1}; f, g] \|_{L^2} \leq C \|f\|_{H^{s-1,0}} \|g\|_{H^{s_0+\frac{3}{2},1}} + C \|g\|_{H^{s-1,0}} \|f\|_{H^{s_0+\frac{3}{2},1}} \quad (A.9)$$

* There exists a constant $C_0$ also depending on $\|f\|_{H^{s_0+\frac{3}{2},2}}$ and $\|g\|_{H^{s_0+\frac{3}{2},2}}$ such that

$$\| [\Lambda^{s-1} \partial_r; f, g] \|_{L^2} \leq C_0 (\|f\|_{H^{s-1,1}} + \|g\|_{H^{s-1,1}}) \quad (A.10)$$
Banach space

$$
\mathcal{B} := \{(V_{\delta}, w_{\delta}, \eta_{\delta}) \in (H^{s,k})^{d+2}, \mathcal{E}(V_{\delta}, w_{\delta}, \eta_{\delta}) \le 2M_0, \frac{1}{2}h_* \le 1 + \epsilon h \le 2h^*, \varrho' \ge c_*\}
$$

where $\mathcal{E}$ is defined in Definition 3.1. We use a continuity argument to show that the
times of existence $T_\delta$ are bounded from below by some $T > 0$ independent of $\delta, \epsilon, \mu$.
Suppose that $T_\delta$ is finite, for a certain $\delta > 0$. Then, by continuity in time of the solution
$(V_\delta, w_\delta, \eta_\delta)$ and continuity of $\mathcal{E}$ and the constant $C$ provided by Proposition 3.4, there
exists a time $T > 0$ such that for all $t < T$, $C(t) < 2C(0)$. Proposition 3.4 then reads

$$
\frac{d}{dt} \mathcal{E}(t) \le 2C(0) \mathcal{E}(t).
$$

Grönwall lemma allows us to bound $\mathcal{E}(t)$ by $e^{2tC(0)}$. Taking $T < \ln(2) \frac{1}{2C(0)}$, we have that
$\mathcal{E}(t) < 2M_0$.

Recall the equation on $h$, obtained by differentiating the equation on $\eta$ in (E):

$$
\epsilon \partial_t h + \nabla_x \cdot ((1 + \epsilon h)(\underline{V} + \epsilon V)) = 0
$$

Integrating this equation in time allows us to bound 1 + ϵh from above and from below by a continuous function of E. Therefore, for T small enough, we still have 1/2 h* ≤ 1 + ϵh(t, x, r) ≤ 2h* for t < T and all (x, r) ∈ Sr. Hence we can use Cauchy-Lipschitz theorem once again to extend the previous solution up to a time T, independent of δ, μ and ϵ.

**Step 4: existence of a solution of (E)**

As for the uniqueness, the $L^2$ energy estimate on (E) yields that $((V, w, \eta)_\delta)$ is a Cauchy sequence with convergence rate $\delta$. Thus, by interpolation (see Lemma A.8), it is also a Cauchy sequence in $H^{s-1}$. The convergence is thus strong in $H^{s_0+\frac{3}{2},2}$, so that we may pass to the limit in every non-linear term in (E) and the limit $(V, w, \eta)$ is indeed a solution of (E). Finally, as the sequence $((V, w, \eta)_\delta)$ is bounded in $H^s$, it has a weak limit by Banach-Alaoglu's theorem, and this weak-limit coincides with the strong limit in $H^{s_0+\frac{3}{2},2}$. □

**Remark 4.1.** Here we crucially used that the only unbounded operator in (E) is $\nabla_x$, which we can easily mollify using Fourier transform as $x \in \mathbb{R}^d$. In Eulerian coordinates, note that there is a term $w \partial_z V$, see for instance $(E^{\text{eul}})$. Thus, as we do not have any boundary condition on $V$, such a regularization is not possible. Therefore, the proof in Eulerian coordinates would require adaptations.

4.2. **Well-posedness in Eulerian coordinates.** We now state the result of Theorem 1.1 in Eulerian coordinates, for the sake of completeness. Let us first detail the assumptions necessary for Corollary 4.2 to hold.

Let $s \in \mathbb{N}$. First, let $\underline{M} > 0$. We assume that if $\underline{V}, \underline{\rho} \in W^{s+1,\infty}$, then

$$
(H.12) \qquad |\underline{V}^{\text{eul}}|_{W^{s+1,\infty}} + |\underline{\rho}|_{W^{s+1,\infty}} \le \underline{M}.
$$

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

$$
(H.13) \qquad \| V_{\text{in}}^{\text{eul}} \|_{H^s}^2 + \mu \| w_{\text{in}}^{\text{eul}} \|_{H^s}^2 + \| \rho_{\text{in}} \|_{H^s} \le M_{\text{in}}.
$$

Moreover, the initial data satisfy the boundary condition and incompressibility condi-
tion :

$$
\begin{align*}
& w_{\text{in}}^{\text{eul}}|_{|r=0} = w_{\text{in}}^{\text{eul}}|_{|r=1} = 0, \\
& \nabla_x \cdot V_{\text{in}}^{\text{eul}} + \partial_z w_{\text{in}}^{\text{eul}} = 0. \tag{H.14}
\end{align*}
$$
*Proof.* Estimate (A.8) is detailed in [BD24, Lemma A.9].

Estimate (A.9) follows from standard symmetric commutator estimates in $\mathbb{R}^d$ and continuous embedding of Lemma A.2.

For estimate (A.10), we start by writing

$$ \begin{aligned} [\Lambda^{s-1} \partial_r; f, g] &= \Lambda^{s-1} \partial_r (fg) - \Lambda^{s-1} \partial_r fg - f \Lambda^{s-1} \partial_r g \\ &= [\Lambda^{s-1}, g] \partial_r f + [\Lambda^{s-1}, f] \partial_r g. \end{aligned} $$

Now, standard commutator estimates in $\mathbb{R}^d$ yield:

$$ \begin{aligned} \|[\Lambda^{s-1} \partial_r; f, g]\|_{L^2(\mathbb{R}^d)} \le c (|g|_{H^{s-1}} |\partial_r f|_{H^{s_0}} + |g|_{H^{s_0+1}} |\partial_r f|_{H^{s-1}} \\ + |f|_{H^{s-1}} |\partial_r g|_{H^{s_0}} + |f|_{H^{s_0+1}} |\partial_r g|_{H^{s-1}}), \end{aligned} $$

where $c$ only depends on $s_0$, $s$. Now taking the $L^2$-norm in $r$ of the previous inequality and applying Lemma A.2 on each term yields (A.10). □

**Lemma A.7 (Composition estimates in $H^{s,k}(S_r)$).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$. Let $s \in \mathbb{R}$ with $s \ge s_0 + \frac{1}{2}$, $k \in \mathbb{N}$ with $1 \le k \le s$, $M > 0$. There exists $C > 0$ such that for any $F \in W^{s,k,\infty}(\mathbb{R} \times (0,1))$ with $F(0, \cdot) = 0$, and any $u \in H^{s,k}(S_r)$ such that $\|u\|_{H^{s,k}} \le M$, the following holds. Define $F \circ u : (x, r) \mapsto F(u(x, r), r)$. Then

$$ (A.11) \qquad \|F \circ u\|_{H^{s,k}} \le C |F|_{W^{s,k,\infty}(\mathbb{R} \times (0,1))} \|u\|_{H^{s,k}}. $$

In the particular case $u = \epsilon h$ and $F(y) = \frac{y}{1+y}$, under assumption (H.5), we get

$$ (A.12) \qquad \| \frac{\epsilon h}{1 + \epsilon h} \|_{H^{s,k}} \le C \|h\|_{H^{s,k}}. $$

*Proof.* The proof is detailed in [BD24, Lemma A.5]. □

We end this appendix with an interpolation result in $H^{s,k}$ spaces:

**Lemma A.8.** Let $s > 1$, $k = s$, and $0 \le s' \le s - 1$, $k' = s'$. Then there exists $C > 0$, $\alpha, \beta \in ]0, 1[$ such that for any $f \in H^{s,k}$,

$$ \|f\|_{H^{s',k'}} \le C \|f\|_{L^2}^{\alpha} \|f\|_{H^{s,k}}^{\beta}. $$

*Proof.* Let $(\chi_0, \chi_1) \in \mathcal{C}^\infty([0,1])$ be a partition of unity of $[0,1]$, namely $\chi_0(r) = 1$ for $r \in [0,1/3]$ and $\chi_0(r) = 0$ for $r \in [2/3,1]$ (mutatis mutandis for $\chi_1$) and $\chi_0 + \chi_1 = 1$.

Let $f \in H^{s,k}$, therefore $f = f_0 + f_1 := f\chi_0 + f\chi_1$, and we can extend $f_0$ to the half space $\mathbb{R}^d \times [0, +\infty[$ in a smooth way. Therefore, classical interpolation results in the half space (see for instance [Tay11, Section 4.4] yield

$$ |f_0(x, \cdot)|_{H^{k'}} \le |f_0(x, \cdot)|_{L^2}^{\gamma_1} |f_0(x, \cdot)|_{H^k}^{\gamma_2}, $$

for any $x$ fixed. Now note that as $\chi_0$ is bounded in $\mathcal{C}^\infty$, we have the bound

$$ f_0(x, \cdot)|_{H^k} \le |\chi_0|_{\mathcal{C}^\infty} |f|_{H^k}. $$

The same bound is true for the $L^2$ norm. Finally, the same reasoning is valid for $f_1$ and as $f = f_0 + f_1$, we get

$$ |f|_{H^{k'}} \le C |f|_{L^2}^{\gamma_1} |f|_{H^k}^{\gamma_2}. $$

Here, $C > 0$ depends on $\chi_0, \chi_1$ but not on $f$.

Let $0 \le l' \le k' \le k - 1$. We apply the previous inequality to $\Lambda^{s'-l'}f$:

$$ \begin{aligned} \|\Lambda^{s'-l'} \partial_r^{l'} f\|_{L^2} &\le C \|\Lambda^{s'-l'} f\|_{L^2}^{\gamma_1} \sum_{l \le l'+1} \|\Lambda^{s'-l'} \partial_r^l f\|_{L^2}^{\gamma_2} \\ &\le C \|f\|_{L^2}^{\alpha} \|f\|_{H^{s'-l'+1,0}}^{\gamma_3} \|f\|_{H^{s'+1,l'+1}}^{\gamma_2}, \end{aligned} $$

by classical interpolation inequality in $\mathbb{R}^d$ (see for instance [Tay11, Section 4.2]). As $s' + 1 \le s$, $k' + 1 \le k$ and the exponents are non-zero, this yields the result. □
The stratification is stable, i.e. there exists $c_* > 0$ such that:

$$ (H.15) \qquad \underline{\rho}' \geq c_* $$

We now state our main result in Eulerian coordinates.

**Corollary 4.2.** Let $d \in \mathbb{N}^*$, $s_0 > \frac{d}{2}$, $s \in \mathbb{N}$ with $s \ge s_0 + \frac{5}{2}$. Let $\underline{V}^{\text{eul}}$, $\underline{\rho} \in W^{s+1,\infty}$ satisfying (H.12), and $(V_{\text{in}}^{\text{eul}}, w_{\text{in}}^{\text{eul}}, \rho_{\text{in}}) \in H^s(S_z)^{d+2}$ satisfying (H.13), (H.14), (H.15). We also make the assumptions

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

Then there exists $T > 0$ depending only on $M, M_{\text{in}}$ such that the following holds. There exists a unique solution $(V^{\text{eul}}, w^{\text{eul}}, \eta) \in L^\infty([0, T], H^s(S_z)^{d+2})$ to (1.7) with initial data $(V_{\text{in}}^{\text{eul}}, w_{\text{in}}^{\text{eul}}, \rho_{\text{in}})$.

*Proof.* Let us define the quantities involved in Theorem 1.1. Let $(t, x, r) \in S_r$ and

$$ \left\{ \begin{array}{l} \underline{\eta}(r) := -r, \\ \underline{\rho}(r) := \underline{\rho}(\underline{\eta}(r)), \\ \underline{V} := \underline{V}^{\text{eul}}(\underline{\eta}(r)), \\ \epsilon \eta_{\text{in}}(x, r) := (\underline{\rho} + \epsilon \rho_{\text{in}})^{-1}(x, r) - \underline{\eta}(r), \\ V_{\text{in}}(x, r) := V_{\text{in}}^{\text{eul}}(\underline{\eta}(r) + \epsilon \eta_{\text{in}}(x, r)), \\ V_{\text{in}}^{\text{eff}}(x, r) := V_{\text{in}}(x, r) + \frac{1}{\epsilon} (\underline{V}^{\text{eul}}(\underline{\eta}(r) + \epsilon \eta_{\text{in}}(x, r)) - \underline{V}(r)), \\ w_{\text{in}}(x, r) := w_{\text{in}}^{\text{eul}}(\underline{\eta}(r) + \epsilon \eta_{\text{in}}(x, r)). \end{array} \right. $$

Note the second term in $V_{\text{in}}^{\text{eff}}$ that, in spite of the factor $\frac{1}{\epsilon}$, is of order 1 (by Taylor expansion, or composition estimates). Because of this term, we can write

$$ (\underline{V} + \epsilon V_{\text{in}}^{\text{eff}})(x, r) = (\underline{V}^{\text{eul}} + \epsilon V_{\text{in}}^{\text{eul}})(t, x, \underline{\eta}(r) + \epsilon \eta_{\text{in}}(x, r)), $$

and $\underline{V}$ is independent of $x$.

