# Long time well-posedness of Whitham-Boussinesq systems

Martin Oen Paulsen  Department of Mathematics
University of Bergen
Postbox 7800
5020 Bergen
Norway<EMAIL_ADDRESS>
###### Abstract.

Consideration is given to three different full dispersion Boussinesq systems
arising as asymptotic models in the bi-directional propagation of weakly
nonlinear surface waves in shallow water. We prove that, under a non-
cavitation condition on the initial data, these three systems are well-posed
on a time scale of order $\mathcal{O}(\frac{1}{{\varepsilon}})$, where
${\varepsilon}$ is a small parameter measuring the weak non-linearity of the
waves. This result seems new for one of these systems, even for short time.
The two other systems involve surface tension, and for one of them, the non-
cavitation condition has to be sharpened when the surface tension is small.
The proof relies on suitable symmetrizers and the classical theory of
hyperbolic systems. However, we have to track the small parameters carefully
in the commutator estimates to get the long time well-posedness.

Finally, combining our results with the recent ones of Emerald provide a full
justification of these systems as water wave models in a larger range of
regimes than the classical $(a,b,c,d)$-Boussinesq systems.

###### Key words and phrases:

Whitham-Boussinesq; Long time well-posedness; Symmetrizers

###### 2010 Mathematics Subject Classification:

Primary: 35Q35; Secondary: 76B15, 76B45

##  1\. Introduction

###  1.1. Full dispersion models

The Korteweg-de Vries (KdV) equation is an asymptotic model for the
unidirectional propagation of small amplitude, long waves on the surface of an
ideal fluid of constant depth. It was introduced in [8, 32] to model the
propagation of solitary waves in shallow water with a wide range of
applications both mathematically and physically. However, its dispersion is
too strong in high frequencies when compared to the full water wave system. In
particular, the KdV equation does not feature wave breaking or peaking waves.
To overcome these shortcomings, Whitham introduced in [52] an equation with an
improved dispersion relation. He replaced the KdV dispersion with the exact
dispersion of the linearized water wave system obtaining the equation

$\partial_{t}\zeta+\sqrt{\mathcal{K}_{\mu}}(D)\partial_{x}\zeta+{\varepsilon}\zeta\partial_{x}\zeta=0,$
(1.1)

for $(x,t)\in{\mathbb{R}}\times{\mathbb{R}}^{+}$, where the function
$\zeta(x,t)\in{\mathbb{R}}$ denotes the surface elevation and the operator
$\sqrt{\mathcal{K}_{\mu}}(D)$ is the square root of the Fourier multiplier
$\mathcal{K}_{\mu}(D)$ defined in frequency by

$K_{\mu}(\xi)=\frac{tanh(\sqrt{\mu}|\xi|)}{\sqrt{\mu}|\xi|}\left(1+\beta\mu\xi^{2}\right).$
(1.2)

Moreover, $\mu$ and ${\varepsilon}$ are small parameters related to the level
of dispersion and nonlinearity, and $\beta$ is a nonnegative parameter related
to the surface tension111Actually, Whitham introduced the equation formally
without the parameters $\mu$ and ${\varepsilon}$.,222 He also did not include
surface tension, i.e. $\beta=0$..

Whitham conjectured in [52] that equation (1.1) would allow, in addition to
the KdV traveling-wave regime, the occurrence of waves of greatest height with
a sharp crest as well as the formation of shocks. However, it was not until
recently that these phenomena were rigorously proved. We mention among others
the existence of periodic waves [18], the existence and stability of traveling
waves [17, 4, 48, 27], the formation of shocks [24, 45], Benjamin-Feir
instabilities [47, 25], the existence of periodic waves of greatest height
[20] and solitary waves of greatest height [50]. Note that in the case of
surface tension ($\beta>0$), the dynamics appear to be rather different (see
e.g. [31] and the references therein).

These results illustrate some mathematical properties uniquely related to an
improved dispersion relation, though there are some phenomena that the Whitham
equation does not feature due to its unidirectionality. For instance, the
Euler equations admit non-modulational instabilities of small-amplitude
periodic traveling waves [36], but the unidirectional nature of the Whitham
equation is believed to prohibit such instabilities [10].

Regarding the two-way propagation of waves at the surface of a fluid and in
the long wave regime, Bona, Chen, and Saut derived a three-parameter family of
Boussinesq systems [5]

$\begin{cases}(1-b\mu\partial_{x}^{2})\partial_{t}\zeta+(1+a\mu\partial_{x}^{2})\partial_{x}v+{\varepsilon}\partial_{x}(\zeta
v)=0\\\
(1-d\mu\partial_{x}^{2})\partial_{t}v+(1+c\mu\partial_{x}^{2})\partial_{x}\zeta+{\varepsilon}v\partial_{x}v=0,\end{cases}$
(1.3)

where $a$, $b$, $c$ and $d$ are real parameters satisfying
$a+b+c+d=\frac{1}{3}$, $\zeta$ is the deviation of the free surface with
respect to its rest state, and $v$ approximates the fluid velocity at some
height in the fluid domain. Like the KdV equation, the Boussinesq systems are
celebrated models for surface waves in coastal oceanography. Analogously to
the unidirectional case, one could replace the dispersion by the linearized
dispersion of the water wave equations in (1.3). These improved dispersion
versions are expected to lead to a more “accurate” description of the full
water wave system. Those systems are commonly referred to as the Whitham-
Boussinesq systems or full dispersion Boussinesq systems.

Actually, there are different possibilities of full dispersion Boussinesq
systems. This paper will focus on three important ones, linking them to some
specific cases of the Boussinesq systems without BBM terms ($b=d=0$). To be
precise, we introduce the operator $\mathcal{T}_{\mu}(D)$ corresponding to
$\mathcal{K}_{\mu}(D)$ for $\beta=0$, and whose Fourier symbol is defined by

$T_{\mu}(\xi)=\frac{tanh(\sqrt{\mu}|\xi|)}{\sqrt{\mu}|\xi|}.$ (1.4)

First, we consider the system

$\begin{cases}\partial_{t}\zeta+\mathcal{K}_{\mu}(D)\partial_{x}v+{\varepsilon}\partial_{x}(\zeta
v)=0\\\
\partial_{t}v+\partial_{x}\zeta+{\varepsilon}v\partial_{x}v=0,\end{cases}$
(1.5)

introduced in [33, 1, 38] without surface tension and in [31] with surface
tension. Here, as above, $\zeta$ denotes the elevation of the surface around
its equilibrium position, while $v$ approximates the fluid velocity at the
free surface. In the case zero surface tension, it is proved that (1.5) models
solitary waves [39] and admit high-frequency (non-modulational) instabilities
of small-amplitude periodic traveling waves [19]. We also observe that (1.5)
is related to (1.3) by expanding (1.2) in low frequencies. Indeed, since
$K_{\mu}(\xi)\simeq 1+\mu(\beta-\frac{1}{3})\xi^{2}$ by a Taylor expansion we
see that (1.5) reduce to (1.3) with $(a,b,c,d)=(\frac{1}{3}-\beta,0,0,0)$.

A second system is obtained by applying the operator (1.4) to
$\partial_{x}\zeta$, which gives

$\begin{cases}\partial_{t}\zeta+\partial_{x}v+{\varepsilon}\partial_{x}(\zeta
v)=0\\\
\partial_{t}v+\mathcal{T}_{\mu}(D)\partial_{x}\zeta+{\varepsilon}v\partial_{x}v=0.\end{cases}$
(1.6)

This system was first introduced in [26], where it is proved that (1.6)
features Benjamin-Feir (modulational) instabilities. Note that while $\zeta$
plays the same role as for system (1.5), it is $\mathcal{T}_{\mu}^{-1}(D)v$
which approximates the velocity potential at the free surface in this case. We
also observe that the system reduces in the formal limit $\sqrt{\mu}|\xi|\to
0$ to the Boussinesq system (1.3) with $(a,b,c,d)=(0,0,\frac{1}{3},0)$ in low
frequencies.

Finally, we will also consider a full dispersion version of (1.3) when
$\mathcal{T}_{\mu}(D)$ is applied to the nonlinear terms, while
$\mathcal{K}_{\mu}(D)$ is applied on the $\partial_{x}\zeta$. This system
reads

$\begin{cases}\partial_{t}\zeta+\partial_{x}v+{\varepsilon}\mathcal{T}_{\mu}(D)\partial_{x}(\zeta
v)=0\\\
\partial_{t}v+\mathcal{K}_{\mu}(D)\partial_{x}\zeta+{\varepsilon}\mathcal{T}_{\mu}(D)(v\partial_{x}v)=0.\end{cases}$
(1.7)

Here $\zeta$ and $v$ play the same roles as for system (1.6). It was
introduced in [13] and has the advantage of being Hamiltonian. Moreover, the
existence of solitary waves is proved in [14].

###  1.2. Full justification

A fundamental question in the derivation of an asymptotic model is whether its
solution converges to the solution of the original physical system. In
particular, we say that an asymptotic model is a valid approximation of the
Euler equations with a free surface if we can answer the following points in
the affirmative [33]:

  1. 1.

The solutions of the water wave equations exist on the relevant scale
$\mathcal{O}(\frac{1}{{\varepsilon}})$.

  2. 2.

The solutions of the asymptotic model exist (at least) on the scale
$\mathcal{O}(\frac{1}{{\varepsilon}})$.

  3. 3.

Lastly, we must establish the consistency between the asymptotic model and the
water wave equations, and then show that the error is of order
$\mathcal{O}(\mu{\varepsilon}t)$ when comparing the two solutions.

The first point was proved by Alvarez-Samaniego and Lannes [2] for surface
gravity waves and Ming, Zhang and Zhang [37] for gravity-capillary waves in
the weakly transverse regime, while points $2.$ and $3.$ are specific to the
asymptotic model under consideration. For instance, in the case of the Whitham
equation, Klein et al. [31] compared its solution rigorously with those of the
KdV equation. In particular, they proved that the difference of two solutions
evolving from the same initial datum is bounded by
$\mathcal{O}({\varepsilon}^{2}t)$ for all $0\leq t\lesssim{\varepsilon}^{-1}$
with ${\varepsilon},\mu$ in the KdV-regime:

$\mathcal{R}_{KdV}=\\{({\varepsilon},\mu)\>:\>0\leq\mu\leq
1,\quad\mu={\varepsilon}\\},$

which justified the Whitham equation as a water wave model in this regime by
relying on the justification of the KdV equation [9, 33].

On the other hand, due to the improved dispersion relation of (1.1), Emerald
[22] was able to decouple the parameters $({\varepsilon},\mu)$ and prove an
error estimate between the Whitham equation and the water wave system with a
precision $\mathcal{O}(\mu{\varepsilon}t)$ for $0\leq
t\lesssim{\varepsilon}^{-1}$ in the shallow water regime:

$\mathcal{R}_{SW}=\\{({\varepsilon},\mu)\>:\>0\leq\mu\leq 1,\quad
0\leq{\varepsilon}\leq 1\\}.$ (1.8)

Consequently, the Witham equation is valid for a larger set of small
parameters when compared to the KdV equation.

In the case of the Boussinesq systems (1.3), consistency was first proved in
[6] for $({\varepsilon},\mu)\in\mathcal{R}_{KdV}$ by relying on intermediate
symmetric systems for which the long time well-posedness follows by classical
arguments. However the long time well-posedness for the $(a,b,c,d)$ Boussinesq
is far from trivial. This result was proved333In the most dispersive case
$(a,b,c,d)=(\frac{1}{6},\frac{1}{6},0,0)$, the relevant time scale
$\mathcal{O}(\varepsilon^{-1})$ is still missing; the best results being on a
time scale $\mathcal{O}(\varepsilon^{-\frac{2}{3}})$ [43, 44], (see also [35]
on a time scale $\mathcal{O}(\varepsilon^{-\frac{1}{2}})$ by using dispersive
techniques). later by Saut, Xu and Wang [42, 46]. The proof relies on suitable
symmetrizers and hyperbolic theory.

The natural next step is to consider the Whitham-Boussinesq systems for
$({\varepsilon},\mu)\in\mathcal{R}_{SW}$. In particular, the goal of this
paper is to establish the well-posedness of (1.5), (1.6) and (1.7), with
uniform bounds, on time intervals of size
$\mathcal{O}(\frac{1}{{\varepsilon}})$. Since point $1.$ of the justification
is already established, the long-time existence and consistency remain. Using
the method of Emerald, one can prove the consistency of any Whitham-Boussinesq
system with the water wave system (see also [21] for other full dispersion
shallow water models). Therefore, having the long time well-posedness theory
for (1.5), (1.6) and (1.7) will provide the final step for the full
justification of these systems.

###  1.3. Former well-posedness results

Regarding system (1.5), we know from previous studies that surface tension
plays a fundamental role in the well-posedness theory. In fact, when $\beta=0$
the initial value problem associated to system (1.5) is probably ill-posed
unless $\zeta>0$ (see the formal argument in Section 4 in [31]). We refer to
[40] for a well-posedness under the non-physical condition $\zeta\geq
c_{0}>0$. When surface tension is taken into account, system (1.5) was proved
to be locally well-posed by Kalisch and Pilod [28] for $(\zeta,v)\in
H^{s}(\mathbb{R})\times H^{s+\frac{1}{2}}(\mathbb{R})$, $s>\frac{5}{2}$, by
using a modified energy method. We also refer to the work by Wang [51] for an
alternative proof using a nonlocal symmetrizer. However, it is worth noting
that all these well-posedness resutls were proved on a short time without
considering the small parameters ${\varepsilon}$ and $\mu$. Finally, in the
formal limit $\sqrt{\mu}|\xi|\rightarrow 0$, one recovers the Boussinesq
system corresponding to $(a,b,c,d)=(\frac{1}{3}-\beta,0,0,0)$. This system has
been proved in [46] to be well-posed on large time for $\beta>\frac{1}{3}$,
while it is known to be ill-posed for $\beta<\frac{1}{3}$ [3]. This is a
formal indication that the threshold $\beta=\frac{1}{3}$ will play an
important role for the long well-posedness of (1.5). We will come back to this
issue in the next section (see Figure 1).

As far as we know, there are no well-posedness results for system (1.6) even
on short time. In the formal limit $\sqrt{\mu}|\xi|\rightarrow 0$, it reduces
to the Boussinesq system corresponding to $(a,b,c,d)=(0,0,\frac{1}{3},0)$,
which is believed to be ill-posed [31].

Lastly, attention is turned to (1.7). There are several results when
$\beta=0$. In this case, Dinvay [12] proved short time local well-posedness
for $(\zeta,v)\in H^{s+\frac{1}{2}}(\mathbb{R})\times H^{s+1}(\mathbb{R})$,
$s\geq 0$. The proof is based on standard hyperbolic theory that involves a
modified energy similar to [28]. This result was then extended in [16] by
exploiting the smoothing effect of the linear flow using dispersive techniques
improving the regularity to $H^{s}(\mathbb{R})\times
H^{s+\frac{1}{2}}(\mathbb{R})$, $s>-\frac{1}{10}$. Furthermore, when
considering small data, the system is globally well-posed due to the control
of the Hamiltonian. The estimates derived in the aforementioned papers are not
uniform444After the completion of this work, we learned from Vincent Duchêne
that Louis Emerald proved in his PhD thesis long time well-posedness results
for a class of Whitham-Boussinesq systems containing (1.7) in the case
$\beta=0$ [23] (see also [15]). His results deal with different systems than
the one in this paper and thus complete each other well. in $\mu$. However, a
recent study by Tesfahun [49] proved that the system corresponding to (1.7) in
the 2-dimensional case and without surface tension is well-posed on a time
interval of order $\mathcal{O}(\frac{1}{\sqrt{{\varepsilon}}})$ in the
KdV$-$regime. Indeed, dispersive techniques are tailored-made for short waves
and therefore seem not to be well suited to capture the long wave regime (see
for instance [35] for similar results for the Boussinesq system in the KdV-KdV
case). Finally, in the case of surface tension $\beta>0$, Dinvay proved in
[11] the short time local well-posedness of the system by using modified
energy techniques. This result also implies the small data global well-
posedness in this case.

