for any initial data $(h_{0},{\bm{u}}_{0},w_{0})\in H^{s,k}(\Omega)^{2+d}$
satisfying the boundary condition $w_{0}|_{\varrho=\rho_{1}}=0$ and the
incompressibility condition

$-(\underline{h}+h_{0})\nabla_{\bm{x}}\cdot{\bm{u}}_{0}-(\nabla_{\bm{x}}\eta_{0})\cdot({\underline{{\bm{u}}}}^{\prime}+\partial_{\varrho}{\bm{u}}_{0})+\partial_{\varrho}w_{0}=0,$

(denoting
$\eta_{0}(\cdot,\varrho)=\int_{\varrho}^{\rho_{1}}h_{0}(\cdot,\varrho^{\prime})\operatorname{d\\!}\varrho^{\prime}$),
the bounds

$\big{\lVert}\eta_{0}\big{\rVert}_{H^{s+p,k+p}}+\big{\lVert}{\bm{u}}_{0}\big{\rVert}_{H^{s+p,k+p}}+\big{\lvert}\eta_{0}\big{|}_{\varrho=\rho_{0}}\big{\rvert}_{H^{s+p}_{\bm{x}}}+\kappa^{1/2}\big{\lVert}h_{0}\big{\rVert}_{H^{s+p,k+p}}=M_{0}\leq
M$

and the stable stratification assumption

$\inf_{({\bm{x}},\varrho)\in\Omega}h_{\star}\leq\underline{h}(\varrho)+h_{0}({\bm{x}},\varrho)\leq
h^{\star},$

the following holds. Denoting

$(T^{\rm h})^{-1}=C^{\rm
h}\,\big{(}1+\kappa^{-1}\big{(}\big{\lvert}\underline{{\bm{u}}}^{\prime}\big{\rvert}_{L^{2}_{\varrho}}^{2}+M_{0}^{2}\big{)}\big{)},$

as in Lemma 5.1 there exists a unique strong solution $(h^{\rm
nh},{\bm{u}}^{\rm nh},w^{\rm nh})\in\mathcal{C}^{0}([0,T^{\rm
h}];H^{s,k}(\Omega)^{1+d})$ to the non-hydrostatic equations (2.1) with
initial data $(h^{\rm nh},{\bm{u}}^{\rm nh},w^{\rm
nh})\big{|}_{t=0}=(h_{0},{\bm{u}}_{0},w_{0})$. Moreover, one has $h^{\rm
nh}\in L^{2}(0,T^{\rm h};H^{s+1,k}(\Omega))$, $\eta^{\rm nh}\in L^{2}(0,T^{\rm
h};H^{s+2,k}(\Omega))$ and, for any $t\in[0,T^{\rm h}]$, the lower and the
upper bounds hold

$\inf_{({\bm{x}},\varrho)\in\Omega}\underline{h}(\varrho)+h^{\rm
nh}(t,{\bm{x}},\varrho)\geq
h_{\star}/3,\quad\sup_{({\bm{x}},\varrho)\in\Omega}\underline{h}(\varrho)+h^{\rm
nh}(t,{\bm{x}},\varrho)\leq 3h^{\star},$

and the estimate below holds true

$\displaystyle\big{\lVert}\eta^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s,k}}+\big{\lVert}{\bm{u}}^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s,k}}+\mu^{1/2}\big{\lVert}w^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s,k}}+\big{\lvert}\eta^{\rm
nh}\big{|}_{\varrho=\rho_{0}}(t,\cdot)\big{\rvert}_{H^{s}_{\bm{x}}}$
$\displaystyle\quad+\kappa^{1/2}\big{\lVert}h^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s,k}}+\mu^{1/2}\kappa^{1/2}\big{\lVert}\nabla_{\bm{x}}\eta^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s,k}}$
$\displaystyle\quad+\kappa^{1/2}\big{\lVert}\nabla_{\bm{x}}\eta^{\rm
nh}\big{\rVert}_{L^{2}(0,t;H^{s,k})}+\kappa^{1/2}\big{\lvert}\nabla_{\bm{x}}\eta^{\rm
nh}\big{|}_{\varrho=\rho_{0}}\big{\rvert}_{L^{2}(0,t;H^{s}_{\bm{x}})}$
$\displaystyle\quad+\kappa\big{\lVert}\nabla_{\bm{x}}h^{\rm
nh}\big{\rVert}_{L^{2}(0,t;H^{s,k})}+\mu^{1/2}\kappa\big{\lVert}\nabla_{\bm{x}}^{2}\eta^{\rm
nh}\big{\rVert}_{L^{2}(0,t;H^{s,k})}\leq C\,M_{0},$ (5.9)

and $(h^{\rm nh},{\bm{u}}^{\rm nh})$ converges strongly in
$L^{\infty}(0,T;H^{s,k}(\Omega)^{1+d})$ towards $(h^{\rm h},{\bm{u}}^{\rm h})$
the corresponding solution to the hydrostatic equations (2.5), as $\mu\searrow
0$.

###### Proof.

We closely follow the proof of Proposition 4.8 and exhibit a bootstrap
argument on the functional

$\displaystyle\mathcal{F}(t):$ $\displaystyle=\big{\lVert}\eta^{\rm
d}(t,\cdot)\big{\rVert}_{H^{s,k}}+\big{\lVert}{\bm{u}}^{\rm
d}(t,\cdot)\big{\rVert}_{H^{s,k}}+\mu^{1/2}\big{\lVert}w^{\rm
d}(t,\cdot)\big{\rVert}_{H^{s,k}}+\big{\lvert}\eta^{\rm
d}\big{|}_{\varrho=\rho_{0}}(t,\cdot)\big{\rvert}_{H^{s}_{\bm{x}}}$
$\displaystyle\quad+\kappa^{1/2}\big{\lVert}h^{\rm
d}(t,\cdot)\big{\rVert}_{H^{s,k}}+\mu^{1/2}\kappa^{1/2}\big{\lVert}\nabla_{\bm{x}}\eta^{\rm
d}(t,\cdot)\big{\rVert}_{H^{s,k}}$
$\displaystyle\quad+\kappa^{1/2}\big{\lVert}\nabla_{\bm{x}}\eta^{\rm
d}\big{\rVert}_{L^{2}(0,t;H^{s,k})}+\kappa^{1/2}\big{\lvert}\nabla_{\bm{x}}\eta^{\rm
d}\big{|}_{\varrho=\rho_{0}}\big{\rvert}_{L^{2}(0,t;H^{s}_{\bm{x}})}$
$\displaystyle\quad+\kappa\big{\lVert}\nabla_{\bm{x}}h^{\rm
d}\big{\rVert}_{L^{2}(0,t;H^{s,k})}+\mu^{1/2}\kappa\big{\lVert}\nabla_{\bm{x}}^{2}\eta^{\rm
d}\big{\rVert}_{L^{2}(0,t;H^{s,k})}$

where we denote

$h^{\rm d}:=h^{\rm nh}-h^{\rm h};\quad\eta^{\rm d}:=\eta^{\rm nh}-\eta^{\rm
h};\quad{\bm{u}}^{\rm d}:={\bm{u}}^{\rm nh}-{\bm{u}}^{\rm h};\quad w^{\rm
d}:=w^{\rm nh}-w^{\rm h}$

with the usual notation for $\eta^{\rm nh}$, $\eta^{\rm h}$, and $w^{\rm h}$
is defined by (5.1). Denoting by $T^{\star}$ the maximal existence time of the
non-hydrostatic solution provided by Proposition 4.3, we set

$T^{\rm nh}=\sup\Big{\\{}0<T<\min(T^{\star},T^{\rm
h})\,:\;\forall\,t\in(0,T),\;h_{\star}/3\leq\underline{h}(\varrho)+h^{\rm
nh}(t,{\bm{x}},\varrho)\leq 3h^{\star}\quad\\\
\text{and}\quad\mathcal{F}(t)\leq\mu^{1/2}M_{0}\exp(C_{0}t),\quad\mathcal{F}(t)\leq\kappa^{1/2}M_{0}\Big{\\}},$
(5.10)

with $C_{0}$ sufficiently large (to be determined later on). We will show by
the standard continuity argument that $T^{\rm nh}=\min(T^{\star},T^{\rm h})$,
which in turns yields $T^{\star}>T^{\rm h}$ and shows the result. Indeed, the
converse inequality $T^{\rm nh}=T^{\star}\leq T^{\rm h}$ yields a
contradiction by Proposition 4.3 and the desired estimates immediately follow
from the control of $\mathcal{F}$, the bound

$\big{\lVert}h^{\rm
h}(t,\cdot)\big{\rVert}_{H^{s+1,k+1}}+\big{\lVert}{\bm{u}}^{\rm
h}(t,\cdot)\big{\rVert}_{H^{s+1,k+1}}+\big{\lVert}\eta^{\rm
h}(t,\cdot)\big{\rVert}_{H^{s+1,k+1}}+\big{\lVert}w^{\rm
h}(t,\cdot)\big{\rVert}_{H^{s+1,k+1}}\leq C^{\rm h}M_{0}$ (5.11)

provided by Lemma 5.1, and triangular inequality (when $C$ is chosen
sufficiently large).