Theorem 1.1 then claims that there exists a unique solution $(V^{\text{eff}}, w, \eta)$ to (E), related to the quantities at equilibrium and initial conditions above. We can then define

$$ \left\{ \begin{array}{l} \epsilon \rho(t, x, z) := (\underline{\eta} + \epsilon \underline{\eta})^{-1}(t, x, z) - \underline{\rho}(z), \\ V^{\text{eul}}(t, x, z) := V^{\text{eff}}(t, x, \underline{\rho}(z) + \epsilon \rho(t, x, z)) - \frac{1}{\epsilon} (\underline{V}^{\text{eul}}(\underline{\eta}(r) + \epsilon \eta(t, x, r)) - \underline{V}(r)), \\ w^{\text{eul}}(t, x, z) := w(t, x, \underline{\rho}(z) + \epsilon \rho(t, x, z)). \end{array} \right. $$

Note that the definition of $V^{\text{eul}}$ is such that $\underline{V}^{\text{eul}} + \epsilon V^{\text{eul}} = (\underline{V} + \epsilon V^{\text{eff}})(t, x, \underline{\rho} + \epsilon \underline{\rho})$.

Changing variables in (E') shows that $(V^{\text{eul}}, w^{\text{eul}}, \rho)$ is indeed a solution to (E'eul) with the desired boundary conditions and initial data.

For the uniqueness, note that two solutions of (E'eul) with the same initial data yields, via change of variables, two solutions of (E). Thus, according to Theorem 1.1, the solutions are the same. $\square$

In order to conclude this study, let us compare Corollary 4.2 with [DLS20, Theorem 2].
APPENDIX B. DEFINITIONS AND RESULTS ON ALINHAC'S GOOD UNKNOWN

The system (E) of the Euler equations in isopycnal coordinates mainly differs from the one in Eulerian coordinates by the operators $\nabla^\varphi$, which arise from the chain rule when differentiating a composition of the unknowns with the change of variables. In other words, the operator $\nabla^\varphi$ is the gradient in Eulerian coordinates. Recall that the change of variables $\varphi$ is

$$ (B.1) \qquad \varphi(t, x, r) = (t, x, -r + \epsilon\eta(t, x, r)). $$

Recalling the notation $h := -\partial_r \eta$, we write its Jacobian matrix $\partial^\varphi$, as defined in (1.16). Recall the definition (1.17) of $\nabla^\varphi, \partial_t^\varphi, \partial_r^\varphi$. The non-linearity of these operators raises an issue when one wants to differentiate the system, say apply an operator $\Lambda$ of the form $\Lambda^{s-k} \partial_r^k$ with $s > 0, 0 \le k \le s$, as $\Lambda$ and $\nabla^\varphi$ do not commute, and the commutator involves higher derivatives in $\eta$ than what the energy method allows us to control. This is detailed in the following lemma.

**Lemma B.1.** Let $s \in \mathbb{R}$ with $s > s_0 + \frac{3}{2}, 0 \le k \le s$, with $k \in \mathbb{N}$. Assume that $\eta \in H^{s+1,2\sqrt{k}+1}$ and there exist $h_*, h^*$ such that $0 < h_* \le 1 + \epsilon h \le h^*$ with $h := -\partial_r \eta$. Then there exists $C > 0$ such that the following holds. We write $\Lambda$ an operator of the form $\Lambda^{s-k} \partial_r^k$ or $|D|\Lambda^{s-1}$ (with $k=0$ by convention). Let $f \in H^{s-1, k\Lambda(s-1)^{1/2}}$. Then

$$ (B.2) \qquad \| [\Lambda, \nabla^\varphi] f \|_{L^2} \le \epsilon C \| \eta \|_{H^{s\sqrt{s_0+\frac{3}{2}}+1, 2\sqrt{k}+1}} \| \nabla f \|_{H^{s-1, k\Lambda(s-1)^{1/2}}}. $$

*Proof.* Let $\Lambda$ an operator of the form $\Lambda^{s-k} \partial_r^k$ or $|D|\Lambda^{s-1}$. Recall the expression (1.17) of $\nabla^\varphi$. As $\Lambda$ and $\nabla^\varphi$ commute, we can write

$$ \| [\Lambda, \nabla^\varphi] f \|_{L^2} = \epsilon \| [\Lambda, \frac{1}{1+\epsilon h} \nabla^\varphi] f \|_{L^2}. $$

Recall indeed that $h = -\partial_r \eta$. Thus the commutator estimate (A.5) yields

$$ \| [\Lambda, \nabla^\varphi] f \|_{L^2} \le C \epsilon \| \frac{1}{1+\epsilon h} \nabla^\eta \|_{H^{s\sqrt{s_0+\frac{3}{2}}+1, 2\sqrt{k}}} \| f \|_{H^{s-1, k\Lambda(s-1)^{1/2}}}. $$

Then, the product estimate (A.2) and the composition estimate (A.11) yield the desired estimate. $\square$

This issue is solved by using Alinhac's good unknowns, introduced in [Ali89]. See also [MR17, Section 2.3] for its use for the free-surface Navier-Stokes equation, and [Lan13] for its use for the water waves equations, as well as [CL15, Proposition 3.17] where the importance of the relation (B.3) is emphasized. Recall the notation

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

**Lemma B.2** (Alinhac's good unknown, [Ali89]). Let $s \in \mathbb{R}$ with $s \ge s_0 + \frac{5}{2}, k, l \in \mathbb{N}$ with $0 \le l \le k \le s$ and $k \ge 2$. Assume that $\eta \in H^{s_0+\frac{3}{2}, 2}$ and there exist $h_*, h^*$ such that $0 < h_* \le 1 + \epsilon h \le h^*$ with $h := -\partial_r \eta$, and that there exists $M > 0$ such that (H.12) holds. Then there exists $C > 0$ depending only on $M$ such that the following holds. We write $\Lambda$ an operator of the form $\Lambda^{s-l} \partial_r^l$ (with $l \ge 1$) or $|D|\Lambda^{s-1}$ (with $l=0$ by convention).

• Let $f \in H^{s,k}$. We have

$$ (B.3) \qquad \begin{gathered} \Lambda \nabla_{t,x,r}^\varphi f = \nabla_{t,x,r}^\varphi f \Lambda + \epsilon (\Lambda \eta \partial_r^\varphi) \nabla_{t,x,r}^\varphi f + \epsilon R_1^\Lambda (\nabla_{t,x,r} f), \\ \text{where } \epsilon R_1^\Lambda (\nabla_{t,x,r} f) := (\partial \varphi)^{-T} [\Lambda; (\partial \varphi)^T, \nabla_{t,x,r}^\varphi f], \text{ and we can write} \end{gathered} $$

$$ (B.4) \qquad \| R_1^\Lambda (\nabla_{t,x,r} f) \|_{L^2} \le C \| \nabla_{t,x,r} \eta \|_{H^{s-1, k\Lambda(s-1)}} \| \nabla_{t,x,r}^\varphi f \|_{H^{s-1, k\Lambda(s-1)}}. $$

Here, Alinhac's good unknown is defined as

$$ (B.5) \qquad f^\Lambda := \Lambda f + \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \partial_r f. $$

• For $f \in (H^{s,k})^{d+1}$ a vector-valued function of $d+1$ components, we can write

$$ (B.6) \qquad \Lambda \nabla^\varphi \cdot \mathbf{f} = \nabla^\varphi \cdot \mathbf{f}^\Lambda + \epsilon \Lambda \eta \partial_r^\varphi (\nabla^\varphi \cdot \mathbf{f}) + \epsilon R_2^\Lambda (\nabla \mathbf{f}), $$

where

$$ \epsilon R_2^\Lambda (\nabla \mathbf{f}) := \sum_{i \in \{1, \dots, d+1\}} R_1^\Lambda (\nabla_{t,x,r} \mathbf{f}_i) \cdot \mathbf{e}_i. $$
**Remark 4.3.** The main novelty with respect to [DLS20, Theorem 2] is the addition of a background current $V$, which is not immediate in Eulerian coordinates. It would indeed make appear a term $wV'$, which is singular with respect to the shallowness parameter $\mu$. In isopycnal coordinates, we do not have this term; note however that $V$ appears in the incompressibility condition, which raises similar difficulties.

Note also that we do not need some assumptions made in [DLS20, Theorem 2], namely the strong Boussinesq assumption (that is, the factor $\frac{1}{\bar{\rho}+\epsilon\rho}$ in front of the pressure terms in (1.7) is constant), the assumption of uniform stratification (constant Brunt-Väisälä frequency $-\frac{\rho'}{\bar{\rho}}$) and the well-prepared initial data. The downside is the times of existence in Theorem 1.1 and Corollary 4.2, that are only uniform in $\epsilon$, whereas in [DLS20, Theorem 2] the time of existence is of order $\frac{\sqrt{\mu}}{\epsilon}$.

## APPENDIX A. PRODUCT, COMMUTATOR AND COMPOSITION ESTIMATES

In this appendix we gather technical results in Sobolev spaces, such as product, commutator and composition estimates. Most of these results has already been derived, see for instance [Lan13], [DB22].

**Lemma A.1 (Integration by parts in isopycnal coordinates).** If $f$ and $g$ are $H^1(S_r)$, then

$$ \int (1 + \epsilon h) g \nabla^\varphi \cdot f dx dr = - \int_{r=0} f \cdot \mathbf{e}_3 g dx + \int_{r=1} f \cdot \mathbf{e}_3 g dx - \int (1 + \epsilon h) f \nabla^\varphi g dx dr, $$
where $\mathbf{e}_3$ is the vertical upward vector, i.e. the normal vector to the bottom and rigid lid, pointing upward.

*Proof.* The proof consists in going back in Eulerian coordinates to use the standard integration by parts formula, and then doing the isopycnal change of coordinates again. Note that $(1 + \epsilon h)$ is the Jacobian of the isopycnal change of coordinates. $\square$

**Lemma A.2 (A continuous embedding for anisotropic spaces).** Let $s \in \mathbb{R}$, and $f \in H^{s+\frac{1}{2},1}$, we can write

$$ \sup_{0 \le r \le 1} |f(\cdot, r)|_{H^s} \lesssim \|f\|_{H^{s+\frac{1}{2},1}}. $$

*Proof.* See [DB22, Lemma A.1] or [Lan13, Proposition 2.12] $\square$

**Lemma A.3 (Standard product estimates in Sobolev spaces $H^s(\mathbb{R}^d)$).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$, $s \ge -s_0$. There exists a constant $C > 0$ such that for any $f \in H^s(\mathbb{R}^d) \cap H^{s_0}(\mathbb{R}^d)$, $g \in H^s(\mathbb{R}^d)$, we can write

$$ |f g|_{H^s(\mathbb{R}^d)} \le C |f|_{H^{s_0}} |g|_{H^s} + C \langle |f|_{H^s} |g|_{H^{s_0}} \rangle_{s>s_0}. $$

**Lemma A.4 (Product estimates in anisotropic spaces, [DB22]).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$, $s \in \mathbb{R}$ and $k \in \mathbb{N}$ with $0 \le k \le s$.

* There exists $C > 0$ such that for any $f \in L^2(S_r)$ and $g \in H^{s_0+\frac{1}{2},1}(S_r)$, we can write

$$ (A.1) \qquad \|fg\|_{L^2} \le C\|f\|_{L^2}\|g\|_{H^{s_0+1/2,1}}. $$

* If $s \ge s_0 + \frac{1}{2}$, $k \ge 1$, then there exists $C > 0$ such that for any $g \in H^{s,k}$, $f \in H^{s,k}$

$$ (A.2) \qquad \|fg\|_{H^{s,k}} \le C\|f\|_{H^{s,k}}\|g\|_{H^{s,k}}. $$

* If $f \in W_r^{k,\infty}$ is independent of $x$ and $g \in H^{s,k}$, then

$$ (A.3) \qquad \|fg\|_{H^{s,k}} \le C|f|_{W^{k,\infty}}\|g\|_{H^{s,k}}. $$
Moreover,

$$
(B.7) \qquad \|R_2^\Lambda(\nabla f)\|_{L^2} \le C \epsilon \|\nabla_{t,x,r} \eta\|_{H^{s-1,k\Lambda(s-1)}} \|\nabla^\varphi f\|_{H^{s-1,k\Lambda(s-1)}}.
$$

Here, Alinhac's good unknown is defined as in (B.5) component by component.