###  1.4. Main results

In the current paper, we take into account the small parameters
$({\varepsilon},\mu)$ and prove the well-posedness of (1.5), (1.6) and (1.7)
on a time scale $\mathcal{O}(\frac{1}{{\varepsilon}})$.

In the case of systems (1.6) and (1.7), we will work under the standard non-
cavitation condition.

######  Definition 1.1 (Non-cavitation condition).

Let $s>\frac{1}{2}$ and ${\varepsilon}\in(0,1)$. We say the initial surface
elevation $\zeta_{0}\in H^{s}(\mathbb{R})$ satisfies the “non-cavitation
condition” if there exist $h_{0}\in(0,1)$ such that

$1+{\varepsilon}\zeta_{0}(x)\geq h_{0},\quad\text{for
all}\>\>\>x\in\mathbb{R}.$ (1.9)

In the case of system (1.5), we will distinguish between the cases
$\beta\geq\frac{1}{3}$ and $0<\beta<\frac{1}{3}$. More precisely, for
$\beta\geq\frac{1}{3}$, we will also assume the non-cavitation condition in
Definition 1.1, while for $0<\beta<\frac{1}{3}$, we have to impose the
following $\beta-$dependent surface condition.

######  Definition 1.2 ($\beta-$dependent surface condition).

Let $s>\frac{1}{2}$, ${\varepsilon}\in(0,1)$ and $\beta\in(0,\frac{1}{3})$. We
say the initial surface elevation $\zeta_{0}\in H^{s}(\mathbb{R})$ satisfy the
“$\beta-$dependent surface condition” if

$1+{\varepsilon}\zeta_{0}(x)\geq h_{\beta},\quad\text{for
all}\>\>\>x\in\mathbb{R},$ (1.10)

where $h_{\beta}=1-\frac{\beta}{2}$.

######  Remark 1.3.

For $0<\beta<\frac{1}{3}$, $K_{\mu}(\xi)$ is not a monotone function for
positive frequencies as we can be seen in the figure below. This is why we
choose to impose condition (1.10) in this case.

Figure 1. The multipiler $K_{1}(\xi)$ in the case when $\beta\geq\frac{1}{3}$
(dash-dot) and $\beta<\frac{1}{3}$ (line). The horizontal line (dashed)
specifies the minimum.

######  Remark 1.4.

One can see the $\beta-$dependent surface condition as a constraint on the
initial data that is related to the minimum of the function $K_{\mu}(\xi)$.
For instance, if we consider the multiplier in Figure $1$, then an admissible
initial datum must satisfy the constraint in the figure below.

Figure 2. The blue line denotes the initial surface elevation
$1+{\varepsilon}\zeta_{0}$, and is restricted by $h_{\beta}$ when
$0<\beta<\frac{1}{3}$.

Before we state the main results, we define a natural solution space for
systems (1.5) and (1.6).

######  Definition 1.5.

We define the norm on the function space $V^{s}_{\sqrt{\mu}}(\mathbb{R})$ to
be

$\|(\zeta,v)\|_{V^{s}_{\sqrt{\mu}}}^{2}:=\|\zeta\|_{H^{s}}^{2}+\|v\|_{H^{s}}^{2}+\sqrt{\mu}\|D^{\frac{1}{2}}v\|_{H^{s}}^{2}.$

######  Theorem 1.6.

Let $s>2$, $\beta>0$ and ${\varepsilon},\mu\in(0,1)$. Assume that
$(\zeta_{0},v_{0})\in V^{s}_{\sqrt{\mu}}(\mathbb{R})$ satisfies either the
non-cavitation condition (1.9) in the case $\beta\geq 1/3$ or the
$\beta-$dependent surface condition (1.10) in the case $0<\beta<\frac{1}{3}$.
Moreover, we assume that

$0<{\varepsilon}\leq\frac{1}{k^{2}_{\beta}\|(\zeta_{0},v_{0})\|_{V^{s}_{\sqrt{\mu}}}}\quad\text{for}\quad
k^{2}_{\beta}=\begin{cases}\frac{c}{\beta}\>\>\quad\text{for}\>\>\>0<\beta<\frac{1}{3}\\\
c\beta\quad\text{for}\>\>\>\beta\geq\frac{1}{3}\end{cases}$ (1.11)

for some $c>0$. Then there exists a positive $T$ given by

$T=\frac{1}{k^{1}_{\beta}\|(\zeta_{0},v_{0})\|_{V^{s}_{\sqrt{\mu}}}}\quad\text{with}\quad
k_{\beta}^{1}=\begin{cases}\frac{c}{\beta}\quad\quad\text{for}\>\>0<\beta<\frac{1}{3}\\\
c\beta^{2}\>\>\>\quad\text{for}\>\>\beta\geq\frac{1}{3}\end{cases}$ (1.12)

such that (1.5) admits a unique solution

$(\zeta,v)\in C([0,T/{\varepsilon}]:V^{s}_{\sqrt{\mu}}(\mathbb{R}))\cap
C^{1}([0,T/{\varepsilon}]:V^{s-\frac{3}{2}}_{\sqrt{\mu}}(\mathbb{R})),$

that satisfies

$\sup\limits_{t\in[0,T/{\varepsilon}]}\|(\zeta,v)\|_{V^{s}_{\sqrt{\mu}}}\lesssim\|(\zeta_{0},v_{0})\|_{V^{s}_{\sqrt{\mu}}}.$
(1.13)

Furthermore, there exist a neighborhood $\mathcal{U}$ of $(\zeta_{0},v_{0})$
such that the flow map

$F_{T,{\varepsilon},\mu}^{s}:V_{\sqrt{\mu}}^{s}({\mathbb{R}})\>\rightarrow\>C([0,\tfrac{T}{2{\varepsilon}}];V^{s}_{\sqrt{\mu}}({\mathbb{R}})),\quad(\zeta_{0},v_{0})\>\>\mapsto\>\>(\zeta,v),$

is continuous.

######  Remark 1.7.

The proof of the continuous dependence on long time of order
$\mathcal{O}(\frac{1}{{\varepsilon}})$ seems to be new for Boussinesq type
systems. It relies on the Bona-Smith argument and could be easily adapted for
the $(a,b,c,d)$-Boussinesq systems.

######  Remark 1.8.

A heuristic argument can be made to argue that the physical solutions appear
when the initial data is of order one in terms of ${\varepsilon}$ [41]. To
illustrate this point, take the Burgers equation

$u_{t}-{\varepsilon}uu_{x}=0,$

a simple model that can describe an inviscid fluid in shallow water theory.
Then by the energy method, it is easy to deduce that the time of existence is
of order $T\sim\frac{1}{{\varepsilon}\|u_{0}\|_{H^{s}}}$ for $s>\frac{3}{2}$.
As a consequence, we have that $T\sim\frac{1}{{\varepsilon}}$ if the initial
data is of size $\mathcal{O}_{{\varepsilon}}(1)$.

######  Remark 1.9.

If $\beta\sim 1$ then ${\varepsilon}\lesssim 1$ by (1.11), and so (1.12)
implies that $T/{\varepsilon}\sim 1/{\varepsilon}$. On the other hand, in the
case of having $\beta<<1$, (1.11) would impose ${\varepsilon}\lesssim\beta$ ,
and by (1.12) we have the existence on the timescale
$T/{\varepsilon}\sim\beta/{\varepsilon}$.

######  Remark 1.10.

Regarding the $\beta-$dependent surface condition, we demonstrate that the
solution will persist for a long time and satisfy
${\varepsilon}\zeta(x,t)\geq-c\beta$ for some constant $c>0$. One should also
note that this is coherent since $0<{\varepsilon}\lesssim\beta$ as explained
in the previous remark. For a related discussion on this physical condition
see Subsection 1.3.

Next, we state a well-posedness result for (1.6). The system does not feature
any surface tension but is well-posed for a long time under the standard non-
cavitation condition.

######  Theorem 1.11.

Let $s>\frac{3}{2}$ and $\mu\in(0,1)$. Assume that $(\zeta_{0},v_{0})\in
V^{s}_{\sqrt{\mu}}(\mathbb{R})$ satisfies the non-cavitation condition (1.9).
Also assume that for some $c>0$ that $0<{\varepsilon}\leq
c\big{(}\|(\zeta_{0},v_{0})\|_{V_{\sqrt{\mu}}^{s}}\big{)}^{-1}$. Then there
exists $T=c\big{(}\|(\zeta_{0},v_{0})\|_{V^{s}_{\sqrt{\mu}}}\big{)}^{-1}$ such
that (1.6) admits a unique solution

$(\zeta,v)\in C([0,T/{\varepsilon}]:V^{s}_{\sqrt{\mu}}(\mathbb{R}))\cap
C^{1}([0,T/{\varepsilon}]:V^{s-1}_{\sqrt{\mu}}(\mathbb{R})),$

that satisfies

$\sup\limits_{t\in[0,T/{\varepsilon}]}\|(\zeta,v)\|_{V^{s}_{\sqrt{\mu}}}\lesssim\|(\zeta_{0},v_{0})\|_{V^{s}_{\sqrt{\mu}}}.$

In addition, the flow map is continuous with respect to the initial data.

######  Remark 1.12.

As far as we know, Theorem 1.11 is the first well-posedness result for system
(1.6).

Similarly, we can combine the techniques used to prove Theorem 1.6 and Theorem
1.11 to establish the long time well-posedness of (1.7) in the space:

######  Definition 1.13.

Define the norm on $X^{s}_{\beta,\mu}({\mathbb{R}})$ to be

$\|(\zeta,v)\|_{X^{s}_{\beta,\mu}}^{2}:=\|\zeta\|_{H^{s}}^{2}+\beta\mu\|D^{1}\zeta\|_{H^{s}}^{2}+\|v\|^{2}_{H^{s}}+\sqrt{\mu}\|D^{\frac{1}{2}}v\|^{2}_{H^{s}}.$

######  Theorem 1.14.

Let $s>\frac{3}{2}$, $\beta>0$ and $\mu\in(0,1)$. Assume that
$(\zeta_{0},v_{0})\in X^{s}_{\beta,\mu}(\mathbb{R})$ satisfies the non-
cavitation condition (1.9). Also assume that for some $c>0$ that
$0<{\varepsilon}\leq
c\big{(}\|(\zeta_{0},v_{0})\|_{X^{s}_{\beta,\mu}}\big{)}^{-1}$. Then there
exists $T=c\big{(}\|(\zeta_{0},v_{0})\|_{X^{s}_{\beta,\mu}}\big{)}^{-1}$ such
that (1.7) admits a unique solution

$(\zeta,v)\in C([0,T/{\varepsilon}]:X^{s}_{\beta,\mu}(\mathbb{R})\cap
C^{1}([0,T/{\varepsilon}]:X^{s-1}_{\beta,\mu}(\mathbb{R})),$

that satisfies

$\sup\limits_{t\in[0,T/{\varepsilon}]}\|(\zeta,v)\|_{X^{s}_{\beta,\mu}}\lesssim\|(\zeta_{0},v_{0})\|_{X^{s}_{\beta,\mu}}.$

In addition, the flow map is continuous with respect to the initial data.

######  Remark 1.15.

Including $\beta>0$ in the norm in the definition of
$X^{s}_{\beta,\mu}({\mathbb{R}})$ will allow us to obtain a long time well-
posedness result under the non-cavitation condition. Additionally, when
$0<\beta<\frac{1}{3}$ then ${\varepsilon}$ is independent from the surface
tension parameter.

######  Remark 1.16.

For the sake of clarity, we have chosen to work in the one dimensional case.
Theorems 1.6, 1.11 and 1.14 could possibly be extended to the $2$-dimensional
case by following the same methods since the symbols $\mathcal{K}_{\mu}(D)$
and $\mathcal{T}_{\mu}(D)$ are radial.

###  1.5. Strategy and outline

The proof of Theorem 1.6 relies mainly on energy estimates similar to the ones
provided in [28] on a fixed time. Though, we use the idea of Wang [51], who
included the nonlocal operator $\mathcal{K}_{\mu}(D)$ in the definition of the
energy555Wang actually used this multiplier in the case $\mu=1$.:

######  Definition 1.17.

Let $(\eta,u)={\varepsilon}(\zeta,v)$ and $J^{s}$ be the bessel potential of
order $-s$. Then we define the energy associated to (1.5):

$E_{s}(\eta,u):=\int_{{\mathbb{R}}}(J^{s}\eta)^{2}\>dx+\int_{{\mathbb{R}}}(\mathcal{K}_{\mu}(D)+\eta)(J^{s}u)^{2}\>dx.$

This energy formulation will free us to cancel out specific nonlinear terms
that appear naturally in the computations yielding the estimate

$\frac{d}{dt}E_{s}(\eta,u)\lesssim_{\beta}\big{(}E_{s}(\eta,u)\big{)}^{\frac{3}{2}}.$
(1.14)

Combined with the coercivity of the energy, then by a standard bootstrap
argument, one deduces a solution with the lifespan of
$T_{0}=\mathcal{O}(\frac{1}{{\varepsilon}})$. We refer the reader to
Proposition 3.1 and Lemma 5.2 for these results. The proof of the energy
estimate is similar to the one presented in [51], but we keep track of the
small parameters. We should also note that estimate (1.14) is applied to a
regularized version of (1.5), where we recover the original system using a
Bona-Smith argument.

To run the Bona-Smith argument for $s>2$, one classically needs to estimate
the difference between two solutions at the
$V^{0}_{\sqrt{\mu}}({\mathbb{R}})-$level. These estimates will be the most
technical point of the paper and are specific to the dependence of the small
parameters. In short, the technical difficulty is related to the apparent need
for ’generalized’ Kato-Ponce type commutator estimates on
$\mathcal{K}_{\mu}(D)$ (see Lemma 2.9 and the generalization for
$\mathcal{K}_{\mu}(D)$ in Lemma 2.11). Whereas for the case $\mu=1$, one can
use Calderón type estimates to simplify $\mathcal{K}_{\mu}(D)$ directly (see
[28] and the reformulated system $(2.1)$). The main idea will be to split
$\mathcal{K}_{\mu}(D)$ in high and low frequencies, and then derive new
commutator estimates that allow us to obtain the necessary order of $\mu$ in
the estimates related to the energy.

