# Relaxing the sharp density stratification and columnar motion assumptions in
layered shallow water systems

Mahieddine Adim  IRMAR, Univ. Rennes , F-35000 Rennes, France.  Roberta
Bianchini  IAC, Consiglio Nazionale delle Ricerche, 00185 Rome, Italy.
Vincent Duchêne  IRMAR, Univ. Rennes , F-35000 Rennes, France.  CNRS.
Corresponding author<EMAIL_ADDRESS>
###### Abstract

We rigorously justify the bilayer shallow-water system as an approximation to
the hydrostatic Euler equations in situations where the flow is density-
stratified with close-to-piecewise constant density profiles, and close-to-
columnar velocity profiles. Our theory accommodates with continuous
stratification, so that admissible deviations from bilayer profiles are not
pointwise small. This leads us to define refined approximate solutions that
are able to describe at first order the flow in the pycnocline. Because the
hydrostatic Euler equations are not known to enjoy suitable stability
estimates, we rely on thickness-diffusivity contributions proposed by Gent and
McWilliams. Our strategy also applies to one-layer and multilayer frameworks.

##  1 Introduction

#### Motivation

The bilayer shallow water system is a standard model for the description of
internal waves in density-stratified flows in situations where the density
distribution is such that the fluid can be approximately described as two
layers with almost-constant densities separated by a thin a pycnocline; see
e.g. [27, Chap. 6]. In addition to this sharp stratification assumption, the
formal derivation of the bilayer shallow water system relies on two additional
ingredients. Firstly, the internal pressure is assumed to be hydrostatic, that
is the pressure-gradient force balances the external force due to gravity.
Secondly, the flow velocity is assumed to be columnar, that is the horizontal
velocity of fluid particles is constant with respect to the vertical variable
within each layer. Of course the validity of bilayer shallow water system
relies on the expectation that, if originally approximately satisfied, these
three assumptions remain accurate on a relevant timescale as the flow evolves.

The rigorous justification of the hydrostatic assumption in the shallow water
regime —that is when the typical horizontal wavelength of the flow is large
with respect to the vertical depth of the layer— has been rigorously analyzed
either in situations of homogeneous density [5, 36, 23, 37], with smooth
density distributions [42, 41], or in the bilayer framework [11, 21, 20]. In
the bilayer framework and assuming that the pressure is hydrostatic, the
columnar assumption is propagated exactly by the flow. By this we mean that
the bilayer shallow water system produces exact solutions to the hydrostatic
(incompressible) Euler equations with a density and horizontal velocity
distributions which are piecewise constant with respect to the vertical
variable. This statement is made explicit below.

In this work we investigate solutions to the hydrostatic Euler equations in
the vicinity of such solutions, that is relaxing the sharp stratification as
well as columnar motion assumptions. We prove that for initial data suitably
close to the bilayer framework, the emerging solutions to the hydrostatic
Euler equations remain close to the solution predicted by the bilayer shallow
water system on a relevant timescale.

This task is made difficult in part because we lack good stability estimates
for the hydrostatic Euler equations in the presence of density stratification.
For that matter, as in our previous works [9, 1], we rely on the regularizing
properties of thickness-diffusivity terms proposed by Gent and McWilliams [25]
so as to model the effective contributions of geostrophic eddy correlations in
non-eddy-resolving systems.

#### Description of our results

Specifically, the hydrostatic Euler equations we consider take the form

$\displaystyle\partial_{t}h+\partial_{x}((1+h)(\underline{u}+u))$
$\displaystyle=\kappa\partial_{x}^{2}h,$ (1.1)
$\displaystyle\partial_{t}u+\left(\underline{u}+u-\kappa\frac{\partial_{x}h}{1+h}\right)\partial_{x}u+\frac{1}{\underline{\rho}}\partial_{x}\Psi$
$\displaystyle=0,$

where the Montgomery potential $\Psi$ is given by

$\Psi(t,x,r)=\underline{\rho}(r)\int_{-1}^{r}h(t,x,r^{\prime})\operatorname{{\rm
d}\\!}r^{\prime}+\int_{r}^{0}\underline{\rho}(r^{\prime})h(t,x,r^{\prime})\operatorname{{\rm
d}\\!}r^{\prime}.$ (1.2)

Here, the equations are formulated using isopycnal coordinates (in particular
we assume that the fluid is stratified in the sense that the two-dimensional
fluid domain is foliated through lines of equal density, namely isopycnals).
The variable $h$ represents the deviation of the infinitesimal depth of
isopycnals from the reference value $1$, and $u$ is the deviation of the
horizontal velocity of the fluid particles from the reference value
$\underline{u}$, and $\underline{\rho}$ their density. The unknowns $h,u$
depend on the time $t$, the horizontal space $x$, and the variable
$r\in(-1,0)$ referring to the isopycnal line at stake, while $\underline{u}$
and $\underline{\rho}$ are given and depend only on $r$. The derivation of
these equations from the more standard formulation in Eulerian coordinates is
described for instance in [9].111Let us point out that in [9] we choose to
label isopycnal lines using the value of the density of fluid particles:
$\varrho=\underline{\rho}(r)$. Here we use a different convention, so as to
set the reference infinitesimal depth of isopycnals at value
$\underline{h}(r)=1$. Notice that the change of variable
$\varrho=\underline{\rho}(r)$ is bijective in the stably stratified situation,
i.e. when $r\mapsto\underline{\rho}(r)$ is strictly decreasing, but that our
choice in this work allows to consider stratifications that are not strictly
monotonic such as homogeneous and layered configurations. Incidentally, let us
mention that in (1.1) we have set the gravity acceleration to $g=1$ and the
total depth of the fluid domain at rest to
$\int_{-1}^{0}\underline{h}(r)\operatorname{{\rm d}\\!}r=1$ through suitable
rescaling.  Finally $\kappa>0$ is the thickness diffusivity coefficient and
$u_{\star}=-\kappa\frac{\partial_{x}h}{1+h}$ is often referred to as the
“bolus velocity”.

The corresponding bilayer shallow water system reads

$\begin{cases}\partial_{t}H_{s}+\partial_{x}\big{(}(\underline{H}_{s}+H_{s})(\underline{U}_{s}+U_{s})\big{)}=\kappa\partial_{x}^{2}H_{s},\\\
\partial_{t}H_{b}+\partial_{x}\big{(}(\underline{H}_{b}+H_{b})(\underline{U}_{b}+U_{b})\big{)}=\kappa\partial_{x}^{2}H_{s},\\\
\partial_{t}U_{s}+(\underline{U}_{s}+U_{s}-\kappa\frac{\partial_{x}H_{s}}{\underline{H}_{s}+H_{s}})\partial_{x}U_{s}+\partial_{x}H_{s}+\partial_{x}H_{b}=0,\\\
\partial_{t}U_{b}+(\underline{U}_{b}+U_{b}-\kappa\frac{\partial_{x}H_{b}}{\underline{H}_{b}+H_{b}})\partial_{x}U_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}H_{s}+\partial_{x}H_{b}=0.\end{cases}$
(1.3)

Here, $H_{s}$ (resp. $H_{b}$) represents the deviation of the depth of the
upper (resp. lower) layer from the reference constant value
$\underline{H}_{s}$ (resp. $\underline{H}_{b}$), and $U_{s}$ (resp. $U_{b}$)
is the deviation of the horizontal velocity within the upper (resp. lower)
layer from the reference constant value $\underline{U}_{s}$ (resp.
$\underline{U}_{b}$). Notice $H_{s},H_{b},U_{s},U_{b}$ depend only on the time
and horizontal space variables, bringing about a relative ease of use of the
bilayer model. We denote $\rho_{s}$ (resp. $\rho_{b}$) the constant density of
fluid particles in the upper (resp. lower) layers. Finally, $\kappa>0$ is
again the thickness diffusivity coefficient.

As mentioned above, solutions to (1.3) provide exact solutions to (1.1).
Specifically, if we denote

$\displaystyle\underline{\rho}_{\rm bl}(r)$ $\displaystyle=\rho_{s}{\bf
1}_{(-\underline{H}_{s},0)}(r)+\rho_{b}{\bf 1}_{(-1,-\underline{H}_{s})}(r),$
(1.4) $\displaystyle\underline{u}_{\rm bl}(r)$
$\displaystyle=\underline{U}_{s}{\bf
1}_{(-\underline{H}_{s},0)}(r)+\underline{U}_{b}{\bf
1}_{(-1,-\underline{H}_{s})}(r),$ $\displaystyle u_{\rm bl}(\cdot,r)$
$\displaystyle=U_{s}{\bf 1}_{(-\underline{H}_{s},0)}(r)+U_{b}{\bf
1}_{(-1,-\underline{H}_{s})}(r),$ $\displaystyle h_{\rm bl}(\cdot,r)$
$\displaystyle=\frac{H_{s}}{\underline{H}_{s}}{\bf
1}_{(-\underline{H}_{s},0)}(r)+\frac{H_{b}}{\underline{H}_{b}}{\bf
1}_{(-1,\underline{H}_{s})}(r),$

where
$(\rho_{s},\rho_{b},\underline{H}_{s},\underline{H}_{b},\underline{U}_{s},\underline{U}_{b},H_{s},H_{b},U_{s},U_{b})$
is a solution to (1.3), then $(\underline{\rho}_{\rm bl},\underline{u}_{\rm
bl},h_{\rm bl},u_{\rm bl})$ is a solution to (1.1)-(1.2). In this work we
shall compare these solutions with the ones emerging from profiles satisfying
(1.4) only approximately. Specifically, our results are twofold.

  1. i.

We prove that strong solutions to the bilayer shallow water system (1.3)
emerge from sufficiently regular initial data satisfying some hyperbolicity
conditions.

  2. ii.

We prove that strong solutions to the hydrostatic Euler equations (1.1) emerge
from profiles close to the piecewise constant profiles given by (1.4), and
that these solutions remain close to the bilayer solutions.

Notice that to accommodate our aim of comparing solutions with a piecewise
constant density distribution with a solution with a continuous density
distribution, we need to consider deviations that can be pointwise large in
small regions, that is the pycnocline. This demands to weaken the topology
measuring the size of deviations in (ii). Yet a control in the strong
$L^{\infty}$ topology associated with a Banach algebra turns out to be
necessary to secure suitable convergence estimates. Our strategy then relies
on the construction of a refined approximate solution, which is proved to be
close to the corresponding solution to the hydrostatic Euler equation in a
strong topology, and close to the bilayer solution in a weaker topology. Hence
this refined approximate solution improves the description of the exact
solution within the pycnocline.

A second important remark is that while the contribution of thickness
diffusivity is essential to our stability estimates, we wish to control and
compare solutions on a time interval which is uniform with respect to the
thickness diffusivity parameter, $0<\kappa\leq 1$. The dependency on the
thickness parameter will appear only as a restriction on the size of
admissible deviations. Concerning (i) this is made possible by the well-known
fact that the bilayer shallow-water system is well-posed (under some
hyperbolicity conditions) when $\kappa=0$ (see [40]). Yet obtaining the
corresponding result for $\kappa>0$ is not straightforward and demands to use
finely the structure of the thickness diffusivity parameters, following the
“two-velocity” strategy developed in the context of the BD-entropy (see e.g.
[16]). Concerning (ii) we use the existence of the bilayer solution and
consequently the existence of the refined approximate solution to bootstrap
the control of sufficiently close solutions to the hydrostatic Euler equations
on the relevant timescale. For that purpose we strongly use the regularizing
effect of thickness diffusivity contributions, but any non-uniformity with
respect to the diffusivity parameter, $\kappa$, can be balanced through the
smallness of the deviations.

Let us point out that the result (ii) applies to any given (sufficiently
regular with respect to the time and horizontal space variables) solution to
the hydrostatic Euler equations. Hence our work provides the same stability
estimates around other solutions, constructed for instance in the framework of
multiple layers and/or simple waves.

#### Related literature

Several existing works discuss the matter of modeling thin pycnoclines through
the bilayer framework. Let us first recall that in the bilayer framework, non-
hydrostatic pressure contributions trigger strong Kelvin–Helmholtz
instabilities that in particular prevent the well-posedness of the initial-
value problem in the absence of any additional regularizing ingredients; see
[32]. Such regularizing ingredients include interfacial tension as proved in
[34] but this is not expected to be te physically relevant mechanism. In [10],
Bogucki and Garrett describe and model a scenario of interface-thickening due
to mixing triggered by shear instabilities, up to a situation where the
Richardson number in the interface becomes compatible with the celebrated
Miles [39] and Howard [30] stability ion. Recall however that the bilayer
shallow water system does not suffer from shear-induced instabilities when
shear velocities are sufficiently small. Consistently, the authors in [19]
discuss the simultaneous limits of sharp stratification together with shallow
water (which can be considered as a low-frequency or hydrostatic pressure
limit). In our work we impose the hydrostatic pressure assumption thus taming
shear instabilities, although as pointed out previously the stability of
continuously stratified hydrostatic Euler equations is poorly understood in
the presence of density variations (see [43, 12, 28, 38] in the homogeneous
framework). Notice also that shear-induced instabilities disappear when
restricting the framework to purely traveling waves. In this framework, James
[31] (improving upon [44, 4]) was able to rigorously justify the sharp
stratification limit: writing the bilayer and continuously stratified problems
in a unified formulation, James proved the existence of internal traveling
waves associated with density stratifications in a small neighborhood
(according to the $L^{2}$ topology) of the bilayer framework which converge
towards the bilayer solution in the limit of sharp stratification. Our results
are in the same spirit: system (1.1) is our unified formulation, and the
topology controlling the limit of sharp stratification in our work is the
$L^{1}_{r}$ topology. However, our results admit non-trivial dynamics thanks
to the hydrostatic assumption and the presence of thickness diffusivity.

The propagation of internal waves with thin pycnoclines in relation with
bilayer models was also investigated through experiments. In particular Grue
et al. [29] set up precise experiments generating large-amplitude solitary
waves and reported the dynamical development of rolls on the trailing side of
the largest considered waves in accordance with the mechanism promoted by
Bogucki and Garrett, while bilayer models provide very accurate predictions
otherwise. Almgren, Camassa and Tiron [3] investigate thoroughly this matter
through careful numerical simulations and analytical results of asymptotic
bilayer models, analyzing the triggering of shear-induced instabilities in the
region of maximal displacement as well as their advection into stable regions
of the flow. White and Helfrich [46] consider internal bores generated by a
dam-break, and compare continuously stratified and bilayer models with
numerical experiments. From their findings they suggest an improvement on
existing bilayer theories. In [18], Camassa and Tiron optimize bilayer models
(specifically calibrating the top and bottom densities and the position of the
sharp interface) and compare the analytical predictions of the optimized
bilayer models with respect to the numerically computed continuously
stratified solutions —considering infinitesimally small waves, internal bores
and solitary waves— showing excellent agreement even in situations of
relatively thick pycnoclines. Furthermore they propose a new asymptotic model
taking into account thin pycnoclines in view of reconstructing analytically
local properties of traveling waves within the pycnocline, which is similar in
spirit with the “refined approximate solution” that we introduce in this work.
Notice the authors consider that “fully time-dependent models governing the
evolution of the pycnocline thickness probably constitute one of the most
relevant extensions of the model [they] have introduced”, and we believe that
our work provides a partial answer in that respect.

#### Outline

This manuscript is structured as follows. In Section 2 we study the bilayer
shallow water systems. We first recall some known results without thickness
diffusivity, and then consider the system with thickness diffusivity
contributions. The main result is Proposition 2.7 which provides for any
sufficiently regular initial data satisfying some hyperbolicity criterion the
existence and control of strong solutions to the bilayer system on a time
interval which is uniform with respect to $0<\kappa\leq 1$, while Proposition
2.8 states the strong convergence as $\kappa\searrow 0$. In Section 3 we study
the hydrostatic Euler equations. We provide first some stability estimates
with respect to perturbations of the equations and of the data using suitable
distances. As a second step we introduce refined approximate solutions
associated with some given reference exact solution and close-by profiles.
Building upon these approximate solutions, we prove Proposition 3.8 which
controls the difference between the reference solution and exact solutions to
the hydrostatic Euler equations emerging from close-by profiles. Together,
Proposition 2.7 and Proposition 3.8 provide the announced result that for
profiles suitably close to the bilayer framework and satisfying some
hyperbolicity criterion the emerging solutions to the hydrostatic Euler
equations remain close to the solution predicted by the bilayer shallow water
system on a relevant timescale. The rigorous statement is displayed in Section
4, completed with a discussion on analogous statements in the one-layer and
multilayer frameworks.

#### Notations

Let us introduce some notations for functional spaces used in this work.

  * •

The spaces $L^{p}(\mathbb{R})$ are the standard Lebesgue spaces endowed with
the usual norms denoted $\|\cdot\|_{L^{p}}$.

  * •

The spaces $W^{k,p}(\mathbb{R})$ for $k\in\mathbb{N}$ are the $L^{p}$-based
Sobolev spaces endowed with the usual norms denoted $\|\cdot\|_{W^{k,p}}$.

  * •

The spaces $H^{s}(\mathbb{R})$ for $s\in\mathbb{R}$ are the $L^{2}$-based
Sobolev spaces endowed with the usual norms denoted $\|\cdot\|_{H^{s}}$.

  * •

Given $I$ a real interval and $X$ a Banach space, $L^{p}(I;X)$ (respectively
$C^{n}(I;X)$) the space of $p$-integrable (respectively $n$-continuously
differentiable) $X$-valued functions, endowed with their usual norms.

  * •

When useful, we provide insights on the variables at stake in aforementioned
functional spaces by means of subscripts. For instance for
$f:(x,r)\in\mathbb{R}\times(-1,0)\mapsto f(x,r)\in\mathbb{R}$ we may denote

$\|f\|_{L^{\infty}_{r}H^{s}_{x}}=\operatorname*{ess\,sup}\big{(}\big{\\{}\|f(\cdot,r)\|_{H^{s}}\
,\ r\in(-1,0)\\}).$

  * •

We sometimes also use subscripts to provide information on the interval at
stake in functional spaces. For instance for $T>0$ and
$f:(t,x)\in[0,T]\times\mathbb{R}\mapsto f(t,x)\in\mathbb{R}$ we may denote

$\|f\|_{L^{\infty}_{T}H^{s}_{x}}=\operatorname*{ess\,sup}\big{(}\big{\\{}\|f(t,\cdot)\|_{H^{s}}\
,\ t\in[0,T]\\}).$

##  2 The bilayer shallow water system

In this section we analyze the bilayer shallow water system (1.3). We first
consider the case without diffusivity ($\kappa=0$) and recall the
hyperbolicity analysis due to Ovsjannikov [40]. We complete it by exhibiting
explicit symmetrizers of the system of conservation laws. The standard theory
for quasilinear systems then provides the local well-posedness of the initial-
value problem, that we state in Proposition 2.4.