Proof. We prove here the first point, as the proof of the second one can be deduced from the first one and the definition

$$
\nabla^{\varphi} \cdot f = \sum_{i \in \{1, \dots, d+1\}} \partial_i^{\varphi} f_i.
$$

We write $\Lambda$ an operator of the form $\Lambda^{s-k} \partial_r^k$ or $|D|\Lambda^{s-1}$ (with $k=0$ by convention). We can write

$$
\nabla^{\varphi} f = (\partial \varphi)^{-T} \nabla f.
$$

Multiplying both sides by $(\partial \varphi)^T$ and applying $\Lambda$ yields

$$
(\partial \varphi)^T \Lambda \nabla^\varphi f + \Lambda \left( (\partial \varphi)^T \right) \nabla^\varphi f + [\Lambda; (\partial \varphi)^T, \nabla^\varphi f] = \nabla \Lambda f.
$$

Applying $(\partial \varphi)^{-1}$ to both sides yields

$$
\Lambda \nabla^\varphi f = -(\partial \varphi)^{-T} \Lambda (\partial \varphi)^T \nabla^\varphi f + \nabla^\varphi \Lambda f + \epsilon R_1^\Lambda (\nabla f),
$$

which we write in such a way that reveals Alinhac's good unknown

$$
(B.8) \qquad \Lambda \nabla^\varphi f = \nabla^\varphi f^\Lambda + \nabla^\varphi (\Lambda \varphi \cdot \nabla^\varphi f) - (\partial \varphi)^{-T} \Lambda (\partial \varphi)^T \nabla^\varphi f + \epsilon R_1^\Lambda (\nabla f).
$$

Here, we wrote

$$
f^{\Lambda} := \Lambda f - (\Lambda \varphi \cdot \nabla^{\varphi}) f.
$$

Now, we need to compute the sum of the second and third terms. To this end, recall the notation $\partial^\varphi \varphi := (\partial_i^\varphi \varphi)_{i,j \in \{1, \dots, d+1\}}$. We then write

$$
\begin{align*}
(\partial \varphi)^T (\partial^\varphi \Lambda \varphi)^T &= (\partial \varphi)^T (\partial_i^\varphi \Lambda \varphi_j)_{(i,j) \in \{1, \dots, d+1\}^2} \\
&= (\partial \varphi)^T \nabla^\varphi (\Lambda \varphi_1 \ \dots \ \Lambda \varphi_{d+1}) \\
&= (\partial \varphi)^T (\partial \varphi)^{-T} \nabla^\varphi (\Lambda \varphi_1 \ \dots \ \Lambda \varphi_{d+1}) \\
&= (\partial \Lambda \varphi)^T \\
&= (\Lambda \partial \varphi)^T.
\end{align*}
$$

Moreover,

$$
\begin{align*}
(\partial^\varphi \Lambda \varphi)^T \nabla^\varphi f &= (\partial_i^\varphi \Lambda \varphi_j)_{(i,j) \in \{1, \dots, d+1\}^2} \nabla^\varphi f \\
&= \nabla^\varphi ((\Lambda \varphi \cdot \nabla^\varphi f) - \Lambda \varphi \cdot \nabla^\varphi (\nabla^\varphi f)).
\end{align*}
$$

Thus we get

$$
\nabla^{\varphi} (\Lambda \varphi \cdot \nabla^{\varphi} f) - (\partial \varphi)^{-T} \Lambda (\partial \varphi)^T \nabla^{\varphi} f = \Lambda \varphi \cdot \nabla^{\varphi} (\nabla^{\varphi} f),
$$

which yields

$$
(B.9) \qquad \Lambda \nabla^\varphi f = \nabla^\varphi (\Lambda f + \Lambda \varphi \cdot \nabla^\varphi f) + (\Lambda \varphi \cdot \nabla^\varphi f) \nabla^\varphi f + \epsilon R_1^\Lambda (\nabla f),
$$

which in turn yields (B.3) once we specify the special form (B.1) of $\varphi$.

For the estimate on $R_1^\Lambda(\nabla f)$, the key point is to notice that

$$
(\partial \varphi)^T = I_{d,1} + \epsilon J^T,
$$

where

$$
J := \begin{pmatrix} 0 & 0 & \partial_t \eta \\ 0 & 0 & \nabla_x \eta \\ 0 & 0 & -h \end{pmatrix}.
$$

As the symmetric commutator $[\Lambda; \cdot, \cdot]$ is bi-linear and $[\Lambda; I_{d,1}, \nabla^\varphi f] = 0$, we get

$$
[\Lambda; (\partial \varphi)^T, \nabla^\varphi f] = \epsilon [\Lambda; J, \nabla^\varphi f].
$$

The result comes from the estimate (A.8) on symmetric commutators. Note that we used the definition of $\nabla^\varphi$,
the product estimate (A.2) as well as (A.12) to write

$$
\| \nabla f \|_{H^{s-1, k\Lambda(s-1)}} \le C \| \nabla \eta \|_{H^{s-1, k\Lambda(s-1)}} \| \nabla^\varphi f \|_{H^{s-1, k\Lambda(s-1)}}.
$$
**Proof.** For the first estimate, we use the standard Sobolev inequality

$$|f|_{L^\infty} \leq c |f|_{H^s},$$

for $s \geq s_0 > d/2$ and where $c$ depends on $s_0$. We can then write

$$\|fg\|_{L^2} = \|fg|_{L_x^2}|_{L_y^2} \le c \|f|_{L_x^2}|g|_{H^{s_0}}|_{L_y^2} \le c \|f|_{L_x^2}|_{L_y^2} \sup_{r \in [0,1]} |g|_{H^{s_0}} \le c \|f\|_{L^2} \|g\|_{H^{s_0+\frac{1}{2},1}}$$

by the embedding of Lemma A.2.

The second inequality is proved by using Leibniz rule for the vertical derivatives and then use the previous estimate. This is thoroughly done in [DB22, Lemma A.3].

For the last one it suffices to estimate terms of the form $\partial_r^l f \partial_r^{l_2} g$ for $l_1 + l_2 = l \leq k$ in $H^{s-l}$. This is done by using $L^\infty - L^2$ Hölder estimate. $\square$

We now state commutator estimates in $H^{s,k}$ spaces. Note the use of $H^{s-1,k\wedge(s-1)}$ norms, so that in the case $k=s$, we gain one derivative.

**Lemma A.5 (Commutator estimates).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$, $s > 0$ and $k \in \mathbb{N}$ with $0 \le k \le s$.

* If $l \in \mathbb{N}$ with $0 \le l \le k$, then there exists $C > 0$ such that for any $f \in H^{s \vee (s_0 + \frac{3}{2}), k \vee 2}$ and $g \in H^{(s-1) \vee (s_0 + \frac{1}{2}), 2 \vee (k \wedge (s-1))}$, we can write

$$(A.4) \qquad \| [ \Lambda^{s-l} \partial_r^l, f ] g \|_{L^2} \le C \| f \|_{H^{s \vee (s_0 + \frac{3}{2}), k \vee 2}} \| g \|_{H^{(s-1) \vee (s_0 + \frac{1}{2}), 2 \vee (k \wedge (s-1))}}.$$

* If $s \ge 1$, then there exists $C > 0$ such that for any $f \in H^{s,0} \cap H^{s_0+\frac{3}{2},1}$, $g \in H^{s-1,0}$, we can write

$$(A.5) \qquad \| [ |D| \Lambda^{s-1}, f ] g \|_{L^2} \le C \| \nabla_x f \|_{H^{s_0+\frac{1}{2},1}} \| g \|_{H^{(s-1)\vee(s_0+\frac{1}{2},0)}} + C \langle \| \nabla_x f \|_{H^{s-1}\vee(s_0+\frac{1}{2},0)} \| g \|_{H^{s_0+\frac{1}{2},1}} \rangle_{s>s_0+1}.$$

* Let $k \ge 1$. There exists $C > 0$ such that if $f \in W_r^{k,\infty}$ and $g \in H^{s,k}$, then

$$(A.6) \qquad \| [ \Lambda^{s-k} \partial_r^k, f ] g \|_{L^2} \le C |f|_{W^{k,\infty}} \|g\|_{H^{s-1,k-1}}.$$

**Proof.** Estimate (A.4) is detailed in [DB22, Lemma A.8].

Estimate (A.5) is done in [Lan13, Corollary B.16], but we sketch the proof for the sake of completeness. We start by writing the standard the commutator estimate in $H^s(\mathbb{R}^d)$ (see for instance [Lan06, Theorem 6, Remark 38] and references therein), namely

$$\| [ |D| \Lambda^{s-1}, f ] g \|_{L^2} \le C |\nabla_x f|_{H^{s_0}} |g|_{H^{s-1}} + \langle |\nabla_x f|_{H^{s-1}} |g|_{H^{s_0}} \rangle_{s>s_0+1}.$$

Thus, taking the $L^2$-norm in the vertical dimension and using the continuous embedding of Lemma A.2 yields the result

$$\| [ |D| \Lambda^{s-1}, f ] g \|_{L^2} \le C \| \nabla_x f \|_{H^{s_0+\frac{1}{2},1}} \| g \|_{H^{s-1,0}} + C \langle \| \nabla_x f \|_{H^{s-1,0}} \| g \|_{H^{s_0+\frac{1}{2},1}} \rangle_{s>s_0+1}.$$

For inequality (A.6), we start by noticing that as $f$ only depends on $r$, it commutes with $\Lambda^{s-k}$. Then the standard the commutator estimate in $L_r^2$ yields the result. $\square$

**Lemma A.6 (Symmetric commutator estimates).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$, $s \in \mathbb{R}$ with $s \ge s_0 + \frac{5}{2}$ and $k \in \mathbb{N}$ with $2 \le k \le s$.

* If $l \in \mathbb{N}$ with $0 \le l \le k$. Then there exists $C > 0$ such that for any $f \in H^{s,k}$, $g \in H^{s-1,k \wedge (s-1)}$, we can write

$$(A.7) \qquad \| [ \Lambda^{s-l} \partial_r^l, f, g ] \|_{L^2} \le C \| f \|_{H^{s-1, k \wedge (s-1)}} \| g \|_{H^{s-1, k \wedge (s-1)}}.$$

* For $s \in \mathbb{R}$ and $s \ge 1$, $f \in H^{s-1,0} \cap H^{s_0+\frac{3}{2},1}$, $g \in H^{s-1,0} \cap H^{s_0+\frac{3}{2},1}$ we can write

$$(A.8) \qquad \| [ |D| \Lambda^{s-1}, f, g ] \|_{L^2} \le C \| f \|_{H^{s-1,0}} \| g \|_{H^{s_0+\frac{3}{2},1}} + C \| g \|_{H^{s-1,0}} \| f \|_{H^{s_0+\frac{3}{2},1}}.$$
Let us finally discuss the particular form of Alinhac's good unknown stated in (B.5). Note that the definition of Alinhac's good unknown derived in (B.8) does not rely on the particular form (B.1) of $\varphi$. In what follows, we write

$$ \varphi_0(t, x, r) := (t, x, -r) $$

so that

$$ (B.10) \qquad \varphi = \varphi_0 + \epsilon(0, 0, \eta). $$

Note that plugging (B.10) into (B.8) does not yield (B.5), due to the fact that $\Lambda\varphi_0 \neq 0$. However, simple computations show that $\nabla_x \Lambda\varphi_0 = 0$ and $\partial_r \Lambda\varphi_0 = 0$, so that we can write

$$ (B.11) \qquad \nabla^{\varphi} (\Lambda\varphi_0 \cdot \nabla^{\varphi} f) = (\Lambda\varphi_0 \cdot \nabla^{\varphi}) \nabla^{\varphi} f. $$

Subtracting (B.11) from (B.9) yields

$$ \Lambda\nabla^{\varphi} f = \nabla^{\varphi} (\Lambda f + \Lambda(\varphi - \varphi_0) \cdot \nabla^{\varphi} f) + \Lambda(\varphi - \varphi_0) \cdot \nabla^{\varphi}(\nabla^{\varphi} f) + R_1^{\Lambda}(\nabla f), $$

which is exactly (B.6), as $\varphi - \varphi_0 = (0, 0, \eta)$. $\square$

Note that using commutator estimates (A.9) or (A.10) instead of (A.8), we have a more precise estimate of $R_1^{\Lambda}(\nabla f)$ in the cases $\Lambda = |D|\Lambda^{s-1}$ and $\Lambda = \Lambda^{s-1}\partial_r$. We state the result in the following lemma.