For the proof of Theorem 1.11, we follow the same strategy, but in this case,
the dispersion operator (1.4) is regularizing. The trick will be to introduce
a scaled Bessel potential in the energy, allowing us to mimic the properties
of (1.2). The energy is given by:

######  Definition 1.18.

Let $(\eta,u)={\varepsilon}(\zeta,v)$ and $J^{\frac{1}{2}}_{\mu}$ be the
scaled Bessel potential defined by the symbol
$\xi\mapsto(1+\mu\xi^{2})^{\frac{1}{4}}$ in frequency. Then the energy
associated to (1.6) reads:

$\mathcal{E}_{s}(\eta,u):=\int_{{\mathbb{R}}}\mathcal{T}_{\mu}(D)(J^{\frac{1}{2}}_{\mu}J^{s}\eta)^{2}\>dx+\int_{{\mathbb{R}}}(1+\eta)(J_{\mu}^{\frac{1}{2}}J^{s}u)^{2}\>dx.$

The energy formulated in Definition 1.18 is new and will require commutator
estimates specific to the equation. This will, in turn, allow us to decouple
the parameters $\mu$ and ${\varepsilon}$ in the estimates and, by extension,
provide an estimate in the form of (1.14).

In the same spirit, we define a modified energy for system (1.7):

######  Definition 1.19.

Let $(\eta,u)={\varepsilon}(\zeta,v)$ and $\beta>0$. Then the energy
associated to (1.7) reads:

$\mathscr{E}_{s}(\eta,u):=\int_{{\mathbb{R}}}(1+\beta\mu
D^{2})(J^{s}\eta)^{2}\>dx+\int_{{\mathbb{R}}}(\mathcal{T}^{-1}_{\mu}(D)+\eta)(J^{s}u)^{2}\>dx.$

Note also that the energy includes the surface tension parameter $\beta$ and
will allow us to deduce an estimate on the form (1.14), where the coercivity
estimate will be uniform in $\beta$. In turn, this will provide the long time
well-posedness for $\beta<<1$ and $T/{\varepsilon}\sim 1/{\varepsilon}$ as
pointed out in Remark 1.15.

The paper is organized as follows. In Section 2, we introduce some important
technical results whose proofs will be postponed to the appendix. In the same
section, we also present new commutator estimates needed to treat the
nonlinear terms when estimating the energy in Sections 3 and 4. Lastly, we
conclude in Section 5 by combining the results obtained in the former sections
to prove Theorem 1.6 in full detail, while the proof of Theorem 1.11 and
Theorem 1.14 will follow by the same arguments.

###  1.6. Notation

  * •

We let $c$ denote a positive constant independent of $\mu,{\varepsilon}$ that
may change from line to line. Also, as a shorthand, we use the notation
$a\lesssim b$ to mean $a\leq c\>b$. Similarly, if the constant depends on
$\beta$, we write $a\lesssim_{\beta}b$. In particular, we define the constants
depending on $\beta$,

$c_{\beta}^{1}=\begin{cases}c\beta\quad\text{for}\>\>0<\beta<\frac{1}{3}\\\
c\>\>\hskip
1.42271pt\quad\text{for}\>\>\beta\geq\frac{1}{3}\end{cases}\quad\text{and}\quad
c_{\beta}^{2}=\begin{cases}c\>\>\hskip
1.42271pt\quad\text{for}\>\>0<\beta<\frac{1}{3}\\\
c\beta\quad\text{for}\>\>\beta\geq\frac{1}{3}\end{cases}$ (1.15)

  * •

Let $(V,\|\cdot\|_{V})$ be a vector space. Then for $\alpha\geq 0$,
$\lambda>0$ and $f_{\lambda}\in V$ be a function depending on $\lambda$, we
define the “big$-\mathcal{O}$” notation to be

$\|f_{\lambda}\|_{V}=\mathcal{O}(\lambda^{\alpha})\iff\lim\limits_{\lambda\rightarrow
0}\lambda^{-\alpha}\|f_{\lambda}\|_{V}<\infty.$

Similarly, we define the “small$-o$” notation to be

$\|f_{\lambda}\|_{V}=o(\lambda^{\alpha})\iff\lim\limits_{\lambda\rightarrow
0}\lambda^{-\alpha}\|f_{\lambda}\|_{V}=0.$

  * •

Let $L^{2}({\mathbb{R}})$ be the usual space of square integrable functions
with norm $\|f\|_{2}=\sqrt{\int_{{\mathbb{R}}}|f(x)|^{2}\>dx}$. Also, for any
$f,g\in L^{2}({\mathbb{R}})$ we denote the scalar product by
$\big{(}f,g\big{)}_{L^{2}}=\int_{{\mathbb{R}}}f(x)\overline{g(x)}\>dx$.

  * •

For any tempered distribution $f$, the operator $\mathcal{F}$ denoting the
Fourier transform, applied to $f$, will be written as $\hat{f}(\xi)$ or
$\mathcal{F}f(\xi)$.

  * •

Let $m:{\mathbb{R}}\rightarrow\mathbb{R}$ be a smooth function. Then we will
use the notation $m(D)$ for a multiplier defined in frequency by
$\widehat{m(D)f}(\xi)=m(\xi)\hat{f}(\xi)$.

  * •

For any $s\in\mathbb{R}$ we call the multiplier
$\widehat{D^{s}f}(\xi)=|\xi|^{s}\hat{f}(\xi)$ the Riesz potential of order
$-s$. One should note that $D^{1}=\mathcal{H}\partial_{x}$, where
$\widehat{\mathcal{H}f}(\xi)=-i\>\text{sgn}(\xi)\hat{f}(\xi)$ is the Hilbert
transform.

  * •

For any $s\in\mathbb{R}$ we call the multiplier
$J^{s}=(1+D^{2})^{\frac{s}{2}}=\langle D\rangle^{s}$ the Bessel potential of
order $-s$. Moreover, the Sobolev space $H^{s}(\mathbb{R})$ is equivalent to
the weighted $L^{2}-$space; $\|f\|_{H^{s}}=\|J^{s}f\|_{L^{2}}$. We also find
it convenient to define $J^{\frac{1}{2}}_{\mu}$ which is a multiplier
assosiated to the symbol:

$\mathcal{F}(J^{\frac{1}{2}}_{\mu}f)(\xi)=(1+\mu\xi^{2})^{\frac{1}{4}}\hat{f}(\xi).$
(1.16)

  * •

We say $f$ is a Schwartz function $\mathscr{S}(\mathbb{R})$, if $f\in
C^{\infty}(\mathbb{R})$ and satisfies for all $\alpha,\beta\in\mathbb{N}$,

$\sup\limits_{x}|x^{\alpha}\partial_{x}^{\beta}f|<\infty.$

  * •

If $A$ and $B$ are two operators, then we denote the commutator between them
to be $[A,B]=AB-BA$.

##  2\. Preliminary results

###  2.1. Pointwise estimates

The first result concerns the properties of the dispersive part of the
equation. Namely, we deduce pointwise estimates for the multipliers (1.2) and
(1.4) that are needed to obtain the coercivity of the energy (see, for
instance, equation (3.7) below). Moreover, these estimates will prove
essential when dealing with the nonlinear parts of the equation that appear in
the energy estimates.

######  Lemma 2.1.

Let $\mu\in(0,1)$. Then we have the following pointwise estimates on the
kernel $K_{\mu}(\xi):$

  * •

For $\beta\geq 0$, we have the upper bound

$K_{\mu}(\xi)\lesssim 1+\beta(1+\beta\sqrt{\mu}|\xi|).$ (2.1)

  * •

If $\beta\geq\frac{1}{3}$, then for all $h_{0}\in(0,1)$ we have the lower
bound

$K_{\mu}(\xi)\geq(1-\frac{h_{0}}{2})+c\sqrt{\mu}|\xi|,$ (2.2)

whereas, if $0<\beta<\frac{1}{3}$, we have the lower bound

$K_{\mu}(\xi)\geq\beta+c\beta\sqrt{\mu}|\xi|.$ (2.3)

  * •

The derivative of the symbol $K_{\mu}(\xi)$ satisfies

$\bigg{|}\frac{d}{d\xi}\sqrt{K_{\mu}(\xi)}\bigg{|}\lesssim\langle\xi\rangle^{-1}+\sqrt{\beta}\mu^{\frac{1}{4}}\langle\xi\rangle^{-\frac{1}{2}}.$
(2.4)

  * •

We have the following comparison of $\sqrt{\mathcal{K}_{\mu}(\xi)}$ by

$\big{|}\sqrt{K_{\mu}(\xi)}-\sqrt{\beta}\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\big{|}\lesssim\sqrt{\beta}+\beta.$
(2.5)

  * •

There holds

$\sqrt{K_{\mu}(\xi)}\langle\xi\rangle^{s-1}|\xi|\lesssim(\sqrt{\beta}+\beta)\langle\xi\rangle^{s}+\sqrt{\beta}\mu^{\frac{1}{4}}\langle\xi\rangle^{s}|\xi|^{\frac{1}{2}}.$
(2.6)

######  Remark 2.2.

For inequality (2.3), it is crucial to specify the dependence in $\beta$ as it
will provide the coercivity of the energy when $0<\beta<\frac{1}{3}$. The same
is true for (2.2), whose importance will be revealed in the proof of
Proposition 3.1 below. Though, we note that (2.3) does not agree with (2.2)
when $\beta=\frac{1}{3}$. This is because the lower bound in (2.3) is not
optimal, but it does not play a role for the overall result.

######  Remark 2.3.

We also trace the dependence in $\beta$ for the first pointwise estimate
(2.1), and it will sometimes be replaced with $c_{\beta}^{2}$ given by (1.15).
This constant will again appear when we prove the energy estimates which will
provide the size of the time of existence (see Lemma 5.2 in the proof Theorem
1.6).

The proof of Lemma 2.1 is technical and postponed to the Appendix in Section
A.2. A corollary of Proposition 2.1 may now be stated.

######  Corollary 2.4.

Take $f\in\mathscr{S}(\mathbb{R})$, $\mu\in(0,1)$ and $s\in\mathbb{R}$. Then
in the case $\beta\geq\frac{1}{3}$ and for all $h_{0}\in(0,1)$ we have

$(1-\frac{h_{0}}{2})\|f\|_{H^{s}}^{2}+c\sqrt{\mu}\|D^{\frac{1}{2}}f\|_{H^{s}}^{2}\leq||\sqrt{\mathcal{K}_{\mu}}(D)f\|_{H^{s}}^{2}\leq
c_{\beta}^{2}\|f\|_{H^{s}}^{2}+c\beta\sqrt{\mu}\|D^{\frac{1}{2}}f\|_{H^{s}}^{2}.$
(2.7)

Similarly, in the case $0<\beta<\frac{1}{3}$ there holds

$\beta\|f\|_{H^{s}}^{2}+c\beta\sqrt{\mu}\|D^{\frac{1}{2}}f\|_{H^{s}}^{2}\leq||\sqrt{\mathcal{K}_{\mu}}(D)f\|_{H^{s}}^{2}\leq
c_{\beta}^{2}\|f\|_{H^{s}}^{2}+c\sqrt{\mu}\|D^{\frac{1}{2}}f\|_{H^{s}}^{2}.$
(2.8)

###### Proof.

The upper bound in (2.7) follows by Plancherel’s identity and the pointwise
estimate (2.1), while the lower bound is a consequence of (2.2).

In the same way, for $0<\beta<\frac{1}{3}$, then (2.8) is deduced from (2.3).
∎

Similarly, we state some useful pointwise estimates on $T_{\mu}(\xi)$ and the
scaled Bessel potential $J^{\frac{1}{2}}_{\mu}$, where the proof is presented
in Appendix A.2.

######  Lemma 2.5.

Let $\mu\in(0,1)$. Then we have the following pointwise estimates on the
kernel $T_{\mu}(\xi):$

  * •

For all $h_{0}\in(0,1)$ there holds

$(1-\frac{h_{0}}{2})+c\sqrt{\mu}|\xi|\leq(T_{\mu}(\xi))^{-1}\lesssim
1+\sqrt{\mu}|\xi|.$ (2.9)

  * •

There holds

$1\lesssim T_{\mu}(\xi)\langle\sqrt{\mu}\xi\rangle\lesssim 1.$ (2.10)

  * •

For $s\in{\mathbb{R}}$ there holds

$\bigg{|}\frac{d}{d\xi}\langle\xi\rangle^{s}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}\bigg{|}\lesssim\langle\xi\rangle^{s-1}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}.$
(2.11)

  * •

For $s\in{\mathbb{R}}$ there holds

$\bigg{|}\frac{d}{d\xi}\sqrt{T_{\mu}(\xi)}\langle\xi\rangle^{s}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}\bigg{|}\lesssim\langle\xi\rangle^{s-1}.$
(2.12)

  * •

There holds

$\bigg{|}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}-\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\bigg{|}\lesssim
1.$ (2.13)

A direct consequence of the above estimates can now be given.

######  Corollary 2.6.

Let $f\in\mathscr{S}({\mathbb{R}})$, $\mu\in(0,1)$, $s\in{\mathbb{R}}$ and
$c>0$. Then for all $h_{0}\in(0,1)$ there holds

$\|\sqrt{\mathcal{T}_{\mu}}(D)f\|_{L^{2}}\leq\|f\|_{L^{2}}.$ (2.14)
$(1-\frac{h_{0}}{2})\|f\|_{H^{s}}^{2}+c\sqrt{\mu}\|D^{\frac{1}{2}}f\|_{H^{s}}^{2}\leq\|\sqrt{\mathcal{T}_{\mu}}^{-1}(D)f\|^{2}_{H^{s}}\lesssim\|f\|_{H^{s}}^{2}+c\sqrt{\mu}\|D^{\frac{1}{2}}f\|_{H^{s}}^{2}.$
(2.15)
$\|f\|_{H^{s}}\lesssim\|\sqrt{\mathcal{T}_{\mu}}(D)J^{\frac{1}{2}}_{\mu}f\|_{H^{s}}\lesssim\|f\|_{H^{s}}.$
(2.16)
$\|f\|_{H^{s}}^{2}+\sqrt{\mu}\|D^{\frac{1}{2}}f\|_{H^{s}}^{2}\lesssim\|J^{\frac{1}{2}}_{\mu}f\|_{H^{s}}^{2}\lesssim\|f\|_{H^{s}}^{2}+\sqrt{\mu}\|D^{\frac{1}{2}}f\|_{H^{s}}^{2}.$
(2.17)

###  2.2. Commutator estimates

To handle derivatives in the nonlinear parts of the equations, we need
commutator estimates on $\mathcal{K}_{\mu}(D)$ and $\mathcal{T}_{\mu}(D)$.

######  Lemma 2.7.

Let $f,g\in\mathscr{S}({\mathbb{R}})$, $\mu\in(0,1)$, $s\geq 1$, and
$t_{0}>\frac{1}{2}$. Then we have the following commutator estimate

$\displaystyle\|[\sqrt{\mathcal{K}_{\mu}}(D)J^{s},f]\partial_{x}g\|_{L^{2}}$
$\displaystyle\lesssim(c_{\beta}^{2}\|f\|_{H^{s}}+\sqrt{\beta}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}f\|_{H^{s}})\|\partial_{x}g\|_{H^{t_{0}}}$
$\displaystyle\hskip
14.22636pt+(c_{\beta}^{2}\|g\|_{H^{s}}+\sqrt{\beta}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}g\|_{H^{s}})\|\partial_{x}f\|_{H^{t_{0}}}.$
(2.18)

In the high regularity setting, the proof will follow the same lines as in
[51], but we track the dependence in $\mu$ and $\beta$ using the pointwise
estimates above.