Extending such result for the system with diffusivity ($\kappa>0$) uniformly
with respect to $\kappa\in(0,1]$ is not as obvious as one could naively think,
because the aforementioned symmetrizer behaves poorly with respect to
diffusivity contributions. In order to deal with this issue, we exhibit
regularization effects stemming from the diffusivity contributions that apply
to the total velocity (that is adding the bolus velocities to the velocity
unknowns). This is in the spirit of the BD entropy that arose in the context
of the barotropic Euler equations with degenerate viscosities (see [16]). We
infer a stability result on the linearized system, Lemma 2.6, which eventually
yields the “large-time” —that is uniform with respect to $\kappa\in(0,1]$—
control of solutions stated in Proposition 2.7, and their strong convergence
towards corresponding solutions to the non-diffusive system as $\kappa\searrow
0$ stated in Proposition 2.8.

###  2.1 The system without thickness diffusivity

We consider the system

$\begin{cases}\partial_{t}H_{s}+\partial_{x}((\underline{H}_{s}+H_{s})(\underline{U}_{s}+U_{s}))=0,\\\
\partial_{t}H_{b}+\partial_{x}((\underline{H}_{b}+H_{b})(\underline{U}_{b}+U_{b}))=0,\\\
\partial_{t}U_{s}+(\underline{U}_{s}+U_{s})\partial_{x}U_{s}+\partial_{x}H_{s}+\partial_{x}H_{b}=0,\\\
\partial_{t}U_{b}+(\underline{U}_{b}+U_{b})\partial_{x}U_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}H_{s}+\partial_{x}H_{b}=0.\end{cases}$
(2.1)

We shall also always assume $\rho_{s}\geq 0$ and $\rho_{b}>0$. Through
rescaling and Galilean invariance we can assume without loss of generality
that ${\underline{H}_{s}+\underline{H}_{b}=1}$ and
${\underline{U}_{s}+\underline{U}_{b}=0}$.

In compact form, the system reads

$\partial_{t}\bm{U}+{\sf A}(\underline{\bm{U}}+\bm{U})\partial_{x}\bm{U}=0$

with $\bm{U}:=(H_{s},H_{b},U_{s},U_{b})$,
$\underline{\bm{U}}:=(\underline{H}_{s},\underline{H}_{b},\underline{U}_{s},\underline{U}_{b})$
and where we introduce the matrix-valued function

${\sf
A}:(H_{s},H_{b},U_{s},U_{b})\in\mathbb{R}^{4}\mapsto\begin{pmatrix}U_{s}&0&H_{s}&0\\\
0&U_{b}&0&H_{b}\\\ 1&1&U_{s}&0\\\
\frac{\rho_{s}}{\rho_{b}}&1&0&U_{b}\end{pmatrix}.$ (2.2)

The following Lemma concerning the hyperbolicity domain of the bilayer shallow
water system is proved in [40, 7, 45].

######  Lemma 2.1.

Let $0<\rho_{s}<\rho_{b}$ and
$\bm{U}:=(H_{s},H_{b},U_{s},U_{b})\in\mathbb{R}^{4}$ be such that that
$H_{s},H_{b}>0$. There exist two values
$0<\operatorname{Fr}_{-}<\operatorname{Fr}_{+}$ such that the following holds:

  1. 1.

If $|U_{b}-U_{s}|<\sqrt{H_{b}}\operatorname{Fr}_{-}$, then there exist four
distinct real eigenvalues of the matrix ${\sf A}(\bm{U})$.

  2. 2.

If
$\sqrt{H_{b}}\operatorname{Fr}_{-}<|U_{b}-U_{s}|<\sqrt{H_{b}}\operatorname{Fr}_{+}$,
then there exist two distinct real eigenvalues of the matrix ${\sf A}(\bm{U})$
and two distinct complex conjugate eigenvalues.

  3. 3.

If $|U_{b}-U_{s}|>\sqrt{H_{b}}\operatorname{Fr}_{+}$, then there exist four
distinct real eigenvalues of the matrix ${\sf A}(\bm{U})$.

Moreover, $\operatorname{Fr}_{-}$ and $\operatorname{Fr}_{+}$ depend only and
smoothly on $H_{s}/H_{b}\in(0,+\infty)$ and $\rho_{s}/\rho_{b}\in(0,1)$.

######  Remark 2.2.

Ovsjannikov [40] —revisited by Barros and Choi [7] and then by Viríssimo and
Milewski [45]— provided a nice geometrical approach to the critical values
$\operatorname{Fr}_{+}$ and $\operatorname{Fr}_{-}$. The characteristic
polynomial associated to ${\sf A}(\bm{U})$ is

$P(\lambda)=\big{(}(U_{b}-\lambda)^{2}-H_{b}\big{)}\big{(}(U_{s}-\lambda)^{2}-H_{s}\big{)}-\frac{\rho_{s}}{\rho_{b}}H_{s}H_{b}.$

Notice that $\lambda\in\mathbb{R}$ is a real root of $P$ if and only if
$(p_{s},p_{b}):=(\frac{U_{s}-\lambda}{\sqrt{H_{s}}},\frac{U_{b}-\lambda}{\sqrt{H_{b}}})$
satisfies the following identities:

$\big{(}p_{s}^{2}-1\big{)}\big{(}p_{b}^{2}-1\big{)}=\frac{\rho_{s}}{\rho_{b}},\qquad
p_{s}\sqrt{H_{s}}-U_{s}=p_{b}\sqrt{H_{b}}-U_{b}.$ (2.3)

The first equality describes a fourth-order curve parametrized by
$\rho_{s}/\rho_{b}$ having four axes of symmetry and consisting of an inner
closed curve and four hyperbolic branches and the second equality describes
the straight line with slope $\sqrt{H_{s}/H_{b}}$ and intercept
$(U_{b}-U_{s})/\sqrt{H_{b}}$. In this geometrical approach,
$\operatorname{Fr}_{-}$ and $\operatorname{Fr}_{+}$ (and their opposite) are
the intercepts of the tangents to the fourth-order curves with slope
$\sqrt{H_{s}/H_{b}}$.

Figure 1 reproduces the aforementioned curves and straight lines for several
parameter values.

(a) $\rho_{s}/\rho_{b}=0.1$

(b) $\rho_{s}/\rho_{b}=0.5$

(c) $\rho_{s}/\rho_{b}=0.9$

Figure 1: Solutions to (2.3) with $H_{s}=1/3$, $H_{b}=2/3$, and differnet
values for $\rho_{s}/\rho_{b}$. Solutions to the quartic equation are in black
(plain). Solutions to the linear equation with
$(U_{b}-U_{s})/\sqrt{H_{b}}=1/2$ (green, plain),
$(U_{b}-U_{s})/\sqrt{H_{b}}=3/2$ (red, dashed) and
$(U_{b}-U_{s})/\sqrt{H_{b}}=5/2$ (blue, dot-dashed).

In this work we restrict our analysis to the hyperbolic domain described by
Lemma 2.1(1). While standard theory for strictly hyperbolic systems guarantees
the existence of a symmetrizer to (2.1) by using spectral projections [8], the
following Lemma provides an (almost) explicit expression for such a
symmetrizer.

######  Lemma 2.3.

Let $0<\rho_{s}<\rho_{b}$ and
$\bm{U}:=(H_{s},H_{b},U_{s},U_{b})\in\mathbb{R}^{4}$ be such that that
$H_{s},H_{b}>0$ and

$|U_{s}-U_{b}|<\sqrt{H_{b}}\operatorname{Fr}_{-}$ (2.4)

where
$\operatorname{Fr}_{-}=\operatorname{Fr}_{-}(H_{s}/H_{b},\rho_{s}/\rho_{b})>0$
has been defined in Lemma 2.1.

There exists $\lambda\in[\min(\\{U_{s},U_{b}\\}),\max(\\{U_{s},U_{b}\\})]$
such that, denoting
$U_{\ell}^{\lambda}:=U_{\ell}-\lambda\in[-|U_{b}-U_{s}|,|U_{b}-U_{s}|]$ for
$\ell\in\\{s,b\\}$) the matrix

${\sf
S}^{\lambda}(\bm{U}):=\begin{pmatrix}\frac{\rho_{s}}{\rho_{b}}&\frac{\rho_{s}}{\rho_{b}}&\frac{\rho_{s}}{\rho_{b}}U_{s}^{\lambda}&0\\\
\frac{\rho_{s}}{\rho_{b}}&1&0&U_{b}^{\lambda}\\\
\frac{\rho_{s}}{\rho_{b}}U_{s}^{\lambda}&0&\frac{\rho_{s}}{\rho_{b}}H_{s}&0\\\
0&U_{b}^{\lambda}&0&H_{b}\end{pmatrix}$

satisfies (i) ${\sf S}^{\lambda}{\sf A}$ is symmetric; and (ii) ${\sf
S}^{\lambda}$ is symmetric, definite positive.

Moreover, $\lambda$ can be chosen so that $\frac{\lambda-
U_{\ell}}{\sqrt{H_{\ell}}}$ (for $\ell\in\\{s,b\\}$) depends only and smoothly
on $H_{s}/H_{b}>0$, $\rho_{s}/\rho_{b}\in(0,1)$, and
$(U_{b}-U_{s})/\sqrt{H_{b}}\in(-\operatorname{Fr}_{-},\operatorname{Fr}_{-})$.

###### Proof.

It is straightforward to check that ${\sf S}^{\lambda}$ and ${\sf
S}^{\lambda}{\sf A}$ are symmetric (and real-valued) for any value of
$\lambda\in\mathbb{R}$. In order to prove that ${\sf S}^{\lambda}$ is definite
positive for a suitable choice of $\lambda$, we rely on Sylvester’s criterion.
We obtain the requirements

$H_{b}>0,\quad\frac{\rho_{s}}{\rho_{b}}H_{s}H_{b}>0,\quad\frac{\rho_{s}}{\rho_{b}}H_{s}H_{b}-\frac{\rho_{s}}{\rho_{b}}H_{s}(U_{b}^{\lambda})^{2}>0$

and

$\big{(}\frac{\rho_{s}}{\rho_{b}}\big{)}^{2}\Big{(}\big{(}(U_{b}^{\lambda})^{2}-H_{b}\big{)}\big{(}(U_{s}^{\lambda})^{2}-H_{s}\big{)}-\frac{\rho_{s}}{\rho_{b}}H_{s}H_{b}\Big{)}>0.$

The last inequality is equivalent to $P(\lambda)>0$ where $P$ is the
aforementioned characteristic polynomial. By Lemma 2.1, under the condition
(2.4) there are four distinct real roots to $P$, which we can denote
$\lambda_{1}<\lambda_{2}<\lambda_{3}<\lambda_{4}$ and $P(\lambda)>0$ for any
$\lambda\in(\lambda_{2},\lambda_{3})$. Moreover for all
$\lambda\in(\lambda_{2},\lambda_{3})$,
$(p_{s}^{\lambda},p_{b}^{\lambda}):=(\frac{U_{s}-\lambda}{\sqrt{H_{s}}},\frac{U_{b}-\lambda}{\sqrt{H_{b}}})$
belongs to the domain delimited by the inner closed curve, and in particular
we have $(p_{\ell}^{\lambda})^{2}<1$. Hence we find that for all
$\lambda\in(\lambda_{2},\lambda_{3})$ all principal minors are positive, and
hence ${\sf S}^{\lambda}$ is definite positive.

By the standard perturbation theory [33], $p_{\ell}^{\lambda}$ for
$\ell\in\\{s,b\\}$ and $\lambda\in\\{\lambda_{2},\lambda_{3}\\}$ depend
smoothly on $H_{s}/H_{b}>0$, $\rho_{s}/\rho_{b}\in(0,1)$ and
$(U_{b}-U_{s})/\sqrt{H_{b}}\in(-\operatorname{Fr}_{-},\operatorname{Fr}_{-})$.
What is more, we can always choose (smoothly)
$\lambda\in(\lambda_{2},\lambda_{3})$ so that $p_{s}^{\lambda}>0$ and
$p_{b}^{\lambda}<0$ when $U_{b}<U_{s}$, or $p_{s}^{\lambda}<0$ and
$p_{b}^{\lambda}>0$ when $U_{b}>U_{s}$, which corresponds to enforcing
$\lambda\in[\min(\\{U_{s},U_{b}\\}),\max(\\{U_{s},U_{b}\\})]$. This concludes
the proof. ∎

The following proposition follows from the standard theory on strictly
hyperbolic systems (see e.g. [8]). For convenience, we define for
$\varsigma\in(0,1)$ a compact subset of the domain of strict hyperbolicity as

$\mathfrak{p}^{\varsigma}:=\big{\\{}(\rho_{s},\rho_{b},H_{s},H_{b},U_{s},U_{b})\in\mathbb{R}^{6}\
:\\\ \varsigma/2\leq\rho_{s}/\rho_{b}\leq 1-\varsigma/2,\quad\varsigma\leq
H_{s}/H_{b}\leq\varsigma^{-1},\quad
H_{s}+H_{b}\geq\varsigma,\quad\operatorname{Fr}_{-}-\tfrac{|U_{b}-U_{s}|}{\sqrt{H_{b}}}\geq\varsigma\big{\\}}$
(2.5)

where
$\operatorname{Fr}_{-}=\operatorname{Fr}_{-}(\rho_{s}/\rho_{b},H_{s}/H_{b})$
is defined in Lemma 2.1.

######  Proposition 2.4 (Well-posedness).

Let $s\geq s_{0}>3/2$, $\varsigma>0$ and $M_{0}>0$. There exist $C>0$ and
$T>0$ such that the following holds.

For all
$(\rho_{s},\rho_{b},\underline{H}_{s},\underline{H}_{b},\underline{U}_{s},\underline{U}_{b})\in\mathbb{R}^{6}$
such that $\underline{H}_{s}+\underline{H}_{b}=1$ and
$\underline{U}_{s}+\underline{U}_{b}=0$ and
$(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\in H^{s}(\mathbb{R})^{4}$ such that

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}^{0}(x),\underline{H}_{b}+H_{b}^{0}(x),\underline{U}_{s}+U_{s}^{0}(x),\underline{U}_{b}+U_{b}^{0}(x))\in\mathfrak{p}^{\varsigma}$

and

$\|(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\|_{H^{s_{0}}}\leq M_{0}$

there exists a unique $(H_{s},H_{b},U_{s},U_{b})\in
C([0,T^{\star});H^{s}(\mathbb{R})^{4})\cap
C^{1}([0,T^{\star});H^{s-1}(\mathbb{R})^{4})$ maximal-in-time (classical)
solution to (2.1) emerging from the initial data
$(H_{s},H_{b},U_{s},U_{b})\big{|}_{t=0}=(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})$.

Moreover, one has $T^{\star}>T/M_{0}$ and for any $t\in[0,T/M_{0}]$ one has

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}(t,x),\underline{H}_{b}+H_{b}(t,x),\underline{U}_{s}+U_{s}(t,x),\underline{U}_{b}+U_{b}(t,x))\in\mathfrak{p}^{\varsigma/2}$

and

$\|(H_{s}(t,\cdot),H_{b}(t,\cdot),U_{s}(t,\cdot),U_{b}(t,\cdot))\|_{H^{s}}\leq
C\|(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\|_{H^{s}}.$

Moreover, the maximal existence time (resp. the emerging solution in
$C([0,T^{\star});H^{s}(\mathbb{R})^{4})$) is a lower semi-continuous (resp.
continuous) function of the initial data in $H^{s}(\mathbb{R})^{4}$ and if
$T^{\star}<\infty$ then

$\|(H_{s}(t,\cdot),H_{b}(t,\cdot),U_{s}(t,\cdot),U_{b}(t,\cdot))\|_{H^{s_{0}}}\to\infty\text{
as }t\to T^{\star}.$

###  2.2 The system with diffusivity

We now consider the system

$\begin{cases}\partial_{t}H_{s}+\partial_{x}((\underline{H}_{s}+H_{s})(\underline{U}_{s}+U_{s}))=\kappa\partial_{x}^{2}H_{s},\\\
\partial_{t}H_{b}+\partial_{x}((\underline{H}_{b}+H_{b})(\underline{U}_{b}+U_{b}))=\kappa\partial_{x}^{2}H_{b},\\\
\partial_{t}U_{s}+(\underline{U}_{s}+U_{s}-\kappa\frac{\partial_{x}H_{s}}{\underline{H}_{s}+H_{s}})\partial_{x}U_{s}+\partial_{x}H_{s}+\partial_{x}H_{b}=0,\\\
\partial_{t}U_{b}+(\underline{U}_{b}+U_{b}-\kappa\frac{\partial_{x}H_{b}}{\underline{H}_{b}+H_{b}})\partial_{x}U_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}H_{s}+\partial_{x}H_{b}=0.\end{cases}$
(2.6)

######  Proposition 2.5 (Small time well-posedness).

Let $s\geq s_{0}>3/2$, $\varsigma\in(0,1)$, $M_{0}>0$ and $c>1$. There exists
$T>0$ such that the following holds.

For all $\kappa>0$, for all
$(\rho_{s},\rho_{b},\underline{H}_{s},\underline{H}_{b},\underline{U}_{s},\underline{U}_{b})\in\mathbb{R}^{6}$
such that $\underline{H}_{s}+\underline{H}_{b}=1$ and
$\underline{U}_{s}+\underline{U}_{b}=0$ and for all
$(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\in H^{s}(\mathbb{R})^{4}$ such that

$0\leq\rho_{s}/\rho_{b}\leq\varsigma^{-1}\quad\text{ and }\quad\forall
x\in\mathbb{R},\quad\underline{H}_{s}+H_{s}^{0}(x)\geq\varsigma,\quad\underline{H}_{b}+H_{b}^{0}(x)\geq\varsigma$

and

$\|(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\|_{H^{s_{0}}}\leq M_{0}$

there exists a unique $(H_{s},H_{b},U_{s},U_{b})\in
C([0,T^{\star});H^{s}(\mathbb{R})^{4})$ maximal-in-time strong solution to
(2.6) emerging from the initial data
$(H_{s},H_{b},U_{s},U_{b})\big{|}_{t=0}=(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})$.