**Lemma B.3.** Let $s \in \mathbb{R}$ with $s \ge 1$. Assume that $\eta \in H^{s_0+\frac{5}{2},3}$ and there exist $h_*, h^*$ such that $0 < h_* \le 1+\epsilon h \le h^*$ with $h := -\partial_r \eta$. Then there exists $C_0 > 0$ that can depend on $\|\eta\|_{H^{s_0+\frac{3}{2},3}}$ such that the following holds.

Let $\Lambda$ be $|D|\Lambda^{s-1}$ (with $k=0$ by convention) or $\Lambda^{s-1}\partial_r$ (with $k=1$ by convention), then

$$ (B.12) \qquad \|R_1^{\Lambda}(\nabla_{t,x,r} f)\|_{L^2} \le C_0 \|\nabla_{t,x,r} \eta\|_{H^{s-1,k}} \|\nabla_{t,x,r}^{\varphi} f\|_{H^{s-1,k}}. $$

Note that Lemma B.2 is still valid when replacing $\nabla^{\varphi}$ by

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

by applying $J_{\mu} := \left( \frac{\sqrt{\mu} I_d}{1} \right)$ to (B.15), (B.4), (B.7), (B.3), (B.6).

We show, in the following lemma, how the control on $\nabla^{\varphi} f^{\Lambda}$ yields a control on $\nabla f$ in Sobolev norms. Note that for this lemma we need to work in $H^s(S_r) := H^{s,s}(S_r)$ as we use commutator estimates in order to gain one derivative (see the estimates of Lemma A.5 and the remark just above).

**Lemma B.4.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s > s_0 + \frac{3}{2}$. Then there exists $C > 0$ such that the following holds. Recall that $h := -\partial_r \eta$. We denote by $\mathcal{D} := \{\Lambda^{s-1} \partial_r^l, l \in \{1, \dots, s\} \} \cup \{|D|\Lambda^{s-1}\}$ the set of the differential operators that will be used in this lemma.

We make the assumption (H.5) as well as

$$ \|\eta\|_{H^{s(\kappa_0+\frac{3}{2})}} \le M, $$

for a constant $M > 0$. Then there exists a constant $C$ depending only on $M, M$ such that the following holds:

* If $f \in H^1$ then there exists $c$ depending only on $s_0, h_*$ such that

$$ (B.13) \qquad \|\nabla^{\varphi} f\|_{L^2} \le c(1 + \|\eta\|_{H^{s_0+\frac{3}{2},2}}) \|\nabla f\|_{L^2}. $$

Conversely,

$$ (B.14) \qquad \|\nabla f\|_{L^2} \le c(1 + \|\eta\|_{H^{s_0+\frac{3}{2},2}}) \|\nabla^{\varphi} f\|_{L^2}. $$

* If $f \in H^s$,

$$ (B.15) \qquad \left\{ \begin{array}{l} \|f\|_{H^s} \le C \left( \sum_{\Lambda \in \mathcal{D}} \|f^{\Lambda}\|_{L^2} + \|f\|_{H^{s_0+\frac{3}{2},2}} \right), \\ \sum_{\Lambda \in \mathcal{D}} \|f^{\Lambda}\|_{L^2} + \|f\|_{H^{s_0+\frac{3}{2},2}} \le C \|f\|_{H^s}. \end{array} \right. $$
• There exists a constant $C_0$ also depending on $\|f\|_{H^{s_0+\frac{3}{2},2}}$ and $\|g\|_{H^{s_0+\frac{3}{2},2}}$ such that

$$ (A.9) \qquad \|[\Lambda^{s-1} \partial_r; f, g]\|_{L^2} \le C_0 (\|f\|_{H^{s-1,1}} + \|g\|_{H^{s-1,1}}). $$

*Proof.* Estimate (A.7) is detailed in [DB22, Lemma A.9].

Estimate (A.8) follows from standard symmetric commutator estimates in $\mathbb{R}^d$ and continuous embedding of Lemma A.2.

For estimate (A.9), we start by writing

$$ \begin{aligned} [\Lambda^{s-1} \partial_r; f, g] &= \Lambda^{s-1} \partial_r (fg) - \Lambda^{s-1} \partial_r f g - f \Lambda^{s-1} \partial_r g \\ &= [\Lambda^{s-1}, g] \partial_r f + [\Lambda^{s-1}, f] \partial_r g. \end{aligned} $$

Now, standard commutator estimates in $\mathbb{R}^d$ yield:

$$ \| [\Lambda^{s-1} \partial_r; f, g] \|_{L^2(\mathbb{R}^d)} \le c (|g|_{H^{s-1}} |\partial_r f|_{H^{s_0}} + |g|_{H^{s_0+1}} |\partial_r f|_{H^{s-1}} \\ + |f|_{H^{s-1}} |\partial_r g|_{H^{s_0}} + |f|_{H^{s_0+1}} |\partial_r g|_{H^{s-1}}), $$

where $c$ only depends on $s_0, s$. Now taking the $L^2$-norm in $r$ of the previous inequality and applying Lemma A.2 on each term yields (A.9). □

**Lemma A.7 (Composition estimates in $H^{s,k}(S_r)$).** Let $d \in \mathbb{N}^*$, $s_0 > d/2$. Let $s \in \mathbb{R}$ with $s \ge s_0 + \frac{1}{2}$, $k \in \mathbb{N}$ with $1 \le k \le s$, $M > 0$. There exists $C > 0$ such that for any $F \in W^{s,k,\infty}(\mathbb{R} \times (0,1))$ with $F(0, \cdot) = 0$, and any $u \in H^{s,k}(S_r)$ such that $\|u\|_{H^{s,k}} \le M$, the following holds. Define $F \circ u : (x, r) \mapsto F(u(x, r), r)$. Then

$$ (A.10) \qquad \|F \circ u\|_{H^{s,k}} \le C |F|_{W^{s,k,\infty}(\mathbb{R} \times (0,1))} \|u\|_{H^{s,k}}. $$

In the particular case $u = \epsilon h$ and $F(y) = \frac{y}{1+y}$, under assumption (H.5), we get

$$ (A.11) \qquad \| \frac{\epsilon h}{1+\epsilon h} \|_{H^{s,k}} \le C \|h\|_{H^{s,k}} $$

*Proof.* The proof is detailed in [DB22, Lemma A.5]

<figure><img src="image_5.png" /></figure>

We end this appendix with an interpolation result in $H^{s,k}$ spaces:

**Lemma A.8.** Let $s > 1$, $k = s$, and $0 \le s' \le s - 1$, $k' = s'$. Then there exists $C > 0$, $\alpha, \beta \in [0,1]$ such that for any $f \in H^{s,k}$,

$$ \|f\|_{H^{s',k'}} \le C \|f\|_{L^2}^{\alpha} \|f\|_{H^{s,k}}^{\beta}. $$

*Proof.* Let $(\chi_0, \chi_1) \in \mathcal{C}^\infty([0,1])$ be a partition of unity of $[0,1]$, namely $\chi_0(r) = 1$ for $r \in [0,1/3]$ and $\chi_0(r) = 0$ for $r \in [2/3, 1]$ (mutatis mutandis for $\chi_1$) and $\chi_0 + \chi_1 = 1$.

Let $f \in H^{s,k}$, therefore $f = f_0 + f_1 := f\chi_0 + f\chi_1$, and we can extend $f_0$ to the half space $\mathbb{R}^d \times [0, +\infty[$ in a smooth way. Therefore, classical interpolation results in the half space (see for instance [Tay11, Section 4.4]) yield

$$ |f_0(x, \cdot)|_{H^{k'}} \le |f_0(x, \cdot)|_{L^2}^{\gamma_1} |f_0(x, \cdot)|_{H^k}^{\gamma_2}, $$

for any $x$ fixed. Now note that as $\chi_0$ is bounded in $\mathcal{C}^\infty$, we have the bound

$$ f_0(x, \cdot)|_{H^k} \le |\chi_0|_{\mathcal{C}^\infty} |f|_{H^k}. $$

The same bound is true for the $L^2$ norm. Finally, the same reasoning is valid for $f_1$ and as $f = f_0 + f_1$, we get

$$ |f|_{H^{k'}} \le C |f|_{L^2}^{\gamma_1} |f|_{H^k}^{\gamma_2}. $$
• If $f \in H^s$,

$$ (B.16) \qquad \left\{ \begin{array}{l} \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} + \sum_{\Lambda \in \mathcal{D}} \|\nabla f^\Lambda\|_{L^2} \le C (\|\nabla^\varphi f\|_{H^s} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}}), \\ \\ \|\nabla^\varphi f\|_{H^s} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} \le C \left( \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} + \sum_{\Lambda \in \mathcal{D}} \|\nabla f^\Lambda\|_{L^2} \right). \end{array} \right. $$

*Proof.* For inequalities (B.13) and (B.14), we use the definition of $\nabla^\varphi$ and the product estimate (A.1). We now write $\Lambda$ an operator of the form $\Lambda^{s-l} \partial_r^l$ for $1 \le l \le s$ or $|D|\Lambda^{s-1}$ (with $l=0$ by convention). For the inequalities in (B.15), we use the definition of Alinhac's good unknown (B.5)

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

We take the $L^2$-norm of this equation, use triangular inequality together with the product estimate (A.1) and $1+\epsilon h \ge h_*$

$$ \sum_{\Lambda \in \mathcal{D}} \|f^\Lambda\|_{L^2} \le c \|f\|_{H^s} + c \epsilon \|\eta\|_{H^s} \|f\|_{H^{s_0+\frac{3}{2},2}}, $$

where $c$ only depends on $s, h_*$. This yields the result, up to a last detail : in the case $l=0$, we use $|D|\Lambda^{s-1}$, so we lack the control of the $L^2$-norm. However the term $\|f\|_{H^{s_0+\frac{3}{2},2}}$ provides this control, and by interpolation we get the result. The converse inequality is proven the same way.
For the last two equations, we use equation (B.3) of Lemma B.2 to write

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

We take the $L^2$-norm of this expression. We use the product estimate (A.1) for the second term and the estimate (B.4) for the last term:

$$ \|\Lambda \nabla^\varphi f\|_{L^2} \le C \|\nabla^\varphi f^\Lambda\|_{L^2} + \epsilon \|\eta\|_{H^s} \|\nabla^\varphi f\|_{H^{s_0+\frac{3}{2},2}} + C \epsilon \|\eta\|_{H^{s,2\nu(s-1)+1}} \|\nabla^\varphi f\|_{H^{s-1,2\nu(s-1)}}. $$

Note that here we use $k=s$ so that $k \wedge (s-1) = k-1$ (with the notation of Lemma A.5). Adding $\|\nabla f\|_{H^{s_0+\frac{3}{2},2}}$ to the previous inequality and summing the results for $\Lambda$ in $\mathcal{D}$ yields the inequality

$$ \|\nabla^\varphi f\|_{H^{s,k}} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} \le C \left( \sum_{\Lambda \in \mathcal{D}} \|\nabla^\varphi f^\Lambda\|_{L^2} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} \right) + C \|\nabla^\varphi f\|_{H^{s-1,2\nu(s-1)}}. $$

The last term is also controlled by $\sum_{\Lambda \in \mathcal{D}} \|\nabla^\varphi f^\Lambda\|_{L^2} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}}$ by induction on $s$, as the base case $s \le s_0 + \frac{3}{2}$ is treated directly by using the product estimate (A.2) in the definition of $\nabla^\varphi$, namely

$$ \|\nabla^\varphi f\|_{H^{s_0+\frac{3}{2},2\nu k}} \le \|\nabla f\|_{H^{s_0+\frac{3}{2},2\nu s}} + c \|\eta\|_{H^{s_0+\frac{5}{2},2\nu s+1}} \|\nabla f\|_{H^{s_0+\frac{3}{2},2\nu s}}, $$

where $c$ only depends on $s, h_*$. The converse inequality is proven in the same way.

□

We can (and need to) treat case of no vertical derivatives (i.e. $H^{s,0}$ instead of $H^s := H^{s,s}$), which we do in the following lemma.