###### Proof.

First, write the commutator as a bilinear form:

$\displaystyle\big{\|}[\sqrt{\mathcal{K}_{\mu}}(D)J^{s},f]\partial_{x}g\big{\|}_{L^{2}}=\bigg{\|}\int_{{\mathbb{R}}}\Big{(}\sqrt{K_{\mu}(\xi)}\langle\xi\rangle^{s}-\sqrt{K_{\mu}(\rho)}\langle\rho\rangle^{s}\Big{)}\hat{f}(\xi-\rho)\widehat{\partial_{x}g}(\rho)\>d\rho\bigg{\|}_{L^{2}_{\xi}}.$

Then if $a=\min\\{|\xi-\rho|,|\rho|\\}$ and $b=\max\\{|\xi-\rho|,|\rho|\\}$,
we can use the mean value theorem, leaving us to estimate the following terms

$\displaystyle\Big{|}\sqrt{K_{\mu}(\xi)}\langle\xi\rangle^{s}-\sqrt{K_{\mu}(\rho)}\langle\rho\rangle^{s}\Big{|}\lesssim\sup\limits_{\omega\in(a,b)}|m(\omega)|\>|\xi-\rho|,$

where

$m(\omega)=m_{1}(\omega)+m_{2}(\omega)=\langle\xi\rangle^{s}\frac{d}{d\xi}\sqrt{K_{\mu}(\xi)}+\langle\xi\rangle^{s-1}\sqrt{K_{\mu}(\xi)}.$

But using (2.5) to estimate $m_{1}(\omega)$ and (2.4) to treat
$m_{2}(\omega)$, we deduce

$m(\omega)\lesssim
c_{\beta}^{2}\langle\omega\rangle^{s-1}+\sqrt{\beta}\mu^{\frac{1}{4}}\langle\omega\rangle^{s-1}|\omega|^{\frac{1}{2}},$
(2.19)

where the upper bound is increasing for $s\geq 1$. Therefore the supremum is
attained at $|\xi-\rho|$ or $|\rho|$ by definition of $b$. In particular, if
$b=|\xi-\rho|$ then we may conclude by Minkowski integral inequality, the
Cauchy-Schwarz inequality and (2.19) that

$\displaystyle\big{\|}[\sqrt{\mathcal{K}_{\mu}}(D)J^{s},f]\partial_{x}g\big{\|}_{L^{2}}$
$\displaystyle\lesssim
c_{\beta}^{2}\bigg{\|}\int_{{\mathbb{R}}}\langle\xi-\rho\rangle^{s-1}|\xi-\rho|\>|\hat{f}(\xi-\rho)|\>|\widehat{\partial_{x}g}(\rho)|\>d\rho\bigg{\|}_{L^{2}_{\xi}}$
$\displaystyle\hskip
14.22636pt+\sqrt{\beta}\mu^{\frac{1}{4}}\bigg{\|}\int_{{\mathbb{R}}}\langle\xi-\rho\rangle^{s-1}|\xi-\rho|^{\frac{1}{2}}|\xi-\rho|\>|\hat{f}(\xi-\rho)|\>|\widehat{\partial_{x}g}(\rho)|\>d\rho\bigg{\|}_{L^{2}_{\xi}}$
$\displaystyle\lesssim(c_{\beta}^{2}\|f\|_{H^{s}}+\sqrt{\beta}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}f\|_{H^{s}})\|\partial_{x}g\|_{H^{t_{0}}},$

for $t_{0}>\frac{1}{2}$. On the other hand, if $b=|\rho|$, then we make a
change of coordinates and argue similarly to deduce,

$\displaystyle\big{\|}[\sqrt{\mathcal{K}_{\mu}}(D)J^{s},f]\partial_{x}g\big{\|}_{L^{2}}$
$\displaystyle\lesssim
c_{\beta}^{2}\bigg{\|}\int_{{\mathbb{R}}}\langle\xi-\gamma\rangle^{s-1}|\widehat{\partial_{x}g}(\xi-\gamma)|\>|\gamma|\>|\hat{f}(\gamma)|\>d\gamma\bigg{\|}_{L^{2}_{\xi}}$
$\displaystyle\hskip
14.22636pt+\sqrt{\beta}\mu^{\frac{1}{4}}\bigg{\|}\int_{{\mathbb{R}}}\langle\gamma\rangle^{s-1}|\xi-\gamma|^{\frac{1}{2}}|\widehat{\partial_{x}g}(\xi-\gamma)|\>|\gamma|\>|\hat{f}(\gamma)|\>d\gamma\bigg{\|}_{L^{2}_{\xi}}$
$\displaystyle\lesssim(c_{\beta}^{2}\|g\|_{H^{s}}+\sqrt{\beta}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}g\|_{H^{s}})\|\partial_{x}f\|_{H^{t_{0}}}.$

Adding the two scenarios, we may conclude that (2.7) holds.

∎

We will also need a commutator estimates on $\mathcal{T}_{\mu}(D)$ and
$J^{\frac{1}{2}}_{\mu}$.

######  Lemma 2.8.

Let $f,g\in\mathscr{S}({\mathbb{R}})$, $s\geq 1$, $t_{0}>\frac{1}{2}$,
$\mu\in(0,1)$ and $J^{\frac{1}{2}}_{\mu}$ as defined in (1.16).

  * •

Then we have a Kato-Ponce type estimate

$\displaystyle\|[J^{s}J^{\frac{1}{2}}_{\mu},f]\partial_{x}g\|_{L^{2}}$
$\displaystyle\lesssim(\|f\|_{H^{s}}+\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}f\|_{H^{s}})\|\partial_{x}g\|_{H^{t_{0}}}$
$\displaystyle\hskip
14.22636pt+(\|g\|_{H^{s}}+\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}g\|_{H^{s}})\|\partial_{x}f\|_{H^{t_{0}}}.$
(2.20)

  * •

There holds

$\|[\sqrt{\mathcal{T}_{\mu}}(D)J^{s}J^{\frac{1}{2}}_{\mu},f]\partial_{x}g\|_{L^{2}}\lesssim\|f\|_{H^{s}}\|g\|_{H^{t_{0}+1}}+\|f\|_{H^{t_{0}+1}}\|g\|_{H^{s}}.$
(2.21)

###### Proof.

The proof is similar to the one of Lemma 2.7 and relies on the pointwise
estimates established in Lemma 2.5. Indeed, for (2.20) we define
$a_{1}(D)(f,g):=[J^{s}J^{\frac{1}{2}}_{\mu},f]\partial_{x}g$ and use the mean
value theorem combined with (2.11) to deduce

$\displaystyle|\hat{a}_{1}(\xi)(f,g)|$
$\displaystyle\leq\int_{{\mathbb{R}}}\Big{|}\langle\xi\rangle^{s}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}-\langle\rho\rangle^{s}\langle\sqrt{\mu}\rho\rangle^{\frac{1}{2}}\Big{|}|\hat{f}(\xi-\rho)|\>|\widehat{\partial_{x}g}(\rho)|\>d\rho$
$\displaystyle\lesssim\int_{{\mathbb{R}}}\langle\xi-\rho\rangle^{s-1}\langle\sqrt{\mu}(\xi-\rho)\rangle^{\frac{1}{2}}|\xi-\rho|\>|\hat{f}(\xi-\rho)|\>|\widehat{\partial_{x}g}(\eta)|\>d\rho$
$\displaystyle\hskip
14.22636pt+\int_{{\mathbb{R}}}\langle\rho\rangle^{s-1}\langle\sqrt{\mu}\rho\rangle^{\frac{1}{2}}|\xi-\rho|\>|\hat{f}(\xi-\rho)|\>|\widehat{\partial_{x}g}(\rho)|\>d\rho.$

Then if we apply the $L^{2}({\mathbb{R}})-$norm with respect to $\xi$, we can
argue as in Lemma 2.7 that

$\displaystyle\|\hat{a}_{1}(\xi)(f,g)\|_{L^{2}_{\xi}}$
$\displaystyle\lesssim\|J^{\frac{1}{2}}_{\mu}f\|_{H^{s}}\int_{{\mathbb{R}}}|\widehat{\partial_{x}g}(\rho)|\>d\rho+\|J^{\frac{1}{2}}_{\mu}g\|_{H^{s}}\int_{{\mathbb{R}}}|\rho|\>|\hat{f}(\rho)|\>d\rho.$

Then use the definiton of $a_{1}(D)(f,g)$ and (2.17) to conclude.

The proof of (2.21) is the same, with
$a_{2}(D)(f,g):=[\sqrt{\mathcal{T}_{\mu}}(D)J^{s}J^{\frac{1}{2}}_{\mu},f]\partial_{x}g$.
We use (2.12) to find that

$\displaystyle|\hat{a}_{2}(\xi)(f,g)|$
$\displaystyle\leq\int_{{\mathbb{R}}}\Big{|}\sqrt{T_{\mu}(\xi)}\langle\xi\rangle^{s}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}-\sqrt{T_{\mu}(\rho)}\langle\rho\rangle^{s}\langle\sqrt{\mu}\rho\rangle^{\frac{1}{2}}\Big{|}|\hat{f}(\xi-\rho)|\>|\widehat{\partial_{x}g}(\rho)|\>d\rho$
$\displaystyle\lesssim\int_{{\mathbb{R}}}\langle\xi-\rho\rangle^{s-1}|\xi-\rho|\>|\hat{f}(\xi-\rho)|\>|\widehat{\partial_{x}g}(\rho)|\>d\rho$
$\displaystyle\hskip
14.22636pt+\int_{{\mathbb{R}}}\langle\rho\rangle^{s-1}|\xi-\rho|\>|\hat{f}(\xi-\rho)|\>|\widehat{\partial_{x}g}(\rho)|\>d\rho,$

and the result follows.

∎

Next, we state the classical Kato-Ponce commutator estimate. We will use it
repeatedly to commute the Bessel potential with functions to obtain the
desired energy estimates in the coming sections.

######  Lemma 2.9 (Kato - Ponce commutator estimates [29]).

Let $s\geq 0$, $p,p_{2},p_{3}\in(1,\infty)$ and $p_{1},p_{4}\in(1,\infty]$
such that
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}$.
Then

$\|J^{s}(fg)\|_{L^{p}}\lesssim\|f\|_{L^{p_{1}}}\|J^{s}g\|_{L^{p_{2}}}+\|J^{s}f\|_{L^{p_{3}}}\|g\|_{L^{p_{4}}}$
(2.22)

and

$\|[J^{s},f]g\|_{L^{p}}\lesssim\|\partial_{x}f\|_{L^{p_{1}}}\|J^{s-1}g\|_{L^{p_{2}}}+\|J^{s}f\|_{L^{p_{3}}}\|g\|_{L^{p_{4}}}.$
(2.23)

Similar commutator estimates also hold for more general multipliers. In fact,
by splitting the frequency domain into two parts using smooth cut-off
functions defined in frequency, we can obtain sharper commutator estimates
specific to equation (1.5).

######  Definition 2.10.

We define the smooth cut-off functions $\chi^{(i)}\in\mathscr{S}(\mathbb{R})$
as Fourier multipliers

$\mathcal{F}(\chi^{(i)}(D)f)(\xi)=\chi^{(i)}(|\xi|)\hat{f}(\xi),$

for any $f\in\mathscr{S}(\mathbb{R})$ with the following properties:

$0\leq\chi^{(i)}(\xi)\leq
1,\qquad(\chi^{(1)}(\xi))^{2}+(\chi^{(2)}(\xi))^{2}=1\quad\text{on}\quad{\mathbb{R}},$

and

$\text{supp}\>\chi^{(1)}\subset[-1,1],\quad\text{supp}\>\chi^{(2)}\>\subset{\mathbb{R}}\backslash\big{[}-\frac{1}{2},\frac{1}{2}\big{]}.$

Moreover, we denote the scaled version in $\mu$ by
$\chi^{(i)}_{\mu}(\xi)=\chi^{(i)}(\sqrt{\mu}\xi)$.

We have the results:

######  Lemma 2.11.

Let $s>\frac{3}{2}$, $\mu\in(0,1)$ and $f,g\in\mathscr{S}(\mathbb{R})$.

  * •

Let $(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)$ be the multiplier of the
symbol $(\chi^{(1)}_{\mu}\sqrt{{K}_{\mu}})(\xi)$. Then

$\|(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)f\|_{L^{2}}\lesssim_{\beta}\|f\|_{L^{2}},$
(2.24)

and

$\|[(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D),f]\partial_{x}g\|_{L^{2}}\lesssim_{\beta}\|f\|_{H^{s}}\|g\|_{L^{2}}.$
(2.25)

  * •

We define the symbol

$\displaystyle\sigma_{\mu,\frac{1}{2}}(D)$
$\displaystyle:=\bigg{(}\frac{1}{\sqrt{\mu}|D|}+\beta\sqrt{\mu}|D|\bigg{)}^{\frac{1}{2}}.$
(2.26)

Then

$\|(\chi^{(2)}_{\mu}\sigma_{\mu,\frac{1}{2}})(D)f\|_{L^{2}}\lesssim_{\beta}\|f\|_{L^{2}}+\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}f\|_{L^{2}}$
(2.27)

and

$\|[(\chi^{(2)}_{\mu}\sigma_{\mu,\frac{1}{2}})(D),f]\partial_{x}g\|_{L^{2}}\lesssim_{\beta}\mu^{\frac{1}{4}}\|f\|_{H^{s}}\|g\|_{H^{\frac{1}{2}}}.$
(2.28)

  * •

Lastly, we define the symbol $\sigma_{\mu,0}(D)$ to be

$\sigma_{\mu,0}(D):=\bigg{(}\frac{1}{\sqrt{\mu}|D|}+\beta\sqrt{\mu}|D|-\mathcal{K}_{\mu}(D)\bigg{)}^{\frac{1}{2}}.$
(2.29)

Then

$\|(\chi^{(2)}_{\mu}\sigma_{\mu,0})(D)f\|_{L^{2}}\lesssim_{\beta}\|f\|_{L^{2}}$
(2.30)

and

$\|[(\chi^{(2)}_{\mu}\sigma_{\mu,0})(D),f]\partial_{x}g\|_{L^{2}}\lesssim_{\beta}\|f\|_{H^{s}}\|g\|_{L^{2}}.$
(2.31)

The proof is postponed to Appendix A.3, where we also will prove the following
commutator estimates at the $L^{2}({\mathbb{R}})-$level:

######  Lemma 2.12.

Let $s>\frac{3}{2}$, $\mu\in(0,1)$ and $f,g\in\mathscr{S}({\mathbb{R}})$.