Moreover, $T^{\star}>\kappa T$ and for any $t\in[0,\kappa T]$ one has

$\forall
x\in\mathbb{R},\quad\underline{H}_{s}+H_{s}(t,x)\geq\varsigma/c,\quad\underline{H}_{b}+H_{b}(t,x)\geq\varsigma/c$

and

$\max(\\{\|(H_{s},H_{b},U_{s},U_{b})\|_{L^{\infty}(0,t;H^{s})},\kappa^{1/2}\|(\partial_{x}H_{s},\partial_{x}H_{b})\|_{L^{2}(0,t;H^{s})}\\})\leq
c\|(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\|_{H^{s}}.$

Moreover, the maximal existence time (resp. the emerging solution in
$C([0,T^{\star});H^{s}(\mathbb{R})^{4})$) is a lower semi-continuous (resp.
continuous) function of the initial data in $H^{s}(\mathbb{R})^{4}$ and if
$T^{\star}<\infty$ then

$\|(H_{s}(t,\cdot),H_{b}(t,\cdot),U_{s}(t,\cdot),U_{b}(t,\cdot))\|_{H^{s_{0}}}\to\infty\text{
as }t\to T^{\star}.$

###### Proof.

The proof has been given in [1], but we sketch it here for convenience. We
view (2.6) as a system of two transport-diffusion equations and two transport
equations, coupled only through order-zero source terms:

$\begin{cases}\partial_{t}H_{s}+(\underline{U}_{s}+U_{s})\partial_{x}H_{s}-\kappa\partial_{x}^{2}H_{s}=-(\underline{H}_{s}+H_{s})\partial_{x}U_{s},\\\
\partial_{t}H_{b}+(\underline{U}_{b}+U_{b})\partial_{x}H_{b}-\kappa\partial_{x}^{2}H_{b}=-(\underline{H}_{b}+H_{b})\partial_{x}U_{b},\\\
\partial_{t}U_{s}+(\underline{U}_{s}+U_{s}-\kappa\frac{\partial_{x}H_{s}}{\underline{H}_{s}+H_{s}})\partial_{x}U_{s}=-\partial_{x}H_{s}-\partial_{x}H_{b},\\\
\partial_{t}U_{b}+(\underline{U}_{b}+U_{b}-\kappa\frac{\partial_{x}H_{b}}{\underline{H}_{b}+H_{b}})\partial_{x}U_{b}=-\frac{\rho_{s}}{\rho_{b}}\partial_{x}H_{s}-\partial_{x}H_{b}.\end{cases}$

The standard theory on transport and transport-diffusion equations (see [6])
allows to bootstrap the standard fixed-point strategy through Picard iterates

$\begin{cases}\partial_{t}H_{s}^{n+1}+(\underline{U}_{s}+U_{s}^{n})\partial_{x}H_{s}^{n+1}-\kappa\partial_{x}^{2}H_{s}^{n+1}=-(\underline{H}_{s}+H_{s}^{n})\partial_{x}U_{s}^{n},\\\
\partial_{t}H_{b}^{n+1}+(\underline{U}_{b}+U_{b}^{n})\partial_{x}H_{b}^{n+1}-\kappa\partial_{x}^{2}H_{b}^{n+1}=-(\underline{H}_{b}+H_{b}^{n})\partial_{x}U_{b}^{n},\\\
\partial_{t}U_{s}^{n+1}+(\underline{U}_{s}+U_{s}^{n}-\kappa\frac{\partial_{x}H_{s}^{n}}{\underline{H}_{s}+H_{s}^{n}})\partial_{x}U_{s}^{n+1}=-\partial_{x}H_{s}^{n}-\partial_{x}H_{b}^{n},\\\
\partial_{t}U_{b}^{n+1}+(\underline{U}_{b}+U_{b}^{n}-\kappa\frac{\partial_{x}H_{b}^{n}}{\underline{H}_{b}+H_{b}^{n}})\partial_{x}U_{b}^{n+1}=-\frac{\rho_{s}}{\rho_{b}}\partial_{x}H_{s}^{n}-\partial_{x}H_{b}^{n},\end{cases}$

which defines a sequence satisfying the following estimates (where $c_{0}$ is
a non-essential constant depending on $s$)

$\max(\\{\|(H_{s}^{n+1},H_{b}^{n+1},U_{s}^{n+1},U_{b}^{n+1})\|_{L^{\infty}(0,t;H^{s})},\kappa^{1/2}\|(\partial_{x}H_{s}^{n+1},\partial_{x}H_{b}^{n+1})\|_{L^{2}(0,t;H^{s})}\\})\\\
\leq\Big{(}\|(H_{s}^{0},H_{b}^{0})\|_{H^{s}}+\kappa^{-1/2}\big{(}(\underline{H}_{s}+\|H_{s}^{n}\|_{L^{\infty}(0,t;H^{s})})\|U_{s}^{n}\|_{L^{2}(0,t;H^{s})}+(\underline{H}_{b}+\|H_{b}^{n}\|_{L^{\infty}(0,t;H^{s})})\|U_{b}^{n}\|_{L^{2}(0,t;H^{s})}\big{)}\\\
+\|(U_{s}^{0},U_{b}^{0})\|_{H^{s}}+(1+\frac{\rho_{s}}{\rho_{b}})\|(\partial_{x}H_{s}^{n},\partial_{x}H_{b}^{n})\|_{L^{1}(0,t;H^{s})}\Big{)}\\\
\times\exp\big{(}c_{0}\|(U_{s}^{n},U_{b}^{n})\|_{L^{1}(0,t;H^{s})}+c_{0}\kappa\|(\tfrac{\partial_{x}H_{s}^{n}}{\underline{H}_{s}+H_{s}^{n}},\tfrac{\partial_{x}H_{b}^{n}}{\underline{H}_{b}+H_{b}^{n}})\|_{L^{1}(0,t;H^{s})}\big{)}$

and converging in $C(0,t;H^{s})$ provided $t\in(0,\kappa T]$ where $T$ is
chosen sufficiently small.

The proof of the continuity of the flow map can be obtained along the same
lines, using the continuity with respect to the initial data and Lipschitz-
continuity with respect to source terms of the transport-diffusion and
transport equations. ∎

The terminology “Small time well-posedness” in Proposition 2.5 refers to the
fact that the time of existence and control of solutions of the above result
is limited to $T^{\star}\gtrsim\kappa$, and in particular may vanish as
$\kappa\searrow 0$. Notice that, differently from the statement of Proposition
2.4, we do not assume that the flow is stably stratified, namely
$\rho_{s}<\rho_{b}$. Assuming additionally that the flow is stably stratified,
the first author improved this result in situations with small shear
velocities and small deviations from the shear equilibrium, and obtains in [1]
the existence and uniform control of solution up to times
${T^{\star}\gtrsim\big{(}1+\kappa^{-1}(|\underline{U}_{b}-\underline{U}_{s}|^{2}+M_{0}^{2})\big{)}^{-1}}$.

In the following results, we complete the picture by showing that, in the
situation where the shear velocity is small enough to guarantee that the flow
is in the hyperbolic domain of the non-diffusive equation, then the time of
existence is uniform with respect to $\kappa\in(0,1]$. In fact we shall prove
the expected property that solutions to the diffusive system (2.6) converge as
$\kappa\searrow 0$ towards corresponding solutions to the non-diffusive system
(2.1) as long as the non-diffusive solution is bounded.

In order to obtain stability estimates that are uniform with respect to
$\kappa\in(0,1]$, we rely on two main ideas. Firstly, we shall use energy
estimates using the explicit symmetrizer adapted to the non-diffusive system
introduced in Lemma 2.3 (while the strategy in [1] used only its block-
diagonal component). Because the non-diagonal components of the symmetrizer
behave poorly with respect to the diffusive contributions, we need another
ingredient. Specifically, we notice that the total velocities
$V_{\ell}:=U_{\ell}-\kappa\frac{\partial_{x}H_{\ell}}{\underline{H}_{\ell}+H_{\ell}}$
($\ell\in\\{s,b\\}$) associated to solutions to (2.6) satisfy the system

$\begin{cases}\partial_{t}H_{s}+\partial_{x}((\underline{H}_{s}+H_{s})(\underline{U}_{s}+V_{s}))=0,\\\
\partial_{t}H_{b}+\partial_{x}((\underline{H}_{b}+H_{b})(\underline{U}_{s}+V_{b}))=0,\\\
\partial_{t}V_{s}+(\underline{U}_{s}+V_{s}-\kappa\frac{\partial_{x}H_{s}}{\underline{H}_{s}+H_{s}})\partial_{x}V_{s}+\partial_{x}H_{s}+\partial_{x}H_{b}=\kappa\partial_{x}^{2}V_{s},\\\
\partial_{t}V_{b}+(\underline{U}_{b}+V_{b}-\kappa\frac{\partial_{x}H_{b}}{\underline{H}_{b}+H_{b}})\partial_{x}V_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}H_{s}+\partial_{x}H_{b}=\kappa\partial_{x}^{2}V_{b}.\end{cases}$
(2.7)

We observe that diffusive terms act as effective viscosity contributions on
the total velocities. The last two equations read equivalently

$\begin{cases}\partial_{t}V_{s}+(\underline{U}_{s}+V_{s})\partial_{x}V_{s}+\partial_{x}H_{s}+\partial_{x}H_{b}=\frac{\kappa}{\underline{H}_{s}+H_{s}}\partial_{x}((\underline{H}_{s}+H_{s})\partial_{x}V_{s}),\\\
\partial_{t}V_{b}+(\underline{U}_{b}+V_{b})\partial_{x}V_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}H_{s}+\partial_{x}H_{b}=\frac{\kappa}{\underline{H}_{b}+H_{b}}\partial_{x}((\underline{H}_{b}+H_{b})\partial_{x}V_{b}),\end{cases}$

and we recognize the shallow-water equations with degenerate viscosity
contributions which were advocated by Gent in [24] and derived from the
Navier–Stokes equations in [26, 17]. In their analysis of such systems (and
generalizations thereof), Bresch and Desjardins introduced the so-called BD
entropy in [15, 13, 14] (see also [16] for a refined analysis), which is based
precisely in the reformulation of (2.7) as (2.6) (in dimension $d=1$).

In the same spirit, we combine the regularizing effects of the effective
diffusivity and viscosity terms with aforementioned energy estimates, which
allows us to obtain suitable stability estimates presented in Lemma 2.6,
below.

Applying Lemma 2.6 to (the derivatives of) solutions to (2.6)-(2.7), we find a
time of existence which is uniform with respect to $\kappa$. We state the
result in forthcoming Proposition 2.7.

Applying Lemma 2.6 to (the derivatives of) the difference between solutions to
(2.6)-(2.7) and corresponding solutions to the non-diffusive system (2.1)
($\kappa=0$) yields the aforementioned convergence of the former towards the
latter as $\kappa\searrow 0$, on a time interval defined by the solutions
without diffusivity. We state the result in forthcoming Proposition 2.8.

######  Lemma 2.6 (Stability).

Let $\varsigma\in(0,1)$ and $M>0$. There exists $c>0$ depending only on
$\varsigma$ and $C>0$ depending also on $M$ such that the following holds.

Let $\kappa\in(0,1]$, $0<\rho_{s}<\rho_{b}$ and
$\bm{U}:=(H_{s},H_{b},U_{s},U_{b}),\
\bm{V}:=(H_{s},H_{b},V_{s},V_{b})\in{C([0,T];W^{1,\infty}(\mathbb{R})^{4})}\cap{C^{1}([0,T];L^{\infty}(\mathbb{R})^{4})}$
be such that for all $t\in[0,T]$, the hyperbolicity condition holds:

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},H_{s}(t,x),H_{b}(t,x),U_{s}(t,x),U_{b}(t,x))\in\mathfrak{p}^{\varsigma},$

where $\mathfrak{p}^{\varsigma}$ is defined in (2.5), and

$\|\bm{U}(t,\cdot)\|_{W^{1,\infty}}+\|\bm{V}(t,\cdot)\|_{W^{1,\infty}}+\kappa\|(\partial_{x}^{2}H_{s}(t,\cdot),\partial_{x}^{2}H_{b}(t,\cdot))\|_{L^{\infty}}+\kappa^{-1}\|(\bm{U}-\bm{V})(t,\cdot)\|_{L^{\infty}}+\|\partial_{t}\bm{U}(t,\cdot)\|_{L^{\infty}}\leq
M.$

Let $\dot{\bm{U}}:=(\dot{H}_{s},\dot{H}_{b},\dot{U}_{s},\dot{U}_{b})$ and
$\dot{\bm{V}}:=(\dot{H}_{s},\dot{H}_{b},\dot{V}_{s},\dot{V}_{b})$ be
sufficiently regular solutions to the linearized equations with remainders

$\displaystyle\partial_{t}\dot{\bm{U}}+{\sf
A}^{\kappa}(\bm{U})\partial_{x}\dot{\bm{U}}=\kappa{\sf
D}_{1}\partial_{x}^{2}\dot{\bm{U}}+\bm{R}_{\bm{U}},$
$\displaystyle\partial_{t}\dot{\bm{V}}+{\sf
A}^{\kappa}(\bm{V})\partial_{x}\dot{\bm{V}}=\kappa{\sf
D}_{2}\partial_{x}^{2}\dot{\bm{V}}+\bm{R}_{\bm{V}},$

where we denote

${\sf
A}^{\kappa}:(H_{s},H_{b},U_{s},U_{b})\mapsto\begin{pmatrix}U_{s}&0&H_{s}&0\\\
0&U_{b}&0&H_{b}\\\ 1&1&U_{s}-\kappa\tfrac{\partial_{x}H_{s}}{H_{s}}&0\\\
\frac{\rho_{s}}{\rho_{b}}&1&0&U_{b}-\kappa\tfrac{\partial_{x}H_{b}}{H_{b}}\end{pmatrix},$

and

${\sf D}_{1}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{pmatrix},\qquad{\sf D}_{2}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{pmatrix}.$

Moreover, denote $R:=(R_{s},R_{b})$ such that

$\dot{V}_{s}=\dot{U}_{s}-\kappa\frac{\partial_{x}\dot{H}_{s}}{H_{s}}+R_{s},\quad\dot{V}_{b}=\dot{U}_{b}-\kappa\frac{\partial_{x}\dot{H}_{b}}{H_{b}}+R_{b}.$
(2.8)

Then, for any $t\in[0,T]$, one has the estimate

$\displaystyle\|\dot{\bm{U}}(t,\cdot)\|_{L^{2}}+\|\dot{\bm{V}}(t,\cdot)\|_{L^{2}}+c\kappa^{1/2}\|\partial_{x}\dot{\bm{V}}(t,\cdot)\|_{L^{2}(0,t;L^{2})}\leq
c^{-1}\,\big{(}\|\dot{\bm{U}}(t=0,\cdot)\|_{L^{2}}+\|\dot{\bm{V}}(t=0,\cdot)\|_{L^{2}}\big{)}\exp(C\,M\,t)$
$\displaystyle\quad+C\int_{0}^{t}\big{(}\|\bm{R}_{\bm{U}}(t^{\prime},\cdot)\|_{L^{2}}+\|\bm{R}_{\bm{V}}(t^{\prime},\cdot)\|_{L^{2}}+M\|R(t^{\prime},\cdot)\|_{H^{1}}\big{)}\exp(C\,M\,(t-t^{\prime}))\operatorname{{\rm
d}\\!}t.$

###### Proof.

Denote

${\sf
S}^{\lambda}:(H_{s},H_{b},U_{s},U_{b})\mapsto\begin{pmatrix}\frac{\rho_{s}}{\rho_{b}}&\frac{\rho_{s}}{\rho_{b}}&\frac{\rho_{s}}{\rho_{b}}U_{s}^{\lambda}&0\\\
\frac{\rho_{s}}{\rho_{b}}&1&0&U_{b}^{\lambda}\\\
\frac{\rho_{s}}{\rho_{b}}U_{s}^{\lambda}&0&\frac{\rho_{s}}{\rho_{b}}H_{s}&0\\\
0&U_{b}^{\lambda}&0&H_{b}\end{pmatrix}$

where $U_{\ell}^{\lambda}:=U_{\ell}-\lambda$ (for $\ell\in\\{s,b\\}$) with
$\lambda$ provided in Lemma 2.3. Using that ${\sf S}^{\lambda}(\cdot)$ and
${\sf S}^{\lambda}(\cdot){\sf A}^{0}(\cdot)$ are symmetric, and integration by
parts, we have the energy identities

$\displaystyle\frac{1}{2}\frac{\operatorname{{\rm d}\\!}}{\operatorname{{\rm
d}\\!}t}\Big{(}\big{(}{\sf
S}^{\lambda}(\bm{U})\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\big{(}{\sf
S}^{\lambda}(\bm{U})\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}\Big{)}$
$\displaystyle=\big{(}{\sf
S}^{\lambda}(\bm{U})\partial_{t}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\big{(}{\sf
S}^{\lambda}(\bm{U})\partial_{t}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}+\frac{1}{2}\big{(}[\partial_{t},{\sf
S}^{\lambda}(\bm{U})]\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\frac{1}{2}\big{(}[\partial_{t},{\sf
S}^{\lambda}(\bm{V})]\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}$
$\displaystyle=-\big{(}{\sf S}^{\lambda}(\bm{U}){\sf
A}^{\kappa}(\bm{U})\partial_{x}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}-\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
A}^{\kappa}(\bm{V})\partial_{x}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
D}_{1}\partial_{x}^{2}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
D}_{2}\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}$
$\displaystyle\qquad+\big{(}{\sf
S}^{\lambda}(\bm{U})\bm{R}_{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\big{(}{\sf
S}^{\lambda}(\bm{U})\bm{R}_{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}+\frac{1}{2}\big{(}[\partial_{t},{\sf
S}^{\lambda}(\bm{U})]\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\frac{1}{2}\big{(}[\partial_{t},{\sf
S}^{\lambda}(\bm{U})]\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}$
$\displaystyle=\frac{1}{2}\big{(}[\partial_{x},{\sf S}^{\lambda}(\bm{U}){\sf
A}^{0}(\bm{U})]\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\frac{1}{2}\big{(}[\partial_{x},{\sf
S}^{\lambda}(\bm{V}){\sf
A}^{0}(\bm{V})]\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}$
$\displaystyle\qquad-\big{(}{\sf S}^{\lambda}(\bm{U})({\sf
A}^{\kappa}(\bm{U})-{\sf
A}^{0}(\bm{U}))\partial_{x}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}-\big{(}{\sf
S}^{\lambda}(\bm{V})({\sf A}^{\kappa}(\bm{V})-{\sf
A}^{0}(\bm{V}))\partial_{x}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}$
$\displaystyle\qquad-\big{(}({\sf S}^{\lambda}(\bm{U})-{\sf
S}^{\lambda}(\bm{V})){\sf
A}^{\kappa}(\bm{V})\partial_{x}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}+\frac{1}{2}\big{(}[\partial_{t},{\sf
S}^{\lambda}(\bm{U})]\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\frac{1}{2}\big{(}[\partial_{t},{\sf
S}^{\lambda}(\bm{U})]\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}$
$\displaystyle\qquad+\big{(}{\sf
S}^{\lambda}(\bm{U})\bm{R}_{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\big{(}{\sf
S}^{\lambda}(\bm{U})\bm{R}_{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
D}_{1}\partial_{x}^{2}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
D}_{2}\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}$
$\displaystyle=:A-\big{(}{\sf S}^{\lambda}(\bm{U})({\sf
A}^{\kappa}(\bm{U})-{\sf
A}^{0}(\bm{U}))\partial_{x}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
D}_{1}\partial_{x}^{2}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
D}_{2}\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}.$

By means of Cauchy–Schwarz inequality we find that

$|A|\leq
C\,\big{(}\|\dot{\bm{U}}\|_{L^{2}}+\|\dot{\bm{V}}\|_{L^{2}}\big{)}\times\big{(}M\|\dot{\bm{U}}\|_{L^{2}}+M\|\dot{\bm{V}}\|_{L^{2}}+\kappa
M\|\partial_{x}\dot{\bm{V}}\|_{L^{2}}+\|\bm{R}_{\bm{U}}\|_{L^{2}}+\|\bm{R}_{\bm{V}}\|_{L^{2}}\big{)},$

where $C$ denotes a multiplicative constant depending only on $\varsigma$ and
$M$, and which may change from line to line. We now focus on the remaining
terms. We first notice that defects of symmetry in ${\sf
S}^{\lambda}(\bm{U})({\sf A}^{\kappa}(\bm{U})-{\sf A}^{0}(\bm{U}))$ arise only
in the first two rows, and that the first two components of $\dot{\bm{U}}$
equal the first two components of $\dot{\bm{V}}$. Hence using integration by
parts and Cauchy–Schwarz inequality we infer