Let us now derive from Lemma 5.2 the necessary estimates for the bootstrap
argument. In the following we repeatedly use the triangular inequality to
infer from (5.10) and (5.11) the corresponding control (5.9) with $C$
depending only $C^{\rm h}$, $T^{\rm h}$ (and $\kappa\leq 1$). We shall denote
by $C$ a constant depending uniquely on
$s,k,\underline{M},M,h_{\star},h^{\star}$ and $C^{\rm h}$, $T^{\rm h}$, but
not on $C_{0}$, and which may change from line to line.

By means of (5.7a)-(5.7b) and Lemma 3.4, we infer from (5.10)-(5.11)

$\kappa^{1/2}\big{\lVert}h^{\rm
d}\big{\rVert}_{L^{\infty}(0,T;H^{s,k})}+\kappa\big{\lVert}\nabla_{\bm{x}}h^{\rm
d}\big{\rVert}_{L^{2}(0,T;H^{s,k})}\leq
C\,\big{(}\big{\lvert}\mathcal{F}\big{\rvert}_{L^{1}_{T}}+\big{\lvert}\mathcal{F}\big{\rvert}_{L^{2}_{T}}\big{)}.$

Next, by differentiating with respect to space the first equation of (5.6a)
using (5.6b) and Lemma 3.4, we infer

$\mu^{1/2}\kappa^{1/2}\big{\lVert}\nabla_{\bm{x}}\eta^{\rm
d}\big{\rVert}_{L^{\infty}(0,T;H^{s,k})}+\mu^{1/2}\kappa\big{\lVert}\nabla_{\bm{x}}^{2}\eta^{\rm
d}\big{\rVert}_{L^{2}(0,T;H^{s,k})}\leq
C\,\big{(}\big{\lvert}\mathcal{F}\big{\rvert}_{L^{1}_{T}}+\big{\lvert}\mathcal{F}\big{\rvert}_{L^{2}_{T}}\big{)}.$

Now, we use (5.6a)-(5.6b)-(5.6c) and proceeding as in the proof of Proposition
4.8 (together with the above estimates) we infer that for any $t\in(0,T)$,

$\mathcal{F}(t)\leq
C_{1}\big{\lvert}\mathcal{F}\big{\rvert}_{L^{1}_{t}}+C_{2}\big{\lvert}\mathcal{F}\big{\rvert}_{L^{2}_{t}}+C_{3}\,\mu^{1/2}M_{0}\,t$

with $C_{i}$ ($i\in\\{1,2,3\\}$) depending uniquely on
$s,k,\underline{M},M,h_{\star},h^{\star}$ and $C^{\rm h}$, $T^{\rm h}$. By
using the inequality $\mathcal{F}(t)\leq\mu^{1/2}M_{0}\exp(C_{0}t)$ from
(5.10) and the inequality $\tau\leq\exp(\tau)$ (for $\tau\geq 0$), we deduce

$\mathcal{F}(t)\leq
C_{1}\mu^{1/2}M_{0}C_{0}^{-1}\exp(C_{0}t)+C_{2}\mu^{1/2}M_{0}(2C_{0})^{-1/2}\exp(C_{0}t)+C_{3}\mu^{1/2}M_{0}C_{0}^{-1}\exp(C_{0}t).$

There remains to choose $C_{0}$ sufficiently large so that
$C_{1}C_{0}^{-1}+C_{2}(2C_{0})^{-1/2}+C_{3}C_{0}^{-1}<1$, and restrict to
$\mu$ sufficiently small so that $\mu^{1/2}M_{0}\exp(C_{0}T^{\rm
h})\leq\mu^{1/2}M_{0}C^{1/2}/2\leq\kappa^{1/2}/2$. The upper and lower bounds
for $\underline{h}+h^{\rm nh}$ follow immediately from the corresponding ones
for $\underline{h}+h^{\rm h}$ provided by Lemma 5.1 and triangular inequality,
augmenting $C$ if necessary. Then the usual continuity argument yields, as
desired, $T^{\rm nh}=\min(T^{\star},T^{\rm h})$. ∎

###  5.4. Improved convergence rate

Proposition 5.3 established the strong convergence for regular well-prepared
initial data of the solution to the non-hydrostatic equations, (2.1), towards
the corresponding solution to the hydrostatic equations, (2.5), as
$\mu\searrow 0$. The convergence rate displayed in the proof is
$\mathcal{O}(\mu^{1/2})$. The aim of this section is to provide an improved
and _optimal_ convergence rate $\mathcal{O}(\mu)$. The strategy is based on
the interpretation of the non-hydrostatic solution as an approximate solution
to the hydrostatic equations (in the sense of consistency) and the use of the
uniform control obtained in Proposition 5.3.

######  Proposition 5.4.

There exists $p\in\mathbb{N}$ such that for any $s,k\in\mathbb{N}$ with
$k=s>\frac{5}{2}+\frac{d}{2}$ and $\underline{M},M,h_{\star},h^{\star}>0$,
there exists $C=C(s,k,\underline{M},M,h_{\star},h^{\star})>0$ such that under
the assumptions of Proposition 5.3 and using the notations therein,

$\big{\lVert}h^{\rm nh}-h^{\rm h}\big{\rVert}_{L^{\infty}(0,T^{\rm
h};H^{s-1,k-1})}+\big{\lVert}\eta^{\rm nh}-\eta^{\rm
h}\big{\rVert}_{L^{\infty}(0,T^{\rm h};H^{s,k})}+\big{\lVert}{\bm{u}}^{\rm
nh}-{\bm{u}}^{\rm h}\big{\rVert}_{L^{\infty}(0,T^{\rm h};H^{s,k})}\leq
C\,\mu.$

######  Corollary 5.5.

Incrementing $p\in\mathbb{N}$, we find that for any $s,k\in\mathbb{N}$ such
that $k=s>\frac{3}{2}+\frac{d}{2}$,

$\big{\lVert}h^{\rm nh}-h^{\rm h}\big{\rVert}_{L^{\infty}(0,T^{\rm
h};H^{s,k})}+\big{\lVert}\eta^{\rm nh}-\eta^{\rm
h}\big{\rVert}_{L^{\infty}(0,T^{\rm h};H^{s+1,k+1})}+\big{\lVert}{\bm{u}}^{\rm
nh}-{\bm{u}}^{\rm h}\big{\rVert}_{L^{\infty}(0,T^{\rm h};H^{s+1,k+1})}\leq
C\,\mu$

with $C=C(s+1,k+1,\underline{M},M,h_{\star},h^{\star})>0$.

###### Proof.

Since all arguments of the proof have been already used in slightly different
contexts, we only quickly sketch the argument.

For any $p^{\prime}\in\mathbb{N}$, we may use Proposition 5.3 with indices
$s+p^{\prime}$ and $k+p^{\prime}$ to infer the existence of the non-
hydrostatic solution $(h^{\rm nh},{\bm{u}}^{\rm nh},w^{\rm
nh})\in\mathcal{C}([0,T^{\rm h}];H^{s+p^{\prime},k+p^{\prime}}(\Omega)^{2+d})$
and the control

$\sup_{t\in[0,T^{\rm h}]}\Big{(}\big{\lVert}\eta^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s+p^{\prime},k+p^{\prime}}}+\big{\lVert}{\bm{u}}^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s+p^{\prime},k+p^{\prime}}}+\big{\lvert}\eta^{\rm
nh}\big{|}_{\varrho=\rho_{0}}(t,\cdot)\big{\rvert}_{H^{s+p^{\prime}}_{\bm{x}}}\Big{)}\leq
C\,M_{0}.$

By using $h^{\rm nh}=-{\partial}_{\varrho}\eta^{\rm nh}$ and the divergence-
free condition

$w^{\rm nh}=({\underline{{\bm{u}}}}+{\bm{u}}^{\rm
nh})\cdot\nabla_{\bm{x}}\eta^{\rm
nh}-\int_{\varrho}^{\rho_{1}}\nabla_{\bm{x}}\cdot((\underline{h}+h^{\rm
nh})({\underline{{\bm{u}}}}+{\bm{u}}^{\rm
nh}))\operatorname{d\\!}\varrho^{\prime}$

we obtain (augmenting $C$ if necessary)