  * •

For the composition of $\sqrt{T_{\mu}}(D)$ and $J^{\frac{1}{2}}_{\mu}$ there
holds,

$\|[\sqrt{T_{\mu}}(D)J^{\frac{1}{2}}_{\mu},f]\partial_{x}g\|_{L^{2}}\lesssim\|f\|_{H^{s}}\|g\|_{L^{2}}.$
(2.32)

  * •

While for the usual Bessel potential there holds,

$\|[\sqrt{T_{\mu}}(D)J^{s},f]\partial_{x}g\|_{L^{2}}\lesssim\|f\|_{H^{s}}\|J^{s}g\|_{L^{2}}.$
(2.33)

  * •

Similarly, when the operator $J^{s}$ is the identity we have

$\|[\sqrt{T_{\mu}}(D),f]\partial_{x}g\|_{L^{2}}\lesssim\|f\|_{H^{s}}\|g\|_{L^{2}}.$
(2.34)

  * •

The derivative of the following commutator satisfies

$\|\partial_{x}[\sqrt{\mathcal{T}_{\mu}}(D),f]g\|_{L^{2}}\lesssim\|f\|_{H^{s}}\|g\|_{L^{2}}.$
(2.35)

  * •

Lastly, we can commute $J^{\frac{1}{2}}_{\mu}$ by

$\|[J^{\frac{1}{2}}_{\mu},f]\partial_{x}g\|_{L^{2}}\lesssim\|f\|_{H^{s}}\|J^{\frac{1}{2}}_{\mu}g\|_{L^{2}}.$
(2.36)

###  2.3. Classical estimates

Before turning to the proof of the energy estimates, we state some necessary
results that will also be used throughout the paper. First, recall the
embeddings (see, for example [34]).

######  Lemma 2.13 (Sobolev embeddings).

Let $f\in\mathscr{S}({\mathbb{R}})$ and $s\in(0,\frac{1}{2})$. Then
$H^{s}({\mathbb{R}})\hookrightarrow L^{p}({\mathbb{R}})$ with
$p=\frac{2}{1-2s}$, and there holds

$\|f\|_{L^{p}}\lesssim\|D^{s}f\|_{L^{2}}.$ (2.37)

Moreover, In the case $s>\frac{1}{2}$, then $H^{s}({\mathbb{R}})$ is
continuously embedded in $L^{\infty}({\mathbb{R}})$.

We also will use the Leibniz rule for the Riesz potential on multiple
occasions.

######  Lemma 2.14 (Fractional Leibniz rule [30]).

Let $\sigma=\sigma_{1}+\sigma_{2}\in(0,1)$ with $\sigma_{i}\in[0,\sigma]$ and
$p,p_{1},p_{2}\in(1,\infty)$ satisfy
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$. Then, for
$f,g\in\mathscr{S}({\mathbb{R}})$

$\|D^{\sigma}(fg)-fD^{\sigma}g-gD^{\sigma}f\|_{L^{p}}\lesssim\|D^{\sigma_{1}}f\|_{L^{p_{1}}}\|D^{\sigma_{2}}g\|_{L^{p_{2}}}.$
(2.38)

Moreover, the case $\sigma_{2}=0$, $p_{2}=\infty$ is also allowed.

Finally, we recall the following results for the Bona-Smith argument (provided
in the classical paper [7]) on the multiplier $\varphi_{\delta}(D)$ defined
by:

######  Definition 2.15.

Let $\varphi\in\mathscr{S}(\mathbb{R})$ such that $\int\varphi=1$ and for
$\delta>0$ define the regularization operators $\varphi_{\delta}(D)$ in
frequency by

$\displaystyle\forall f\in
L^{2}(\mathbb{R}),\quad\forall\xi\in\mathbb{R},\quad\widehat{\varphi_{\delta}f}(\xi):=\varphi(\delta\xi)\hat{f}(\xi),$

where $\varphi$ is a real valued and $\varphi(0)=1$.

We give the version of the regularization estimates as presented in [34]
(Proposition $9.1$).

######  Proposition 2.16.

Let $s>0$, $\delta>0$ and $f\in\mathscr{S}(\mathbb{R})$. Then

$\|\varphi_{\delta}(D)f\|_{H^{s+\alpha}}\lesssim\delta^{-\alpha}\|f\|_{H^{s}},\quad\forall\alpha>0,$
(2.39)

and

$\|\varphi_{\delta}(D)f-f\|_{H^{s-\beta}}\lesssim\delta^{\beta}\|f\|_{H^{s}},\quad\forall\beta\in[0,s].$
(2.40)

Moreover, there holds

$\|\varphi_{\delta}(D)f-f\|_{H^{s-\beta}}\underset{\delta\rightarrow
0}{=}o(\delta^{\beta}),\>\>\>\quad\forall\beta\in[0,s].$ (2.41)

##  3\. A priori estimates

In this section, we give a priori estimates for solutions of the three systems
(1.5), (1.6), and (1.7).

###  3.1. Estimates for system (1.5)

As noted in the introduction, we revisit the energy estimate in [51] to keep
track of the parameters $\beta,{\varepsilon}$ and $\mu$. For simplicity, we
adopt the notation $\mathbb{U}=(\eta,u)^{T}={\varepsilon}(\zeta,v)^{T}$, where
we write (1.5) on the compact form:

$\partial_{t}\mathbb{U}+M(\mathbb{U},D)\mathbb{U}=\mathbb{0},$ (3.1)

with

$M(\mathbb{U},D)=\begin{pmatrix}u\partial_{x}&(\mathcal{K}_{\mu}(D)+\eta)\partial_{x}\\\
\partial_{x}&u\partial_{x}\end{pmatrix}.$ (3.2)

Also, we simplify the notation for the energy given in Definition 1.17 by
introducing the symmetrizer

$Q(\mathbb{U},D)=Q^{(1)}(\mathbb{U},D)+Q^{(2)}(\mathbb{U},D)=\begin{pmatrix}1&0\\\
0&\eta\end{pmatrix}+\begin{pmatrix}0&0\\\
0&\mathcal{K}_{\mu}(D)\end{pmatrix}.$ (3.3)

Then the energy given in Definition 1.17 can be rewritten as

$E_{s}(\mathbb{U},D)=\big{(}J^{s}\mathbb{U},Q(\mathbb{U},D)J^{s}\mathbb{U}\big{)}_{L^{2}}.$

######  Proposition 3.1.

Let $s>2$, ${\varepsilon},\mu\in(0,1)$ and $(\eta,u)={\varepsilon}(\zeta,v)\in
C([0,T_{0}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ be a solution to (3.1) on a time
interval $[0,T_{0}]$ for some $T_{0}>0$. Moreover, assume there exist
$h_{0}\in(0,1)$ and $h_{1}>0$ such that

$h_{0}-1\leq\eta(x,t),\quad\forall(x,t)\in\mathbb{R}\times[0,T_{0}]\quad\text{and}\quad\sup\limits_{t\in[0,T_{0}]}\|(\eta,u)\|_{H^{s}\times
H^{s}}\leq h_{1},$ (3.4)

when $\beta\geq\frac{1}{3}$, and that

$-\frac{\beta}{2}\leq\eta(x,t),\quad\forall(x,t)\in\mathbb{R}\times[0,T_{0}]\quad\text{and}\quad\sup\limits_{t\in[0,T_{0}]}\|(\eta,u)\|_{H^{s}\times
H^{s}}\leq h_{1},$ (3.5)

when $0<\beta<\frac{1}{3}$.

Then, for the energy given in Definition 1.17 and $c_{\beta}^{i}$ defined by
(1.15),

$\frac{d}{dt}E_{s}(\mathbb{U})\leq
c_{\beta}^{2}\>\big{(}E_{s}(\mathbb{U})\big{)}^{\frac{3}{2}},$ (3.6)

for all $0<t<T_{0}$, and

$c_{\beta}^{1}\|(\eta,u)\|^{2}_{V^{s}_{\sqrt{\mu}}}\leq E_{s}(\mathbb{U})\leq
c_{\beta}^{2}\|(\eta,u)\|^{2}_{V^{s}_{\sqrt{\mu}}},$ (3.7)

for all $0<t<T_{0}$.

######  Remark 3.2.

Note that we aim to prove (3.6) with power $\frac{3}{2}$ on the right-hand
side. This result will prove essential in getting the time of existence
$T\sim\frac{1}{{\varepsilon}}$ in the proof of Theorem 1.6. One should also
note that if we have (3.7), then it is enough to show

$\frac{d}{dt}E_{s}(\mathbb{U})\lesssim_{\beta}\|(\eta,u)\|^{3}_{V^{s}_{\sqrt{\mu}}},$

to obtain (3.6). With this in mind, in the proof of the proposition, we will
repeatedly use assumption (3.4)$-$(3.5) to discard higher powers in the norm
of the solution than $3$. Meaning the terms of form
$\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3+n}$ for $n\in{\mathbb{N}}$ will be
bounded by $\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}$ since this seems to be the
best we can hope for when using the current method.

###### Proof of Proposition 3.1.

We first prove estimate (3.7) in the case $\beta\geq 1/3$. By definition, we
have that

$\displaystyle
E_{s}(\mathbb{U})=\|J^{s}\eta\|_{L^{2}}^{2}+\big{(}J^{s}u,(\mathcal{K}_{\mu}(D)+\eta)J^{s}u\big{)}_{L^{2}}.$

Thus, as a result of the non-cavitation condition (3.4) and the estimate
(2.7), there holds

$\displaystyle\big{(}J^{s}u,(\mathcal{K}_{\mu}(D)+\eta)J^{s}u\big{)}_{L^{2}}$
$\displaystyle\geq\frac{h_{0}}{2}\|u\|_{H^{s}}^{2}+c\sqrt{\mu}\|D^{\frac{1}{2}}u\|_{H^{s}}^{2}.$

The reverse inequality holds for any $\beta>0$ and is a consequence of (2.7),
Hölder’s inequality, the Sobolev embedding with $s>\frac{3}{2}$, and
conditions (3.4)$-$(3.5). Indeed, we observe that

$E_{s}(\mathbb{U})\leq\|\eta\|_{H^{s}}^{2}+\|\sqrt{\mathcal{K}_{\mu}}(D)u\|_{H^{s}}^{2}+\|\eta\|_{L^{\infty}}\|u\|_{H^{s}}^{2}\leq
c_{\beta}\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{2}.$

In the case $0<\beta<\frac{1}{3}$, we impose the $\beta-$dependent surface
condition (3.5), leaving less to be absorbed for the coercivity and in
conjunction with (2.8). This implies

$\displaystyle\big{(}J^{s}u,(\mathcal{K}_{\mu}(D)+\eta)J^{s}u\big{)}_{L^{2}}\geq\frac{\beta}{2}\|u\|^{2}_{H^{s}}+c\sqrt{\mu}\|D^{\frac{1}{2}}u\|_{H^{s}}^{2}.$

As a consequence, we have that (3.7) is established for all $\beta>0$.

Next, we prove (3.6). By using (3.1) and the fact that $Q(\mathbb{U},D)$ is
self-adjoint, we compute

$\displaystyle\frac{1}{2}\frac{d}{dt}E_{s}(\mathbb{U})$
$\displaystyle=\big{(}J^{s}\partial_{t}\mathbb{U},Q(\mathbb{U},D)J^{s}\mathbb{U}\big{)}_{L^{2}}+\frac{1}{2}\big{(}J^{s}\mathbb{U},(\partial_{t}Q(\mathbb{U},D))J^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=-\big{(}J^{s}M(\mathbb{U},D)\mathbb{U},Q(\mathbb{U},D)J^{s}\mathbb{U}\big{)}_{L^{2}}+\frac{1}{2}\big{(}J^{s}\mathbb{U},(\partial_{t}Q(\mathbb{U},D))J^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=:-I+II.$

Control of $I$. We may write

$\displaystyle I$
$\displaystyle=\big{(}[J^{s},M(\mathbb{U},D)]\mathbb{U},Q^{(1)}(\mathbb{U},D)J^{s}\mathbb{U}\big{)}_{L^{2}}+\big{(}Q^{(1)}(\mathbb{U},D)M(\mathbb{U},D)J^{s}\mathbb{U},J^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}J^{s}M(\mathbb{U},D)\mathbb{U},Q^{(2)}(\mathbb{U},D)J^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=:I_{1}+I_{2}+I_{3}.$

Control of $I_{1}$. It follows from the Cauchy-Schwarz inequality that

$|I_{1}|\leq\|[J^{s},M(\mathbb{U},D)]\mathbb{U}\|_{L^{2}}\|Q^{(1)}(\mathbb{U},D)J^{s}\mathbb{U}\|_{L^{2}}.$

The second term is easily treated,

$\displaystyle\|Q^{(1)}(\mathbb{U},D)J^{s}\mathbb{U}\|_{L^{2}}\lesssim\|J^{s}\eta\|_{L^{2}}+\|\eta\|_{L^{\infty}}\|J^{s}u\|_{L^{2}}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}},$

by Hölder’s inequality, the Sobolev embedding with $s>\frac{1}{2}$, and
assumption (3.4). Furthermore, using the Kato-Ponce commutator estimate (2.23)
yields

$\displaystyle\|[J^{s},M(\mathbb{U},D)]\mathbb{U}\|_{L^{2}}$
$\displaystyle\leq\|[J^{s},u]\partial_{x}\eta\|_{L^{2}}+\|[J^{s},\eta]\partial_{x}u\|_{L^{2}}+\|[J^{s},u]\partial_{x}u\|_{L^{2}}$
$\displaystyle\leq\|\eta\|_{H^{s}}\|u\|_{H^{s}}+\|u\|_{H^{s}}^{2}$
$\displaystyle\leq\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{2}.$

The desired bound on $I_{1}$ follows:

$|I_{1}|\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$

Control of $I_{2}+I_{3}$. First note that
$(a_{ij})=Q^{(1)}(\mathbb{U},D)M(\mathbb{U},D)$ is given by,

$\displaystyle(a_{ij})=\begin{pmatrix}u\partial_{x}&(\mathcal{K}_{\mu}(D)+\eta)\partial_{x}\\\
\eta\partial_{x}&\eta u\partial_{x}\end{pmatrix}.$

We must estimate each piece below,

$\displaystyle\big{(}Q^{(1)}(\mathbb{U},D)M(\mathbb{U},D)J^{s}\mathbb{U},J^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=\big{(}a_{11}J^{s}\eta,J^{s}\eta\big{)}_{L^{2}}+\big{(}a_{12}J^{s}u,J^{s}\eta\big{)}_{L^{2}}+\big{(}a_{21}J^{s}\eta,J^{s}u\big{)}_{L^{2}}+\big{(}a_{22}J^{s}u,J^{s}u\big{)}_{L^{2}}$
$\displaystyle=:A_{11}+A_{12}+A_{21}+A_{22}.$

As we will shortly see, $A_{12}+A_{21}$ needs to be compensated by $B_{21}$,
that is defined by the remaining part:

$\displaystyle\big{(}J^{s}M(\mathbb{U},D)\mathbb{U},Q^{(2)}(\mathbb{U},D)J^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=\big{(}\partial_{x}J^{s}\eta,\mathcal{K}_{\mu}(D)J^{s}u\big{)}_{L^{2}}+\big{(}J^{s}(u\partial_{x}u),\mathcal{K}_{\mu}(D)J^{s}u\big{)}_{L^{2}}$
$\displaystyle=:B_{21}+B_{22},$

while $B_{22}$ is the price we pay for symmetry.