$|\big{(}{\sf S}^{\lambda}(\bm{U})({\sf A}^{\kappa}(\bm{U})-{\sf
A}^{0}(\bm{U}))\partial_{x}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}|\leq\kappa\,C\,M\,\|\dot{\bm{U}}\|_{L^{2}}\times\big{(}\|\dot{\bm{U}}\|_{L^{2}}+\|\partial_{x}\dot{\bm{V}}\|_{L^{2}}\big{)}.$

Then, again making use of the identity ${\sf
D}_{1}\partial_{x}^{2}\dot{\bm{U}}=(\partial_{x}^{2}\dot{H}_{1},\partial_{x}^{2}\dot{H}_{2},0,0)={\sf
D}_{1}\partial_{x}^{2}\dot{\bm{V}}$ we infer that

$\kappa\big{(}{\sf S}^{\lambda}(\bm{U}){\sf
D}_{1}\partial_{x}^{2}\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
D}_{2}\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}=\kappa\big{(}{\sf
S}^{\lambda}(\bm{U})({\sf D}_{1}+{\sf
D}_{2})\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U}){\sf
D}_{1}\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{U}}-\dot{\bm{V}}\big{)}_{L^{2}}.$

After integration by parts, and since ${\sf D}_{1}+{\sf
D}_{2}=\operatorname{Id}$, we find that

$\kappa\big{(}{\sf S}^{\lambda}(\bm{U})({\sf D}_{1}+{\sf
D}_{2})\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}}\leq-\kappa\big{(}{\sf
S}^{\lambda}(\bm{U})\partial_{x}\dot{\bm{V}},\partial_{x}\dot{\bm{V}}\big{)}_{L^{2}}+\kappa\,C\,M\,\|\partial_{x}\dot{\bm{V}}\|_{L^{2}}\|\dot{\bm{V}}\|_{L^{2}}.$

Then, using from (2.8) that
$\kappa\partial_{x}\dot{H}_{\ell}=H_{\ell}((\dot{U}_{\ell}-\dot{V}_{\ell})+R_{\ell})$
(where $\ell\in\\{s,b\\}$), we obtain the identities

$\displaystyle\kappa\big{(}{\sf S}^{\lambda}(\bm{U}){\sf
D}_{1}\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{U}}-\dot{\bm{V}}\big{)}_{L^{2}}$
$\displaystyle=\kappa\sum_{\ell\in\\{s,b\\}}\big{(}\rho_{\ell}U_{\ell}^{\lambda}\partial_{x}^{2}\dot{H}_{\ell},\dot{U}_{\ell}-\dot{V}_{\ell}\big{)}_{L^{2}}$
$\displaystyle=\sum_{\ell\in\\{s,b\\}}\big{(}\rho_{\ell}U_{\ell}^{\lambda}\partial_{x}(H_{\ell}(\dot{U}_{\ell}-\dot{V}_{\ell}+R_{\ell})),\dot{U}_{\ell}-\dot{V}_{\ell}\big{)}_{L^{2}}$
$\displaystyle=\sum_{\ell\in\\{s,b\\}}\frac{\rho_{\ell}}{2}\big{(}(U_{\ell}^{\lambda}\partial_{x}H_{\ell}-H_{\ell}\partial_{x}U_{\ell}^{\lambda})(\dot{U}_{\ell}-\dot{V}_{\ell}),\dot{U}_{\ell}-\dot{V}_{\ell}\big{)}_{L^{2}}$
$\displaystyle\quad+\kappa\sum_{\ell\in\\{s,b\\}}\rho_{\ell}\big{(}U_{\ell}^{\lambda}\partial_{x}(H_{\ell}R_{\ell}),\dot{U}_{\ell}-\dot{V}_{\ell}\big{)}_{L^{2}},$

where we used integration by parts in the last line. We infer

$\kappa\big{(}{\sf S}^{\lambda}(\bm{U}){\sf
D}_{1}\partial_{x}^{2}\dot{\bm{V}},\dot{\bm{U}}-\dot{\bm{V}}\big{)}_{L^{2}}\leq\,C\,M\,\big{(}\|\dot{\bm{U}}\|_{L^{2}}+\|\dot{\bm{V}}\|_{L^{2}}\big{)}^{2}+C\,M\,\big{(}\|\dot{\bm{U}}\|_{L^{2}}+\|\dot{\bm{V}}\|_{L^{2}}\big{)}\|R\|_{H^{1}}.$

Combining all these estimate and denoting

${\mathcal{E}}:=\big{(}{\sf
S}^{\lambda}(\bm{U})\dot{\bm{U}},\dot{\bm{U}}\big{)}_{L^{2}}+\big{(}{\sf
S}^{\lambda}(\bm{U})\dot{\bm{V}},\dot{\bm{V}}\big{)}_{L^{2}},$

one has

$\frac{1}{2}\frac{\operatorname{{\rm d}\\!}}{\operatorname{{\rm
d}\\!}t}{\mathcal{E}}+\kappa\big{(}{\sf
S}^{\lambda}(\bm{U})\partial_{x}\dot{\bm{V}},\partial_{x}\dot{\bm{V}}\big{)}_{L^{2}}\leq
C\,M\,\big{(}\|\dot{\bm{U}}\|_{L^{2}}+\|\dot{\bm{V}}\|_{L^{2}}+\kappa\|\partial_{x}\dot{\bm{V}}\|_{L^{2}}\big{)}\big{(}\|\dot{\bm{U}}\|_{L^{2}}+\|\dot{\bm{V}}\|_{L^{2}}\big{)}\\\
+C\big{(}\|\bm{R}_{\bm{U}}\|_{L^{2}}+\|\bm{R}_{\bm{V}}\|_{L^{2}}+M\|R\|_{H^{1}}\big{)}\big{(}\|\dot{\bm{U}}\|_{L^{2}}+\|\dot{\bm{V}}\|_{L^{2}}\big{)}.$
(2.9)

By using that ${\sf S}^{\lambda}(\bm{U})$ is definite positive, we find that
there exists $c>0$ depending only on $\varsigma$ such that

${\mathcal{E}}\geq
c^{2}\|\dot{\bm{U}}\|_{L^{2}}^{2}+c^{2}\|\dot{\bm{V}}\|_{L^{2}}^{2},\quad\big{(}{\sf
S}^{\lambda}(\bm{U})\partial_{x}\dot{V},\partial_{x}\dot{V}\big{)}_{L^{2}}\geq
c^{2}\|\partial_{x}\dot{\bm{V}}\|_{L^{2}}^{2}.$

Hence we find (using the Peter Paul inequality and augmenting $C$) that

$\frac{1}{2}\frac{\operatorname{{\rm d}\\!}}{\operatorname{{\rm
d}\\!}t}{\mathcal{E}}+\tfrac{1}{2}c^{2}\kappa\|\partial_{x}\dot{\bm{V}}\|_{L^{2}}^{2}\leq
C\,M\,{\mathcal{E}}+C\big{(}\|\bm{R}_{\bm{U}}\|_{L^{2}}+\|\bm{R}_{\bm{V}}\|_{L^{2}}+M\|R\|_{H^{1}}\big{)}\,{\mathcal{E}}^{1/2},$

and the result follows by Gronwall’s Lemma. ∎

######  Proposition 2.7 (Large-time well-posedness).

Let $s\geq s_{0}>3/2$, $\varsigma\in(0,1)$ and $M_{0}>0$. There exists $C>0$
and $T>0$ such that the following holds.

Let $\kappa\in(0,1]$,
$(\rho_{s},\rho_{b},\underline{H}_{s},\underline{H}_{b},\underline{U}_{s},\underline{U}_{b})\in\mathbb{R}^{6}$
such that $\underline{H}_{s}+\underline{H}_{b}=1$ and
$\underline{U}_{s}+\underline{U}_{b}=0$, and let
$(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\in H^{s+1}(\mathbb{R})^{2}\times
H^{s}(\mathbb{R})^{2}$ such that the hyperbolicity condition holds:

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}^{0}(x),\underline{H}_{b}+H_{b}^{0}(x),\underline{U}_{s}+U_{s}^{0}(x),\underline{U}_{b}+U_{b}^{0}(x))\in\mathfrak{p}^{\varsigma},$

where $\mathfrak{p}^{\varsigma}$ is defined in (2.5), and

$\|(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0},\kappa\partial_{x}H_{s}^{0},\kappa\partial_{x}H_{b}^{0})\|_{H^{s_{0}}}\leq
M_{0}.$

Denote $(H_{s},H_{b},U_{s},U_{b})\in C([0,T^{\star});H^{s}(\mathbb{R})^{4})$
the maximal-in-time solution to (2.6) emerging from the initial data
$(H_{s},H_{b},U_{s},U_{b})\big{|}_{t=0}=(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})$
as defined in Proposition 2.5.

One has $T^{\star}>T/M_{0}$ and for any $t\in[0,T/M_{0}]$,

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}(t,x),\underline{H}_{b}+H_{b}(t,x),\underline{U}_{s}+U_{s}(t,x),\underline{U}_{b}+U_{b}(t,x))\in\mathfrak{p}^{\varsigma/2},$

and

$\|(H_{s},H_{b},U_{s},U_{b},\kappa\partial_{x}H_{s},\kappa\partial_{x}H_{b})(t,\cdot)\|_{H^{s}}+\|(\partial_{t}H_{s},\partial_{t}H_{b},\partial_{t}U_{s},\partial_{t}U_{b})(t,\cdot)\|_{H^{s-1}}\\\
\leq
C\|(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0},\kappa\partial_{x}H_{s}^{0},\kappa\partial_{x}H_{b}^{0})\|_{H^{s}}.$

###### Proof.

We assume that the initial data is smooth, so that
$(H_{s},H_{b},U_{s},U_{b},V_{s},V_{b})$ are smooth on their domain of
existence. The general case is obtained by regularizing the initial data and
passing to the limit, thanks to the persistence of regularity and continuity
of the flow map stated in Proposition 2.5.

Denote
$V_{\ell}:=U_{\ell}-\kappa\frac{\partial_{x}H_{\ell}}{\underline{H}_{\ell}+H_{\ell}}$
($\ell\in\\{s,b\\}$),
$\Lambda^{s}:=(\operatorname{Id}-\partial_{x}^{2})^{s/2}$ and

$(\dot{H}_{s},\dot{H}_{b},\dot{U}_{s},\dot{U}_{b},\dot{V}_{s},\dot{V}_{b}):=(\Lambda^{s}H_{s},\Lambda^{s}H_{b},\Lambda^{s}U_{s},\Lambda^{s}U_{b},\Lambda^{s}V_{s},\Lambda^{s}V_{b}).$

Applying the operator $\Lambda^{s}$ to (2.6) and (2.7), we obtain

$\begin{cases}\partial_{t}\dot{H}_{s}+(\underline{H}_{s}+H_{s})\partial_{x}\dot{U}_{s}+(\underline{U}_{s}+U_{s})\partial_{x}\dot{H}_{s}=\kappa\partial_{x}^{2}\dot{H}_{s}+R_{H}(H_{s},U_{s}),\\\
\partial_{t}\dot{H}_{b}+(\underline{H}_{b}+H_{b})\partial_{x}\dot{U}_{b}+(\underline{U}_{b}+U_{b})\partial_{x}\dot{H}_{b}=\kappa\partial_{x}^{2}\dot{H}_{b}+R_{H}(H_{b},U_{b}),\\\
\partial_{t}\dot{U}_{s}+(\underline{U}_{s}+U_{s}-\kappa\frac{\partial_{x}H_{s}}{\underline{H}_{s}+H_{s}})\partial_{x}\dot{U}_{s}+\partial_{x}\dot{H}_{s}+\partial_{x}\dot{H}_{b}=R_{U}(H_{s},U_{s}),\\\
\partial_{t}\dot{U}_{b}+(\underline{U}_{b}+U_{b}-\kappa\frac{\partial_{x}H_{b}}{\underline{H}_{b}+H_{b}})\partial_{x}\dot{U}_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}\dot{H}_{s}+\partial_{x}\dot{H}_{b}=R_{U}(H_{b},U_{b}),\end{cases}$

and

$\begin{cases}\partial_{t}\dot{H}_{s}+(\underline{H}_{s}+H_{s})\partial_{x}\dot{V}_{s}+(\underline{U}_{s}+V_{s})\partial_{x}\dot{H}_{s}=R_{H}(H_{s},V_{s}),\\\
\partial_{t}\dot{H}_{b}+(\underline{H}_{b}+H_{b})\partial_{x}\dot{V}_{b}+(\underline{U}_{b}+V_{b})\partial_{x}\dot{H}_{b}=R_{H}(H_{b},V_{b}),\\\
\partial_{t}\dot{V}_{s}+(\underline{U}_{s}+V_{s}-\kappa\frac{\partial_{x}H_{s}}{\underline{H}_{s}+H_{s}})\partial_{x}\dot{V}_{s}+\partial_{x}\dot{H}_{s}+\partial_{x}\dot{H}_{b}=\kappa\partial_{x}^{2}\dot{V}_{s}+R_{U}(H_{s},V_{s}),\\\
\partial_{t}\dot{V}_{b}+(\underline{U}_{b}+V_{b}-\kappa\frac{\partial_{x}H_{b}}{\underline{H}_{b}+H_{b}})\partial_{x}\dot{V}_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}\dot{H}_{s}+\partial_{x}\dot{H}_{b}=\kappa\partial_{x}^{2}\dot{V}_{s}+R_{U}(H_{b},V_{b}),\end{cases}$

with remainders $R_{H}$ and $R_{V}$ defined as

$R_{H}(H,V):=-[\Lambda^{s},H]\partial_{x}V-[\Lambda^{s},V]\partial_{x}H\quad\text{
and }\quad
R_{U}(H,V):=-[\Lambda^{s},V-\kappa\tfrac{\partial_{x}H}{\underline{H}+H}]\partial_{x}V.$

Moreover, applying the operator $\Lambda^{s}$ to the identity
$V_{\ell}=U_{\ell}-\kappa\frac{\partial_{x}H_{\ell}}{\underline{H}_{\ell}+H_{\ell}}$
($\ell\in\\{s,b\\}$) yields

$\dot{V}_{\ell}=\dot{U}_{\ell}-\kappa\frac{\partial_{x}\dot{H}_{\ell}}{\underline{H}_{\ell}+H_{\ell}}+R(H_{\ell}),$

with

$R(H)=-\kappa[\Lambda^{s},\tfrac{1}{\underline{H}+H}]\partial_{x}H.$

Standard commutator estimates and composition estimates in Sobolev spaces; see
e.g. [35, Appendix B] yield

$\displaystyle\|R_{H}(H,V)\|_{L^{2}}$ $\displaystyle\leq
C\big{(}\|H\|_{H^{s_{0}}}+\|V\|_{H^{s_{0}}}\big{)}\big{(}\|H\|_{H^{s}}+\|V\|_{H^{s}}\big{)},$
$\displaystyle\|R_{U}(H,V)\|_{L^{2}}$ $\displaystyle\leq
C\big{(}\kappa\|\partial_{x}H\|_{H^{s_{0}}}+\|V\|_{H^{s_{0}}}\big{)}\big{(}\kappa\|\partial_{x}H\|_{H^{s}}+\|V\|_{H^{s}}\big{)},$
$\displaystyle\|R(H)\|_{H^{1}}$ $\displaystyle\leq
C\big{(}\|H\|_{H^{s_{0}}}+\kappa\|\partial_{x}H\|_{H^{s_{0}}}\big{)}\big{(}\|H\|_{H^{s}}+\kappa\|\partial_{x}H\|_{H^{s}}\big{)},$

where $C$ is a positive constant depending only on $s,s_{0}$,
$\|H\|_{H^{s_{0}}}$ and $\inf_{\mathbb{R}}(\underline{H}+H)>0$.

Moreover, notice that by using the equations (2.6), (2.7) and the identity
$\kappa\partial_{x}H_{\ell}=(\underline{H}_{\ell}+H_{\ell})(U_{\ell}-V_{\ell})$
for $\ell\in\\{s,b\\}$, we have

$\|(\partial_{t}H_{s},\partial_{t}H_{b},\partial_{t}U_{s},\partial_{t}U_{b})\|_{H^{s-1}}+\kappa\|(\partial_{x}H_{s},\partial_{x}H_{b})\|_{H^{s}}+\kappa^{-1}\|(U_{s}-V_{s},U_{b}-V_{b})\|_{H^{s-1}}\\\
\leq C\,\|(H_{s},H_{b},U_{s},U_{b},V_{s},V_{b})\|_{H^{s}},$

where the multiplicative constant $C$ depends on
$\|(H_{s},H_{b},U_{s},U_{b},V_{s},V_{b})\|_{H^{s_{0}}}$ and
$\inf_{\mathbb{R}}(\underline{H}_{\ell}+H_{\ell})>0$ (for $\ell\in\\{s,b\\}$).

We may thus apply Lemma 2.6, and infer that we can set $C$ depending on
$s_{0},M_{0},\varsigma$, and $c$ depending only on $\varsigma$, so that as
long as

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}(t,x),\underline{H}_{b}+H_{b}(t,x),\underline{U}_{s}+U_{s}(t,x),\underline{U}_{b}+U_{b}(t,x))\in\mathfrak{p}^{\varsigma/2},$
(2.10)

and

$\|(H_{s},H_{b},U_{s},U_{b},V_{s},V_{b})(t,\cdot)\|_{H^{s_{0}}}\leq 2cM_{0}$
(2.11)

one has (using the standard $H^{s_{0}}\subset W^{1,\infty}$ continuous
embedding and the fact that $0\leq\underline{H}_{s},\underline{H}_{b}\leq 1$
and $\underline{U}_{s}+\underline{U}_{b}=0$)

$\|(H_{s},H_{b},U_{s},U_{b},V_{s},V_{b})(t,\cdot)\|_{H^{s_{0}}}\leq
c\,M_{0}\exp(C\,M_{0}\,t)\\\
+CM_{0}\int_{0}^{t}\|(H_{s},H_{b},U_{s},U_{b},V_{s},V_{b})(t^{\prime},\cdot)\|_{H^{s_{0}}}\exp(C\,M_{0}\,(t-t^{\prime}))\operatorname{{\rm
d}\\!}t.$

Applying Gronwall’s Lemma, we find that if $CM_{0}t$ is smaller than a
universal constant, then

$\|(H_{s},H_{b},U_{s},U_{b},V_{s},V_{b})(t,\cdot)\|_{H^{s}}\leq\frac{3c}{2}\,M_{0}.$

What is more we have (augmenting $C$ if necessary)

$|(H_{s}(t,\cdot)-H_{s}^{0},H_{b}(t,\cdot)-H_{b}^{0},U_{s}(t,\cdot)-U_{s}^{0},U_{b}(t,\cdot)-U_{b}^{0})|\leq\int_{0}^{t}\|(\partial_{t}H_{s},\partial_{t}H_{b},\partial_{t}U_{s},\partial_{t}U_{b})\|_{L^{\infty}}\leq
CM_{0}t.$

Hence lowering further $CM_{0}t$, we infer that
$(\rho_{s},\rho_{b},H_{s}(t,x),H_{b}(t,x),U_{s}(t,x),U_{b}(t,x))\in\mathfrak{p}^{\varsigma/4}$
for all $x\in\mathbb{R}$. By the usual continuity argument we infer that the
assumptions (2.10) and (2.11) do hold for $t\in[0,T/M_{0}]$ with $T$ depending
on $s,s_{0},M_{0},\varsigma$. This yields the lower bound on the maximal time
of existence and the claimed upper bound on the solution follows from the
above estimates replacing $s_{0}$ with $s$. ∎

Combined together, Proposition 2.5 and Proposition 2.7 yield a time of
existence for solutions to (2.6) emerging from sufficiently regular initial
data which is independent of $\kappa\in(0,1]$. The following result describes
the behavior of these solutions as $\kappa\searrow 0$.