$\sup_{t\in[0,T^{\rm h}]}\Big{(}\big{\lVert}h^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s+p^{\prime}-1,k+p^{\prime}-1}}+\big{\lVert}w^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s+p^{\prime}-1,k+p^{\prime}-1}}\Big{)}\leq
C\,M_{0}$

and hence, by Corollary 4.2 (specifically (4.19)), Poincaré inequality (4.10)
and choosing $p^{\prime}$ sufficiently large, that $P_{\rm
nh}(\cdot,\varrho):=P^{\rm
nh}(\cdot,\varrho)-\int_{\rho_{0}}^{\varrho}\varrho^{\prime}h^{\rm
nh}(\cdot,\varrho^{\prime})\operatorname{d\\!}\varrho^{\prime}$ satisfies

$\sup_{t\in[0,T^{\rm h}]}\big{\lVert}P_{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s+1,k+1}}\leq C\,\mu\,M_{0}.$

From this estimate we infer (by Lemma 5.1) that $h^{\rm d}:=h^{\rm nh}-h^{\rm
h}$ and ${\bm{u}}^{\rm d}:={\bm{u}}^{\rm nh}-{\bm{u}}^{\rm h}$ satisfies

$\displaystyle{\partial}_{t}\eta^{\rm d}+(\underline{{\bm{u}}}+{\bm{u}}^{\rm
nh})\cdot\nabla_{\bm{x}}\eta^{\rm
d}+\int_{\varrho}^{\rho_{1}}(\underline{{\bm{u}}}^{\prime}+{\partial}_{\varrho}{\bm{u}}^{\rm
nh})\cdot\nabla_{\bm{x}}\eta^{\rm
d}\,\operatorname{d\\!}\varrho^{\prime}+\int_{\varrho}^{\rho_{1}}(\underline{h}+h^{\rm
nh})\nabla_{\bm{x}}\cdot{\bm{u}}^{\rm d}\,\operatorname{d\\!}\varrho^{\prime}$
(5.12) $\displaystyle+\int_{\varrho}^{\rho_{1}}{\bm{u}}^{\rm
d}\cdot\nabla_{\bm{x}}h^{\rm h}+h^{\rm d}\nabla_{\bm{x}}\cdot{\bm{u}}^{\rm
h}\,\operatorname{d\\!}\varrho^{\prime}$
$\displaystyle=\kappa\Delta_{\bm{x}}\eta^{\rm d},$
$\displaystyle{\partial}_{t}{\bm{u}}^{\rm
d}+\big{(}(\underline{{\bm{u}}}+{\bm{u}}^{\rm
nh}-\kappa\tfrac{\nabla_{\bm{x}}h^{\rm nh}}{\underline{h}+h^{\rm
nh}})\cdot\nabla_{\bm{x}}\big{)}{\bm{u}}^{\rm
d}+{\frac{\rho_{0}}{\varrho}\nabla_{\bm{x}}\eta^{\rm
d}|_{\varrho=\rho_{0}}}+{\frac{1}{\varrho}\int_{\rho_{0}}^{\varrho}\nabla_{\bm{x}}\eta^{\rm
d}\,\operatorname{d\\!}\varrho^{\prime}}$ $\displaystyle+\big{(}({\bm{u}}^{\rm
d}-\kappa(\tfrac{\nabla_{\bm{x}}h^{\rm nh}}{\underline{h}+h^{\rm
nh}}-\tfrac{\nabla_{\bm{x}}h^{\rm h}}{\underline{h}+h^{\rm
h}}))\cdot\nabla_{\bm{x}}\big{)}{\bm{u}}^{\rm h}$ $\displaystyle={\bm{R}}^{\rm
nh},$

with ${\bm{R}}^{\rm nh}:=-\frac{\nabla_{\bm{x}}{{P_{\rm
nh}}}}{\varrho}-\frac{\nabla_{\bm{x}}\eta^{\rm
nh}}{\varrho(\underline{h}+h^{\rm nh})}{\partial}_{\varrho}{{P_{\rm nh}}}$
satisfying (by Lemma A.3 and Lemma A.6) the bound

$\sup_{t\in[0,T^{\rm h}]}\big{\lVert}{\bm{R}}^{\rm
nh}(t,\cdot)\big{\rVert}_{H^{s,k}}\leq C\,\mu\,M_{0}.$

From this, inspecting the proof of Lemma 5.2, we infer that as long as

$\mathcal{F}_{s,k}:=\big{\lVert}h^{\rm
d}\big{\rVert}_{H^{s-1,k-1}}+\big{\lVert}\eta^{\rm
d}\big{\rVert}_{H^{s,k}}+\big{\lvert}\eta^{\rm
d}\big{|}_{\varrho=\rho_{0}}\big{\rvert}_{H^{s}_{\bm{x}}}+\big{\lVert}{\bm{u}}^{\rm
d}\big{\rVert}_{H^{s,k}}+\kappa^{1/2}\big{\lVert}h^{\rm
d}\big{\rVert}_{H^{s,k}}\leq\kappa^{1/2}M_{0},$

one has for any ${\bm{\alpha}}\in\mathbb{N}^{d}$ and $j\in\mathbb{N}$ such
that $|{\bm{\alpha}}|+j\leq s$ that
$\eta^{({\bm{\alpha}},j)}:={\partial}_{\bm{x}}^{\bm{\alpha}}{\partial}_{\varrho}^{j}\eta^{\rm
d}$,
${\bm{u}}^{({\bm{\alpha}},j)}:={\partial}_{\bm{x}}^{\bm{\alpha}}{\partial}_{\varrho}^{j}{\bm{u}}^{\rm
d}$ and
$h^{({\bm{\alpha}},j)}:={\partial}_{\bm{x}}^{\bm{\alpha}}{\partial}_{\varrho}^{j}h^{\rm
d}$ satisfy

$\displaystyle\partial_{t}\eta^{({\bm{\alpha}},j)}+(\underline{{\bm{u}}}+{\bm{u}}^{\rm
nh})\cdot\nabla_{\bm{x}}\eta^{({\bm{\alpha}},j)}+\Big{\langle}\int_{\varrho}^{\rho_{1}}(\underline{{\bm{u}}}^{\prime}+{\partial}_{\varrho}{\bm{u}}^{\rm
nh})\cdot\nabla_{\bm{x}}\eta^{({\bm{\alpha}},j)}\,\operatorname{d\\!}\varrho^{\prime}$
$\displaystyle+\int_{\varrho}^{\rho_{1}}(\underline{h}+h^{\rm
nh})\nabla_{\bm{x}}\cdot{\bm{u}}^{({\bm{\alpha}},j)}\operatorname{d\\!}\varrho^{\prime}\Big{\rangle}_{j=0}-\kappa\Delta_{\bm{x}}\eta^{({\bm{\alpha}},j)}$
$\displaystyle=R_{{\bm{\alpha}},j},$
$\displaystyle\partial_{t}{\bm{u}}^{({\bm{\alpha}},j)}+\big{(}({\underline{{\bm{u}}}}+{\bm{u}}^{\rm
nh}-\kappa\tfrac{\nabla_{\bm{x}}h^{\rm nh}}{\underline{h}+h^{\rm
nh}})\cdot\nabla_{\bm{x}}\big{)}{\bm{u}}^{({\bm{\alpha}},j)}+\Big{\langle}\frac{\rho_{0}}{\varrho}\nabla_{\bm{x}}\eta^{({\bm{\alpha}},j)}\big{|}_{\varrho=\rho_{0}}$
$\displaystyle+\frac{1}{\varrho}\int_{\rho_{0}}^{\varrho}\nabla_{\bm{x}}\eta^{({\bm{\alpha}},j)}\operatorname{d\\!}\varrho^{\prime}\Big{\rangle}_{j=0}$
$\displaystyle={\bm{R}}^{\rm nh}_{{\bm{\alpha}},j},$

and

$\partial_{t}h^{({\bm{\alpha}},j)}+(\underline{{\bm{u}}}+{\bm{u}})\cdot\nabla_{\bm{x}}h^{({\bm{\alpha}},j)}=\kappa\Delta_{\bm{x}}\partial_{\varrho}^{j}h^{({\bm{\alpha}},j)}+r_{{\bm{\alpha}},j}+\nabla_{\bm{x}}\cdot{\bm{r}}_{{\bm{\alpha}},j},$

with

$\big{\lVert}R_{{\bm{\alpha}},j}\big{\rVert}_{L^{2}(\Omega)}+\big{\lVert}{\bm{R}}^{\rm
nh}_{{\bm{\alpha}},j}\big{\rVert}_{L^{2}(\Omega)}\leq
C\,\big{(}\mathcal{F}_{s,k}+M_{0}\kappa\big{\lVert}\nabla_{\bm{x}}h^{\rm
d}\big{\rVert}_{H^{s,k}}\big{)}+C\,\mu\,M_{0}$

and

$\kappa^{1/2}\big{\lVert}r_{{\bm{\alpha}},j}\big{\rVert}_{L^{2}(\Omega)}+\big{\lVert}{\bm{r}}_{{\bm{\alpha}},j}\big{\rVert}_{L^{2}(\Omega)}\leq
C\,\mathcal{F}_{s,k}.$