Control of $A_{11}$. Integration by part and the Sobolev embedding yields

$\displaystyle|A_{11}|$
$\displaystyle\leq\frac{1}{2}\big{|}\big{(}\partial_{x}uJ^{s}\eta,J^{s}\eta\big{)}_{L^{2}}\big{|}\leq\frac{1}{2}\|\partial_{x}u\|_{L^{\infty}}\|\eta\|_{H^{s}}^{2}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$

Control of $A_{12}+A_{21}+B_{21}$. By definition, consideration is given to
the expression

$\displaystyle A_{12}+A_{21}+B_{21}$
$\displaystyle=\big{(}(\mathcal{K}_{\mu}(D)+\eta)\partial_{x}J^{s}u,J^{s}\eta\big{)}_{L^{2}}+\big{(}(\mathcal{K}_{\mu}(D)+\eta)\partial_{x}J^{s}\eta,J^{s}u\big{)}_{L^{2}}.$

Observe, after integration by parts that

$\displaystyle A_{12}$
$\displaystyle=-\big{(}J^{s}u,(\mathcal{K}_{\mu}(D)+\eta)\partial_{x}J^{s}\eta\big{)}_{L^{2}}-\big{(}J^{s}u,\partial_{x}\eta
J^{s}\eta\big{)}_{L^{2}}.$

The first term cancels with $(A_{21}+B_{21})$, while the Sobolev embedding
easily controls the remaining part,

$|\big{(}J^{s}u,\partial_{x}\eta
J^{s}\eta\big{)}_{L^{2}}|\leq\|\partial_{x}\eta\|_{L^{\infty}}\|\eta\|_{H^{s}}\|u\|_{H^{s}}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$

Control of $A_{22}$. We simply use integration by parts as above together with
(3.4)$-$(3.5) to deduce

$|A_{22}|\leq|\big{(}\eta u\partial_{x}J^{s}u,J^{s}u\big{)}_{L^{2}}|\leq
c_{\beta}^{2}\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$

Control of $B_{22}$. We observe, after integrating by parts that

$\displaystyle B_{22}$
$\displaystyle=\big{(}J^{s}(u\partial_{x}u),\mathcal{K}_{\mu}(D)J^{s}u\big{)}_{L^{2}}$
$\displaystyle=\big{(}[\sqrt{\mathcal{K}_{\mu}}(D)J^{s},u]\partial_{x}u,\sqrt{\mathcal{K}_{\mu}}(D)J^{s}u\big{)}_{L^{2}}-\frac{1}{2}\big{(}(\partial_{x}u)\sqrt{\mathcal{K}_{\mu}}(D)J^{s}u,\sqrt{\mathcal{K}_{\mu}}(D)J^{s}u\big{)}_{L^{2}}.$

Thus, we deduce by using Hölder’s inequality, estimates (2.7) and (2.7) that

$\displaystyle|B_{22}|\leq
c_{\beta}^{2}\|(\eta,u)\|_{V_{\sqrt{\mu}}^{s}}^{3}.$

Control of $II$. First we claim that
$\|\mathcal{K}_{\mu}(D)\partial_{x}u\|_{L^{\infty}}\lesssim_{\beta}\|(\eta,u)\|_{V_{\sqrt{\mu}}^{s}}$
for $s>2$. Indeed, it follows from (2.1) and the Sobolev embedding
$H^{\frac{1}{2}^{+}}({\mathbb{R}})\hookrightarrow L^{\infty}({\mathbb{R}})$
that

$\displaystyle\|\mathcal{K}_{\mu}(D)\partial_{x}u\|_{L^{\infty}}$
$\displaystyle\leq\|\partial_{x}u\|_{H^{s-\frac{3}{2}}}+\beta\sqrt{\mu}\|D^{1}\partial_{x}u\|_{H^{s-\frac{3}{2}}}$
$\displaystyle\leq
c_{\beta}^{2}(\|u\|_{H^{s}}+\sqrt{\mu}\|D^{\frac{1}{2}}u\|_{H^{s}}).$ (3.8)

Then we observe by using equation (3.1) yields,

$\displaystyle II$
$\displaystyle=\big{(}J^{s}u,(\partial_{t}\eta)J^{s}u\big{)}_{L^{2}}=-\big{(}J^{s}u,(\mathcal{K}_{\mu}(D)\partial_{x}u)J^{s}u\big{)}_{L^{2}}-\big{(}J^{s}u,(\partial_{x}(\eta
u))J^{s}u\big{)}_{L^{2}}.$

Consequently, the desired estimate follows from Hölder’s inequality, the
Sobolev embedding, and the above claim that,

$\displaystyle\
II\lesssim\|\mathcal{K}_{\mu}(D)\partial_{x}u\|_{L^{\infty}}\|u\|_{H^{s}}^{2}+\|\partial_{x}(\eta
u)\|_{L^{\infty}}\|u\|_{H^{s}}^{2}\lesssim_{\beta}\|(\eta,u)\|_{V_{\sqrt{\mu}}^{s}}^{3}.$
(3.9)

Adding together all the estimates, combined with (3.7) yields,

$\frac{d}{dt}E_{s}(\mathbb{U})\leq
c_{\beta}^{2}\>\big{(}E_{s}(\mathbb{U})\big{)}^{\frac{3}{2}},$

and completes the proof of Proposition 3.1.

∎

###  3.2. Estimates for system (1.6)

As in the former subsection we define
$\mathbb{U}=(\eta,u)^{T}={\varepsilon}(\zeta,v)^{T}$ and we write the system
on a compact form:

$\partial_{t}\mathbb{U}+\mathcal{M}(\mathbb{U},D)\mathbb{U}=\mathbb{0},$
(3.10)

with

$\mathcal{M}(\mathbb{U},D)=\begin{pmatrix}u\partial_{x}&(1+\eta)\partial_{x}\\\
\mathcal{T}_{\mu}(D)\partial_{x}&u\partial_{x}\end{pmatrix}.$ (3.11)

We define the symmetrizer associated to (3.10) to be

$\mathcal{Q}(\mathbb{U},D)=\begin{pmatrix}\mathcal{T}_{\mu}(D)&0\\\
0&1+\eta\end{pmatrix}.$ (3.12)

Then the energy given in Definition 1.18 can be written as

$\mathcal{E}_{s}(\mathbb{U})=\big{(}J^{s}J^{\frac{1}{2}}_{\mu}\mathbb{U},\mathcal{Q}(\mathbb{U},D)J^{s}J^{\frac{1}{2}}_{\mu}\mathbb{U}\big{)}_{L^{2}},$
(3.13)

and the a priori estimate for (1.6) is stated in the following proposition.

######  Proposition 3.3.

Let $s>\frac{3}{2}$, ${\varepsilon},\mu\in(0,1)$ and
$(\eta,u)={\varepsilon}(\zeta,v)\in
C([0,T_{0}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ be a solution to (1.6) on a time
interval $[0,T_{0}]$ for some $T_{0}>0$. Moreover, assume there exist
$h_{0}\in(0,1)$ and $h_{1}>0$ such that

$h_{0}-1\leq\eta(x,t),\quad\forall(x,t)\in\mathbb{R}\times[0,T_{0}]\quad\text{and}\quad\sup\limits_{t\in[0,T_{0}]}\|(\eta,u)\|_{H^{s}\times
H^{s}}\leq h_{1}.$ (3.14)

Then, for the energy given in Definition 1.18, there holds

$\frac{d}{dt}\mathcal{E}_{s}(\mathbb{U})\lesssim\big{(}\mathcal{E}_{s}(\mathbb{U})\big{)}^{\frac{3}{2}},$
(3.15)

for all $0<t<T_{0}$, and

$\|(\eta,u)\|^{2}_{V^{s}_{\sqrt{\mu}}}\lesssim\mathcal{E}_{s}(\mathbb{U})\lesssim\|(\eta,u)\|^{2}_{V^{s}_{\sqrt{\mu}}},$
(3.16)

for all $0<t<T_{0}$.

###### Proof of Proposition 3.3.

We begin by proving (3.16). By Definition (1.18) of the energy, the non-
cavitation condition (3.14), (2.17), and (2.16) we obtain the lower bound

$\displaystyle\mathcal{E}_{s}(\mathbb{U})$
$\displaystyle=\|\sqrt{\mathcal{T}_{\mu}}(D)J^{\frac{1}{2}}_{\mu}\eta\|_{H^{s}}^{2}+\big{(}J^{\frac{1}{2}}_{\mu}J^{s}u,(1+\eta)J^{\frac{1}{2}}_{\mu}J^{s}u\big{)}_{L^{2}}$
$\displaystyle\geq
c\|\eta\|_{H^{s}}^{2}+h_{0}\|J^{\frac{1}{2}}_{\mu}u\|_{H^{s}}^{2}$
$\displaystyle\geq c\|\eta\|_{H^{s}}^{2}+c\cdot
h_{0}(\|u\|_{H^{s}}^{2}+\sqrt{\mu}\|D^{\frac{1}{2}}u\|_{H^{s}}^{2}),$

for some $c>0$. The reverse inequality follows by the estimates (2.16),
(2.17), Hölder’s inequality, the Sobolev embedding, and (3.14):

$\displaystyle\mathcal{E}_{s}(\mathbb{U})\leq\|\sqrt{\mathcal{T}_{\mu}}(D)J^{\frac{1}{2}}_{\mu}\eta\|_{H^{s}}^{2}+\|J^{\frac{1}{2}}_{\mu}u\|_{H^{s}}^{2}+\|\eta\|_{L^{\infty}}\|J^{\frac{1}{2}}_{\mu}u\|^{2}_{H^{s}}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{2}.$

Next, we prove (3.15). There follows by using (3.10) and the self-adjointness
of $\mathcal{Q}(U,D)$ that

$\displaystyle\frac{1}{2}\frac{d}{dt}\mathcal{E}_{s}(\mathbb{U})$
$\displaystyle=-\big{(}J^{s}J_{\mu}^{\frac{1}{2}}\mathcal{M}(\mathbb{U},D)\mathbb{U},\mathcal{Q}(\mathbb{U},D)J^{s}J_{\mu}^{\frac{1}{2}}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle\hskip
28.45274pt+\frac{1}{2}\big{(}J^{s}J_{\mu}^{\frac{1}{2}}\mathbb{U},(\partial_{t}\mathcal{Q}(\mathbb{U},D))J^{s}J_{\mu}^{\frac{1}{2}}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=:-\mathcal{I}+\mathcal{II}.$

Control of $\mathcal{I}$. By definition of (3.13) we decompose $\mathcal{I}$
in four pieces,

$\displaystyle\mathcal{I}$
$\displaystyle=\big{(}J^{s}J^{\frac{1}{2}}_{\mu}(u\partial_{x}\eta),J^{s}J^{\frac{1}{2}}_{\mu}\mathcal{T}_{\mu}(D)\eta\big{)}_{L^{2}}+\big{(}J^{s}J^{\frac{1}{2}}_{\mu}\big{(}(1+\eta)\partial_{x}u\big{)},J^{s}J^{\frac{1}{2}}_{\mu}\mathcal{T}_{\mu}(D)\eta\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}J^{s}J^{\frac{1}{2}}_{\mu}\mathcal{T}_{\mu}(D)\partial_{x}\eta,(1+\eta)J^{s}J^{\frac{1}{2}}_{\mu}u\big{)}_{L^{2}}+\big{(}J^{s}J^{\frac{1}{2}}_{\mu}u\partial_{x}u,(1+\eta)J^{s}J^{\frac{1}{2}}_{\mu}u\big{)}_{L^{2}}$
$\displaystyle=:\mathcal{A}_{11}+\mathcal{A}_{12}+\mathcal{A}_{21}+\mathcal{A}_{22}.$

Control of $\mathcal{A}_{11}$. We aim to exploit symmetries, and we first
write $\mathcal{A}_{11}$ as

$\displaystyle\mathcal{A}_{11}$
$\displaystyle=\big{(}[J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D),u]\partial_{x}\eta,J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}uJ^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}\eta,J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta\big{)}_{L^{2}}$
$\displaystyle=:\mathcal{A}_{11}^{1}+\mathcal{A}_{11}^{2}.$

The first term is treated by the commutator estimate (2.21) with
$s>\frac{3}{2}$, the Cauchy-Schwarz inequality and (2.16). Thus, there holds

$|\mathcal{A}_{11}^{1}|\leq\|[J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D),u]\partial_{x}\eta\|_{L^{2}}\|J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta\|_{L^{2}}\lesssim\|u\|_{H^{s}}\|\eta\|_{H^{s}}^{2}.$

Similar to previous estimates, we use integration by parts and exploit the
symmetries of $\mathcal{A}_{11}^{2}$, then conclude by (2.16), and the Sobolev
embedding $H^{\frac{1}{2}^{+}}({\mathbb{R}})\hookrightarrow
L^{\infty}({\mathbb{R}})$ that

$|\mathcal{A}_{11}|\leq\frac{1}{2}\big{|}\big{(}(\partial_{x}u)J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta,J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta\big{)}_{L^{2}}\big{|}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$

Control of $\mathcal{A}_{12}+\mathcal{A}_{21}$. We first decompose
$\mathcal{A}_{12}$ in two parts

$\displaystyle\mathcal{A}_{12}$
$\displaystyle=\big{(}[J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D),\eta]\partial_{x}u,J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}(1+\eta)J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}u,J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta\big{)}_{L^{2}}$
$\displaystyle=\mathcal{A}_{12}^{1}+\mathcal{A}_{12}^{2}.$

We estimate $\mathcal{A}_{12}^{1}$ the same way we did for
$\mathcal{A}_{11}^{1}$ and obtain

$|\mathcal{A}_{12}^{1}|\lesssim\|u\|_{H^{s}}\|\eta\|_{H^{s}}^{2}.$

For the second term, after integration by parts, we find

$\displaystyle\mathcal{A}_{12}^{2}$
$\displaystyle=-\big{(}(\partial_{x}\eta)J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)u,J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-\big{(}(1+\eta)J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)u,J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}\eta\big{)}_{L^{2}}$
$\displaystyle=\mathcal{A}_{12}^{2,1}+\mathcal{A}_{12}^{2,2}.$

By using the Sobolev embedding and (2.16), we find that

$|\mathcal{A}_{12}^{2,1}|\leq\|\partial_{x}\eta\|_{L^{\infty}}\|J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}u\|_{H^{s}}\|J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}\eta\|_{H^{s}}\lesssim\|u\|_{H^{s}}\|\eta\|_{H^{s}}^{2}.$

On the other hand, we cannot estimate $\mathcal{A}_{12}^{2,2}$ on its own. We
must therefore cancel it with $\mathcal{A}_{21}$. Observe

$\displaystyle\mathcal{A}_{21}$
$\displaystyle=\big{(}J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}\eta,[\sqrt{\mathcal{T}_{\mu}}(D),\eta]J^{s}J^{\frac{1}{2}}_{\mu}u\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}\eta,(1+\eta)J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)u\big{)}_{L^{2}}$
$\displaystyle=\mathcal{A}_{21}^{1}+\mathcal{A}_{21}^{2}.$

First, by using integration by parts, the Cauchy-Schwarz inequality, (2.16),
(2.35) and (2.17) we find that

$\displaystyle|\mathcal{A}_{21}^{1}|=\|J^{s}J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\eta\|_{L^{2}}\|\partial_{x}[\sqrt{\mathcal{T}_{\mu}}(D),\eta]J^{s}J^{\frac{1}{2}}_{\mu}u\|_{L^{2}}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$
(3.17)

On the other hand, we observe that
$\mathcal{A}_{21}^{2}=-\mathcal{A}_{21}^{2,2}$. We may therefore conclude that
the sum satisfies:

$|\mathcal{A}_{12}+\mathcal{A}_{21}|\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$

Control of $\mathcal{A}_{22}$. Similar to $\mathcal{A}_{11}$ we write the
expression with the good commutator:

$\displaystyle\mathcal{A}_{22}$
$\displaystyle=\big{(}[J^{s}J^{\frac{1}{2}}_{\mu},u]\partial_{x}u,(1+\eta)J^{s}J^{\frac{1}{2}}_{\mu}u\big{)}_{L^{2}}+\big{(}uJ^{s}J^{\frac{1}{2}}_{\mu}\partial_{x}u,(1+\eta)J^{s}J^{\frac{1}{2}}_{\mu}u\big{)}_{L^{2}}$
$\displaystyle=\mathcal{A}_{22}^{1}+\mathcal{A}_{22}^{2}.$

Then use the Cauchy-Schwarz inequality, (3.14), (2.20) with $s>\frac{3}{2}$,
and the Sobolev embedding to get

$|\mathcal{A}_{22}^{1}|\lesssim\|[J^{s}J^{\frac{1}{2}}_{\mu},u]\partial_{x}u\|_{L^{2}}(1+\|\eta\|_{L^{\infty}})\|J^{s}J^{\frac{1}{2}}_{\mu}u\|_{L^{2}}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$

While for $\mathcal{A}_{22}^{2}$ we integrate by parts, apply the Sobolev
embedding, and again bound each term by the $V^{s}_{\sqrt{\mu}}-$norm of
$(\eta,u)$ to obtain that

$|\mathcal{A}_{22}^{2}|\lesssim(\|\partial_{x}u\|_{L^{\infty}}+\|\partial_{x}\eta\|_{L^{\infty}})\|J^{\frac{1}{2}}_{\mu}u\|_{H^{s}}^{2}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$

Gathering all these estimates, we conclude that

$|\mathcal{I}|\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$ (3.18)

Control of $\mathcal{II}$. By defintion of (3.12) and (3.10) we get that,

$\displaystyle\mathcal{II}$
$\displaystyle=\big{(}J^{s}J^{\frac{1}{2}}_{\mu}u,(\partial_{t}\eta)J^{s}J^{\frac{1}{2}}_{\mu}u\big{)}_{L^{2}}$
$\displaystyle=-\big{(}J^{s}J^{\frac{1}{2}}_{\mu}u,(\partial_{x}u)J^{s}J^{\frac{1}{2}}_{\mu}u\big{)}_{L^{2}}-\big{(}J^{s}J^{\frac{1}{2}}_{\mu}u,(\partial_{x}(\eta
u))J^{s}J^{\frac{1}{2}}_{\mu}u\big{)}_{L^{2}}$

Then, by using Hölder’s inequality, the Sobolev embedding, (3.14) and (2.17),
we deduce that

$|\mathcal{II}|\lesssim(\|\partial_{x}u\|_{L^{\infty}}+\|\partial_{x}(\eta
u)\|_{L^{\infty}})\|J^{\frac{1}{2}}_{\mu}u\|_{H^{s}}^{2}\lesssim\|(\eta,u)\|_{V^{s}_{\sqrt{\mu}}}^{3}.$
(3.19)

Consequently, we may add (3.18) and (3.19), then apply (3.16) to conclude the
proof of estimate (3.15).

∎

###  3.3. Estimates for system (1.7)

As in the former subsections we let
$\mathbb{U}=(\eta,u)^{T}={\varepsilon}(\zeta,v)^{T}$ and write the system on
the form

$\partial_{t}\mathbb{U}+\mathscr{M}(\mathbb{U},D)\mathbb{U}=\mathbb{0},$
(3.20)

with

$\mathscr{M}(\mathbb{U},D)=\begin{pmatrix}\mathcal{T}_{\mu}(D)(u\partial_{x}\boldsymbol{\cdot})&\partial_{x}+\mathcal{T}_{\mu}(D)(\eta\partial_{x}\boldsymbol{\cdot})\\\
\mathcal{K}_{\mu}(D)\partial_{x}&\mathcal{T}_{\mu}(D)(u\partial_{x}\boldsymbol{\cdot})\end{pmatrix}.$
(3.21)

The symmetrizer is defined by

$\mathscr{Q}(\mathbb{U},D)=\begin{pmatrix}\mathcal{T}_{\mu}^{-1}(D)\mathcal{K}_{\mu}(D)&0\\\
0&\mathcal{T}_{\mu}^{-1}(D)+\eta\end{pmatrix}.$ (3.22)

Then the energy given in Definition 1.19 can be written as

$\mathscr{E}_{s}(\mathbb{U})=\big{(}J^{s}\mathbb{U},\mathscr{Q}(\mathbb{U},D)J^{s}\mathbb{U}\big{)}_{L^{2}},$
(3.23)

and the a priori estimate for (1.7) is stated in the following proposition.

######  Proposition 3.4.

Let $s>\frac{3}{2}$, ${\varepsilon},\mu\in(0,1)$, $\beta>0$, and let
$(\eta,u)={\varepsilon}(\zeta,v)\in
C([0,T_{0}];X^{s}_{\beta,\mu}(\mathbb{R}))$ be a solution to (1.7) on a time
interval $[0,T_{0}]$ for some $T_{0}>0$. Moreover, assume that there exist
$h_{0}\in(0,1)$ and $h_{1}>0$ such that

$h_{0}-1\leq\eta(x,t),\quad\forall(x,t)\in\mathbb{R}\times[0,T_{0}]\quad\text{and}\quad\sup\limits_{t\in[0,T_{0}]}\|(\eta,u)\|_{H^{s}\times
H^{s}}\leq h_{1}.$ (3.24)

Then, for the energy given in Definition 1.19, there holds,

$\frac{d}{dt}\mathscr{E}_{s}(\mathbb{U})\lesssim
c_{\beta}^{2}\big{(}\mathscr{E}_{s}(\mathbb{U})\big{)}^{\frac{3}{2}},$ (3.25)

and the energy is coercive:

$\|(\eta,u)\|^{2}_{X^{s}_{\beta,\mu}}\lesssim\mathscr{E}_{s}(\mathbb{U})\lesssim\|(\eta,u)\|^{2}_{X^{s}_{\beta,\mu}}.$
(3.26)

###### Proof of Proposition 3.4.

We will first provide the coercivity estimate (3.26). By Definition 1.19 for
the energy, the non-cavitation condition (3.24) and (2.15) we obtain the lower
bound

$\displaystyle\mathscr{E}_{s}(\mathbb{U})$
$\displaystyle=\|\langle\sqrt{\beta\mu}D\rangle\eta\|_{H^{s}}^{2}+\big{(}J^{s}u,(\mathcal{T}^{-1}_{\mu}(D)+\eta)J^{s}u\big{)}_{L^{2}}$
$\displaystyle\geq\|\eta\|_{H^{s}}^{2}+\beta\mu\|D^{1}\eta\|_{H^{s}}^{2}+\frac{h_{0}}{2}\|u\|_{H^{s}}^{2}+c\sqrt{\mu}\|D^{\frac{1}{2}}u\|_{H^{s}}^{2},$

for some $c>0$. The reverse inequality follows by the upper bound in (2.15),
the Sobolev embedding and (3.24):

$\displaystyle\mathscr{E}_{s}(\mathbb{U})\leq\|\langle\sqrt{\beta\mu}D\rangle\eta\|_{H^{s}}^{2}+\|\mathcal{T}^{-\frac{1}{2}}_{\mu}(D)J^{s}u\|^{2}+\|\eta\|_{L^{\infty}}\|J^{s}u\|^{2}\lesssim\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{2}.$

We may now prove (3.25). To do so, we use (3.20) and the self-adjointness of
$\mathscr{Q}(\mathbb{U},D)$ to write

$\displaystyle\frac{1}{2}\frac{d}{dt}\mathscr{E}_{s}(\mathbb{U})$
$\displaystyle=-\big{(}J^{s}\mathscr{M}(\mathbb{U},D)\mathbb{U},\mathscr{Q}(\mathbb{U},D)J^{s}\mathbb{U}\big{)}_{L^{2}}+\frac{1}{2}\big{(}J^{s}\mathbb{U},(\partial_{t}\mathscr{Q}(\mathbb{U},D))J^{s}\mathbb{U}\big{)}_{L^{2}}$
$\displaystyle=:\mathscr{I}+\mathscr{II}.$

Control of $\mathscr{I}$. By definition of (3.23) we must estimate the
following terms:

$\displaystyle\mathscr{I}$
$\displaystyle=\big{(}J^{s}(u\partial_{x}\eta),\mathcal{K}_{\mu}(D)J^{s}\eta\big{)}_{L^{2}}+\big{(}J^{s}\partial_{x}u+J^{s}\mathcal{T}_{\mu}(D)(\eta\partial_{x}u),\mathcal{T}_{\mu}^{-1}(D)\mathcal{K}_{\mu}(D)J^{s}\eta\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}J^{s}\mathcal{K}_{\mu}(D)\partial_{x}\eta,(\mathcal{T}_{\mu}^{-1}(D)+\eta)J^{s}u\big{)}_{L^{2}}+\big{(}J^{s}\mathcal{T}_{\mu}(D)(u\partial_{x}u),(\mathcal{T}_{\mu}^{-1}(D)+\eta)J^{s}u\big{)}_{L^{2}}$
$\displaystyle=:\mathscr{A}_{11}+\mathscr{A}_{12}+\mathscr{A}_{21}+\mathscr{A}_{22}.$

Control of $\mathscr{A}_{11}$. We rewrite $\mathscr{A}_{11}$ as

$\displaystyle\mathscr{A}_{11}$
$\displaystyle=\big{(}[J^{s}\sqrt{\mathcal{K}_{\mu}}(D),u]\partial_{x}\eta,J^{s}\sqrt{\mathcal{K}_{\mu}}(D)\eta\big{)}_{L^{2}}+\big{(}uJ^{s}\sqrt{\mathcal{K}_{\mu}}(D)\partial_{x}\eta,J^{s}\sqrt{\mathcal{K}_{\mu}}(D)\eta\big{)}_{L^{2}}$
$\displaystyle=:\mathscr{A}_{11}^{1}+\mathscr{A}_{11}^{2}.$

For the first term, we first note by interpolation and Young’s inequality that

$\sqrt{\beta}\mu^{\frac{1}{4}}\|\eta\|_{H^{s+\frac{1}{2}}}\leq\|\eta\|_{H^{s}}^{\frac{1}{2}}\big{(}\beta\sqrt{\mu}\|\eta\|_{H^{s+1}}\big{)}^{\frac{1}{2}}\lesssim_{\beta}\|\eta\|_{H^{s}}+\beta\sqrt{\mu}\|D^{1}\eta\|_{H^{s}},$
(3.27)

and thus $\mathscr{A}_{11}^{1}$ is treated by the Cauchy-Schwarz inequality,
the commutator estimate (2.7) with $s>\frac{3}{2}$, (2.7), and (3.27):

$|\mathscr{A}_{11}^{1}|\lesssim
c_{\beta}^{2}(\|u\|_{H^{s}}+\mu^{\frac{1}{2}}\|u\|_{H^{s+\frac{1}{2}}})(\|\eta\|_{H^{s}}+\sqrt{\beta}\mu^{\frac{1}{4}}\|\eta\|_{H^{s+\frac{1}{2}}})^{2}\lesssim
c_{\beta}^{2}\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{3}.$

On the other hand, for $\mathscr{A}_{11}^{2}$ we conclude by integration by
parts, (2.7), (3.27), and the Sobolev embedding with $s>\frac{3}{2}$ that

$|\mathscr{A}_{11}|\lesssim|\mathscr{A}_{11}^{1}|+\|\partial_{x}u\|_{L^{\infty}}\|\sqrt{\mathcal{K}_{\mu}}(D)\eta\|_{H^{s}}\|\sqrt{\mathcal{K}_{\mu}}(D)\eta\|_{H^{s}}\lesssim
c_{\beta}^{2}\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{3}.$

Control of $\mathscr{A}_{12}+\mathscr{A}_{21}$. By using integration by parts
we write,

$\displaystyle\mathscr{A}_{12}$
$\displaystyle=\big{(}J^{s}(\eta\partial_{x}u),\mathcal{K}_{\mu}(D)J^{s}\eta\big{)}_{L^{2}}-\big{(}J^{s}u,\mathcal{T}_{\mu}^{-1}(D)\mathcal{K}_{\mu}(D)J^{s}\partial_{x}\eta\big{)}_{L^{2}}$
$\displaystyle=\mathscr{A}_{12}^{1}+\mathscr{A}_{12}^{2}.$

For $\mathscr{A}_{12}^{1}$, observe

$\displaystyle\mathscr{A}_{12}^{1}$
$\displaystyle=\big{(}[J^{s},\eta]\partial_{x}u,\mathcal{K}_{\mu}(D)J^{s}\eta\big{)}_{L^{2}}-\big{(}(\partial_{x}\eta)J^{s}u,\mathcal{K}_{\mu}(D)J^{s}\eta\big{)}_{L^{2}}-\big{(}\eta
J^{s}u,\mathcal{K}_{\mu}(D)J^{s}\partial_{x}\eta\big{)}_{L^{2}}$
$\displaystyle=\mathscr{A}_{12}^{1,1}+\mathscr{A}_{12}^{1,2}+\mathscr{A}_{12}^{1,3}.$

Then the Kato-Ponce commutator estimate (2.9), the Sobolev embedding, and the
pointwise estimate (2.1) combined with Plancherel imply that

$\displaystyle|\mathscr{A}_{12}^{1,1}+\mathscr{A}_{12}^{1,2}|\lesssim
c_{\beta}^{2}\|u\|_{H^{s}}\|\eta\|_{H^{s}}(\|\eta\|_{H^{s}}+\beta\sqrt{\mu}\|D^{1}\eta\|_{H^{s}})\lesssim
c_{\beta}^{2}\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{3}.$

While the contribution of remaining terms,
$\mathscr{A}_{12}^{1,3}+\mathscr{A}_{12}^{2}$, will be canceled by
$\mathscr{A}_{21}$. Indeed, we observe that

$\displaystyle\mathscr{A}_{21}=\big{(}J^{s}\mathcal{K}_{\mu}(D)\partial_{x}\eta,\eta
J^{s}u\big{)}_{L^{2}}+\big{(}J^{s}\mathcal{K}_{\mu}(D)\partial_{x}\eta,\mathcal{T}^{-1}_{\mu}(D)J^{s}u\big{)}_{L^{2}}=-\mathscr{A}_{12}^{1,3}-\mathscr{A}_{12}^{2}.$

Hence, combining these identities and estimates gives the bound

$|\mathscr{A}_{12}+\mathscr{A}_{21}|\lesssim
c_{\beta}^{2}\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{3}.$

Control of $\mathscr{A}_{22}$. Consider the two terms:

$\displaystyle\mathscr{A}_{22}=\big{(}J^{s}(u\partial_{x}u),J^{s}u\big{)}_{L^{2}}+\big{(}J^{s}\mathcal{T}_{\mu}(D)(u\partial_{x}u),\eta
J^{s}u\big{)}_{L^{2}}=\mathscr{A}_{22}^{1}+\mathscr{A}_{22}^{2}.$

The control of $\mathscr{A}_{22}^{1}$ is a direct consequence of the Kato-
Ponce commutator estimate (2.9) and integration by parts. Since
$s>\frac{3}{2}$, we have that