######  Proposition 2.8 (Convergence).

Let $s\geq s_{0}>3/2$, $\varsigma\in(0,1)$ and $M_{0}>0$.

Let $\kappa\in(0,1]$,
$(\rho_{s},\rho_{b},\underline{H}_{s},\underline{H}_{b},\underline{U}_{s},\underline{U}_{b})\in\mathbb{R}^{6}$
such that $\underline{H}_{s}+\underline{H}_{b}=1$ and
$\underline{U}_{s}+\underline{U}_{b}=0$, and
$(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\in H^{s+2}(\mathbb{R})^{4}$ be such
that the hyperbolicity condition holds

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}^{0}(x),\underline{H}_{b}+H_{b}^{0}(x),\underline{U}_{s}+U_{s}^{0}(x),\underline{U}_{b}+U_{b}^{0}(x))\in\mathfrak{p}^{\varsigma},$

where $\mathfrak{p}^{\varsigma}$ is defined in (2.5), and

$\|(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0},\kappa\partial_{x}H_{s}^{0},\kappa\partial_{x}H_{b}^{0})\|_{H^{s_{0}}}\leq
M_{0}.$

Denote

  * •

$(H_{s},H_{b},U_{s},U_{b})$ the maximal solution to the non-diffusive system
(2.1) emerging from the initial data

$(H_{s},H_{b},U_{s},U_{b})\big{|}_{t=0}=(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})$

as defined in Proposition 2.4;

  * •

for all $\kappa>0$,
$(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa})$ the maximal
solution to system (2.6) emerging from the initial data

$(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa})\big{|}_{t=0}=(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})$

as defined in Proposition 2.5;

  * •

$T_{0}>0$ and $c_{0}>1$ such that for all $t\in[0,T_{0}]$,

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}(t,x),\underline{H}_{b}+H_{b}(t,x),\underline{U}_{s}+U_{s}(t,x),\underline{U}_{b}+U_{b}(t,x))\in\mathfrak{p}^{\varsigma/c_{0}}$

and

$\|(H_{s},H_{b},U_{s},U_{b})(t,\cdot)\|_{H^{s+2}}\leq c_{0}M_{0}.$

Then there exists $\kappa_{0}>0$ and $C>0$, both depending only on
$s,s_{0},\varsigma,M_{0},T_{0},c_{0}$, such that for all
$\kappa\in(0,\kappa_{0}]$,
$(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa})(t,\cdot)$ is
well-defined for all $t\in[0,T_{0}]$ and satisfies the hyperbolicity condition

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}^{\kappa}(t,x),\underline{H}_{b}+H_{b}^{\kappa}(t,x),\underline{U}_{s}+U_{s}^{\kappa}(t,x),\underline{U}_{b}+U_{b}^{\kappa}(t,x))\in\mathfrak{p}^{\varsigma/(2c_{0})}$

and the upper bound

$\|(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa})(t,\cdot)\|_{H^{s}}\leq
2c_{0}M_{0}.$

Moreover, one has for all $t\in[0,T_{0}]$

$\|(H_{s}^{\kappa}-H_{s},H_{b}^{\kappa}-H_{b},U_{s}^{\kappa}-U_{s},U_{b}^{\kappa}-U_{b})(t,\cdot)\|_{H^{s}}\leq\kappa\,CM_{0}.$

###### Proof.

Denote $V_{\ell}:=U_{\ell}$ and
$V_{\ell}^{\kappa}:=U_{\ell}^{\kappa}-\kappa\frac{\partial_{x}H_{\ell}^{\kappa}}{\underline{H}_{\ell}+H_{\ell}^{\kappa}}$
($\ell\in\\{s,b\\}$),
$\Lambda^{s}:=(\operatorname{Id}-\partial_{x}^{2})^{s/2}$, and

$(\dot{H}_{s},\dot{H}_{b},\dot{U}_{s},\dot{U}_{b},\dot{V}_{s},\dot{V}_{b}):=(\Lambda^{s}(H_{s}^{\kappa}-H_{s}),\Lambda^{s}(H_{b}^{\kappa}-H_{b}),\Lambda^{s}(U_{s}^{\kappa}-U_{s}),\Lambda^{s}(U_{b}^{\kappa}-U_{b}),\Lambda^{s}(V_{s}^{\kappa}-V_{s}),\Lambda^{s}(V_{b}^{\kappa}-V_{b})).$

Substracting (2.6), (2.7) and (2.1) we obtain

$\begin{cases}\partial_{t}\dot{H}_{s}+(\underline{H}_{s}+H_{s}^{\kappa})\partial_{x}\dot{U}_{s}+(\underline{U}_{s}+U_{s}^{\kappa})\partial_{x}\dot{H}_{s}=\kappa\partial_{x}^{2}\dot{H}_{s}+\kappa\Lambda^{s}\partial_{x}^{2}H_{s}+R_{H}(H_{s}^{\kappa},U_{s}^{\kappa},H_{s},U_{s}),\\\
\partial_{t}\dot{H}_{b}+(\underline{H}_{b}+H_{b}^{\kappa})\partial_{x}\dot{U}_{b}+(\underline{U}_{b}+U_{b}^{\kappa})\partial_{x}\dot{H}_{b}=\kappa\partial_{x}^{2}\dot{H}_{b}+\kappa\Lambda^{s}\partial_{x}^{2}H_{b}+R_{H}(H_{b}^{\kappa},U_{b}^{\kappa},H_{b},U_{b}),\\\
\partial_{t}\dot{U}_{s}+(\underline{U}_{s}+U_{s}^{\kappa}-\kappa\frac{\partial_{x}H_{s}^{\kappa}}{\underline{H}_{s}+H_{s}^{\kappa}})\partial_{x}\dot{U}_{s}+\partial_{x}\dot{H}_{s}+\partial_{x}\dot{H}_{b}=\kappa\Lambda^{s}\big{(}\frac{\partial_{x}H_{s}^{\kappa}}{\underline{H}_{s}+H_{s}^{\kappa}}\partial_{x}U_{s}\big{)}+R_{U}(H_{s}^{\kappa},U_{s}^{\kappa},H_{s},U_{s}),\\\
\partial_{t}\dot{U}_{b}+(\underline{U}_{b}+U_{b}^{\kappa}-\kappa\frac{\partial_{x}H_{b}^{\kappa}}{\underline{H}_{b}+H_{b}^{\kappa}})\partial_{x}\dot{U}_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}\dot{H}_{s}+\partial_{x}\dot{H}_{b}=\kappa\Lambda^{s}\big{(}\frac{\partial_{x}H_{b}^{\kappa}}{\underline{H}_{b}+H_{b}^{\kappa}}\partial_{x}U_{b}\big{)}+R_{U}(H_{b}^{\kappa},U_{b}^{\kappa},H_{b},U_{b}),\end{cases}$

and

$\begin{cases}\partial_{t}\dot{H}_{s}+(\underline{H}_{s}+H_{s}^{\kappa})H_{s}^{\kappa}\partial_{x}\dot{V}_{s}+(\underline{U}_{s}+V_{s}^{\kappa})\partial_{x}\dot{H}_{s}=R_{H}(H_{s}^{\kappa},V_{s}^{\kappa},H_{s},V_{s}),\\\
\partial_{t}\dot{H}_{b}+(\underline{H}_{b}+H_{b}^{\kappa})H_{b}^{\kappa}\partial_{x}\dot{V}_{b}+(\underline{U}_{b}+V_{b}^{\kappa})\partial_{x}\dot{H}_{b}=R_{H}(H_{b}^{\kappa},V_{b}^{\kappa},H_{b},V_{b}),\\\
\partial_{t}\dot{V}_{s}+(\underline{U}_{s}+V_{s}^{\kappa}-\kappa\frac{\partial_{x}H_{s}^{\kappa}}{\underline{H}_{s}+H_{s}^{\kappa}})\partial_{x}\dot{V}_{s}+\partial_{x}\dot{H}_{s}+\partial_{x}\dot{H}_{b}\\\
\hskip
142.26378pt=\kappa\partial_{x}^{2}\dot{V}_{s}+\kappa\Lambda^{s}\partial_{x}^{2}V_{s}+\kappa\Lambda^{s}\big{(}\frac{\partial_{x}H_{s}^{\kappa}}{\underline{H}_{s}+H_{s}^{\kappa}}\partial_{x}V_{s}\big{)}+R_{U}(H_{s}^{\kappa},V_{s}^{\kappa},H_{s},V_{s}),\\\
\partial_{t}\dot{V}_{b}+(\underline{U}_{b}+V_{b}^{\kappa}-\kappa\frac{\partial_{x}H_{b}^{\kappa}}{\underline{H}_{b}+H_{b}^{\kappa}})\partial_{x}\dot{V}_{b}+\frac{\rho_{s}}{\rho_{b}}\partial_{x}\dot{H}_{s}+\partial_{x}\dot{H}_{b}\\\
\hskip
142.26378pt=\kappa\partial_{x}^{2}\dot{V}_{b}+\kappa\Lambda^{s}\partial_{x}^{2}V_{b}+\kappa\Lambda^{s}\big{(}\frac{\partial_{x}H_{b}^{\kappa}}{\underline{H}_{b}+H_{b}^{\kappa}}\partial_{x}V_{b}\big{)}+R_{U}(H_{b}^{\kappa},V_{b}^{\kappa},H_{b},V_{b}),\end{cases}$

where, for any $\ell\in\\{s,b\\}$, we denote

$\displaystyle R_{H}(H_{\ell}^{\kappa},U_{\ell}^{\kappa},H_{\ell},U_{\ell})$
$\displaystyle=-\Lambda^{s}\big{(}(H_{\ell}^{\kappa}-H_{\ell})\partial_{x}U_{\ell}+(U_{\ell}^{\kappa}-U_{\ell})\partial_{x}H_{\ell}\big{)}$
$\displaystyle\quad-[\Lambda^{s},H_{\ell}^{\kappa}](\partial_{x}U_{\ell}^{\kappa}-\partial_{x}U_{\ell})-[\Lambda^{s},U_{\ell}^{\kappa}](\partial_{x}H_{\ell}^{\kappa}-\partial_{x}H_{\ell}),$
$\displaystyle R_{U}(H_{\ell}^{\kappa},U_{\ell}^{\kappa},H_{\ell},U_{\ell})$
$\displaystyle=-\Lambda^{s}\big{(}(U_{\ell}^{\kappa}-U_{\ell})\partial_{x}U_{\ell}\big{)}-[\Lambda^{s},U_{\ell}^{\kappa}-\kappa\tfrac{\partial_{x}H_{\ell}^{\kappa}}{\underline{H}_{\ell}+H_{\ell}^{\kappa}}](\partial_{x}U_{\ell}^{\kappa}-\partial_{x}U_{\ell}).$

Moreover, one has by definition

$\dot{V}_{\ell}=\dot{U}_{\ell}-\kappa\frac{\partial_{x}\dot{H}_{\ell}}{\underline{H}_{\ell}+H_{\ell}^{\kappa}}+R(H_{\ell}^{\kappa},H_{\ell})$

where

$R(H_{\ell}^{\kappa},H_{\ell})=-\kappa[\Lambda^{s},\tfrac{1}{\underline{H}_{\ell}+H_{\ell}^{\kappa}}]\partial_{x}H_{\ell}^{\kappa}-\kappa\frac{\partial_{x}\Lambda^{s}H_{\ell}}{\underline{H}_{\ell}+H_{\ell}^{\kappa}}.$

Standard commutator estimates and composition estimates in Sobolev spaces; see
e.g. [35, Appendix B] yield

$\displaystyle\|R_{H}(H_{\ell}^{\kappa},V_{\ell}^{\kappa},H_{\ell},V_{\ell})\|_{L^{2}}$
$\displaystyle\leq
C\big{(}\|\partial_{x}H_{\ell}\|_{H^{s}}+\|\partial_{x}V_{\ell}\|_{H^{s}}+\|\partial_{x}H_{\ell}^{\kappa}\|_{H^{s-1}}+\|\partial_{x}V_{\ell}^{\kappa}\|_{H^{s-1}}\big{)}\big{(}\|\dot{H}_{\ell}\|_{L^{2}}+\|\dot{V}_{\ell}\|_{L^{2}}\big{)},$
$\displaystyle\|R_{U}(H_{\ell}^{\kappa},V_{\ell}^{\kappa},H_{\ell},V_{\ell})\|_{L^{2}}$
$\displaystyle\leq
C\big{(}\|\partial_{x}V_{\ell}\|_{H^{s}}+\kappa\|\partial_{x}H_{\ell}^{\kappa}\|_{H^{s}}+\|\partial_{x}V_{\ell}^{\kappa}\|_{H^{s-1}}\big{)}\|\dot{V}_{\ell}\|_{L^{2}},$
$\displaystyle\|R(H_{\ell}^{\kappa},H_{\ell})\|_{H^{1}}$ $\displaystyle\leq
C\kappa\big{(}\|\partial_{x}H_{\ell}^{\kappa}\|_{H^{s}}+\|\partial_{x}H_{\ell}\|_{H^{s+1}}\big{)},$

where $C$ is a positive constant depending only on $s$, $s_{0}$,
$\|H_{\ell}^{\kappa}\|_{H^{s_{0}}}$ and
$\inf_{\mathbb{R}}\underline{H}_{\ell}+H_{\ell}^{\kappa}>0$ for any
$\ell\in\\{s,b\\}$.

Moreover notice that by the equations (2.6) and (2.7) and using the identity
$\kappa\partial_{x}H_{\ell}^{\kappa}=(\underline{H}_{\ell}+H_{\ell}^{\kappa})(U_{\ell}^{\kappa}-V_{\ell}^{\kappa})$
we have

$\|(\partial_{t}H_{s}^{\kappa},\partial_{t}H_{b}^{\kappa},\partial_{t}U_{s}^{\kappa},\partial_{t}U_{b}^{\kappa})\|_{H^{s-1}}+\kappa\|(\partial_{x}H_{s}^{\kappa},\partial_{x}H_{b}^{\kappa})\|_{H^{s}}+\kappa^{-1}\|(U_{s}^{\kappa}-V_{s}^{\kappa},U_{b}^{\kappa}-V_{b}^{\kappa})\|_{H^{s-1}}\\\
\leq
C\|(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa},V_{s}^{\kappa},V_{b}^{\kappa})\|_{H^{s}},$

where the multiplicative constant $C$ depends on
$\|(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa},V_{s}^{\kappa},V_{b}^{\kappa})\|_{H^{s_{0}}}$
and $\inf_{\mathbb{R}}\underline{H}_{\ell}+H_{\ell}^{\kappa}>0$ for
${\ell\in\\{s,b\\}}$.

We may thus apply Lemma 2.6, and infer that we can set $C$ depending on
$s,s_{0},M_{0},\varsigma$, and $c$ depending only on $\varsigma$, so that as
long as

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}^{\kappa}(t,x),\underline{H}_{b}+H_{b}^{\kappa}(t,x),\underline{U}_{s}+U_{s}^{\kappa}(t,x),\underline{U}_{b}+U_{b}^{\kappa}(t,x))\in\mathfrak{p}^{\varsigma/(2c_{0})},$
(2.12)

and

$\|(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa},V_{s}^{\kappa},V_{b}^{\kappa})(t,\cdot)\|_{H^{s}}\leq
2c_{0}M_{0}$ (2.13)

one has, using that
$(\dot{H}_{s},\dot{H}_{b},\dot{U}_{s},\dot{U}_{b},\dot{V}_{s},\dot{V}_{b})|_{t=0}=(0,0,0,0,-\kappa\Lambda^{s}(\frac{\partial_{x}H_{s}^{\kappa}}{\underline{H}_{s}+H_{s}^{\kappa}})|_{t=0},-\kappa\Lambda^{s}(\frac{\partial_{x}H_{b}^{\kappa}}{\underline{H}_{b}+H_{b}^{\kappa}})|_{t=0})$,

$\|(\dot{H}_{s},\dot{H}_{b},\dot{U}_{s},\dot{U}_{b},\dot{V}_{s},\dot{V}_{b})(t,\cdot)\|_{L^{2}}+c\kappa^{1/2}\|(\partial_{x}\dot{H}_{s},\partial_{x}\dot{H}_{b},\partial_{x}\dot{V}_{s},\partial_{x}\dot{V}_{b})\|_{L^{2}(0,t;L^{2})}\leq\kappa
CM_{0}\exp(Cc_{0}M_{0}t)\\\
+C\int_{0}^{t}\Big{(}c_{0}M_{0}\|(\dot{H}_{s},\dot{H}_{b},\dot{U}_{s},\dot{U}_{b},\dot{V}_{s},\dot{V}_{b},\kappa\partial_{x}\dot{H}_{s},\kappa\partial_{x}\dot{H}_{b})(t^{\prime},\cdot)\|_{L^{2}}\\\
+\kappa\|(\partial_{x}H_{s}^{\kappa},\partial_{x}H_{b}^{\kappa},\partial_{x}^{2}H_{s},\partial_{x}^{2}H_{b},\partial_{x}^{2}V_{s},\partial_{x}^{2}V_{b},\partial_{x}U_{s},\partial_{x}U_{b},\partial_{x}V_{s},\partial_{x}V_{b})(t^{\prime},\cdot)\|_{H^{s}}\Big{)}\exp(C\,c_{0}M_{0}\,(t-t^{\prime}))\operatorname{{\rm
d}\\!}t.$

Now, we use in the integrand the triangle inequality

$\|(\partial_{x}H_{s}^{\kappa},\partial_{x}H_{b}^{\kappa})(t^{\prime},\cdot)\|_{H^{s}}\leq\|(\partial_{x}\dot{H}_{s},\partial_{x}\dot{H}_{b})(t^{\prime},\cdot)\|_{L^{2}}+\|(\partial_{x}H_{s},\partial_{x}H_{b})(t^{\prime},\cdot)\|_{H^{s}}.$

The first contribution may be absorbed by the left-hand side if $\kappa$ is
sufficiently small (depending on $c,C,c_{0}M_{0},T_{0}$), and the second
contribution is estimated, as other terms, using the assumption

$\sup\big{(}\big{\\{}\|(H_{s},H_{b},U_{s},U_{b})(t,\cdot)\|_{H^{s+2}}\ :\
t\in[0,T_{0}]\big{\\}}\big{)}\leq c_{0}M_{0}.$

Applying then Gronwall’s Lemma, we find that

$\|(\dot{H}_{s},\dot{H}_{b},\dot{U}_{s},\dot{U}_{b},\dot{V}_{s},\dot{V}_{b})(t,\cdot)\|_{L^{2}}\leq\kappa
Cc_{0}M_{0}K$ (2.14)

where $K$ depends only on $Cc_{0}M_{0}T_{0}$.