**Lemma B.5.** Let $s_0 > d/2$, $s \in \mathbb{R}$ with $s > s_0 + \frac{3}{2}$. Recall that $h := -\partial_r \eta$. We denote by $\Lambda := |D|\Lambda^{s-1}$. We make the assumption (H.5) as well as

$$ \|\eta\|_{H^{s\nu(s_0+\frac{5}{2}),3}} \le M, $$

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

$$ (B.17) \qquad \left\{ \begin{array}{l} \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} + \|\nabla f^\Lambda\|_{L^2} \le C (\|\nabla^\varphi f\|_{H^{s,0}} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}}), \\ \\ \|\nabla^\varphi f\|_{H^{s,0}} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} \le C (\|\nabla f\|_{H^{s_0+\frac{3}{2},2}} + \|\nabla f^\Lambda\|_{L^2}). \end{array} \right. $$
Here, $C > 0$ depends on $\chi_0, \chi_1$ but not on $f$.

Let $0 \le l' \le k' \le k - 1$. We apply the previous inequality to $\Lambda^{s'-l'} f$:

$$ \begin{aligned} \|\Lambda^{s'-l'} \partial_r^{l'} f\|_{L^2} &\le C \|\Lambda^{s'-l'} f\|_{L^2}^{\gamma_1} \sum_{l \le l'+1} \|\Lambda^{s'-l'} \partial_r^l f\|_{L^2}^{\gamma_2} \\ &\le C \|f\|_{L^2}^a \|f\|_{H^{s'-l'+1,0}}^{\gamma_3} \|f\|_{H^{s'+1,l'+1}}^{\gamma_2}, \end{aligned} $$

by classical interpolation inequality in $\mathbb{R}^d$ (see for instance [Tay11, Section 4.2]). As $s' + 1 \le s, k' + 1 \le k$ and the exponents are non-zero, this yields the result. $\square$

## APPENDIX B. DEFINITIONS AND RESULTS ON ALINHAC'S GOOD UNKNOWN

The system (E) of the Euler equations in isopycnal coordinates mainly differs from the one in Eulerian coordinates by the operators $\nabla^\varphi$, which arise from the chain rule when differentiating a composition of the unknowns with the change of variables. In other words, the operator $\nabla^\varphi$ is the gradient in Eulerian coordinates. Recall that the change of variables $\varphi$ is

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

Recalling the notation $h := -\partial_r \eta$, we write its Jacobian matrix

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

Recall that, by the chain-rule :

$$ \left\{ \begin{aligned} \partial_t^\varphi v &:= (\partial_t(v \circ \varphi)) \circ \varphi^{-1} = \partial_t v + \epsilon \frac{\partial_t \eta}{1 + \epsilon h} \partial_r v, \\ \nabla_x^\varphi v &:= (\nabla_x(v \circ \varphi)) \circ \varphi^{-1} = \nabla_x v + \epsilon \frac{\nabla_x \eta}{1 + \epsilon h} \partial_r v, \\ \partial_r^\varphi v &:= (\partial_z(v \circ \varphi)) \circ \varphi^{-1} = -\frac{1}{1 + \epsilon h} \partial_r v = -\partial_r v + \epsilon \frac{h}{1 + \epsilon h} \partial_r v, \end{aligned} \right. $$

for $v$ a function defined on $S_r$.

This raises an issue when one wants to differentiate the system, say apply an operator $\Lambda$ of the form $\Lambda^{s-k} \partial_r^k$ with $s > 0, 0 \le k \le s$, as $\Lambda$ and $\nabla^\varphi$ do not commute, and the commutator involves higher derivatives in $\eta$ than what the energy method allows us to control. This is detailed in the following lemma.

**Lemma B.1.** Let $s \in \mathbb{R}$ with $s > s_0 + \frac{3}{2}, 0 \le k \le s$, with $k \in \mathbb{N}$. Assume that $\eta \in H^{s+1,2\vee k+1}$ and there exist $h_*, h^*$ such that $0 < h_* \le 1 + \epsilon h \le h^*$ with $h := -\partial_r \eta$. Then there exists $C > 0$ such that the following holds. We write $\Lambda$ an operator of the form $\Lambda^{s-k} \partial_r^k$ or $|D|\Lambda^{s-1}$ (with $k = 0$ by convention). Let $f \in H^{s-1,k\Lambda(s-1)\vee 2}$. Then

$$ (B.1) \qquad \| [ \Lambda, \nabla^\varphi ] f \|_{L^2} \le \epsilon C \| \eta \|_{H^{s\vee(s_0+\frac{3}{2})+1,2\vee k+1}} \| \nabla f \|_{H^{s-1,k\Lambda(s-1)\vee 2}}. $$

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

Recall the above expression of $\nabla^\varphi$. As $\Lambda$ and $\nabla$ commute, we can write

$$ \| [ \Lambda, \nabla^\varphi ] f \|_{L^2} = \epsilon \| [ \Lambda, \frac{1}{1 + \epsilon h} \nabla \eta ] f \|_{L^2}. $$

Recall indeed that $h = -\partial_r \eta$. Thus the commutator estimate (A.4) yields

$$ \| [ \Lambda, \nabla^\varphi ] f \|_{L^2} \le C \epsilon \left\| \frac{1}{1 + \epsilon h} \nabla \eta \right\|_{H^{s\vee(s_0+\frac{3}{2})+2\vee k}} \| f \|_{H^{s-1, k\Lambda(s-1)\vee 2}}. $$
We first provide in this appendix estimates on the non-linear term and buoyancy term b in (E'). Note that we also have, as in the Eulerian case,

$$ \nabla^{\varphi} \cdot (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U}) = \partial_{i}^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_{j} \partial_{j}^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_{i}, $$

where, on the right-hand side, a sum is taken over the indices i, j.

Note that the conditions on s and k in the following lemma are such that $H^{s,k}$ is an algebra.

**Lemma C.1 (Control of the non-linear term and the buoyancy term).** Let $\mu, \epsilon \le 1$, $M, \underline{M} > 0$, $s_0 > d/2$, $s \in \mathbb{R}$ with $s \ge s_0 + \frac{1}{2}$ and $1 \le k \le s$ with $k \in \mathbb{N}$. We make the assumption (H.5), as well as $\epsilon \le \sqrt{\mu}$ and

$$ \begin{cases} |\varrho|_{W_r^{s,\infty}} + |\underline{V}|_{W_r^{k+1,\infty}} \le \underline{M}, \\ \|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s+1,k+1}} + \sqrt{\mu}\|w\|_{H^{s+1,k+1}} \le M. \end{cases} $$

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

$$ \left\{ \begin{array}{l} \| (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \|_{H^{s,k}} \le \frac{\epsilon}{\sqrt{\mu}} C (\|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s+1,k+1}} + \sqrt{\mu}\|w\|_{H^{s+1,k+1}}), \\ \| \partial_{i}^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_{j} \partial_{j}^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U})_{i} \|_{H^{s,k}} \le \frac{\epsilon}{\sqrt{\mu}} C (\|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s+1,k+1}} + \sqrt{\mu}\|w\|_{H^{s+1,k+1}}), \\ \| b \|_{H^{s,k}} \le C \|\eta\|_{H^{sV(s_0+\frac{3}{2}),kV2}}. \end{array} \right. $$

We can also write

$$ \| (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^{\varphi} (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \|_{L^2} \le \frac{\epsilon}{\sqrt{\mu}} C (\|\eta\|_{H^{s_0+\frac{3}{2},2}} + \|V\|_{H^{s_0+\frac{1}{2},1}} + \sqrt{\mu}\|w\|_{H^{s_0+\frac{1}{2},1}}). $$

*Proof.* We start with the horizontal component of the first estimate. We write

$$ \begin{aligned} (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^{\varphi} (\underline{V} + \epsilon V) &= \underline{V} \cdot \nabla_{x}^{\varphi} \underline{V} + \epsilon \underline{V} \cdot \nabla_{x}^{\varphi} V + \epsilon \mathbf{U} \cdot \nabla^{\varphi} (\underline{V} + \epsilon \underline{V}) \\ &= \epsilon \frac{1}{1 + \epsilon h} (\underline{V} \cdot \nabla_{x} \eta) \underline{V}' + \epsilon \underline{V} \cdot \nabla_{x}^{\varphi} V \\ &\quad + \epsilon V \cdot \nabla_{x}^{\varphi} (\underline{V} + \epsilon V) + \epsilon w \cdot \partial_{r}^{\varphi} (\underline{V} + \epsilon V). \end{aligned} $$

Note the particular expression of the term $\nabla_x^\varphi \underline{V}$ since $\underline{V}$ only depends on $r$. Using $\frac{1}{1+\epsilon h} = 1 - \epsilon \frac{h}{1+\epsilon h}$, this yields

$$ \begin{aligned} (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^{\varphi} (\underline{V} + \epsilon V) ={}& \epsilon (\underline{V} \cdot \nabla_{x} \eta) \underline{V}' - \epsilon^2 \frac{h}{1 + \epsilon h} (\underline{V} \cdot \nabla_{x} \eta) \underline{V}' + \epsilon \underline{V} \cdot \nabla_{x}^{\varphi} V \\ &+ \epsilon V \cdot \nabla_{x}^{\varphi} (\underline{V} + \epsilon V) + \epsilon w \cdot \partial_{r}^{\varphi} (\underline{V} + \epsilon V). \end{aligned} \quad (C.1) $$

We take the $H^{s,k}$-norm of the previous expression (C.1), with $s \ge s_0 + \frac{1}{2}$ and $k \ge 1$. We only treat the second and last terms, as the other ones can be treated the same way. Using product estimates (A.3) and (A.2), we bound the second term by

$$ \| \epsilon^2 \frac{h}{1 + \epsilon h} (\underline{V} \cdot \nabla_x \eta) \underline{V}' \|_{H^{s,k}} \le \epsilon^2 c \| \frac{h}{1 + \epsilon h} \|_{H^{s,k}} \| \nabla_x \eta \|_{H^{s,k}}, $$

where c depends on $\underline{M}$. Using the composition estimate (A.11) and the definition $h := -\partial_r \eta$, we can write

$$ \| \frac{h}{1 + \epsilon h} \|_{H^{s,k}} \le c \|\eta\|_{H^{s+1,k+1}}. $$

For the last term, we write, by definition of $\partial_r^\varphi$:

$$ \epsilon^2 w \partial_r^\varphi V = \epsilon^2 w \partial_r V - \epsilon^3 w \frac{h}{1 + \epsilon h} \partial_r V. $$

So that, by product estimates (A.2) and (A.3)

$$ \begin{aligned} \epsilon^2 \|w \partial_r^\varphi V\|_{H^{s,k}} &\le c \epsilon^2 \|w\|_{H^{s,k}} \|\partial_r V\|_{H^{s,k}} - \epsilon^3 \|w\|_{H^{s,k}} \|\frac{h}{1+\epsilon h}\|_{H^{s,k}} \|\partial_r V\|_{H^{s,k}} \\ &\le \frac{\epsilon^2}{\sqrt{\mu}} C (\|V\|_{H^{s+1,k+1}} + \sqrt{\mu}\|w\|_{H^{s+1,k+1}}). \end{aligned} $$
Then, the product estimate (A.2) and the composition estimate (A.10) yield the desired estimate. □

This issue is solved by using Alinhac's good unknowns, introduced in [Ali89]. See also [Lan13] for its use for the water waves equations, as well as [CL14, Proposition 3.17] where the importance of the relation (B.2) is emphasized. Recall the notation

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

**Lemma B.2** (Alinhac's good unknown, [Ali89]). Let $s \in \mathbb{R}$ with $s > s_0 + \frac{5}{2}$, $k \in \mathbb{N}$ with $0 \le k \le s$. Assume that $\eta \in H^{s_0 + \frac{3}{2}, 2}$ and there exist $h_*, h^*$ such that $0 < h_* \le 1 + \epsilon h \le h^*$ with $h := -\partial_r \eta$. Then there exists $C > 0$ depending only on $\|\eta\|_{H^{s_0 + \frac{3}{2}, 2}}$ such that the following holds. We write $\Lambda$ an operator of the form $\Lambda^{s-k} \partial_r^k$ (with $k \ge 1$) or $|D|\Lambda^{s-1}$ (with $k = 0$ by convention).