We may then proceed as in the proof of Proposition 3.6, and bootstrap the
control

$\mathcal{F}_{s,k}(t)+\kappa^{1/2}\big{\lVert}\nabla_{\bm{x}}\eta^{\rm
d}\big{\rVert}_{L^{2}(0,t;H^{s,k})}+\kappa^{1/2}\big{\lvert}\nabla_{\bm{x}}\eta^{\rm
d}\big{|}_{\varrho=\rho_{0}}\big{\rvert}_{L^{2}(0,t;H^{s}_{\bm{x}})}+\kappa\big{\lVert}\nabla_{\bm{x}}h^{\rm
d}\big{\rVert}_{L^{2}(0,t;H^{s,k})}\leq C\,\mu\,M_{0}$

(choosing $C$ large enough) on the time interval $[0,T^{\rm h}]$. This
concludes the proof. ∎

##  Appendix A Product, composition and commutator estimates

In this section we collect useful estimates in the spaces $H^{s,k}(\Omega)$
introduced in (2.8). Our results will follow from standard estimates in
Sobolev spaces $H^{s}(\mathbb{R}^{d})$ (see e.g. [Lannesbook]*Appendix B and
references therein), and the following continuous embedding. Henceforth we
denote $\Omega=\mathbb{R}^{d}\times(\rho_{0},\rho_{1})$.

######  Lemma A.1.

For any $s\in\mathbb{R}$ and $\rho_{0}<\rho_{1}$,
$H^{s+1/2,1}(\Omega)\subset\mathcal{C}^{0}([\rho_{0},\rho_{1}];H^{s}(\mathbb{R}^{d}))$
and there exists $C>0$ such that for any $F\in H^{s+1/2,1}(\Omega)$,

$\max_{\varrho\in[\rho_{0},\rho_{1}]}\big{\lvert}F(\cdot,\varrho)\big{\rvert}_{H^{s}_{\bm{x}}}\leq
C\big{\lVert}F\big{\rVert}_{H^{s+1/2,1}}.$

More generally, for any $k\geq 1$,
$H^{s+1/2,1}(\Omega)\subset\bigcap_{j=0}^{k-1}\mathcal{C}^{j}([\rho_{0},\rho_{1}];H^{s-j}(\mathbb{R}^{d}))$,
and in particular, for any $s_{0}>d/2$ and $j\in\mathbb{N}$,
$H^{j+s_{0}+\frac{1}{2},j+1}(\Omega)\subset\big{(}\mathcal{C}^{j}(\Omega)\cap
W^{j,\infty}(\Omega)\big{)}$.

###### Proof.

By a density argument, we only need to prove the inequality for smooth
functions $F$. Set $\phi:[\rho_{0},\rho_{1}]\to\mathbb{R}^{+}$ a smooth
function such that $\phi(\rho_{0})=0$ and $\phi(\varrho)=1$ if
$\varrho\geq\tfrac{\rho_{0}+\rho_{1}}{2}$, and deduce that for any
$\varrho\geq\tfrac{\rho_{0}+\rho_{1}}{2}$, recalling the notation
$\Lambda^{s}:=(\operatorname{Id}-\Delta_{\bm{x}})^{s/2}$,

$\displaystyle\int_{\mathbb{R}^{d}}(\Lambda^{s}F)^{2}({\bm{x}},\varrho)\operatorname{d\\!}{\bm{x}}$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{\rho_{0}}^{\varrho}\partial_{\varrho}\big{(}\phi(\varrho^{\prime})(\Lambda^{s}F)^{2}({\bm{x}},\varrho^{\prime})\big{)}\operatorname{d\\!}\varrho^{\prime}\operatorname{d\\!}{\bm{x}}$
$\displaystyle\leq
2\big{\lvert}\phi\big{\rvert}_{L^{\infty}_{\varrho}}\int_{\rho_{0}}^{\rho_{1}}\big{\lvert}\Lambda^{s}F(\cdot,\varrho)\big{\rvert}_{H^{1/2}_{\bm{x}}}\big{\lvert}\Lambda^{s}\partial_{\varrho}F(\cdot,\varrho)\big{\rvert}_{H^{-1/2}_{\bm{x}}}\operatorname{d\\!}\varrho$
$\displaystyle\quad+\big{\lvert}\phi^{\prime}\big{\rvert}_{L^{\infty}}\int_{\rho_{0}}^{\rho_{1}}\big{\lvert}\Lambda^{s}F(\cdot,\varrho)\big{\rvert}_{L^{2}}\big{\lvert}\Lambda^{s}F(\cdot,\varrho)\big{\rvert}_{L^{2}}\operatorname{d\\!}\varrho$
$\displaystyle\lesssim\big{\lVert}F\big{\rVert}_{H^{s+1/2,0}}^{2}+\big{\lVert}\partial_{\varrho}F\big{\rVert}_{H^{s-1/2,0}}^{2}$

Using symmetrical considerations when $\varrho<\tfrac{\rho_{0}+\rho_{1}}{2}$,
we prove the claimed inequality, which yields the first continuous embedding.
Higher-order embeddings follow immediately. ∎

Recall the notation

$A_{s}+\big{\langle}B_{s}\big{\rangle}_{s>s_{\star}}=\begin{cases}A_{s}&\text{
if }s\leq s_{\star}\,,\\\ A_{s}+B_{s}&\text{otherwise.}\end{cases}$

### Product estimates

Recall the standard product estimates in Sobolev spaces
$H^{s}(\mathbb{R}^{d})$.

######  Lemma A.2.

Let $d\in\mathbb{N}^{\star}$, $s_{0}>d/2$.

  1. (1)

For any $s,s_{1},s_{2}\in\mathbb{R}$ such that $s_{1}\geq s$, $s_{2}\geq s$
and $s_{1}+s_{2}\geq s+s_{0}$, there exists $C>0$ such that for any $f\in
H^{s_{1}}(\mathbb{R}^{d})$ and $g\in H^{s_{2}}(\mathbb{R}^{d})$, $fg\in
H^{s}(\mathbb{R}^{d})$ and

$\big{\lvert}fg\big{\rvert}_{H^{s}}\leq
C\big{\lvert}f\big{\rvert}_{H^{s_{1}}}\big{\lvert}g\big{\rvert}_{H^{s_{2}}}.$

  2. (2)

For any $s\geq-s_{0}$, there exists $C>0$ such that for any $f\in
H^{s}(\mathbb{R}^{d})$ and $g\in H^{s}(\mathbb{R}^{d})\cap
H^{s_{0}}(\mathbb{R}^{d})$, $fg\in H^{s}(\mathbb{R}^{d})$ and

$\big{\lvert}fg\big{\rvert}_{H^{s}}\leq
C\big{\lvert}f\big{\rvert}_{H^{s_{0}}}\big{\lvert}g\big{\rvert}_{H^{s}}+C\left\langle\big{\lvert}f\big{\rvert}_{H^{s}}\big{\lvert}g\big{\rvert}_{H^{s_{0}}}\right\rangle_{s>s_{0}}.$

  3. (3)

For any $s_{1},\dots,s_{n}\in\mathbb{R}$ such that $s_{i}\geq 0$ and
$s_{1}+\dots+s_{n}\geq(n-1)s_{0}$, there exists $C>0$ such that for any
$(f_{1},\dots,f_{n})\in H^{s_{1}}(\mathbb{R}^{d})\times\cdots\times
H^{s_{n}}(\mathbb{R}^{d})$, $\prod_{i=1}^{n}f_{i}\in L^{2}(\mathbb{R}^{d})$
and

$\big{\lvert}\prod_{i=1}^{n}f_{i}\big{\rvert}_{L^{2}}\leq
C\prod_{i=1}^{n}\big{\lvert}f_{i}\big{\rvert}_{H^{s_{i}}}.$

Let us turn to product estimates in $H^{s,k}(\Omega)$ spaces.