By using again the triangle inequality, we can lower further $\kappa$
(depending on $c_{0}$) so that (2.14) implies

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},H_{s}^{\kappa}(t,x),H_{b}^{\kappa}(t,x),U_{s}^{\kappa}(t,x),U_{b}^{\kappa}(t,x))\in\mathfrak{p}^{2\varsigma/(3c_{0})}$

and

$\|(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa},V_{s}^{\kappa},V_{b}^{\kappa})(t,\cdot)\|_{H^{s}}\leq\frac{3}{2}c_{0}M_{0}.$

Hence by the usual continuity argument we infer that
$(H_{s}^{\kappa},H_{b}^{\kappa},U_{s}^{\kappa},U_{b}^{\kappa})(t,\cdot)$ is
well-defined for all $t\in[0,T_{0}]$, and (2.12)-(2.13)-(2.14) hold. This
concludes the proof. ∎

##  3 The hydrostatic Euler equations

In this section we study the stability of the hydrostatic Euler equations for
stratified flows:

$\displaystyle\partial_{t}h+\partial_{x}((1+h)(\underline{u}+u))$
$\displaystyle=\kappa\partial_{x}^{2}h,$ (3.1)
$\displaystyle\partial_{t}u+\left(\underline{u}+u-\kappa\frac{\partial_{x}h}{1+h}\right)\partial_{x}u+\frac{1}{\underline{\rho}}\partial_{x}\Psi$
$\displaystyle=0,$

where the Montgomery potential $\Psi$ is given by

$\Psi(\cdot,r)=\underline{\rho}(r)\int_{-1}^{r}h(\cdot,r^{\prime})\operatorname{{\rm
d}\\!}r^{\prime}+\int_{r}^{0}\underline{\rho}(r^{\prime})h(\cdot,r^{\prime})\operatorname{{\rm
d}\\!}r^{\prime}=:({\sf M}[\underline{\rho}]h)(r).$ (3.2)

We recall that our stability results must accommodate with solutions generated
by the bilayer system that are piecewise constant and allow the comparison
with continuously stratified flows. Hence we must allow for deviations that
can be large pointwise, while smallness stems from integration (with respect
to the $r$-variable). In practice we shall manipulate simultaneously the
pointwise as well as the $L^{1}_{r}$ topologies depending on the need. This is
the case for instance in the following Lemma, measuring the Lipschitz
continuity with respect to the density variable of the Montgomery operator
${\sf M}[\underline{\rho}]$ defined (3.2).

######  Lemma 3.1.

Let $\underline{M}>0$. There exists $C>0$ such that for any
$\underline{\rho}_{\ell}$ ($\ell\in\\{1,2\\}$) such that

$\|(\underline{\rho}_{\ell},\tfrac{1}{\underline{\rho}_{\ell}})\|_{L^{1}_{r}\times
L^{\infty}_{r}}\leq\underline{M},$

and for any $h\in L^{\infty}_{r}$, one has for almost any $r\in(-1,0)$,

$\Big{|}\Big{(}\frac{1}{\underline{\rho}_{1}}{\sf
M}[\underline{\rho}_{1}]h-\frac{1}{\underline{\rho}_{2}}{\sf
M}[\underline{\rho}_{2}]h\Big{)}(r)\Big{|}\leq\Big{(}\underline{M}^{3}|\underline{\rho}_{1}-\underline{\rho}_{2}|(r)+\underline{M}\|\underline{\rho}_{1}-\underline{\rho}_{2}\|_{L^{1}_{r}}\Big{)}\|h\|_{L^{\infty}_{r}}$

###### Proof.

$\Big{(}\frac{1}{\underline{\rho}_{1}}{\sf
M}[\underline{\rho}_{1}]h-\frac{1}{\underline{\rho}_{2}}{\sf
M}[\underline{\rho}_{2}]h\Big{)}(r)=\big{(}\frac{1}{\underline{\rho}_{1}(r)}-\frac{1}{\underline{\rho}_{2}(r)}\big{)}\int_{r}^{0}\underline{\rho}_{1}(r^{\prime})h(r^{\prime})\operatorname{{\rm
d}\\!}r^{\prime}+\frac{1}{\underline{\rho}_{2}(r)}\int_{r}^{0}(\underline{\rho}_{1}(r^{\prime})-\underline{\rho}_{2}(r^{\prime}))h(r^{\prime})\operatorname{{\rm
d}\\!}r^{\prime}.$

∎

Note also that we are seeking stability estimates for the system (3.1)-(3.2)
with respect to perturbations of the equations —in particular through
$\underline{\rho}\approx\underline{\rho}_{\rm bl}$ and
$\underline{u}\approx\underline{u}_{\rm bl}$— and with respect to
perturbations of the initial data.

In Section 3.1, we first state a local well-posedness result associated with
the initial-value problem for the system (3.1)-(3.2), and then provide some
stability estimates. These two results by themselves are not sufficient to
bootstrap in a standard manner the strong convergence of solutions as the size
of deviations shrink, because the topology involved in the first result,
namely $L^{\infty}_{r}$, is stronger than the topology used in the second
result, which is roughly speaking $L^{1}_{r}$. For that matter we introduce
and study in Section 3.2 a refined approximate solution which, compared with
the original reference solution, improves the description of solutions
associated with nearby profiles. Specifically, the refined approximate
solution satisfies the following three properties:

  1. i.

it is well-defined and controlled on a time interval which is uniform with
respect to $\kappa\in(0,1]$;

  2. ii.

the difference with respect to the nearby solution is controlled for the
strong norm associated with $L^{\infty}_{r}$;

  3. iii.

the difference with respect to the reference solution is controlled for the
weak norm associated with $L^{1}_{r}$.

The resulting convergence result, Proposition 3.8, is stated and proved in
Section 3.3.

###  3.1 Stability estimates

######  Proposition 3.2 (Well-posedness).

Let $s\geq s_{0}>3/2$, $\varsigma\in(0,1)$, $\underline{M},M_{0}>0$ and $c>1$.
There exists $T>0$ such that the following holds.

For all $\kappa\in(0,1]$, all $(\underline{\rho},\underline{u})\in
L^{\infty}((-1,0))$ such that

$\|(\underline{u},\underline{\rho},\tfrac{1}{\underline{\rho}})\|_{L^{\infty}_{r}}\leq\underline{M},$

and all $(h^{0},u^{0})\in L^{\infty}((-1,0);H^{s}(\mathbb{R})^{2})$ such that
for almost all $r\in(-1,0)$,

$\forall x\in\mathbb{R},\quad 1+h^{0}\geq\varsigma,$

and

$\|(h^{0},u^{0})\|_{L^{\infty}_{r}H^{s_{0}}_{x}}\leq M_{0}$

there exists a unique $(h,u)\in
C([0,T^{\star});L^{\infty}((-1,0);H^{s}(\mathbb{R})^{2})$ maximal-in-time
strong solution to (3.1)–(3.2) emerging from the initial data
$(h,u)\big{|}_{t=0}=(h^{0},u^{0})$.

Moreover, $T^{\star}>\kappa T$ and for any $t\in[0,\kappa T]$ and almost all
$r\in(-1,0)$ one has

$\forall x\in\mathbb{R},\quad 1+h(t,x,r)\geq\varsigma/c$

and

$\max(\\{\|(h,u)\|_{L^{\infty}(0,t;L^{\infty}_{r}H^{s}_{x})},\kappa^{1/2}\|\partial_{x}h\|_{L^{2}(0,t;L^{\infty}_{r}H^{s}_{x})}\\})\leq
c\|(h^{0},u^{0})\|_{L^{\infty}_{r}H^{s}_{x}}.$

Moreover, the maximal existence time (resp. the emerging solution in
$C([0,T^{\star});L^{\infty}((-1,0);H^{s}(\mathbb{R})^{2})$) is a lower semi-
continuous (resp. continuous) function of the initial data in
$L^{\infty}((-1,0);H^{s}(\mathbb{R})^{2}$ and if $T^{\star}<\infty$ then

$\|(h(t,\cdot),u(t,\cdot))\|_{L^{\infty}_{r}H^{s_{0}}_{x}}\to\infty\text{ as
}t\to T^{\star}.$

###### Proof.

The proof is very similar to the proof of Proposition 2.5, using estimates for
transport and transport-diffusion equations pointwisely with respect to the
variable $r\in(-1,0)$. The essential arguments are that $L^{\infty}((-1,0))$
is a Banach algebra and that differentiation with respect to the space
variable $\partial_{x}$ as well as the Fourier multiplier
${\Lambda^{s}=(\operatorname{Id}-\partial_{x}^{2})^{s/2}}$ commute with the
operator $\tfrac{1}{\underline{\rho}}{\sf M}[\underline{\rho}]$, and that the
linear operator $\tfrac{1}{\underline{\rho}}{\sf
M}[\underline{\rho}]:L^{\infty}_{r}L^{2}_{x}\to L^{\infty}_{r}L^{2}_{x}$ is
bounded for any $\underline{\rho}\in L^{\infty}((-1,0))$. ∎

######  Proposition 3.3 (Stability).

Let $s>3/2$, $\varsigma\in(0,1)$ and $\underline{M},M_{1},M_{2}>0$. There
exists $C>0$ such that the following holds.

Let $\kappa\in(0,1]$ and $T>0$ be such that

$CT\leq\kappa.$

Let $\underline{\rho}_{\ell}$, $\underline{u}_{\ell}$ (for $\ell\in\\{1,2\\}$)
be such that

$\|(\underline{\rho}_{\ell},\tfrac{1}{\underline{\rho}_{\ell}})\|_{L^{\infty}_{r}}\leq\underline{M}.$

Let $(h_{\ell},u_{\ell})$ be solutions to (3.1)-(3.2) (with
$\underline{\rho}=\underline{\rho}_{\ell}$ and
$\underline{u}=\underline{u}_{\ell}$) defined on the interval $[0,T]$ and
satisfying

$\|(\partial_{x}h_{1},\partial_{x}u_{1})\|_{L^{\infty}_{T}L^{\infty}_{r}H^{s}_{x}}\leq
M_{1},\qquad\|(h_{2},u_{2})\|_{L^{\infty}_{T}L^{\infty}_{r}H^{s}_{x}}+\kappa^{1/2}\,\|\partial_{x}h_{2}\|_{L^{\infty}_{r}L^{2}_{T}H^{s}_{x}}\leq
M_{2},$

and

$\operatorname*{ess\,inf}_{(t,x,r)\in[0,T]\times\mathbb{R}\times(-1,0)}1+h_{1}(t,x,r)\geq\varsigma,\qquad\operatorname*{ess\,inf}_{(t,x,r)\in[0,T]\times\mathbb{R}\times(-1,0)}1+h_{2}(t,x,r)\geq\varsigma.$

Then one has

$\max(\\{\|(h_{1}-h_{2},u_{1}-u_{2})\|_{L^{\infty}_{T}L^{1}_{r}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}(h_{1}-h_{2})\|_{L^{2}_{T}L^{1}_{r}H^{s}_{x}}\\})\\\
\leq
2\|(h_{1}^{0}-h_{2}^{0},u_{1}^{0}-u_{2}^{0})\|_{L^{1}_{r}H^{s}_{x}}+CT\,\|(\underline{\rho}_{1}-\underline{\rho}_{2},\underline{u}_{1}-\underline{u}_{2})\|_{L^{1}_{r}}$
(3.3)

and for almost any $r\in(-1,0)$,

$\max(\\{\|(h_{1}-h_{2},u_{1}-u_{2})(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}(h_{1}-h_{2})(\cdot,r)\|_{L^{2}_{T}H^{s}_{x}}\\})\\\
\leq
2\big{(}\|(h_{1}^{0}-h_{2}^{0},u_{1}^{0}-u_{2}^{0})(r)\|_{H^{s}_{x}}+\|(h_{1}^{0}-h_{2}^{0},u_{1}^{0}-u_{2}^{0})\|_{L^{1}_{r}H^{s}_{x}}\big{)}\\\
+CT\,\big{(}|(\underline{\rho}_{1}-\underline{\rho}_{2},\underline{u}_{1}-\underline{u}_{2})(r)|+\|(\underline{\rho}_{1}-\underline{\rho}_{2},\underline{u}_{1}-\underline{u}_{2})\|_{L^{1}_{r}}\big{)}.$
(3.4)

###### Proof.

Let us denote $\dot{h}:=h_{1}-h_{2}$, $\dot{u}:=u_{1}-u_{2}$,
$\dot{}\underline{u}:=\underline{u}_{1}-\underline{u}_{2}$ and
$\dot{\underline{\rho}}=\underline{\rho}_{1}-\underline{\rho}_{2}$. We have on
the time interval $I:=[0,T]$

$\begin{cases}\partial_{t}\dot{h}+(\underline{u}_{2}+u_{2})\partial_{x}\dot{h}-\kappa\partial_{x}^{2}\dot{h}=r_{1}+r_{2},\\\
\partial_{t}\dot{u}+\left(\underline{u}_{2}+u_{2}+u_{2}^{\star}\right)\partial_{x}\dot{u}=r_{3},\end{cases}$

where we denote $u_{2}^{\star}:=-\kappa\tfrac{\partial_{x}h_{2}}{1+h_{2}}$,

$r_{1}:=-(\dot{}\underline{u}+\dot{u})\partial_{x}h_{1}-\dot{h}\partial_{x}u_{1},\quad
r_{2}:=-(1+h_{2})\partial_{x}\dot{u}$

and

$r_{3}:=-\left(\dot{}\underline{u}+\dot{u}-\kappa\tfrac{\partial_{x}\dot{h}}{1+h_{2}}+\kappa\dot{h}\tfrac{\partial_{x}h_{1}}{(1+h_{1})(1+h_{2})}\right)\partial_{x}u_{1}-\left(\tfrac{1}{\underline{\rho}_{1}}{\sf
M}[\underline{\rho}_{1}]-\tfrac{1}{\underline{\rho}_{2}}{\sf
M}[\underline{\rho}_{2}]\right)\partial_{x}h_{1}-\tfrac{1}{\underline{\rho}_{2}}{\sf
M}[\underline{\rho}_{2}]\partial_{x}\dot{h}.$

We can now use standard estimates and transport-diffusion and transport
equations ([6]) to infer that there exists $c_{0}$ depending only on $s$ such
that for almost any $r\in(-1,0)$, one has

$\max(\\{\|\dot{h}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}\dot{h}(\cdot,r)\|_{L^{2}_{T}H^{s}_{x}}\\})\leq\Big{(}\|\dot{h}(t=0,\cdot,r)\|_{H^{s}_{x}}+\|r_{1}(\cdot,r)\|_{L^{1}_{T}H^{s}_{x}}+\kappa^{-1/2}\|r_{2}(\cdot,r)\|_{L^{2}_{T}H^{s-1}_{x}}\Big{)}\\\
\times\exp(c_{0}\|\partial_{x}u_{2}(\cdot,r)\|_{L^{1}_{T}H^{s-1}})$

and

$\|\dot{u}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}\leq\Big{(}\|\dot{u}(t=0,\cdot,r)\|_{H^{s}_{x}}+\|r_{3}(\cdot,r)\|_{L^{1}_{T}H^{s}_{x}}\Big{)}\times\exp(c_{0}\|\partial_{x}u_{2}(\cdot,r)+\partial_{x}u_{2}^{\star}(\cdot,r)\|_{L^{1}_{T}H^{s-1}_{x}}).$

Using that $H^{s^{\prime}}(\mathbb{R})$ is a Banach algebra for all
$s^{\prime}>1/2$, we find that for any $t\in[0,T]$

$\|r_{1}(t,\cdot,r)\|_{H^{s}_{x}}\lesssim
M_{1}\times\big{(}|\dot{}\underline{u}(r)|+\|\dot{u}(t,\cdot,r)\|_{H^{s}_{x}}+\|\dot{h}(t,\cdot,r)\|_{H^{s}_{x}}\big{)},\quad\|r_{2}(t,\cdot,r)\|_{H^{s-1}_{x}}\lesssim(1+M_{2})\|\dot{u}(t,\cdot,r)\|_{H^{s}_{x}},$

and, making additionally use of standard composition estimates in Sobolev
spaces ([35, Appendix B]) and Lemma 3.1,

$\|r_{3}(t,\cdot,r)\|_{H^{s}_{x}}\leq
C(s,\underline{M},M_{1},M_{2},\varsigma)\Big{(}|\dot{}\underline{u}(r)|+\|\dot{u}(t,\cdot,r)\|_{H^{s}_{x}}+\kappa\|\dot{h}(t,\cdot,r)\|_{H^{s+1}_{x}}+|\dot{\underline{\rho}}(r)|+\|\dot{\underline{\rho}}\|_{L^{1}_{r}}\Big{)}M_{1}\\\
+\underline{M}^{2}\|\partial_{x}\dot{h}(t,\cdot)\|_{L^{1}_{r}H^{s}_{x}}.$

Finally, using product and composition estimates on
$u_{2}^{\star}=-\kappa\partial_{x}h_{2}+\kappa\tfrac{h_{2}}{1+h_{2}}\partial_{x}h_{2}$,
we have

$\|\partial_{x}u_{2}^{\star}(t,\cdot,r)\|_{H^{s-1}_{x}}\leq
C(M_{2},\varsigma^{-1})\,\kappa\,\|\partial_{x}h_{2}(t,\cdot,r)\|_{H^{s}_{x}}.$

Collecting these estimates we infer that there exists $C>0$ depending only on
$s,\underline{M},M_{1},M_{2},\varsigma$ such that

$\max(\\{\|\dot{h}(\cdot,r),\dot{u}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}\dot{h}(\cdot,r)\|_{L^{2}_{T}H^{s}_{x}}\\})\leq\exp(CM_{2}(T+\kappa^{1/2}T^{1/2}))\\\
\times\Big{(}\|(\dot{h}(t=0,\cdot,r),\dot{u}(t=0,\cdot,r))\|_{H^{s}_{x}}+C\kappa^{-1/2}T^{1/2}\big{(}\|\dot{u}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}+\kappa^{1/2}\|\partial_{x}\dot{h}\|_{L^{2}_{T}L^{1}_{r}H^{s}_{x}}\big{)}\\\
+CM_{1}T\times\big{(}|\dot{}\underline{u}(r)|+|\dot{\underline{\rho}}(r)|+\|\dot{\underline{\rho}}\|_{L^{1}_{r}}+\|\dot{h}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}+\|\dot{u}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}+\kappa
T^{-1/2}\|\partial_{x}\dot{h}(\cdot,r)\|_{L^{2}_{T}H^{s}_{x}}\big{)}\Big{)}.$

Hence there exists $C>0$, depending only on
$s,\underline{M},M_{1},M_{2},\varsigma$ such that for any $T$ sufficiently
small so that one has $M_{1}T\leq(36C)^{-1}$,
$(\kappa^{-1/2}+\kappa^{1/2}M_{1})T^{1/2}\leq(18C)^{-1}$ and
$M_{2}T+M_{2}\kappa^{1/2}T^{1/2}\leq C^{-1}\ln(3/2)$ one has

$\frac{5}{6}\max(\\{\|(\dot{h}(\cdot,r),\dot{u}(\cdot,r))\|_{L^{\infty}_{T}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}\dot{h}(\cdot,r)\|_{L^{2}_{T}H^{s}_{x}}\\})\\\
\leq\frac{3}{2}\Big{(}\|(\dot{h}(t=0,\cdot,r),\dot{u}(t=0,\cdot,r))\|_{H^{s}_{x}}+\tfrac{1}{18}\kappa^{1/2}\|\partial_{x}\dot{h}\|_{L^{2}_{T}L^{1}_{r}H^{s}_{x}}\big{)}+CM_{1}T\,\big{(}|\dot{}\underline{u}(r)|+|\dot{\underline{\rho}}(r)|+\|\dot{\underline{\rho}}\|_{L^{1}_{r}}\big{)}\Big{)}.$
(3.5)

Integrating this inequality with respect to the variable $r$ and using
Minkowski’s inequality, we infer the first stability estimate, (3.3). Plugging
(3.3) in the right-hand side of (3.5), the second stability estimate, (3.4),
follows immediately. ∎

######  Remark 3.4.

The restriction $T\lesssim\kappa$ in Propositions 3.2 and 3.3 is quite
stringent. In [9], some improved stability estimates concerning the system
(3.1)-(3.2) were derived by the authors. The latter estimates exploit a
partial symmetric structure of the equations, and demand some extra regularity
with respect to the variable $r$. Because we cannot afford such regularity
since our stability estimates will be used with piecewise constant functions,
we use in the proof of Proposition 3.3 in a stronger way the parabolic
regularization of thickness diffusivity.