* Let $f \in H^{s,k}$. We have

$$ \begin{array}{l} \Lambda \nabla_{t,x,r}^{\varphi} f = \nabla_{t,x,r}^{\varphi} f \Lambda + \epsilon (\Lambda \eta \partial_r^{\varphi}) \nabla_{t,x,r}^{\varphi} f + \epsilon R_1^{\Lambda} (\nabla_{t,x,r} f), \\ \text{where } \epsilon R_1^{\Lambda} (\nabla_{t,x,r} f) := (\partial \varphi)^{-T} [\Lambda; (\partial \varphi)^T, \nabla_{t,x,r}^{\varphi} f], \text{ and we can write} \end{array} $$

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

Here, Alinhac's good unknown is defined as

$$ (B.4) \qquad f^{\Lambda} := \Lambda f + \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \partial_r f. $$

* For $f \in (H^{s,k})^{d+1}$ a vector-valued function of $d+1$ components, we can write

$$ (B.5) \qquad \Lambda \nabla^{\varphi} \cdot \mathbf{f} = \nabla^{\varphi} \cdot \mathbf{f}^{\Lambda} + \epsilon \Lambda \eta \partial_r^{\varphi} (\nabla^{\varphi} \cdot \mathbf{f}) + \epsilon R_2^{\Lambda} (\nabla \cdot \mathbf{f}), $$

where

$$ \epsilon R_2^{\Lambda} (\nabla \cdot \mathbf{f}) := [\Lambda; (\partial \varphi)^{-T}, \nabla \cdot \mathbf{f}] - (\partial \varphi)^{-T} [\Lambda; (\partial \varphi)^T, (\partial \varphi)^{-T}] \nabla \cdot \mathbf{f}. $$

Moreover,

$$ (B.6) \qquad \| R_2^{\Lambda} (\nabla \cdot \mathbf{f}) \|_{L^2} \le C \epsilon \| \nabla_{t,x,r} \eta \|_{H^{s,k\Lambda(s-1)}} \| \nabla^{\varphi} \cdot \mathbf{f} \|_{H^{s-1,k\Lambda(s-1)}} + C \epsilon \| \nabla_{t,x,r} \eta \|_{H^{s,k\Lambda(s-1)}}^2 \| \nabla^{\varphi} \cdot \mathbf{f} \|_{H^{s_0+\frac{1}{2}, 1}}. $$

Here, Alinhac's good unknown is defined as

$$ (B.7) \qquad \mathbf{f}^{\Lambda} := \Lambda \mathbf{f} + \epsilon \frac{\Lambda \eta}{1 + \epsilon h} \partial_r \mathbf{f}. $$

*Proof.* We prove here the second point, as the proof of the first one is similar. We write $\Lambda$ an operator of the form $\Lambda^{s-k} \partial_r^k$ or $|D|\Lambda^{s-1}$ (with $k = 0$ by convention). We can write

$$ \nabla^{\varphi} \cdot \mathbf{f} = (\partial \varphi)^{-T} \nabla \cdot \mathbf{f}. $$

Now, we apply $\Lambda$

$$ \Lambda \nabla^{\varphi} \cdot \mathbf{f} = \Lambda (\partial \varphi)^{-T} \nabla \cdot \mathbf{f} + (\partial \varphi)^{-T} \nabla \cdot \Lambda \mathbf{f} + [\Lambda; (\partial \varphi)^{-T} \nabla \cdot \mathbf{f}]. $$

From $(\partial \varphi)^T (\partial \varphi)^{-T} = I$ we get

$$ \Lambda (\partial \varphi)^{-T} = -(\partial \varphi)^{-T} \Lambda (\partial \varphi)^T (\partial \varphi)^{-T} - (\partial \varphi)^{-T} [\Lambda; (\partial \varphi)^T, (\partial \varphi)^{-T}]. $$

We can thus write

$$ \Lambda \nabla^{\varphi} \cdot \mathbf{f} = -(\partial \varphi)^{-T} \Lambda (\partial \varphi)^T \nabla^{\varphi} \cdot \mathbf{f} + \nabla^{\varphi} \cdot \Lambda \mathbf{f} + \epsilon R_2^{\Lambda} (\nabla \cdot \mathbf{f}), $$

which we write in such a way that reveals Alinhac's good unknown

$$ (B.8) \qquad \Lambda \nabla^{\varphi} \cdot \mathbf{f} = \nabla^{\varphi} \cdot \mathbf{f}^{\Lambda} + \nabla^{\varphi} \cdot (\Lambda \varphi \cdot \nabla^{\varphi} \mathbf{f}) - (\partial \varphi)^{-T} \Lambda (\partial \varphi)^T \nabla^{\varphi} \cdot \mathbf{f} + \epsilon R_2^{\Lambda} (\nabla \cdot \mathbf{f}). $$
Here, C depends on M. We used the composition estimate (A.11).

Note that the factor $\frac{1}{\sqrt{\mu}}$ comes from the fact that the term $\|w\|_{H^{s+1,k+1}}$ in M comes with a factor $\sqrt{\mu}$. In order to bound the last term in (C.1), it remains to estimate the following term:

$$ \epsilon w \partial_r^\varphi \underline{V} = \epsilon w \partial_r \underline{V} - \epsilon^2 w \frac{h}{1+\epsilon h} \partial_r \underline{V}. $$

We use product estimates (A.3) and (A.2) as well as the composition estimate (A.11), to get

$$ \|\epsilon w \partial_r^\varphi \underline{V}\|_{H^{s,k}} \le c \epsilon \|w\|_{H^{s,k}} + c \epsilon^2 \|w\|_{H^{s,k}} \|h\|_{H^{s,k}}. $$

The vertical component of the term in the first estimate reads

$$ \epsilon (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi w = \epsilon \underline{V} \cdot \nabla_x^\varphi w + \epsilon^2 \underline{V} \cdot \nabla_x^\varphi w + \epsilon^2 w \partial_r^\varphi w $$

The first two terms are treated as before. For the last term, the product estimate (A.2) and the composition estimate (A.11) yield

$$ \|\epsilon^2 w \partial_r^\varphi w\|_{H^{s,k}} \le \frac{\epsilon^2}{\mu} C(\sqrt{\mu} \|w\|_{H^{s+1,k+1}})^2, $$

which is bounded by $\frac{\epsilon}{\sqrt{\mu}}CM$ thanks to the assumption $\epsilon \le \sqrt{\mu}$.

For the second estimate, we write

$$ \begin{aligned} \partial_i^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U})_j \partial_j^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U})_i = & \frac{\epsilon^2}{(1+\epsilon h)^2} \sum_{i,j \le d} \partial_i \eta \underline{V}'_j \partial_j \eta \underline{V}'_i + 2\epsilon^2 \frac{1}{1+\epsilon h} \sum_{i,j \le d} \partial_i \eta \underline{V}'_j \partial_j^\varphi \mathbf{U}_i \\ & -2\epsilon \frac{1}{1+\epsilon h} \sum_{j \le d+1} \underline{V}'_j \partial_j^\varphi w + \epsilon^2 \sum_{i,j \le d+1} \partial_i^\varphi \mathbf{U}_j \partial_j^\varphi \mathbf{U}_i. \end{aligned} \quad (C.2) $$

Each term in (C.2) is treated as before with product estimates (A.2) and (A.3) as well as the composition estimate (A.11). For the last term in (C.2), we need the assumption $\epsilon \le \sqrt{\mu}$, as for the last term in (C.1).

For the last estimate, the proof is very similar and we only treat one term for the sake of conciseness. We write, thanks to Hölder inequality and embedding of Lemma A.2

$$ \begin{aligned} \|V \cdot \left( \frac{\nabla_x \eta}{1+\epsilon h} \partial_r V \right)\|_{L^2} & \le \|V\|_{L^\infty} \|\nabla_x \eta\|_{L^\infty} \frac{1}{1+\epsilon h} \|\partial_r V\|_{L^2} \\ & \le C \|V\|_{H^{\eta+\frac{1}{2},1}} \|\eta\|_{H^{\eta+\frac{3}{2},2}} \|V\|_{H^{1,1}} \\ & \le C \|V\|_{H^{\eta+\frac{1}{2},1}}. \end{aligned} $$

For the last inequality, we used Assumption (H.5), as well as $s_0 + \frac{1}{2} \ge \frac{d}{2} + \frac{1}{2} \ge 1$. $\square$

We now show how to use the incompressibility constraint in order to estimate $\|w\|_{H^s}$ (for s large enough) by terms depending only on V and $\eta$, but with the loss of one derivative.

**Lemma C.2.** Let $s \ge s_0 + \frac{3}{2}$ and $(V, w, \eta) \in (H^s)^{d+2}$ satisfying the incompressibility constraint in (E), as well as the boundary conditions (1.4). Then, there exists $C > 0$ such that

$$ \text{(C.3)} \qquad \|w\|_{H^{s-1}} \le C \|V\|_{H^s} (1 + \|\eta\|_{H^s}) + C \|\eta\|_{H^s} |\underline{V}|_{W^{s,\infty}}. $$

*Proof.* Recall the incompressibility constraint

$$ \nabla_x^\varphi \cdot V + \partial_r^\varphi w + \frac{\nabla_x}{1+\epsilon h} \partial_r \underline{V} = 0, $$

where

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

We thus write

$$ w(t, x, r) = \int_{0}^{r} \partial_{r} w(t, x, r') d r' = \int_{0}^{r} (1 + \epsilon h) \left( \nabla_{x}^{\varphi} \cdot V + \frac{\nabla_{x} \eta}{1 + \epsilon h} \partial_{r} \underline{V} \right). $$

We thus have (C.3), by product estimates (A.2) and (A.3). $\square$
Here, we wrote

$$ \mathbf{f}^{\wedge} := \mathbf{A}\mathbf{f} - (\mathbf{A}\varphi \cdot \nabla^{\varphi})\mathbf{f}. $$

Now, we need to compute the sum of the second and third terms. To this end, recall the notation $\partial^{\varphi} \varphi := (\partial_{i}^{\varphi} \varphi_{j})_{i,j \in \llbracket 0,d+1 \rrbracket}$. We then write

$$ \begin{aligned} (\partial \varphi)^T (\partial^{\varphi} \mathbf{A} \varphi)^T &= (\partial \varphi)^T \begin{pmatrix} \partial_1^{\varphi} \mathbf{A} \varphi_1 & \partial_1^{\varphi} \mathbf{A} \varphi_2 \\ \partial_2^{\varphi} \mathbf{A} \varphi_1 & \partial_2^{\varphi} \mathbf{A} \varphi_2 \end{pmatrix} \\ &= (\partial \varphi)^T \begin{pmatrix} \partial_1^{\varphi} \\ \partial_2^{\varphi} \end{pmatrix} (\mathbf{A} \varphi_1 \quad \mathbf{A} \varphi_2) \\ &= (\partial \varphi)^T (\partial \varphi)^{-T} \begin{pmatrix} \partial_1 \\ \partial_2 \end{pmatrix} (\mathbf{A} \varphi_1 \quad \mathbf{A} \varphi_2) \\ &= (\partial \mathbf{A} \varphi)^T \\ &= (\mathbf{A} \partial \varphi)^T. \end{aligned} $$

Moreover,

$$ \begin{aligned} (\partial^{\varphi} \mathbf{A} \varphi)^T \nabla^{\varphi} \cdot \mathbf{f} &= \begin{pmatrix} \partial_1^{\varphi} \mathbf{A} \varphi_1 & \partial_1^{\varphi} \mathbf{A} \varphi_2 \\ \partial_2^{\varphi} \mathbf{A} \varphi_1 & \partial_2^{\varphi} \mathbf{A} \varphi_2 \end{pmatrix} \nabla^{\varphi} \cdot \mathbf{f} \\ &= \begin{pmatrix} \partial_1^{\varphi} (\mathbf{A} \varphi) \cdot \nabla^{\varphi} \\ \partial_2^{\varphi} (\mathbf{A} \varphi) \cdot \nabla^{\varphi} \end{pmatrix} \cdot \mathbf{f} \\ &= \partial_1^{\varphi} (\mathbf{A} \varphi) \cdot \nabla^{\varphi} f_1 + \partial_2^{\varphi} (\mathbf{A} \varphi) \cdot \nabla^{\varphi} f_2 \\ &= \nabla^{\varphi} \cdot ((\mathbf{A} \varphi \cdot \nabla^{\varphi})\mathbf{f}) - \mathbf{A} \varphi \cdot \nabla^{\varphi} (\nabla^{\varphi} \cdot \mathbf{f}). \end{aligned} $$

Thus we get

$$ \nabla^{\varphi} \cdot (\mathbf{A} \varphi \cdot \nabla^{\varphi} \mathbf{f}) - (\partial \varphi)^{-T} \mathbf{A} (\partial \varphi)^T \nabla^{\varphi} \cdot \mathbf{f} = \mathbf{A} \varphi \cdot \nabla^{\varphi} (\nabla^{\varphi} \cdot \mathbf{f}), $$

which yields

$$ (B.9) \qquad \mathbf{A} \nabla^{\varphi} \cdot \mathbf{f} = \nabla^{\varphi} \cdot (\mathbf{A} \mathbf{f} + \mathbf{A} \varphi \cdot \nabla^{\varphi} \mathbf{f}) + (\mathbf{A} \varphi \cdot \nabla^{\varphi}) \nabla^{\varphi} \cdot \mathbf{f} + \epsilon R_2^{\wedge} (\nabla \cdot \mathbf{f}). $$