######  Lemma A.3.

Let $d\in\mathbb{N}^{\star}$, $s_{0}>d/2$. Let $s,k\in\mathbb{N}$ such that
$s\geq s_{0}+\frac{1}{2}$ and $1\leq k\leq s$. Then $H^{s,k}(\Omega)$ is a
Banach algebra and there exists $C>0$ such that for any $F,G\in
H^{s,k}(\Omega)$,

$\big{\lVert}FG\big{\rVert}_{H^{s,k}}\leq
C\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}G\big{\rVert}_{H^{s,k}}.$

Moreover, if $s\geq s_{0}+\frac{3}{2}$ and $2\leq k\leq s$, then there exists
$C^{\prime}>0$ such that for any $F,G\in H^{s,k}(\Omega)$,

$\big{\lVert}FG\big{\rVert}_{H^{s,k}}\leq
C^{\prime}\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}G\big{\rVert}_{H^{s-1,k-1}}+C^{\prime}\big{\lVert}F\big{\rVert}_{H^{s-1,k-1}}\big{\lVert}G\big{\rVert}_{H^{s,k}},$

and if $s\geq s_{0}+\frac{3}{2}$ and $k=1$, then there exists
$C^{\prime\prime}>0$ such that for any $F,G\in H^{s,k}(\Omega)$,

$\big{\lVert}FG\big{\rVert}_{H^{s,1}}\leq
C^{\prime\prime}\big{\lVert}F\big{\rVert}_{H^{s,1}}\big{\lVert}G\big{\rVert}_{H^{s-1,1}}+C^{\prime\prime}\big{\lVert}F\big{\rVert}_{H^{s-1,1}}\big{\lVert}G\big{\rVert}_{H^{s,1}}.$

###### Proof.

We set two multi-indices
${\bm{\beta}}=({\bm{\beta}}_{\bm{x}},{\bm{\beta}}_{\varrho})\in\mathbb{N}^{d+1}$
and
${\bm{\gamma}}=({\bm{\gamma}}_{\bm{x}},{\bm{\gamma}}_{\varrho})\in\mathbb{N}^{d+1}$
being such that $|{\bm{\beta}}|+|{\bm{\gamma}}|\leq s$ and
${\bm{\beta}}_{\varrho}+{\bm{\gamma}}_{\varrho}\leq k$. Let us first assume
furthermore that ${\bm{\gamma}}_{\varrho}\leq k-1$ and $|{\bm{\gamma}}|\leq
s-1$. Then

$\displaystyle\big{\lVert}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)\big{\rVert}_{L^{2}(\Omega)}^{2}$
$\displaystyle\lesssim\int_{\rho_{0}}^{\rho_{1}}\big{\lvert}\partial^{\bm{\beta}}F(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\beta}}|}_{\bm{x}}}^{2}\big{\lvert}\partial^{\bm{\gamma}}G(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\gamma}}|-\frac{1}{2}}_{\bm{x}}}^{2}\operatorname{d\\!}\varrho$
$\displaystyle\lesssim\big{\lVert}\partial^{\bm{\beta}}F\big{\rVert}_{H^{s-|{\bm{\beta}}|,0}}^{2}\big{\lVert}\partial^{\bm{\gamma}}G\big{\rVert}_{H^{s-|{\bm{\gamma}}|,1}}^{2}\leq\big{\lVert}F\big{\rVert}_{H^{s,k}}^{2}\big{\lVert}G\big{\rVert}_{H^{s,k}}^{2}.$

where we used Lemma A.2(1) with
$(s,s_{1},s_{2})=(0,s-|{\bm{\beta}}|,s-|{\bm{\gamma}}|-\frac{1}{2})$, and
Lemma A.1. If ${\bm{\gamma}}_{\varrho}=k$ or $|{\bm{\gamma}}|=s$, and since
$1\leq k\leq s$, we have ${\bm{\beta}}_{\varrho}\leq k-1$ and
$|{\bm{\beta}}|\leq s-1$ and we may make use of the symmetric estimate. Hence
the proof of the first statement follows from Leibniz rule.

For the second statement, we assume first that
$\max(\\{{\bm{\beta}}_{\varrho},{\bm{\gamma}}_{\varrho}\\})\leq k-1$ and
$\max(\\{|{\bm{\beta}}|,|{\bm{\gamma}}|\\})\leq s-1$. Then, using Lemma A.2
with $(s,s_{1},s_{2})=(0,s-|{\bm{\beta}}|-\frac{1}{2},s-|{\bm{\gamma}}|-1)$
(recall $s\geq s_{0}+\frac{3}{2}$), and Lemma A.1,

$\displaystyle\big{\lVert}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)\big{\rVert}_{L^{2}(\Omega)}$
$\displaystyle\lesssim\big{\lvert}\big{\lvert}\partial^{\bm{\beta}}F(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\beta}}|-\frac{1}{2}}_{\bm{x}}}\big{\lvert}\partial^{\bm{\gamma}}G(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\gamma}}|-1}_{\bm{x}}}\big{\rvert}_{L^{2}_{\varrho}}$
$\displaystyle\lesssim\big{\lVert}F\big{\rVert}_{H^{s,{\bm{\beta}}_{\varrho}+1}}\big{\lVert}G\big{\rVert}_{H^{s-1,{\bm{\gamma}}_{\varrho}}}\leq\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}G\big{\rVert}_{H^{s-1,k-1}},$

Then if ${\bm{\beta}}_{\varrho}=k$ or $|{\bm{\beta}}|=s$, we have (since
$s\geq k\geq 2$) ${\bm{\gamma}}_{\varrho}\leq k-2$ and $|{\bm{\gamma}}|\leq
s-2$, and we infer

$\displaystyle\big{\lVert}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)\big{\rVert}_{L^{2}(\Omega)}$
$\displaystyle\lesssim\big{\lvert}\big{\lvert}\partial^{\bm{\beta}}F(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\beta}}|}_{\bm{x}}}\big{\lvert}\partial^{\bm{\gamma}}G(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\gamma}}|-\frac{3}{2}}_{\bm{x}}}\big{\rvert}_{L^{2}_{\varrho}}$
$\displaystyle\lesssim\big{\lVert}F\big{\rVert}_{H^{s,{\bm{\beta}}_{\varrho}}}\big{\lVert}G\big{\rVert}_{H^{s-1,{\bm{\gamma}}_{\varrho}+1}}\leq\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}G\big{\rVert}_{H^{s-1,k-1}}.$

Of course we have the symmetrical result when ${\bm{\gamma}}_{\varrho}=k$ or
$|{\bm{\gamma}}|=s$, which complete the proof.

Finally, for the last statement, we consider first the case
${\bm{\beta}}_{\varrho}=0$ and $\max(\\{|{\bm{\beta}}|,|{\bm{\gamma}}|\\})\leq
s-1$, and infer as above

$\big{\lVert}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)\big{\rVert}_{L^{2}(\Omega)}\lesssim\big{\lvert}\big{\lvert}\partial^{\bm{\beta}}F(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\beta}}|-\frac{1}{2}}_{\bm{x}}}\big{\lvert}\partial^{\bm{\gamma}}G(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\gamma}}|-1}_{\bm{x}}}\big{\rvert}_{L^{2}_{\varrho}}\lesssim\big{\lVert}F\big{\rVert}_{H^{s,1}}\big{\lVert}G\big{\rVert}_{H^{s-1,1}}.$

The case ${\bm{\beta}}_{\varrho}=1$ (and hence ${\bm{\gamma}}_{\varrho}=0$)
and $\max(\\{|{\bm{\beta}}|,|{\bm{\gamma}}|\\})\leq s-1$ is treated
symmetrically. Then if $|{\bm{\beta}}|=s$ we have
${\bm{\gamma}}_{\varrho}=|{\bm{\gamma}}|=0$, and we infer

$\big{\lVert}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)\big{\rVert}_{L^{2}(\Omega)}\lesssim\big{\lvert}\big{\lvert}\partial^{\bm{\beta}}F(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\beta}}|}_{\bm{x}}}\big{\lvert}\partial^{\bm{\gamma}}G(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\gamma}}|-\frac{3}{2}}_{\bm{x}}}\big{\rvert}_{L^{2}_{\varrho}}\lesssim\big{\lVert}F\big{\rVert}_{H^{s,1}}\big{\lVert}G\big{\rVert}_{H^{s-1,1}}.$

The case $|{\bm{\gamma}}|=s$ is treated symmetrically, and the proof is
complete. ∎

### Composition estimates

Let us recall the standard composition estimate in Sobolev spaces
$H^{s}(\mathbb{R}^{d})$.

######  Lemma A.4.

Let $d\in\mathbb{N}^{\star}$, $s_{0}>d/2$. For any
$\varphi\in\mathcal{C}^{\infty}(\mathbb{R};\mathbb{R})$ such that
$\varphi(0)=0$, and any $M>0$, there exists $C>0$ such that for any $f\in
H^{s_{0}}(\mathbb{R}^{d})\cap H^{s}(\mathbb{R}^{d})$ with
$\big{\lvert}f\big{\rvert}_{H^{s_{0}}}\leq M$, one has $\varphi(f)\in
H^{s}(\mathbb{R}^{d})$ and

$\big{\lvert}\varphi(f)\big{\rvert}_{H^{s}}\leq
C\big{\lvert}f\big{\rvert}_{H^{s}}.$

We now consider composition estimates in $H^{s,k}(\Omega)$.