In order to obtain a final result on a timescale which is independent of the
parameter $\kappa\in(0,1]$ we shall exploit some a priori control on a
reference solution and restrict to initial data as well as shear velocity and
density distributions that are close to the reference data.

###  3.2 Refined approximation

In this section we consider a given reference solution to the hydrostatic
Euler equation (3.1)-(3.2) and build from it a refined approximate solution
associated with nearby profiles. Specifically, let $\underline{\rho}_{\rm
ref},\underline{u}_{\rm ref},h_{\rm ref},u_{\rm ref}$ be a solution to

$\displaystyle\partial_{t}h_{\rm ref}+\partial_{x}((1+h_{\rm
ref})(\underline{u}_{\rm ref}+u_{\rm ref}))$
$\displaystyle=\kappa\partial_{x}^{2}h_{\rm ref},$ (3.6)
$\displaystyle\partial_{t}u_{\rm ref}+\left(\underline{u}_{\rm ref}+u_{\rm
ref}-\kappa\frac{\partial_{x}h_{\rm ref}}{1+h_{\rm
ref}}\right)\partial_{x}u_{\rm ref}+\frac{1}{\underline{\rho}_{\rm ref}}{\sf
M}[\underline{\rho}_{\rm ref}]\partial_{x}h_{\rm ref}$ $\displaystyle=0,$

where we recall that the operator ${\sf M}$ is defined in (3.2). Considering
profiles $(\underline{\rho},\underline{u})$ which are in some sense close to
$(\underline{\rho}_{\rm ref},\underline{u}_{\rm ref})$ we construct
approximate solutions $(h_{\rm app},u_{\rm app})$ as the solutions to

$\displaystyle\partial_{t}h_{\rm app}+\partial_{x}((1+h_{\rm
app})(\underline{u}+u_{\rm app}))$ $\displaystyle=\kappa\partial_{x}^{2}h_{\rm
app},$ (3.7) $\displaystyle\partial_{t}u_{\rm app}+\left(\underline{u}+u_{\rm
app}-\kappa\frac{\partial_{x}h_{\rm app}}{1+h_{\rm
app}}\right)\partial_{x}u_{\rm app}$
$\displaystyle=-\frac{1}{\underline{\rho}}{\sf
M}[\underline{\rho}]\partial_{x}h_{\rm ref}.$

Notice first that $(h_{\rm app},u_{\rm app})$ satisfies approximately the
hydrostatic Euler equations associated with profiles
$(\underline{\rho},\underline{u})$.

######  Proposition 3.5.

For any $s\geq 0$ the refined approximate solution $(h^{\rm app},u^{\rm app})$
satisfies

$\displaystyle\partial_{t}h_{\rm app}+\partial_{x}((1+h_{\rm
app})(\underline{u}+u_{\rm app}))$ $\displaystyle=\kappa\partial_{x}^{2}h_{\rm
app},$ (3.8) $\displaystyle\partial_{t}u_{\rm app}+\left(\underline{u}+u_{\rm
app}-\kappa\frac{\partial_{x}h_{\rm app}}{1+h_{\rm
app}}\right)\partial_{x}u_{\rm app}+\frac{1}{\underline{\rho}}{\sf
M}[\underline{\rho}]\partial_{x}h_{\rm app}$ $\displaystyle=r_{\rm rem}$

with

$\|r_{\rm
rem}(t,\cdot)\|_{L^{\infty}_{r}H^{s}_{x}}\leq\|\underline{\rho}\|_{L^{\infty}_{r}}\|\tfrac{1}{\underline{\rho}}\|_{L^{\infty}_{r}}\|(h_{\rm
ref}-h_{\rm app})(t,\cdot)\|_{L^{1}_{r}H^{s+1}_{x}}$

###### Proof.

We have

$\displaystyle r_{\rm rem}(\cdot,r)$
$\displaystyle=\Big{(}\frac{1}{\underline{\rho}}{\sf
M}[\underline{\rho}](\partial_{x}h_{\rm app}-\partial_{x}h_{\rm
ref})\Big{)}(\cdot,r)$ $\displaystyle=\int_{-1}^{r}(\partial_{x}h_{\rm
app}-\partial_{x}h_{\rm ref})(\cdot,r^{\prime})\operatorname{{\rm
d}\\!}r^{\prime}+\frac{1}{\underline{\rho}(r)}\int_{r}^{0}\underline{\rho}(r^{\prime})(\partial_{x}h_{\rm
app}-\partial_{x}h_{\rm ref})(\cdot,r^{\prime})\operatorname{{\rm
d}\\!}r^{\prime},$

and the result follows since
$1\leq\sup(\\{\underline{\rho}(r^{\prime})/\underline{\rho}(r)\ :\
(r,r^{\prime})\in(-1,0)^{2}\\})\leq\|\underline{\rho}\|_{L^{\infty}_{r}}\|\tfrac{1}{\underline{\rho}}\|_{L^{\infty}_{r}}$.
∎

That the above remainder term $r_{\rm app}$ is small is a consequence of the
subsequent Proposition 3.7. We first prove that for any initial data $(h_{\rm
app},u_{\rm app})|_{t=0}=(h^{0},u^{0})$, the emerging solution $(h_{\rm
app},u_{\rm app})$ is well-defined and controlled on a time interval uniform
with respect to $\kappa\in(0,1]$.

######  Proposition 3.6.

Let $s>3/2$, $\varsigma\in(0,1)$, $\underline{M},M_{0},M_{\rm ref}>0$ and
$c>1$. There exists $C>0$ and $T>0$ such that the following holds.

Let $h_{\rm ref}\in C([0,T_{\rm ref});L^{1}((0,1);H^{s+1}_{x}(\mathbb{R})))$
be such that

$\|\partial_{x}h_{\rm ref}\|_{L^{1}_{T_{\rm ref}}L^{1}_{r}H^{s}_{x}}\leq
M_{\rm ref}.$

For all $\kappa\in(0,1]$, all $(\underline{\rho},\underline{u})\in
L^{\infty}((-1,0))$ such that

$\|(\underline{\rho},\tfrac{1}{\underline{\rho}})\|_{L^{\infty}_{r}}\leq\underline{M},$

and all $(h^{0},u^{0})\in L^{\infty}((0,1);H^{s}(\mathbb{R})^{2})$ such that
for almost all $r\in(-1,0)$

$\forall x\in\mathbb{R},\quad 1+h^{0}\geq\varsigma$

and

$\max(\\{\|h^{0}\|_{L^{\infty}_{r}H^{s-1}_{x}},\kappa^{1/2}\|h^{0}\|_{L^{\infty}_{r}H^{s}_{x}},\|u^{0}\|_{L^{\infty}_{r}H^{s}_{x}}\\})\leq
M_{0},$

there exists a unique $(h_{\rm app},u_{\rm app})\in
C([0,T^{\star});L^{\infty}((0,1);H^{s}_{x}(\mathbb{R})^{2}))$ maximal solution
to (3.7) emerging from the initial data $(h_{\rm app},u_{\rm
app})|_{t=0}=(h^{0},u^{0})$. Moreover one has $T^{\star}\geq T_{\rm
app}:=\min(\\{T_{\rm ref},T\\})$ and for any $t\in[0,T_{\rm app}]$ and almost
any $r\in(-1,0)$ one has the upper bound

$\max(\\{\|h_{\rm app}(\cdot,r)\|_{L^{\infty}_{T_{\rm
app}}H^{s-1}_{x}},\kappa^{1/2}\|h_{\rm app}(\cdot,r)\|_{L^{\infty}_{T_{\rm
app}}H^{s}_{x}},\kappa\|\partial_{x}h_{\rm app}(\cdot,r)\|_{L^{2}_{T_{\rm
app}}H^{s}_{x}}\\})+\|u_{\rm app}(\cdot,r)\|_{L^{\infty}_{T_{\rm
app}}H^{s}_{x}}\\\ \leq
c\max(\\{\|h^{0}(\cdot,r)\|_{H^{s-1}_{x}},\kappa^{1/2}\|h^{0}(t=0,\cdot,r)\|_{H^{s}_{x}},\|u^{0}(\cdot,r)\|_{H^{s}_{x}}\\})+CM_{\rm
ref}.$

###### Proof.

The existence and uniqueness of $(h_{\rm app},u_{\rm app})\in
C([0,T^{\star});L^{\infty}((0,1);H^{s}_{x}(\mathbb{R})^{2}))$ maximal solution
to (3.7) is obtained as in Proposition 3.2. We set $T\in[0,T^{\star})$. By
standard estimates on transport and transport-diffusion equations ([6])
applied to (3.7) we have

$\max(\\{\|h_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s-1}_{x}},\kappa^{1/2}\|\partial_{x}h_{\rm
app}(\cdot,r)\|_{L^{2}_{T}H^{s-1}_{x}}\\})\leq\Big{(}\|h^{0}(\cdot,r)\|_{H^{s-1}_{x}}+\|r_{\rm
app}(\cdot,r)\|_{L^{1}_{T}H^{s-1}_{x}}\Big{)}\\\
\times\exp(c_{0}\|\partial_{x}u_{\rm app}(\cdot,r)\|_{L^{1}_{T}H^{s-1})}),$
$\max(\\{\|h_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}h_{\rm
app}(\cdot,r)\|_{L^{2}_{T}H^{s}_{x}}\\})\leq\Big{(}\|h^{0}(\cdot,r)\|_{H^{s}_{x}}+\kappa^{-1/2}\|r_{\rm
app}(\cdot,r)\|_{L^{2}_{T}H^{{s}-1}_{x}}\Big{)}\\\
\times\exp(c_{0}\|\partial_{x}u_{\rm app}(\cdot,r)\|_{L^{1}_{T}H^{s-1})}),$

and

$\|u_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}\leq\Big{(}\|u^{0}(\cdot,r)\|_{H^{s}_{x}}+\|r_{\rm
ref}(\cdot,r)\|_{L^{1}_{T}H^{s}_{x}}\Big{)}\times\exp(c_{0}\|\partial_{x}u_{\rm
app}(\cdot,r)+\partial_{x}u_{\rm
app}^{\star}(\cdot,r)\|_{L^{1}_{T}H^{s-1}_{x}}),$

where we denote $u_{\rm app}^{\star}:=-\kappa\tfrac{\partial_{x}h_{\rm
app}}{1+h_{\rm app}}$, $r_{\rm app}:=-(1+h_{\rm app})\partial_{x}u_{\rm app}$,
$r_{\rm ref}=-\frac{1}{\underline{\rho}}{\sf
M}[\underline{\rho}]\partial_{x}h_{\rm ref}$, and the constant $c_{0}$ depends
only on $s$.

Now we notice that for almost any $r\in(-1,0)$,

$\displaystyle\|\partial_{x}u_{\rm app}^{\star}(t,\cdot,r)\|_{H^{s-1}_{x}}$
$\displaystyle\leq C(\|h_{\rm
app}(t,\cdot,r)\|_{H^{s-1}_{x}},\varsigma^{-1})\,\kappa\,\|\partial_{x}h_{\rm
app}(t,\cdot,r)\|_{H^{s}_{x}},$ $\displaystyle\|r_{\rm
app}(t,\cdot,r)\|_{H^{s-1}_{x}}$ $\displaystyle\leq C(\|h_{\rm
app}(t,\cdot,r)\|_{H^{s-1}_{x}})\|u_{\rm app}(t,\cdot,r)\|_{H^{s}_{x}},$
$\displaystyle\|r_{\rm ref}(t,\cdot,r)\|_{H^{s}_{x}}$ $\displaystyle\leq
C(\|\underline{\rho}\|_{L^{\infty}_{r}}\|\tfrac{1}{\underline{\rho}}\|_{L^{\infty}_{r}})\|\partial_{x}h_{\rm
ref}(t,\cdot)\|_{L^{1}_{r}H^{s}_{x}}.$

From this we infer that there exists $C$, depending only on $s,\|h_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s-1}_{x}},\varsigma^{-1},\|\underline{\rho}\|_{L^{\infty}_{r}}\|\tfrac{1}{\underline{\rho}}\|_{L^{\infty}_{r}}$
such that

$\max(\\{\|h_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s-1}_{x}},\kappa^{1/2}\|h_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}},\kappa\|\partial_{x}h_{\rm
app}(\cdot,r)\|_{L^{2}_{T}H^{s}_{x}},\|u_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}\\})\\\
\leq\Big{(}\max(\\{\|h^{0}(\cdot,r)\|_{H^{s-1}_{x}},\kappa^{1/2}\|h^{0}(t=0,\cdot,r)\|_{H^{s}_{x}},\|u^{0}(\cdot,r)\|_{H^{s}_{x}}\\})\\\
+C(T+T^{1/2})\|u_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}+C\|\partial_{x}h_{\rm
ref}\|_{L^{1}_{T}L^{1}_{r}H^{s}_{x}}\Big{)}\\\ \times\exp(CT\|u_{\rm
app}(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}+CT^{1/2}\kappa\|\partial_{x}h_{\rm
app}(\cdot,r)\|_{L^{2}_{T}H^{s}_{x}}).$

By the standard continuity argument, we find that

$\max(\\{\|h_{\rm app}(\cdot,r)\|_{L^{\infty}_{T_{\rm
app}}H^{s-1}_{x}},\kappa^{1/2}\|h_{\rm app}(\cdot,r)\|_{L^{\infty}_{T_{\rm
app}}H^{s}_{x}},\kappa\|\partial_{x}h_{\rm app}(\cdot,r)\|_{L^{2}_{T_{\rm
app}}H^{s}_{x}},\|u_{\rm app}(\cdot,r)\|_{L^{\infty}_{T_{\rm
app}}H^{s}_{x}}\\})\\\ \leq
c\,\max(\\{\|h^{0}(\cdot,r)\|_{H^{s-1}_{x}},\kappa^{1/2}\|h^{0}(t=0,\cdot,r)\|_{H^{s}_{x}},\|u^{0}(\cdot,r)\|_{H^{s}_{x}}\\})+CM_{\rm
ref}$

for almost all $r\in(0,1)$ and for all $T_{\rm app}\in[0,T]$ such that

$(T_{\rm app}+T_{\rm app}^{1/2})M_{0}\leq C^{-1}$

where $C$ depends only on $s,\varsigma,\underline{M},cM_{0}$. This concludes
the proof. ∎

We conclude this section by investigating the difference between the reference
solution and the refined approximate solution.

######  Proposition 3.7.

Let $s>3/2$, $\varsigma\in(0,1)$ and $\underline{M},M_{\rm ref},M_{\rm
app}>0$. There exists $C>0$ such that the following holds.

Let $\kappa\in(0,1]$, $T>0$ and let $\underline{\rho}_{\rm ref}$,
$\underline{u}_{\rm ref},\underline{\rho},\underline{u}\in L^{\infty}_{r}$ be
such that

$\|(\underline{\rho}_{\rm ref},\tfrac{1}{\underline{\rho}_{\rm
ref}},\underline{\rho},\tfrac{1}{\underline{\rho}})\|_{L^{\infty}_{r}}\leq\underline{M}.$

Let $(h_{\rm ref},u_{\rm ref})\in
C([0,T];L^{\infty}((-1,0);H^{s+1}(\mathbb{R})^{2})$ be a solution to (3.6)
(that is (3.1)-(3.2) with
$(\underline{\rho},\underline{u})=(\underline{\rho}_{\rm
ref},\underline{u}_{\rm ref})$) defined on the time interval $[0,T]$ and
satisfying

$\|(h_{\rm ref},u_{\rm ref})\|_{L^{\infty}_{T}L^{\infty}_{r}H^{s+1}_{x}}\leq
M_{\rm ref}.$

Let $(h_{\rm app},u_{\rm app})\in
C([0,T];L^{\infty}((-1,0);H^{s+1}(\mathbb{R})^{2})$ be solution to (3.7)
defined on the time interval $[0,T]$ and satisfying

$\|(h_{\rm app},u_{\rm
app})\|_{L^{\infty}_{T}L^{\infty}_{r}H^{s}_{x}}+\kappa^{1/2}\,\|\partial_{x}h_{\rm
app}\|_{L^{\infty}_{r}L^{2}_{T}H^{s}_{x}}\leq M_{\rm app},$

and such that for all $t\in[0,T]$ and almost all $r\in(-1,0)$

$\inf_{x\in\mathbb{R}}1+h_{\rm
ref}(t,x,r)\geq\varsigma,\qquad\inf_{x\in\mathbb{R}}1+h_{\rm
app}(t,x,r)\geq\varsigma.$

Then one has

$\|(h_{\rm ref}-h_{\rm app},u_{\rm ref}-u_{\rm
app})(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}\leq\Big{(}\|(h_{\rm ref}-h_{\rm
app},u_{\rm ref}-u_{\rm app})(t=0,\cdot,r)\|_{H^{s}_{x}}\\\
+C\kappa\,\big{(}|(\underline{\rho}_{\rm
ref}-\underline{\rho},\underline{u}_{\rm
ref}-\underline{u})(r)|+\|(\underline{\rho}_{\rm
ref}-\underline{\rho})\|_{L^{1}_{r}}\big{)}\Big{)}\exp(CT/\kappa).$ (3.9)

###### Proof.

Let us denote $\dot{h}:=h_{\rm ref}-h_{\rm app}$, $\dot{u}:=u_{\rm ref}-u_{\rm
app}$, $\dot{}\underline{u}:=\underline{u}_{\rm ref}-\underline{u}$ and
$\dot{\underline{\rho}}=\underline{\rho}_{\rm ref}-\underline{\rho}$. We have
on the time interval $I:=[0,T]$ for which both function are well-defined

$\begin{cases}\partial_{t}\dot{h}+(\underline{u}+u_{\rm
app})\partial_{x}\dot{h}-\kappa\partial_{x}^{2}\dot{h}=r_{1}+r_{2},\\\
\partial_{t}\dot{u}+\left(\underline{u}+u_{\rm app}+u_{\rm
app}^{\star}\right)\partial_{x}\dot{u}=r_{3},\end{cases}$

where $u_{\rm app}^{\star}:=-\kappa\tfrac{\partial_{x}h_{\rm app}}{1+h_{\rm
app}}$,

$r_{1}:=-(\dot{}\underline{u}+\dot{u})\partial_{x}h_{\rm
ref}-\dot{h}\partial_{x}u_{\rm ref},\quad r_{2}:=-(1+h_{\rm
app})\partial_{x}\dot{u}$

and

$r_{3}:=-\left(\dot{}\underline{u}+\dot{u}-\kappa\tfrac{\partial_{x}\dot{h}}{1+h_{\rm
app}}+\kappa\dot{h}\tfrac{\partial_{x}h_{\rm ref}}{(1+h_{\rm ref})(1+h_{\rm
app})}\right)\partial_{x}u_{\rm ref}-\left(\tfrac{1}{\underline{\rho}_{\rm
ref}}{\sf M}[\underline{\rho}_{\rm ref}]-\tfrac{1}{\underline{\rho}}{\sf
M}[\underline{\rho}]\right)\partial_{x}h_{\rm ref}.$

We can proceed as in the proof of Proposition 3.3 with some straightforward
adjustments as for the contributions of $r_{3}$ since the contribution
$-\tfrac{1}{\underline{\rho}}{\sf M}[\underline{\rho}]\partial_{x}\dot{h}$ is
nonexistent.