For the estimates on $R_1^{\wedge}(\nabla f)$ and $R_2^{\wedge}(\nabla \cdot \mathbf{f})$, the key point is to notice that

$$ (\partial \varphi)^T = I_{d,1} + \epsilon J^T, $$

where

$$ J := \begin{pmatrix} 0 & 0 & \partial_t \eta \\ 0 & 0 & \nabla_x \eta \\ 0 & 0 & -h \end{pmatrix}. $$

As the symmetric commutator $[\mathbf{A}; \cdot, \cdot]$ is bi-linear and $[\mathbf{A}; I_{d,1}, \nabla^{\varphi} f] = 0$, we get

$$ [\mathbf{A}; (\partial \varphi)^T, \nabla^{\varphi} f] = \epsilon [\mathbf{A}; J, \nabla^{\varphi} f]. $$

The result comes from the estimate (A.7) on symmetric commutators. Note that we used the definition of $\nabla^{\varphi}$, the product estimate (A.2) as well as (A.11) to write

$$ \|\nabla \cdot \mathbf{f}\|_{H^{s-1, k\wedge(s-1)}} \le C \|\nabla \eta\|_{H^{s-1, k\wedge(s-1)}} \|\nabla^{\varphi} \cdot \mathbf{f}\|_{H^{s-1, k\wedge(s-1)}}. $$

Let us finally discuss the particular form of Alinhac's good unknown stated in (B.7). Note that the definition of Alinhac's good unknown derived in (B.8) does not rely on the particular form of $\varphi$, that is

$$ \varphi(t, x, r) := (t, x, -r) + \epsilon(0, 0, \eta(t, x, r)). $$

In what follows, we write

$$ \varphi_0(t, x, r) := (t, x, -r) $$
ACKNOWLEDGMENTS

The author thanks his PhD supervisors Vincent Duchêne and David Lannes for their valuable advice.
This work was supported by the BOURGEONS project, grant ANR-23-CE40-0014-01 of the French National Re-
search Agency (ANR).
This work is licensed under CC BY 4.0.

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UNIV. BORDEAUX, CNRS, BORDEAUX INP, IMB, UMR 5251, F-33400 TALENCE, FRANCE
*Email address: theo.fradin@math.u-bordeaux.fr*
so that

$$
(B.10) \qquad \varphi = \varphi_0 + (0, 0, \eta).
$$

Note that plugging (B.10) into (B.8) does not yield (B.7), due to the fact that $\mathbb{A}\varphi_0 \neq 0$.
However, simple computations show that $\nabla_x \mathbb{A}\varphi_0 = 0$ and $\partial_r \mathbb{A}\varphi_0 = 0$, so that we can
write

$$
(B.11) \qquad \nabla^{\varphi} \cdot (\mathbb{A}\varphi \cdot \nabla^{\varphi} \mathbf{f}) = (\mathbb{A}\varphi \cdot \nabla^{\varphi}) \nabla^{\varphi} \cdot \mathbf{f}.
$$

Subtracting (B.11) from (B.9) yields

$$
\mathbb{A} \nabla^{\varphi} \cdot \mathbf{f} = \nabla^{\varphi} \cdot (\mathbb{A} \mathbf{f} + \mathbb{A}(\varphi - \varphi_0) \cdot \nabla^{\varphi} \mathbf{f}) + \mathbb{A}(\varphi - \varphi_0) \cdot \nabla^{\varphi} (\nabla^{\varphi} \cdot \mathbf{f}) + R_2^{\mathbb{A}} (\nabla \cdot \mathbf{f}),
$$

which is exactly (B.5), as $\varphi - \varphi_0 = (0, 0, \eta)$.

<figure><img src="image_6.png" /></figure>

Note that using commutator estimates (A.8) or (A.9) instead of (A.7), we have a more
precise estimate of $R_1^{\mathbb{A}}(\nabla f)$ in the cases $\mathbb{A} = |D|\Lambda^{s-1}$ and $\mathbb{A} = \Lambda^{s-1}\partial_r$. We state the result
in the following lemma.

**Lemma B.3.** Let $s \in \mathbb{R}$ with $s \ge 1$. Assume that $\eta \in H^{s_0+\frac{5}{2},3}$ and there exist $h_*, h^*$ such that $0 < h_* \le 1 + \epsilon h \le h^*$ with $h := -\partial_r \eta$. Then there exists $C_0 > 0$ that can depend on $\|\eta\|_{H^{s_0+\frac{3}{2},3}}$ as well as $\|\nabla_{t,x,r}^\varphi f\|_{H^{s_0+\frac{3}{2},2}}$ such that the following holds.

Let $\mathbb{A}$ be $|D|\Lambda^{s-1}$ (with $k=0$ by convention) or $\Lambda^{s-1}\partial_r$ (with $k=1$ by convention), then

$$
(B.12) \quad \| R_1^{\mathbb{A}}(\nabla_{t,x,r} f) \|_{L^2} \le C_0 \| \nabla_{t,x,r} \eta \|_{H^{s-1,k}} + C_0 \| \nabla_{t,x,r}^{\varphi} f \|_{H^{s-1,k}}.
$$

Note that Lemma B.2 is still valid when replacing $\nabla^\varphi$ by

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

by applying $J_\mu := \left( \sqrt{\mu} I_d \quad 1 \right)$ to (B.15), (B.3), (B.6), (B.2), (B.5).

We show, in the following lemma, how the control on $\nabla^\varphi f^\Lambda$ yields a control on $\nabla f$
in Sobolev norms. Note that for this lemma we need to work in $H^s(S_r) := H^{s,s}(S_r)$ as we
use commutator estimates in order to gain one derivative (see the estimates of Lemma
A.5 and the remark just above).

**Lemma B.4.** Let $s_0 > d/2$, $s \in \mathbb{N}$ with $s > s_0 + \frac{3}{2}$. Then there exists $C > 0$ such that the following holds. Recall that $h := -\partial_r \eta$. We denote by $\mathcal{D} := \{\Lambda^{s-l} \partial_r^l, l \in \llbracket 1, s \rrbracket\} \cup \{|D|\Lambda^{s-1}\}$ the set of the differential operators that will be used in this lemma. We make the assumption (H.5) as well as

$$
\|\eta\|_{H^{s(\kappa_0 + \frac{5}{2})}} \leq M,
$$

for a constant $M > 0$. Then there exists a constant $C$ depending only on $M$, $M$ such that
the following holds:

* If $f \in H^1$ then there exists $c$ depending only on $s_0, h_*$ such that

$$
(B.13) \qquad \| \nabla^\varphi f \|_{L^2} \le c (1 + \| \eta \|_{H^{s_0 + \frac{3}{2}, 2}}) \| \nabla f \|_{L^2}.
$$

Conversely,

$$
(B.14) \qquad \| \nabla f \|_{L^2} \le c (1 + \| \eta \|_{H^{s_0 + \frac{3}{2}, 2}}) \| \nabla^\varphi f \|_{L^2}.
$$
• If $f \in H^s$,

$$ (B.15) \qquad \left\{ \begin{array}{l} \|f\|_{H^s} \le C \left( \sum_{\Lambda \in \mathcal{D}} \|f^{\Lambda}\|_{L^2} + \|f\|_{H^{s_0+\frac{3}{2},2}} \right), \\ \sum_{\Lambda \in \mathcal{D}} \|f^{\Lambda}\|_{L^2} + \|f\|_{H^{s_0+\frac{3}{2},2}} \le C \|f\|_{H^s}. \end{array} \right. $$

• If $f \in H^s$,

$$ (B.16) \qquad \left\{ \begin{array}{l} \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} + \sum_{\Lambda \in \mathcal{D}} \|\nabla f^{\Lambda}\|_{L^2} \le C (\|\nabla^{\varphi} f\|_{H^s} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}}), \\ \|\nabla^{\varphi} f\|_{H^s} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} \le C \left( \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} + \sum_{\Lambda \in \mathcal{D}} \|\nabla f^{\Lambda}\|_{L^2} \right). \end{array} \right. $$

*Proof.* For inequalities (B.13) and (B.14), we use the definition of $\nabla^{\varphi}$ and the product estimate (A.1).

We now write $\Lambda$ an operator of the form $\Lambda^{s-l} \partial_r^l$ for $1 \le l \le s$ or $|D|\Lambda^{s-1}$ (with $l = 0$ by convention).

For the inequalities in (B.15), we use the definition of Alinhac's good unknown (B.4)

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

We take the $L^2$-norm of this equation, use triangular inequality together with the product estimate (A.1) and $1 + \epsilon h \ge h_*$

$$ \sum_{\Lambda \in \mathcal{D}} \|f^{\Lambda}\|_{L^2} \le c \|f\|_{H^s} + c \epsilon \|\eta\|_{H^s} \|f\|_{H^{s_0+\frac{3}{2},2}}, $$

where $c$ only depends on $s, h_*$. This yields the result, up to a last detail : in the case $l = 0$, we use $|D|\Lambda^{s-1}$, so we lack the control of the $L^2$-norm. However the term $\|f\|_{H^{s_0+\frac{3}{2},2}}$ provides this control, and by interpolation we get the result. The converse inequality is proven the same way.

For the last two equations, we use equation (B.2) of Lemma B.2 to write

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

We take the $L^2$-norm of this expression. We use the product estimate (A.1) for the second term and the estimate (B.3) for the last term:

$$ \|\Lambda \nabla^{\varphi} f\|_{L^2} \le C \|\nabla^{\varphi} f^{\Lambda}\|_{L^2} + \epsilon \|\eta\|_{H^s} \|\nabla^{\varphi} f\|_{H^{s_0+\frac{3}{2},2}} + C \epsilon \|\eta\|_{H^{s,2\nu(s-1)+1}} \|\nabla^{\varphi} f\|_{H^{s-1,2\nu(s-1)}}. $$

Note that here we use $k = s$ so that $k \wedge (s-1) = k-1$ (with the notation of Lemma A.5). Adding $\|\nabla f\|_{H^{s_0+\frac{3}{2},2}}$ to the previous inequality and summing the results for $\Lambda$ in $\mathcal{D}$ yields the inequality

$$ \|\nabla^{\varphi} f\|_{H^{s,k}} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} \le C \left( \sum_{\Lambda \in \mathcal{D}} \|\nabla^{\varphi} f^{\Lambda}\|_{L^2} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} \right) + C \|\nabla^{\varphi} f\|_{H^{s-1,2\nu(s-1)}}. $$

The last term is also controlled by $\sum_{\Lambda \in \mathcal{D}} \|\nabla^{\varphi} f^{\Lambda}\|_{L^2} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}}$ by induction on $s$, as the base case $s \le s_0 + \frac{3}{2}$ is treated directly by using the product estimate (A.2) in the definition of $\nabla^{\varphi}$, namely

$$ \|\nabla^{\varphi} f\|_{H^{s_0+\frac{3}{2},2\nu s}} \le \|\nabla f\|_{H^{s_0+\frac{3}{2},2\nu s}} + c \|\eta\|_{H^{s_0+\frac{3}{2},2\nu s+1}} \|\nabla f\|_{H^{s_0+\frac{3}{2},2\nu s}}, $$

where $c$ only depends on $s, h_*$. The converse inequality is proven in the same way.

□
We can (and need to) treat case of no vertical derivatives (i.e. $H^{s,0}$ instead of $H^s := H^{s,s}$), which we do in the following lemma.

**Lemma B.5.** Let $s_0 > d/2$, $s \in \mathbb{R}$ with $s > s_0 + \frac{3}{2}$. Recall that $h := -\partial_r \eta$. We denote by $\Lambda := |D|\Lambda^{s-1}$. We make the assumption (H.5) as well as

$$ \|\eta\|_{H^{s(\eta(s_0+\frac{3}{2}),1)}} \le M, $$

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

$$ (B.17) \qquad \left\{ \begin{array}{l} \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} + \|\nabla f^\Lambda\|_{L^2} \le C (\|\nabla^\varphi f\|_{H^{s,0}} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}}), \\ \|\nabla^\varphi f\|_{H^{s,0}} + \|\nabla f\|_{H^{s_0+\frac{3}{2},2}} \le C (\|\nabla f\|_{H^{s_0+\frac{3}{2},2}} + \|\nabla f^\Lambda\|_{L^2}). \end{array} \right. $$

## APPENDIX C. SOME USEFUL ESTIMATES

We first provide in this appendix estimates on the non-linear term and buoyancy term $b$ in (E'). Note that we also have, as in the Eulerian case,

$$ \nabla^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U}) = \partial_i^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U})_j \partial_j^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U})_i, $$

where, on the right-hand side, a sum is taken over the indices $i, j$.

Note that the conditions on $s$ and $k$ in the following lemma are such that $H^{s,k}$ is an algebra.