######  Lemma A.5.

Let $d\in\mathbb{N}^{\star}$, $s_{0}>d/2$. Let $s,k\in\mathbb{N}$ with $s\geq
s_{0}+\frac{1}{2}$ and $1\leq k\leq s$, and $M>0$. There exists $C>0$ such
that for any $\varphi\in
W^{s,\infty}(\mathbb{R};W^{k,\infty}((\rho_{0},\rho_{1})))$ with
$\varphi(0;\cdot)\equiv 0$, and any $F\in H^{s,k}(\Omega)$ such that
$\big{\lVert}F\big{\rVert}_{H^{s,k}}\leq M$, then $\varphi\circ
F:({\bm{x}},\varrho)\mapsto\varphi(F({\bm{x}},\varrho);\varrho)\in
H^{s,k}(\Omega)$ and

$\big{\lVert}\varphi\circ F\big{\rVert}_{H^{s,k}}\leq
C\big{\lvert}\varphi\big{\rvert}_{W^{s,\infty}(\mathbb{R};W^{k,\infty}((\rho_{0},\rho_{1})))}\big{\lVert}F\big{\rVert}_{H^{s,k}}.$

If moreover $s\geq s_{0}+\frac{3}{2}$ and $2\leq k\leq s$, then there exists
$C^{\prime}>0$ such that for any $F\in H^{s,k}(\Omega)$ such that
$\big{\lVert}F\big{\rVert}_{H^{s-1,k-1}}\leq M$,

$\big{\lVert}\varphi\circ F\big{\rVert}_{H^{s,k}}\leq
C^{\prime}\big{\lvert}\varphi\big{\rvert}_{W^{s,\infty}(\mathbb{R};W^{k,\infty}((\rho_{0},\rho_{1})))}\big{\lVert}F\big{\rVert}_{H^{s,k}}.$

###### Proof.

Let
${\bm{\alpha}}=({\bm{\alpha}}_{\bm{x}},{\bm{\alpha}}_{\varrho})\in\mathbb{N}^{d+1}\setminus\\{{\bf
0}\\}$ with $0\leq|{\bm{\alpha}}|\leq s$ and $0\leq{\bm{\alpha}}_{\varrho}\leq
k$. We have by Faà di Bruno’s formula

$\big{\lVert}\partial^{\bm{\alpha}}(\varphi\circ
F)\big{\rVert}_{L^{2}(\Omega)}\lesssim\sum\big{\lVert}\big{(}({\partial}_{1}^{i}{\partial}_{2}^{j}\varphi)\circ
F\big{)}\,(\partial^{{\bm{\alpha}}^{i,j}_{1}}F)\cdots(\partial^{{\bm{\alpha}}^{i,j}_{i}}F)\big{\rVert}_{L^{2}(\Omega)},$

where $i,j\in\mathbb{N}$ with $i+j\leq|{\bm{\alpha}}|\leq s$, and the multi-
indices
${\bm{\alpha}}^{i,j}_{\ell}=({\bm{\alpha}}^{i,j}_{\ell,{\bm{x}}},{\bm{\alpha}}^{i,j}_{\ell,\varrho})\in\mathbb{N}^{d+1}\setminus\\{{\bf
0}\\}$ satisfy
$\sum_{\ell=1}^{i}{\bm{\alpha}}^{i,j}_{\ell,{\bm{x}}}={\bm{\alpha}}_{\bm{x}}$
and
$j+\sum_{\ell=1}^{i}{\bm{\alpha}}^{i,j}_{\ell,\varrho}={\bm{\alpha}}_{\varrho}$.
If $i=0$ then we have from the mean value theorem that for any $0\leq j\leq k$

$\big{\lVert}({\partial}_{2}^{j}\varphi)\circ
F\big{\rVert}_{L^{2}(\Omega)}=\big{\lVert}({\partial}_{2}^{j}\varphi)\circ
F-({\partial}_{2}^{j}\varphi)\circ
0\big{\rVert}_{L^{2}(\Omega)}\leq\big{\lvert}{\partial}_{1}{\partial}_{2}^{j}\varphi\big{\rvert}_{L^{\infty}(\mathbb{R}\times(\rho_{0},\rho_{1}))}\big{\lVert}F\big{\rVert}_{L^{2}(\Omega)}.$

The case $i=1$ is straightforward, and we now focus on the case $i\geq 2$. We
assume without loss of generality that
$|{\bm{\alpha}}^{i,j}_{1,\varrho}|\geq|{\bm{\alpha}}^{i,j}_{2,\varrho}|\geq\cdots\geq|{\bm{\alpha}}^{i,j}_{i,\varrho}|$
and remark that for $\ell\neq 1$, $|{\bm{\alpha}}^{i,j}_{\ell,\varrho}|\leq
k-1$ (otherwise
$|{\bm{\alpha}}^{i,j}_{1,\varrho}|+|{\bm{\alpha}}^{i,j}_{\ell,\varrho}|=2k>k\geq|{\bm{\alpha}}_{\varrho}|$)
and $|{\bm{\alpha}}^{i,j}_{\ell}|\leq s-|{\bm{\alpha}}^{1,j}_{1}|\leq s-1$.
Hence we have

$\displaystyle\textstyle\big{\lVert}\prod_{\ell=1}^{i}(\partial^{{\bm{\alpha}}^{i,j}_{\ell}}F)\big{\rVert}_{L^{2}(\Omega)}$
$\displaystyle\textstyle\lesssim\big{\lvert}\big{\lvert}\partial^{{\bm{\alpha}}^{i,j}_{1}}F\big{\rvert}_{H^{s-|{\bm{\alpha}}^{i,j}_{1}|}_{\bm{x}}}\big{(}\prod_{\ell=2}^{i}\big{\lvert}\partial^{{\bm{\alpha}}^{i,j}_{\ell}}F\big{\rvert}_{H^{s-|{\bm{\alpha}}^{i,j}_{2}|-\frac{1}{2}}_{\bm{x}}}\big{)}\big{\rvert}_{L^{2}_{\varrho}}$
$\displaystyle\textstyle\lesssim\big{\lVert}F\big{\rVert}_{H^{s,{\bm{\alpha}}^{i,j}_{1,\varrho}}}\big{(}\prod_{\ell=2}^{i}\big{\lVert}F\big{\rVert}_{H^{s,{\bm{\alpha}}^{i,j}_{\ell,\varrho}+1}}\big{)}\leq\big{\lVert}F\big{\rVert}_{H^{s,k}}^{i}$

where we used Lemma A.2(3) and $(i-1)(s-\frac{1}{2})\geq(i-1)s_{0}$ and Lemma
A.1. The first claim follows.

Now we assume additionally that $k\geq 2$ and $s\geq s_{0}+\frac{3}{2}$. The
cases $i\in\\{0,1\\}$ can be treated exactly as above and we deal only with
the case $i\geq 2$, ordering
$|{\bm{\alpha}}^{i,j}_{1,\varrho}|\geq|{\bm{\alpha}}^{i,j}_{2,\varrho}|\geq\cdots\geq|{\bm{\alpha}}^{i,j}_{i,\varrho}|$
as above. Assume first that $|{\bm{\alpha}}^{i,j}_{1,\varrho}|=k\geq 2$. Then
for all $\ell\neq 1$, $|{\bm{\alpha}}^{i,j}_{\ell,\varrho}|=0$ and
$|{\bm{\alpha}}^{i,j}_{\ell}|\leq s-2$, and we conclude as before with

$\textstyle\big{\lVert}\prod_{\ell=1}^{i}(\partial^{{\bm{\alpha}}^{i,j}_{\ell}}F)\big{\rVert}_{L^{2}(\Omega)}\lesssim\big{\lvert}\big{\lvert}\partial^{{\bm{\alpha}}^{i,j}_{1}}F\big{\rvert}_{H^{s-|{\bm{\alpha}}^{i,j}_{1}|}_{\bm{x}}}\big{(}\prod_{\ell=1}^{i}\big{\lvert}\partial^{{\bm{\alpha}}^{i,j}_{\ell}}F\big{\rvert}_{H^{s-|{\bm{\alpha}}^{i,j}_{\ell}|-\frac{3}{2}}_{\bm{x}}}\big{)}\big{\rvert}_{L^{2}_{\varrho}}\lesssim\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}F\big{\rVert}_{H^{s-1,1}}^{i-1}.$