We infer that there exists $C$ depending only on
$s,\varsigma,\underline{M},M_{\rm app},M_{\rm ref}$ such that for all
$\kappa\in(0,1]$ and $T_{0}\in(0,T]$ such that $CT_{0}\leq\kappa$, one has for
almost any $r\in(-1,0)$,

$\max(\\{\|(\dot{h},\dot{u})(\cdot,r)\|_{L^{\infty}_{T_{0}}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}\dot{h}(\cdot,r)\|_{L^{2}_{T_{0}}H^{s}_{x}}\\})\leq
2\|(\dot{h},\dot{u})(t=0,\cdot,r)\|_{H^{s}_{x}}+C^{2}T_{0}\,\big{(}|\dot{}\underline{\rho}(r)|+|\dot{}\underline{u}(r)|+\|\dot{}\underline{\rho}\|_{L^{1}_{r}}\big{)}.$

Iterating this control on $T_{n}:=\min(\\{n\kappa/C,T\\})$, we find that

$\|(\dot{h},\dot{u})(\cdot,r)\|_{L^{\infty}_{T_{n}}H^{s}_{x}}\leq
2^{n+1}\|(\dot{h},\dot{u})(t=0,\cdot,r)\|_{H^{s}_{x}}+C2^{n}\kappa\,\big{(}|\dot{}\underline{\rho}(r)|+|\dot{}\underline{u}(r)|+\|\dot{}\underline{\rho}\|_{L^{1}_{r}}\big{)},$

which yields the claimed estimate. ∎

###  3.3 Convergence

We now conclude our analysis with the following stability result for solutions
to the hydrostatic Euler equations.

######  Proposition 3.8 (Convergence).

Let $s>3/2$, $\underline{M},M_{\rm ref},M_{0}>0$, and $\kappa\in(0,1]$. Then
there exists $C,T$ independent of $\kappa$ and $\delta_{0}>0$ (depending on
$\kappa$) such that the following holds.

Let $(\underline{\rho}_{\rm ref},\underline{u}_{\rm ref})\in
L^{\infty}((-1,0))^{2}$ and $(h_{\rm ref},u_{\rm ref})\in
C([0,T];L^{\infty}((-1,0);H^{s+2}(\mathbb{R})^{2})$ be a solution to (3.6)
(that is (3.1)-(3.2) with
$(\underline{\rho},\underline{u})=(\underline{\rho}_{\rm
ref},\underline{u}_{\rm ref})$) defined on the time interval $[0,T]$ such that

$\|(\underline{\rho}_{\rm ref},\tfrac{1}{\underline{\rho}_{\rm
ref}})\|_{L^{\infty}_{r}}\leq\underline{M},\quad\|(h_{\rm ref},u_{\rm
ref})\|_{L^{\infty}_{T}L^{\infty}_{r}H^{s+2}_{x}}+\|\partial_{x}h_{\rm
ref}\|_{L^{1}_{T}L^{1}_{r}H^{s+2}_{x}}\leq M_{\rm ref}.$

Let $(\underline{\rho},\underline{u})\in L^{\infty}((-1,0))^{2}$ and
$(h^{0},u^{0})\in L^{\infty}((-1,0);H^{s+2}(\mathbb{R})^{2})$ be such that

$\|(\underline{\rho},\tfrac{1}{\underline{\rho}})\|_{L^{\infty}_{r}}\leq\underline{M},\quad\|(h^{0},\kappa^{1/2}\partial_{x}h^{0})\|_{L^{\infty}_{r}H^{s+1}_{x}}+\|u^{0}\|_{L^{\infty}_{r}H^{s+2}_{x}}\leq
M_{0}$

and

$\|(\underline{\rho}_{\rm ref}-\underline{\rho},\underline{u}_{\rm
ref}-\underline{u})\|_{L^{1}_{r}}+\|(h_{\rm ref}|_{t=0}-h^{0},u_{\rm
ref}|_{t=0}-u^{0})\|_{L^{1}_{r}H^{s+1}}\leq\delta_{0}.$

Then $(h,u)\in C([0,T^{\star});L^{\infty}((-1,0);H^{s+2}(\mathbb{R})^{2})$ the
maximal-in-time solution to (3.1)-(3.2) emerging from initial data
$(h,u)|_{t=0}=(h^{0},u^{0})$ is defined on the time interval $[0,T]$ and we
have for almost all $r\in(-1,0)$,

$\|(h_{\rm ref}-h,u_{\rm
ref}-u)(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}\leq\Big{(}\|(h_{\rm
ref}-h_{,}u_{\rm ref}-u)(t=0,\cdot,r)\|_{H^{s}_{x}}+\|(h_{\rm ref}-h,u_{\rm
ref}-u)(t=0,\cdot)\|_{L^{1}_{r}H^{s+1}_{x}}\\\ +|(\underline{\rho}_{\rm
ref}-\underline{\rho},\underline{u}_{\rm
ref}-\underline{u})(r)|+\|(\underline{\rho}_{\rm
ref}-\underline{\rho},\underline{u}_{\rm
ref}-\underline{u})\|_{L^{1}_{r}}\Big{)}\times C\exp(CT/\kappa).$ (3.10)

###### Proof.

We first recall that Proposition 3.7 provides an estimate on the difference
between the reference solution $(h_{\rm ref},u_{\rm ref})\in
C([0,T];L^{\infty}((-1,0);H^{s+2}(\mathbb{R})^{2})$ and the corresponding
approximate solution defined as the solution to the system (3.7) with initial
data $(h_{\rm app},u_{\rm app})|_{t=0}=(h^{0},u^{0})$, $(h_{\rm app},u_{\rm
app})\in C([0,T];L^{\infty}((-1,0);H^{s+2}_{x}(\mathbb{R})^{2}))$, whose
existence and control on the time interval $[0,T]$ (lessening $T$ if
necessary) is provided by Proposition 3.6. Specifically we have for almost any
$r\in(-1,0)$

$\|(h_{\rm ref}-h_{\rm app},u_{\rm ref}-u_{\rm
app})(\cdot,r)\|_{L^{\infty}_{T}H^{s+1}_{x}}\leq\Big{(}\|(h_{\rm ref}-h_{\rm
app},u_{\rm ref}-u_{\rm app})(t=0,\cdot,r)\|_{H^{s+1}_{x}}\\\
+C\kappa\,\big{(}|(\underline{\rho}_{\rm
ref}-\underline{\rho},\underline{u}_{\rm
ref}-\underline{u})(r)|+\|(\underline{\rho}_{\rm
ref}-\underline{\rho})\|_{L^{1}_{r}}\big{)}\Big{)}\exp(CT/\kappa),$

where $C$ depends only on $s,\varsigma,\underline{M},M_{\rm ref}$ and $M_{0}$.

We then consider the difference between the exact solution $(h,u)\in
C([0,T^{\star});L^{\infty}((-1,0);H^{s+2}_{x}(\mathbb{R})^{2}))$ —whose
existence is provided by Proposition 3.2— and the approximate solution. By
means of the consistency result, Proposition 3.5, we can adapt the proof of
Proposition 3.3 and we find that under the assumptions that

$\|(h,u)\|_{L^{\infty}_{T}L^{\infty}_{r}H^{s}_{x}}+\kappa^{1/2}\,\|\partial_{x}h\|_{L^{\infty}_{r}L^{2}_{T}H^{s}_{x}}\leq
M\quad\text{ and
}\quad\operatorname*{ess\,inf}_{(t,x,r)\in[0,T]\times\mathbb{R}\times(-1,0)}1+h(t,x,r)\geq\varsigma$
(3.11)

there exists $C>0$ depending only on $s,\varsigma,\underline{M},M_{\rm ref},M$
such that for all $\kappa\in(0,1]$ and $t\in(0,\min(\\{T,T^{\star}\\}))$ such
that $Ct\leq\kappa$ and for almost any $r\in(-1,0)$ one has

$\max(\\{\|(h-h_{\rm app},u-u_{\rm
app})(\cdot,r)\|_{L^{\infty}_{t}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}(h-h_{\rm
app})(\cdot,r)\|_{L^{2}_{t}H^{s}_{x}}\\})\\\ \leq 2\|(h-h_{\rm app},u-u_{\rm
app})(t=0,\cdot,r)\|_{L^{\infty}_{r}H^{s}_{x}}+Ct\,\|(h_{\rm ref}-h_{\rm
app})(t,\cdot)\|_{L^{1}_{r}H^{s+1}_{x}}.$

Iterating the stability estimate on $T_{n}:=\min(\\{nT_{0},T\\})$ we deduce
that (augmenting $C$ if necessary)

$\max(\\{\|(h-h_{\rm app},u-u_{\rm
app})\|_{L^{\infty}_{t}L^{\infty}_{r}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}(h-h_{\rm
app})\|_{L^{\infty}_{r}L^{2}_{t}H^{s}_{x}}\\})\\\
\leq\exp(CT/\kappa)\|(h-h_{\rm app},u-u_{\rm
app})(t=0,\cdot)\|_{L^{\infty}_{r}H^{s}_{x}}+C\kappa\exp(CT/\kappa)\,\|h_{\rm
ref}-h_{\rm app}\|_{L^{\infty}_{T}L^{1}_{r}H^{s+1}_{x}}.$

Since by construction $(h_{\rm app},u_{\rm app})|_{t=0}=(h,u)|_{t=0}$ and
using the above control on $h_{\rm ref}-h_{\rm app}$ we infer (augmenting $C$
if necessary)

$\max(\\{\|(h-h_{\rm app},u-u_{\rm
app})\|_{L^{\infty}_{t}L^{\infty}_{r}H^{s}_{x}},\kappa^{1/2}\|\partial_{x}(h-h_{\rm
app})\|_{L^{\infty}_{r}L^{2}_{t}H^{s}_{x}}\\})\\\ \leq
C\kappa\exp(CT/\kappa)\,\|(h_{\rm ref}-h,u_{\rm
ref}-u)(t=0,\cdot)\|_{L^{1}_{r}H^{s+1}_{x}}+C\big{(}\kappa\exp(CT/\kappa)\big{)}^{2}\,\|(\underline{\rho}_{\rm
ref}-\underline{\rho},\underline{u}_{\rm ref}-\underline{u})\|_{L^{1}_{r}}.$

Notice that this estimate implies by triangle inequality an upper bound on
$\|(h,u)\|_{L^{\infty}_{t}L^{\infty}_{r}H^{s}_{x}}+\kappa^{1/2}\,\|\partial_{x}h\|_{L^{\infty}_{r}L^{2}_{T}H^{s}_{x}}$
which (choosing $M$ sufficiently large and $\delta_{0}$ sufficiently small)
enforces strictly the condition (3.11), so that by the standard continuity
argument the above holds without restriction on
$t\in(0,\min(\\{T,T^{\star}\\}))$. By the persistence of regularity stated in
Proposition 3.2 we infer that $T^{\star}>T$ and the above holds for any
$t\in(0,T]$. The estimate (3.10) then immediately follows from the triangle
inequality. ∎

##  4 Conclusion

Thanks to Proposition 2.7 (concerning the well-posedness and control of
solutions to the bilayer system) on one hand and Proposition 3.8 (concerning
the control of the deviations of nearby solutions to some given reference
solutions) on the other hand, one infers immediately the announced rigorous
justification of the propagation in time of the columnar motion and sharp
stratification assumptions in near-bilayer situations (within the hydrostatic
framework). Specifically, we have the following result.

######  Theorem 4.1.

Let $s\geq s_{0}>3/2$, $\underline{M}>0$ ,$M_{0}>0$, $\varsigma\in(0,1)$, and
$\kappa\in(0,1]$. Then there exist $C>0,T>0$ independent of $\kappa$ and
$\delta_{0}>0$ (depending on $\kappa$) such that the following holds.

Let $(\underline{\rho},\underline{u})\in L^{\infty}((-1,0))^{2}$ and
$(h^{0},u^{0})\in L^{\infty}((-1,0);H^{s+2}(\mathbb{R})^{2})$ be such that

$\|(\underline{\rho},\tfrac{1}{\underline{\rho}})\|_{L^{\infty}_{r}}\leq\underline{M},\quad\|(h^{0},\kappa^{1/2}\partial_{x}h^{0})\|_{L^{\infty}_{r}H^{s+1}_{x}}+\|u^{0}\|_{L^{\infty}_{r}H^{s+2}_{x}}\leq
M_{0}$

and there exists
$(\rho_{s},\rho_{b},\underline{H}_{s},\underline{H}_{b},\underline{U}_{s},\underline{U}_{b})\in\mathbb{R}^{6}$
such that $\underline{H}_{s}+\underline{H}_{b}=1$ and
$\underline{U}_{s}+\underline{U}_{b}=0$ as well as
$(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0})\in H^{s+4}(\mathbb{R})^{2}\times
H^{s+3}(\mathbb{R})^{2}$ such that the hyperbolicity condition holds:

$\forall
x\in\mathbb{R},\quad(\rho_{s},\rho_{b},\underline{H}_{s}+H_{s}^{0}(x),\underline{H}_{b}+H_{b}^{0}(x),\underline{U}_{s}+U_{s}^{0}(x),\underline{U}_{b}+U_{b}^{0}(x))\in\mathfrak{p}^{\varsigma},$

where $\mathfrak{p}^{\varsigma}$ is defined in (2.5), and

$\|(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0},\kappa\partial_{x}H_{s}^{0},\kappa\partial_{x}H_{b}^{0})\|_{H^{s+3}}\leq
M_{0}$

and such that denoting $(\underline{\rho}_{\rm bl}^{0},\underline{u}_{\rm
bl}^{0},h_{\rm bl}^{0},u_{\rm bl}^{0})$ through (1.4) we have

$\|(h_{\rm bl}^{0}-h^{0},u_{\rm
bl}^{0}-u^{0})\|_{L^{1}_{r}H^{s+1}_{x}}+\|(\underline{\rho}_{\rm
bl}-\underline{\rho},\underline{u}_{\rm
bl}-\underline{u})\|_{L^{1}_{r}}\leq\delta_{0},$

then

  1. 1.

there exists $(H_{s},H_{b},U_{s},U_{b})\in C([0,T];H^{s+3}(\mathbb{R})^{4})$
solution to (1.3) emerging from the initial data

$(H_{s},H_{b},U_{s},U_{b})\big{|}_{t=0}=(H_{s}^{0},H_{b}^{0},U_{s}^{0},U_{b}^{0});$

  2. 2.

there exists $(h,u)\in C([0,T];L^{\infty}((-1,0);H^{s+2}(\mathbb{R})^{2})$
solution to (1.1) emerging from initial data

$(h,u)|_{t=0}=(h^{0},u^{0});$

  3. 3.

denoting $(\underline{\rho}_{\rm bl},\underline{u}_{\rm bl},h_{\rm bl},u_{\rm
bl})$ through (1.4) we have for almost all $r\in(-1,0)$,

$\|(h_{\rm bl}-h,u_{\rm
bl}-u)(\cdot,r)\|_{L^{\infty}_{T}H^{s}_{x}}\leq\Big{(}\|(h_{\rm
bl}^{0}-h^{0},u_{\rm bl}^{0}-u^{0})(\cdot,r)\|_{H^{s}_{x}}+\|(h_{\rm
bl}^{0}-h^{0},u_{\rm bl}^{0}-u^{0})\|_{L^{1}_{r}H^{s+1}_{x}}\\\
+|(\underline{\rho}_{\rm bl}-\underline{\rho},\underline{u}_{\rm
bl}-\underline{u})(r)|+\|(\underline{\rho}_{\rm
bl}-\underline{\rho},\underline{u}_{\rm
bl}-\underline{u})\|_{L^{1}_{r}}\Big{)}\times C\exp(Ct/\kappa).$

######  Remark 4.2.

Proposition 3.8 is not limited to the bilayer framework and applies to any
suitably regular reference solution. Hence it can be combined with results
analogous to Proposition 2.7 to provide results analogous to Theorem 4.1 in
the one-layer and multilayer frameworks.

The result analogous to Proposition 2.7 in the one-layer framework (that is
associated with the standard shallow water equation with thickness diffusivity
that was discussed for instance in [24]) is stated and proved in [2, Sect.
2.2]. Notice it requires neither the discussion on the hyperbolicity domain of
the non-diffusive equations (since the standard non-cavitation assumption
guarantees hyperbolicity) nor the discussion on the parabolic regularization
of the total velocity (since the natural symmetrizer behaves well with
diffusivity contributions).

A result analogous to Proposition 2.7 in the multilayer framework follows from
combining the result of [22] with the analysis of Section 2.2. In the former,
it is proved that assuming sufficiently small shear velocities is a sufficient
condition for the (strict) hyperbolicity of the multilayer system in the
stably stratified situation. Notice however that this smallness condition is
implicit, and not uniform with respect to the number of layers.

#### Acknowledgements

The authors thank the Institute for applied mathematics “Mauro Picone” (IAC)
for its hospitality when this work was initiated, and Laboratoire Ypatia des
Sciences Mathématiques LYSM CNRS-INdAM International Research Laboratory as
well as Univ. Rennes funding for its financial support. MA and VD thank the
program Centre Henri Lebesgue ANR-11-LABX-0020-0, for fostering an attractive
mathematical environment. RB acknowledges fundings from the Italian Ministry
of University and Research, PRIN 2022HSSYPN (Teseo).

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