**Lemma C.1 (Control of the non-linear term and the buoyancy term).** Let $\mu, \epsilon \le 1$, $M, \underline{M} > 0$, $s_0 > d/2$, $s \in \mathbb{R}$ with $s \ge s_0 + \frac{1}{2}$ and $1 \le k \le s$ with $k \in \mathbb{N}$. We make the assumption (H.5), as well as $\epsilon \le \sqrt{\mu}$ and

$$ \left\{ \begin{array}{l} |\varrho|_{W_r^{s,\infty}} + |\underline{V}|_{W_r^{k+1,\infty}} \le \underline{M}, \\ \|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s+1,k+1}} + \sqrt{\mu}\|w\|_{H^{s+1,k+1}} \le M. \end{array} \right. $$

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

$$ \left\{ \begin{array}{l} \| (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \|_{H^{s,k}} \le \frac{\epsilon}{\sqrt{\mu}} C (\|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s+1,k+1}} + \sqrt{\mu}\|w\|_{H^{s+1,k+1}}), \\ \| \partial_i^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U})_j \partial_j^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U})_i \|_{H^{s,k}} \le \frac{\epsilon}{\sqrt{\mu}} C (\|\eta\|_{H^{s+1,k+1}} + \|V\|_{H^{s+1,k+1}} + \sqrt{\mu}\|w\|_{H^{s+1,k+1}}), \\ \| b \|_{H^{s,k}} \le C \|\eta\|_{H^{s(\eta(s_0+\frac{3}{2}),1)}}. \end{array} \right. $$

We can also write

$$ \|(\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi (\underline{\mathbf{U}} + \epsilon \mathbf{U})\|_{L^2} \le \frac{\epsilon}{\sqrt{\mu}} C \left( \|\eta\|_{H^{s_0+\frac{3}{2},2}} + \|V\|_{H^{s_0+\frac{1}{2},1}} + \sqrt{\mu}\|w\|_{H^{s_0+\frac{1}{2},1}} \right). $$

*Proof.* We start with the horizontal component of the first estimate. We write

$$ \begin{aligned} (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi (\underline{V} + \epsilon V) &= \underline{V} \cdot \nabla_x^\varphi \underline{V} + \epsilon \underline{V} \cdot \nabla_x^\varphi \underline{V} + \epsilon \mathbf{U} \cdot \nabla^\varphi (\underline{V} + \epsilon V) \\ &= \epsilon \frac{1}{1 + \epsilon h} (\underline{V} \cdot \nabla_x \eta) \underline{V}' + \epsilon \underline{V} \cdot \nabla_x^\varphi \underline{V} \\ &\quad + \epsilon V \cdot \nabla_x^\varphi (\underline{V} + \epsilon V) + \epsilon w \cdot \partial_r^\varphi (\underline{V} + \epsilon V). \end{aligned} $$

Note the particular expression of the term $\nabla_x^\varphi \underline{V}$ since $\underline{V}$ only depends on $r$. Using $\frac{1}{1+\epsilon h} = 1 - \epsilon \frac{h}{1+\epsilon h}$, this yields

$$ (C.1) \qquad (\underline{\mathbf{U}} + \epsilon \mathbf{U}) \cdot \nabla^\varphi (\underline{V} + \epsilon V) = \epsilon (\underline{V} \cdot \nabla_x \eta) \underline{V}' - \epsilon^2 \frac{h}{1+\epsilon h} (\underline{V} \cdot \nabla_x \eta) \underline{V}' + \epsilon \underline{V} \cdot \nabla_x^\varphi \underline{V} \\ \qquad \qquad \qquad + \epsilon V \cdot \nabla_x^\varphi (\underline{V} + \epsilon V) + \epsilon w \cdot \partial_r^\varphi (\underline{V} + \epsilon V). $$
We take the $H^{s,k}$-norm of the previous expression (C.1), with $s \ge s_0 + \frac{1}{2}$ and $k \ge 1$. We only treat the second and last terms, as the other ones can be treated the same way. Using product estimates (A.3) and (A.2), we bound the second term by

$$ \|\epsilon^2 \frac{h}{1+\epsilon h} (\underline{V} \cdot \nabla_x \eta) \underline{V}'\|_{H^{s,k}} \le \epsilon^2 c \|\frac{h}{1+\epsilon h}\|_{H^{s,k}} \|\nabla_x \eta\|_{H^{s,k}}, $$

where $c$ depends on $M$. Using the composition estimate (A.10) and the definition $h := -\partial_r \eta$, we can write

$$ \|\frac{h}{1+\epsilon h}\|_{H^{s,k}} \le c \|\eta\|_{H^{s+1,k+1}}. $$

For the last term, we write, by definition of $\partial_r^\varphi$:

$$ \epsilon^2 w \partial_r^\varphi V = \epsilon^2 w \partial_r V - \epsilon^3 w \frac{h}{1+\epsilon h} \partial_r V. $$

So that, by product estimates (A.2) and (A.3)

$$ \begin{aligned} \epsilon^2 \|w \partial_r^\varphi V\|_{H^{s,k}} &\le c \epsilon^2 \|w\|_{H^{s,k}} \|\partial_r V\|_{H^{s,k}} - \epsilon^3 \|w\|_{H^{s,k}} \|\frac{h}{1+\epsilon h}\|_{H^{s,k}} \|\partial_r V\|_{H^{s,k}} \\ &\le \frac{\epsilon^2}{\sqrt{\mu}} C (\|V\|_{H^{s+1,k+1}} + \sqrt{\mu} \|w\|_{H^{s+1,k+1}}). \end{aligned} $$

Here, $C$ depends on $M$. We used the composition estimate (A.10).

Note that the factor $\frac{1}{\sqrt{\mu}}$ comes from the fact that the term $\|w\|_{H^{s+1,k+1}}$ in $M$ comes with a factor $\sqrt{\mu}$. In order to bound the last term in (C.1), it remains to estimate the following term:

$$ \epsilon w \partial_r^\varphi \underline{V} = \epsilon w \partial_r \underline{V} - \epsilon^2 w \frac{h}{1+\epsilon h} \partial_r \underline{V}. $$

We use product estimates (A.3) and (A.2) as well as the composition estimate (A.10), to get

$$ \|\epsilon w \partial_r^\varphi \underline{V}\|_{H^{s,k}} \le c \epsilon \|w\|_{H^{s,k}} + c \epsilon^2 \|w\|_{H^{s,k}} \|h\|_{H^{s,k}}. $$

The vertical component of the term in the first estimate reads

$$ \epsilon (\underline{U} + \epsilon \underline{U}) \cdot \nabla^\varphi w = \epsilon \underline{V} \cdot \nabla_x^\varphi w + \epsilon^2 \underline{V} \cdot \nabla_x^\varphi w + \epsilon^2 w \partial_r^\varphi w $$

The first two terms are treated as before. For the last term, the product estimate (A.2) and the composition estimate (A.10) yield

$$ \|\epsilon^2 w \partial_r^\varphi w\|_{H^{s,k}} \le \frac{\epsilon^2}{\mu} C(\sqrt{\mu} \|w\|_{H^{s+1,k+1}})^2, $$

which is bounded by $\frac{\epsilon}{\sqrt{\mu}} CM$ thanks to the assumption $\epsilon \le \sqrt{\mu}$.

For the second estimate, we write

(C.2)

$$ \begin{aligned} \partial_i^\varphi (\underline{U} + \epsilon \underline{U})_j \partial_j^\varphi (\underline{U} + \epsilon \underline{U})_i = & \frac{\epsilon^2}{(1+\epsilon h)^2} \sum_{i,j \le d} \partial_i \eta \underline{V}'_j \partial_j \eta \underline{V}'_i + 2\epsilon^2 \frac{1}{1+\epsilon h} \sum_{i,j \le d} \partial_i \eta \underline{V}'_j \partial_j^\varphi \underline{U}_i \\ & - 2\epsilon \frac{1}{1+\epsilon h} \sum_{j \le d+1} \underline{V}'_j \partial_j^\varphi w + \epsilon^2 \sum_{i,j \le d+1} \partial_i^\varphi \underline{U}_j \partial_j^\varphi \underline{U}_i. \end{aligned} $$

Each term in (C.2) is treated as before with product estimates (A.2) and (A.3) as well as the composition estimate (A.10). For the last term in (C.2), we need the assumption $\epsilon \le \sqrt{\mu}$, as for the last term in (C.1).
For the last estimate, the proof is very similar and we only treat one term for the sake of conciseness. We write, thanks to Hölder inequality and embedding of Lemma A.2

$$ \begin{aligned} \|V \cdot \left( \frac{\nabla_x \eta}{1 + \epsilon h} \partial_r V \right)\|_{L^2} & \le \|V\|_{L^\infty} \|\nabla_x \eta\|_{L^\infty} \frac{1}{1 + \epsilon h} \|\partial_r V\|_{L^2} \\ & \le C \|V\|_{H^{s_0+\frac{1}{2},1}} \|\eta\|_{H^{s_0+\frac{3}{2},2}} \|V\|_{H^{1,1}} \\ & \le C \|V\|_{H^{s_0+\frac{1}{2},1}}. \end{aligned} $$

For the last inequality, we used Assumption (H.5), as well as $s_0 + \frac{1}{2} \ge \frac{d}{2} + \frac{1}{2} \ge 1$. □

We now show how to use the incompressibility constraint in order to estimate $\|w\|_{H^s}$ (for $s$ large enough) by terms depending only on $V$ and $\eta$, but with the loss of one derivative. This is used in the energy estimates, as we only have a control on $\|V\|_{H^s}$, $\|\eta\|_{H^s}$, but only on $\|\sqrt{\mu}w\|_{H^s}$. Therefore, the following lemma yields a control on $\|w\|_{H^{s-1}}$. In other words, the following lemma shows how to trade one derivative for a factor $\sqrt{\mu}$:

**Lemma C.2.** Let $s \ge s_0 + \frac{3}{2}$ and $(V, w, \eta) \in (H^s)^{d+2}$ satisfying the incompressibility constraint in (E), as well as the boundary conditions (1.4). Then, there exists $C > 0$ such that

$$ (C.3) \qquad \|w\|_{H^{s-1}} \le C \|V\|_{H^s} (1 + \|\eta\|_{H^s}) + C \|\eta\|_{H^s} |\underline{V}|_{W^{s,\infty}}. $$

*Proof.* Recall the incompressibility constraint

$$ \nabla_x^\varphi \cdot V + \partial_r^\varphi w + \frac{\nabla_x}{1 + \epsilon h} \partial_r \underline{V} = 0, $$

where

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

We thus write

$$ w(t, x, r) = \int_{0}^{r} \partial_{r} w(t, x, r') dr' = \int_{0}^{r} (1 + \epsilon h) \left( \nabla_{x}^{\varphi} \cdot V + \frac{\nabla_{x} \eta}{1 + \epsilon h} \partial_{r} \underline{V} \right). $$

We thus have, by product estimates (A.2) and (A.3)

$$ \|w\|_{H^{s-1}} \le C(1 + \epsilon \|h\|_{H^{s-1}}) \|\nabla_x^\varphi \cdot V\|_{H^{s-1}} + \|\nabla_x \eta\|_{H^{s-1}} |\partial_r \underline{V}|_{W^{s-1,\infty}}. $$

It remains to estimate the term

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

We start by writing

$$ \frac{1}{1 + \epsilon h} = 1 - \frac{\epsilon h}{1 + \epsilon h}. $$

The composition estimate (A.7) then yields

$$ \| \frac{\epsilon h}{1 + \epsilon h} \|_{H^s} \le C \epsilon \|h\|_{H^{s-1}} \le C \epsilon \|\eta\|_{H^s}, $$

as $h := -\partial_r \eta$. We can thus write:

$$ \begin{aligned} \|\nabla_x^\varphi \cdot V\|_{H^{s-1}} &\le C \left( \|\nabla_x \cdot V\|_{H^{s-1}} + \epsilon \left(1 + \frac{\epsilon h}{1 + \epsilon h}\right) \|\nabla_x \eta\|_{H^{s-1}} \|\partial_r V\|_{H^{s-1}} \right) \\ &\le C(1 + \|\eta\|_{H^s}) \|V\|_{H^s}. \end{aligned} $$

We can finally write:

$$ \|w\|_{H^{s-1}} \le C \|V\|_{H^s} (1 + \|\eta\|_{H^s}) + C \|\eta\|_{H^s} |\underline{V}|_{W^{s,\infty}}. $$
<figure><img src="image_7.png" /></figure>

ACKNOWLEDGMENTS

The author thanks his PhD supervisors Vincent Duchêne and David Lannes for their
valuable advice.

This work was supported by the BOURGEONS project, grant ANR-23-CE40-0014-01 of
the French National Research Agency (ANR).

This work is licensed under CC BY 4.0.

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