Otherwise we have
$|{\bm{\alpha}}^{i,j}_{2,\varrho}|\leq|{\bm{\alpha}}^{i,j}_{1,\varrho}|\leq
k-1$ and $|{\bm{\alpha}}^{i,j}_{2}|\leq s-|{\bm{\alpha}}^{i,j}_{1}|\leq s-1$
and notice that for $\ell\geq 3$, $|{\bm{\alpha}}^{i,j}_{\ell,\varrho}|\leq
k-2$ (since otherwise we have the contradiction
$|{\bm{\alpha}}^{i,j}_{1,\varrho}|+|{\bm{\alpha}}^{i,j}_{2,\varrho}|+|{\bm{\alpha}}^{i,j}_{3,\varrho}|\geq
3(k-1)\geq k+1\geq|{\bm{\alpha}}_{\varrho}|+1$) and
$|{\bm{\alpha}}^{i,j}_{\ell}|\leq
s-|{\bm{\alpha}}^{i,j}_{1}|-|{\bm{\alpha}}^{i,j}_{2}|\leq s-2$. Hence

$\displaystyle\textstyle\big{\lVert}\prod_{\ell=1}^{i}(\partial^{{\bm{\alpha}}^{i,j}_{\ell}}F)\big{\rVert}_{L^{2}(\Omega)}$
$\displaystyle\textstyle\lesssim\big{\lvert}\big{\lvert}\partial^{{\bm{\alpha}}^{i,j}_{1}}F\big{\rvert}_{H^{s-|{\bm{\alpha}}^{i,j}_{1}|-\frac{1}{2}}_{\bm{x}}}\big{\lvert}\partial^{{\bm{\alpha}}^{i,j}_{2}}F\big{\rvert}_{H^{s-|{\bm{\alpha}}^{i,j}_{2}|-1}_{\bm{x}}}\big{(}\prod_{\ell=3}^{i}\big{\lvert}\partial^{{\bm{\alpha}}^{i,j}_{\ell}}F\big{\rvert}_{H^{s-|{\bm{\alpha}}^{i,j}_{\ell}|-\frac{3}{2}}_{\bm{x}}}\big{)}\big{\rvert}_{L^{2}_{\varrho}}$
$\displaystyle\textstyle\lesssim\big{\lVert}F\big{\rVert}_{H^{s,{\bm{\alpha}}^{i,j}_{1,\varrho}+1}}\big{\lVert}F\big{\rVert}_{H^{s-1,{\bm{\alpha}}^{i,j}_{2,\varrho}}}\big{(}\prod_{\ell=3}^{i}\big{\lVert}F\big{\rVert}_{H^{s-1,{\bm{\alpha}}^{i,j}_{\ell,\varrho}+1}}\big{)}\lesssim\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}F\big{\rVert}_{H^{s-1,k-1}}^{i-1}.$

This concludes the proof. ∎

We shall apply the above to estimate quantities such as (but not restricted
to)

$\Phi:({\bm{x}},\varrho)\in\Omega\mapsto\frac{h({\bm{x}},\varrho)}{\underline{h}(\varrho)+h({\bm{x}},\varrho)},$

with $\underline{h}\in W^{k,\infty}((\rho_{0},\rho_{1}))$ and $h\in
H^{s,k}(\mathbb{R}^{d})$ satisfying the condition
$\inf_{({\bm{x}},\varrho)\in\Omega}\underline{h}(\varrho)+h({\bm{x}},\varrho)\geq
h_{\star}>0$. Let us detail the result and its proof for this specific
example.

######  Lemma A.6.

Let $d\in\mathbb{N}^{\star}$, $s_{0}>d/2$. Let $s,k\in\mathbb{N}$ with $s\geq
s_{0}+\frac{1}{2}$ and $1\leq k\leq s$, and $M,\underline{M},h_{\star}>0$.
There exists $C>0$ such that for any $\underline{h}\in
W^{k,\infty}((\rho_{0},\rho_{1}))$ with
$\big{\lvert}\underline{h}\big{\rvert}_{W^{k,\infty}_{\varrho}}\leq\underline{M}$
and any $h\in H^{s,k}(\Omega)$ with $\big{\lVert}h\big{\rVert}_{H^{s,k}}\leq
M$ and satisfying the condition
$\inf_{({\bm{x}},\varrho)\in\Omega}\underline{h}(\varrho)+h({\bm{x}},\varrho)\geq
h_{\star}$, then

$\Phi:({\bm{x}},\varrho)\mapsto\frac{h({\bm{x}},\varrho)}{\underline{h}(\varrho)+h({\bm{x}},\varrho)}\in
H^{s,k}(\Omega),$

and

$\big{\lVert}\Phi\big{\rVert}_{H^{s,k}}\leq
C\big{\lVert}h\big{\rVert}_{H^{s,k}}.$

If moreover $s>\frac{d}{2}+\frac{3}{2}$ and $2\leq k\leq s$, then the above
holds for any $h\in H^{s,k}(\Omega)$ with
$\big{\lVert}h\big{\rVert}_{H^{s-1,k-1}}\leq M$.

###### Proof.

We can write $\Phi=\varphi\circ h$ with
$\varphi(\cdot,\varrho)=f(\cdot,\underline{h}(\varrho))$ where
$f\in\mathcal{C}^{\infty}(\mathbb{R}^{2})$ is set such that
$f(y,z)=\frac{y}{y+z}$ on $\omega:=\\{(y,z)\ :\ \
|y|\leq\big{\lVert}h\big{\rVert}_{L^{\infty}(\Omega)},\
|z|\leq\big{\lvert}\underline{h}\big{\rvert}_{L^{\infty}((\rho_{0},\rho_{1}))},\
y+z\geq h_{\star}\\}$. We can construct $f$ as above such that the control of
$\big{\lvert}\varphi\big{\rvert}_{W^{s,\infty}(\mathbb{R};W^{k,\infty}((\rho_{0},\rho_{1})))}$
depends only on $\big{\lVert}h\big{\rVert}_{L^{\infty}(\Omega)}$ (which is
bounded appealing to Lemma A.1, if $h\in H^{s,k}$ with
$s>\frac{d}{2}+\frac{1}{2},\,1\leq k\leq s$),
$\big{\lvert}\underline{h}\big{\rvert}_{W^{k,\infty}((\rho_{0},\rho_{1}))}$
and $h_{\star}>0$. The result is now a direct application of Lemma A.5.∎

### Commutator estimates

We now recall standard commutator estimates in $H^{s}(\mathbb{R}^{d})$.

######  Lemma A.7.

Let $d\in\mathbb{N}^{\star}$, $s_{0}>d/2$ and $s\geq 0$.

  1. (1)

For any $s_{1},s_{2}\in\mathbb{R}$ such that $s_{1}\geq s$, $s_{2}\geq s-1$
and $s_{1}+s_{2}\geq s+s_{0}$, there exists $C>0$ such that for any $f\in
H^{s_{1}}(\mathbb{R}^{d})$ and $g\in H^{s_{2}}(\mathbb{R}^{d})$,
$[\Lambda^{s},f]g:=\Lambda^{s}(fg)-f\Lambda^{s}g\in L^{2}(\mathbb{R}^{d})$ and

$\big{\lvert}[\Lambda^{s},f]g\big{\rvert}_{L^{2}}\leq
C\big{\lvert}f\big{\rvert}_{H^{s_{1}}}\big{\lvert}g\big{\rvert}_{H^{s_{2}}}.$

  2. (2)

There exists $C>0$ such that for any $f\in L^{\infty}(\mathbb{R}^{d})$ such
that $\nabla f\in H^{s-1}(\mathbb{R}^{d})\cap H^{s_{0}}(\mathbb{R}^{d})$ and
for any $g\in H^{s-1}(\mathbb{R}^{d})$, one has $[\Lambda^{s},f]g\in
L^{2}(\mathbb{R}^{d})$ and

$\big{\lvert}[\Lambda^{s},f]g\big{\rvert}_{L^{2}}\leq C\big{\lvert}\nabla
f\big{\rvert}_{H^{s_{0}}}\big{\lvert}g\big{\rvert}_{H^{s-1}}+C\left\langle\big{\lvert}\nabla
f\big{\rvert}_{H^{s-1}}\big{\lvert}g\big{\rvert}_{H^{s_{0}}}\right\rangle_{s>s_{0}+1}.$

  3. (3)

There exists $C>0$ such that for any $f,g\in H^{s}(\mathbb{R}^{d})\cap
H^{s_{0}+1}(\mathbb{R}^{d})$, the symmetric commutator
$[\Lambda^{s};f,g]:=\Lambda^{s}(fg)-f\Lambda^{s}g-g\Lambda^{s}f\in
L^{2}(\mathbb{R}^{d})$ and

$\big{\lvert}[\Lambda^{s};f,g]\big{\rvert}_{L^{2}}\leq
C\big{\lvert}f\big{\rvert}_{H^{s_{0}+1}}\big{\lvert}g\big{\rvert}_{H^{s-1}}+C\big{\lvert}f\big{\rvert}_{H^{s-1}}\big{\lvert}g\big{\rvert}_{H^{s_{0}+1}}.$

The validity of the above estimates persist when replacing the operator
$\Lambda^{s}$ with the operator $\partial^{\bm{\alpha}}$ with
${\bm{\alpha}}\in\mathbb{N}^{d}$ a multi-index such that $|{\bm{\alpha}}|\leq
s$.

We conclude with commutator estimates in the spaces $H^{s,k}(\Omega)$.

######  Lemma A.8.

Let $d\in\mathbb{N}^{\star}$, $s_{0}>d/2$. Let $s\geq s_{0}+\frac{3}{2}$ and
$k\in\mathbb{N}$ such that $2\leq k\leq s$. Then there exists $C>0$ such that
for any
${\bm{\alpha}}=({\bm{\alpha}}_{\bm{x}},{\bm{\alpha}}_{\varrho})\in\mathbb{N}^{d+1}$
with $|{\bm{\alpha}}|\leq s$ and ${\bm{\alpha}}_{\varrho}\leq k$, one has

$\big{\lVert}[\partial^{\bm{\alpha}},F]G\big{\rVert}_{L^{2}(\Omega)}\leq
C\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}G\big{\rVert}_{H^{s-1,\min(\\{k,s-1\\})}}.$

###### Proof.

We set two multi-indices
${\bm{\beta}}=({\bm{\beta}}_{\bm{x}},{\bm{\beta}}_{\varrho})\in\mathbb{N}^{d+1}$
and
${\bm{\gamma}}=({\bm{\gamma}}_{\bm{x}},{\bm{\gamma}}_{\varrho})\in\mathbb{N}^{d+1}$
with ${\bm{\beta}}+{\bm{\gamma}}={\bm{\alpha}}$, and $|{\bm{\gamma}}|\leq
s-1$. Assume first that ${\bm{\beta}}_{\varrho}\leq k-1$ and
$|{\bm{\beta}}|\leq s-1$. Then

$\displaystyle\big{\lVert}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)\big{\rVert}_{L^{2}(\Omega)}$
$\displaystyle\lesssim\big{\lvert}\big{\lvert}\partial^{\bm{\beta}}F(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\beta}}|-\frac{1}{2}}_{\bm{x}}}\big{\lvert}\partial^{\bm{\gamma}}G(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\gamma}}|-1}_{\bm{x}}}\big{\rvert}_{L^{2}_{\varrho}}$
$\displaystyle\lesssim\big{\lVert}F\big{\rVert}_{H^{s,{\bm{\beta}}_{\varrho}+1}}\big{\lVert}G\big{\rVert}_{H^{s-1,{\bm{\gamma}}_{\varrho}}}\leq\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}G\big{\rVert}_{H^{s-1,\min(\\{k,s-1\\})}},$

where we used Lemma A.2(1) with
$(s,s_{1},s_{2})=(0,s-|{\bm{\beta}}|-\frac{1}{2},s-|{\bm{\gamma}}|-1)$, and
Lemma A.1. Otherwise ${\bm{\gamma}}_{\varrho}=0$ and $|{\bm{\gamma}}|\leq
s-|{\bm{\beta}}|\leq s-2$, and we have

$\displaystyle\big{\lVert}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)\big{\rVert}_{L^{2}(\Omega)}$
$\displaystyle\lesssim\big{\lvert}\big{\lvert}\partial^{\bm{\beta}}F(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\beta}}|}_{\bm{x}}}\big{\lvert}\partial^{\bm{\gamma}}G(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\gamma}}|-\frac{3}{2}}_{\bm{x}}}\big{\rvert}_{L^{2}_{\varrho}}$
$\displaystyle\lesssim\big{\lVert}F\big{\rVert}_{H^{s,{\bm{\beta}}_{\varrho}}}\big{\lVert}G\big{\rVert}_{H^{s-1,1}}\leq\big{\lVert}F\big{\rVert}_{H^{s,k}}\big{\lVert}G\big{\rVert}_{H^{s-1,\min(\\{k,s-1\\})}}.$

The claim follows from decomposing $[\partial^{\bm{\alpha}},F]G$ as a sum of
products as above. ∎

######  Lemma A.9.

Let $d\in\mathbb{N}^{\star}$, $s_{0}>d/2$. Let $s\geq s_{0}+\frac{5}{2}$ and
$k\in\mathbb{N}$ such that $2\leq k\leq s$. Then there exists $C>0$ such that
for any
${\bm{\alpha}}=({\bm{\alpha}}_{\bm{x}},{\bm{\alpha}}_{\varrho})\in\mathbb{N}^{d+1}$
with $|{\bm{\alpha}}|\leq s$ and ${\bm{\alpha}}_{\varrho}\leq k$, one has

$\big{\lVert}[\partial^{\bm{\alpha}};F,G]\big{\rVert}_{L^{2}(\Omega)}\leq
C\big{\lVert}F\big{\rVert}_{H^{s-1,\min(\\{k,s-1\\})}}\big{\lVert}G\big{\rVert}_{H^{s-1,\min(\\{k,s-1\\})}}.$

###### Proof.

We can decompose

$[\partial^{\bm{\alpha}};F,G]=\sum_{{\bm{\beta}}+{\bm{\gamma}}={\bm{\alpha}}}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)$

with multi-indices
${\bm{\beta}}=({\bm{\beta}}_{\bm{x}},{\bm{\beta}}_{\varrho})\in\mathbb{N}^{d+1}$
and
${\bm{\gamma}}=({\bm{\gamma}}_{\bm{x}},{\bm{\gamma}}_{\varrho})\in\mathbb{N}^{d+1}$
such that $|{\bm{\beta}}|+|{\bm{\gamma}}|\leq s$ and
${\bm{\beta}}_{\varrho}+{\bm{\gamma}}_{\varrho}\leq k$, and
$1\leq|{\bm{\beta}}|,|{\bm{\gamma}}|\leq s-1$. Assume furthermore that
${\bm{\beta}}_{\varrho}\leq k-1$ and $|{\bm{\beta}}|\leq s-2$. Then

$\displaystyle\big{\lVert}(\partial^{\bm{\beta}}F)(\partial^{\bm{\gamma}}G)\big{\rVert}_{L^{2}(\Omega)}$
$\displaystyle\lesssim\big{\lvert}\big{\lvert}\partial^{\bm{\beta}}F(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\beta}}|-\frac{3}{2}}_{\bm{x}}}\big{\lvert}\partial^{\bm{\gamma}}G(\cdot,\varrho)\big{\rvert}_{H^{s-|{\bm{\gamma}}|-1}_{\bm{x}}}\big{\rvert}_{L^{2}_{\varrho}}$
$\displaystyle\lesssim\big{\lVert}F\big{\rVert}_{H^{s-1,{\bm{\beta}}_{\varrho}+1}}\big{\lVert}G\big{\rVert}_{H^{s-1,{\bm{\gamma}}_{\varrho}}}\leq\big{\lVert}F\big{\rVert}_{H^{s-1,\min(\\{k,s-1\\})}}\big{\lVert}G\big{\rVert}_{H^{s-1,\min(\\{k,s-1\\})}},$

where we used Lemma A.2(1) with
$(s,s_{1},s_{2})=(0,s-|{\bm{\beta}}|-\frac{3}{2},s-|{\bm{\gamma}}|-1)$, and
Lemma A.1. By symmetry, the result holds if ${\bm{\gamma}}_{\varrho}\leq k-1$
and $|{\bm{\gamma}}|\leq s-2$. Hence there remains to consider the situation
where (${\bm{\beta}}_{\varrho}=k$ or $|{\bm{\beta}}|=s-1$) and
(${\bm{\gamma}}_{\varrho}=k$ or $|{\bm{\gamma}}|=s-1$). Since $s>2$ and
$|{\bm{\beta}}|+|{\bm{\gamma}}|\leq s$, we cannot have
$|{\bm{\beta}}|=|{\bm{\gamma}}|=s-1$. In the same way, we cannot have
${\bm{\beta}}_{\varrho}={\bm{\gamma}}_{\varrho}=k$ since $k>0$. Furthermore ,
we cannot have ${\bm{\beta}}_{\varrho}=k$ and $|{\bm{\gamma}}|=s-1$, since the
former implies $|{\bm{\beta}}|\geq{\bm{\beta}}_{\varrho}=k\geq 2$ and the
latter implies $|{\bm{\beta}}|\leq 1$. Symmetrically, we cannot have
${\bm{\gamma}}_{\varrho}=k$ and $|{\bm{\beta}}|=s-1$. This concludes the
proof. ∎

Acknowledgments VD thanks Eric Blayo for revealing to him the work of Gent and
McWilliams, Mahieddine Adim for his careful proofreading, as well as the
Centre Henri Lebesgue, program ANR-11-LABX-0020-0. RB is partially supported
by the GNAMPA group of INdAM. VD and RB thank Charlotte Perrin for identifying
the relationship between Gent and McWilliams eddy-diffusivity contributions
and the BD entropy, and the anonymous referee for pointing out many additional
relevant references.

## References