$|\mathscr{A}_{22}^{1}|\leq\big{|}\big{(}[J^{s},u]\partial_{x}u,J^{s}u\big{)}_{L^{2}}\big{|}+\frac{1}{2}\big{|}\big{(}(\partial_{x}u)J^{s}u,J^{s}u\big{)}_{L^{2}}\big{|}\lesssim\|u\|_{H^{s}}^{3}.$

To deal with $\mathscr{A}_{22}^{2}$, we make the decomposition

$\displaystyle\mathscr{A}_{22}^{2}$
$\displaystyle=\big{(}[J^{s}\sqrt{\mathcal{T}_{\mu}}(D),u]\partial_{x}u,\sqrt{\mathcal{T}_{\mu}}(D)\eta
J^{s}u\big{)}_{L^{2}}+\big{(}uJ^{s}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}u,[\sqrt{\mathcal{T}_{\mu}}(D),\eta]J^{s}u\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}uJ^{s}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}u,\eta\sqrt{\mathcal{T}_{\mu}}(D)J^{s}u\big{)}_{L^{2}}$
$\displaystyle=\mathscr{A}_{22}^{2,1}+\mathscr{A}_{22}^{2,2}+\mathscr{A}_{22}^{2,3}.$

Then for $\mathscr{A}_{22}^{2,1}$ we employ the Cauchy-Schwarz inequality,
(2.33), (3.24), (2.14), and the Sobolev embedding to deduce

$\displaystyle|\mathscr{A}_{22}^{2,1}|\leq\|[J^{s}\sqrt{\mathcal{T}_{\mu}}(D),u]\partial_{x}u\|_{L^{2}}\|\sqrt{\mathcal{T}_{\mu}}(D)(\eta
J^{s}u)\|_{L^{2}}\lesssim\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{3}.$

Before we treat $\mathscr{A}_{22}^{2,2,1}$, we note that
$\|[\sqrt{\mathcal{T}_{\mu}}(D),\eta]J^{s}u\|_{L^{2}}\lesssim\|\eta\|_{H^{s}}\|u\|_{H^{s}}$.
Indeed, using (2.14) and the Sobolev embedding we find that

$\displaystyle\|[\sqrt{\mathcal{T}_{\mu}}(D),\eta]J^{s}u\|_{L^{2}}\lesssim\|\eta\|_{L^{\infty}}\|J^{s}u\|_{L^{2}}\lesssim\|\eta\|_{H^{s}}\|u\|_{H^{s}}.$
(3.28)

Consequently, using integration by parts, the Cauchy-Schwarz inequality,
(3.28) and (2.35) we get

$\displaystyle|\mathscr{A}_{22}^{2,2}|$
$\displaystyle=\big{|}\big{(}(\partial_{x}u)J^{s}\sqrt{\mathcal{T}_{\mu}}(D)u,[\sqrt{\mathcal{T}_{\mu}}(D),\eta]J^{s}u\big{)}_{L^{2}}\big{|}$
$\displaystyle\hskip
14.22636pt+\big{|}\big{(}uJ^{s}\sqrt{\mathcal{T}_{\mu}}(D)u,\partial_{x}[\sqrt{\mathcal{T}_{\mu}}(D),\eta]J^{s}u\big{)}_{L^{2}}\big{|}$
$\displaystyle\lesssim\|\eta\|_{H^{s}}\|u\|_{H^{s}}^{3},$

then use (3.24) on one term. Similarly, for $\mathscr{A}_{22}^{2,1}$ we use
integration by parts, the Sobolev embedding, and (2.14) to get the bound

$|\mathscr{A}_{22}^{2,3}|\lesssim\|\partial_{x}(\eta
u)\|_{L^{\infty}}\|u\|_{H^{s}}^{2}.$

Therefore, we conclude by (3.24) and gathering all these estimates that

$|\mathscr{A}_{22}|\lesssim\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{3},$

and by extension, we have the bound

$|\mathscr{I}|\lesssim_{\beta}\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{3}.$

Control of $\mathscr{II}$. By defintion of (3.22) and (3.20) we get that,

$\displaystyle\mathscr{II}$
$\displaystyle=\big{(}J^{s}u,(\partial_{t}\eta)J^{s}u\big{)}_{L^{2}}$
$\displaystyle=-\big{(}J^{s}u,(\partial_{x}u)J^{s}u\big{)}_{L^{2}}-\big{(}J^{s}u,(\mathcal{T}_{\mu}(D)\partial_{x}(\eta
u))J^{s}u\big{)}_{L^{2}},$

Then the final estimate follows by the Cauchy-Schwarz inequality, (3.24) and
the fact that $\mathcal{T}_{\mu}(D)$ is bounded on $L^{2}({\mathbb{R}})$, then
apply Hölder’s inequality, and the Sobolev embedding to deduce

$|\mathscr{II}|\leq\|\partial_{x}u\|_{L^{\infty}}\|u\|_{H^{s}}^{2}+\|\mathcal{T}_{\mu}(D)\partial_{x}(\eta
u)\|_{L^{\infty}}\|u\|_{H^{s}}^{2}\lesssim\|(\eta,u)\|_{X^{s}_{\beta,\mu}}^{3}.$

∎

##  4\. Estimates for the difference of two solutions

###  4.1. Estimates for system (1.5)

We will now estimate the difference between two solutions of (1.5) given by
$\mathbb{U}_{1}=(\eta_{1},u_{1})^{T}={\varepsilon}(\zeta_{1},v_{1})^{T}$ and
$\mathbb{U}_{2}=(\eta_{2},u_{2})^{T}={\varepsilon}(\zeta_{2},v_{2})^{T}$. For
convenience, we define $(\psi,w)=(\eta_{1}-\eta_{2},u_{1}-u_{2})$. Then
$\mathbb{W}=(\psi,w)^{T}$ solves

$\partial_{t}\mathbb{W}+M(\mathbb{U}_{1},D)\mathbb{W}=\mathbb{F},$ (4.1)

with $M$ defined as in (3.2) and
$\mathbb{F}=-\big{(}M(\mathbb{U}_{1},D)-M(\mathbb{U}_{2},D)\big{)}\mathbb{U}_{2}$.
Specifically, the source term is given by

$\mathbb{F}=-\begin{pmatrix}w\partial_{x}\eta_{2}+\psi\partial_{x}u_{2}\\\
w\partial_{x}u_{2}\end{pmatrix}.$ (4.2)

The energy associated to (4.1) is given in terms of the symmetrizer
$Q(\mathbb{U}_{1},D)$ defined in (3.3) and reads

$\tilde{E}_{s}(\mathbb{W}):=\big{(}J^{s}\mathbb{W},Q(\mathbb{U}_{1},D)J^{s}\mathbb{W}\big{)}_{L^{2}}.$
(4.3)

The main result of this section reads:

######  Proposition 4.1.

Take $s>2$ and ${\varepsilon},\mu\in(0,1)$. Let
$(\eta_{1},u_{1}),(\eta_{2},u_{2})\in
C([0,T_{0}]:V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ be two solutions of (1.5) on a
time interval $[0,T_{0}]$ for some $T_{0}>0$. Moreover, assume there exist
$h_{0}\in(0,1)$ and $h_{1}>0$ such that

$h_{0}-1\leq\eta_{1}(x,t),\quad\forall(x,t)\in\mathbb{R}\times[0,T_{0}]\quad\text{and}\quad\sup\limits_{t\in[0,T_{0}]}\|(\eta_{1},u_{1})\|_{H^{s}\times
H^{s}}\leq h_{1},$ (4.4)

when $\beta\geq\frac{1}{3}$, and that

$-\frac{\beta}{2}\leq\eta_{1}(x,t),\quad\forall(x,t)\in\mathbb{R}\times[0,T_{0}]\quad\text{and}\quad\sup\limits_{t\in[0,T_{0}]}\|(\eta_{1},u_{1})\|_{H^{s}\times
H^{s}}\leq h_{1},$ (4.5)

when $0<\beta<\frac{1}{3}$.

Define the difference to be
$\mathbb{W}=(\psi,w)=(\eta_{1}-\eta_{2},u_{1}-u_{2})$. Then, for the energy
defined by (4.3), there holds

$\frac{d}{dt}\tilde{E}_{0}(\mathbb{W})\lesssim_{\beta}\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2},$
(4.6)

and

$\|(\psi,w)\|^{2}_{V^{0}_{\sqrt{\mu}}}\lesssim_{\beta}\tilde{E}_{0}(\mathbb{W})\lesssim_{\beta}\|(\psi,w)\|^{2}_{V^{0}_{\sqrt{\mu}}}.$
(4.7)

Furthermore, we have the following estimate at the $V^{s}_{\sqrt{\mu}}-$
level:

$\frac{d}{dt}\tilde{E}_{s}(\mathbb{W})\lesssim_{\beta}|\big{(}J^{s}\mathbb{F},Q(\mathbb{U}_{1},D)J^{s}\mathbb{W}\big{)}_{L^{2}}|+\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{s}_{\sqrt{\mu}}}^{2},$
(4.8)

and

$\|(\psi,w)\|^{2}_{V^{s}_{\sqrt{\mu}}}\lesssim_{\beta}\tilde{E}_{s}(\mathbb{W})\lesssim_{\beta}\|(\psi,w)\|^{2}_{V^{s}_{\sqrt{\mu}}}.$
(4.9)

######  Remark 4.2.

The source term corresponding to $\mathbb{F}$ given by (4.8) will be treated
in the proof of Theorem 1.6 by using regularisation estimates and a classical
Bona-Smith argument.

###### Proof of Proposition 4.1.

First, the proofs of (4.7) and (4.9) are similar to the one of (3.7).

Next, we only prove (4.6), where (4.8) is more straightforward and follows the
same line, utilizing similar estimates to those applied for the proof of
Proposition 3.1.

To prove (4.6), we use (4.1) and the self-adjointness of $Q(\mathbb{U}_{1},D)$
to write

$\displaystyle\frac{1}{2}\frac{d}{dt}\tilde{E}_{0}(\mathbb{W})$
$\displaystyle=\frac{1}{2}\big{(}\mathbb{W},(\partial_{t}Q(\mathbb{U}_{1},D))\mathbb{W}\big{)}_{L^{2}}+\big{(}\mathbb{F},Q(\mathbb{U}_{1},D)\mathbb{W}\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt-\big{(}M(\mathbb{U}_{1},D)\mathbb{W},Q(\mathbb{U}_{1},D)\mathbb{W}\big{)}_{L^{2}}$
$\displaystyle=:I-II-III.$

Control of $I$. We estimate the first term for $s>2$ by arguing similarly to
estimate (3.9). Indeed, we have that

$\displaystyle
I=(w,(\partial_{t}\eta_{1})w)\lesssim\|\partial_{t}\eta_{1}\|_{L^{\infty}}\|w\|_{L^{2}}^{2}\lesssim_{\beta}\|(u_{1},\eta_{1})\|_{V_{\sqrt{\mu}}^{s}}\|(\psi,w)\|_{V_{\sqrt{\mu}}^{0}}^{2}.$

Control of $II$. For $II$, we write

$\displaystyle II$
$\displaystyle=\big{(}w\partial_{x}\eta_{2},\psi\big{)}_{L^{2}}+\big{(}\psi\partial_{x}\eta_{2},\psi\big{)}_{L^{2}}+\big{(}w\partial_{x}u_{2},\eta_{1}w\big{)}_{L^{2}}+\big{(}w\partial_{x}u_{2},\mathcal{K}_{\mu}(D)w\big{)}_{L^{2}}$
$\displaystyle=:II_{1}+II_{2}+II_{3}+II_{4}.$

The first three terms are treated by the Cauchy-Schwarz inequality and the
Sobolev embedding. Take, for instance, $II_{1}$:

$\displaystyle|II_{1}|\lesssim\|w\partial_{x}\eta_{2}\|_{L^{2}}\|\psi\|_{L^{2}}\lesssim\|\eta_{2}\|_{H^{s}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2},$

for $s>\frac{3}{2}$. Then estimating $II_{2}+II_{3}$ similarly gives

$|II_{1}+II_{2}+II_{3}|\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$

Regarding the term containing the multiplier $\mathcal{K}_{\mu}(D)$, we write

$\displaystyle
II_{4}\leq\|\sqrt{\mathcal{K}_{\mu}}(D)(w\partial_{x}u_{2})\|_{L^{2}}\|\sqrt{\mathcal{K}_{\mu}}(D)w\|_{L^{2}}=:II_{4}^{1}\cdot
II_{4}^{2},$

and make the observation

$\displaystyle II_{4}^{1}$
$\displaystyle\leq\|(\sqrt{\mathcal{K}_{\mu}}(D)-\sqrt{\beta}\mu^{\frac{1}{4}}D^{\frac{1}{2}})(w\partial_{x}u_{2})\|_{L^{2}}+\sqrt{\beta}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}(w\partial_{x}u_{2})\|_{L^{2}}$
$\displaystyle=:II_{4}^{1,1}+\sqrt{\beta}II_{4}^{1,2}.$

For the first term, we note that
$(\sqrt{\mathcal{K}_{\mu}}(D)-\sqrt{\beta}\mu^{\frac{1}{4}}D^{\frac{1}{2}})$
is bounded on $L^{2}(\mathbb{R})$ by (2.5), and we can conclude by the Sobolev
embedding that

$\displaystyle
II_{4}^{1,1}\lesssim_{\beta}\|w\partial_{x}\eta_{2}\|_{L^{2}}\lesssim_{\beta}\|\eta_{2}\|_{H^{s}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}.$

For the remaining term, $II_{4}^{1,2}$, we first make an observation. Let
$\nu=\frac{1}{2}^{-}$ and $(p_{1},p_{2})=(\frac{1}{\nu},\frac{2}{1-2\nu})$
then by (2.37) there holds

$\|D^{\frac{1}{2}}\partial_{x}u_{2}\|_{L^{p_{1}}}\mu^{\frac{1}{4}}\|w\|_{L^{p_{2}}}\lesssim\|u_{2}\|_{H^{2-\nu}}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}w\|_{L^{2}}.$
(4.10)

Moreover, by the fractional Leibniz rule (2.38), the triangle inequality and
Hölder’s inequality yields the bound

$\displaystyle II^{1,2}_{4}$
$\displaystyle\lesssim\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}(w\partial_{x}u_{2})-wD^{\frac{1}{2}}\partial_{x}u_{2}-(\partial_{x}u_{2})D^{\frac{1}{2}}w\|_{L^{2}}+\mu^{\frac{1}{4}}\|wD^{\frac{1}{2}}\partial_{x}u_{2}\|_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\mu^{\frac{1}{4}}\|(\partial_{x}u_{2})D^{\frac{1}{2}}w\|_{L^{2}}$
$\displaystyle\lesssim\|D^{\frac{1}{2}}\partial_{x}u_{2}\|_{L^{p_{1}}}\mu^{\frac{1}{4}}\|w\|_{L^{p_{2}}}+\|w\|_{L^{2}}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}\partial_{x}u_{2}\|_{L^{\infty}}+\|\partial_{x}u_{2}\|_{L^{\infty}}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}w\|_{L^{2}}.$

Now, since $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\nu+\frac{1-2\nu}{2}=\frac{1}{2}$,
we may apply (4.10) to deal with the first term, and combined with the Sobolev
embedding $H^{\frac{1}{2}^{+}}({\mathbb{R}})\hookrightarrow
L^{\infty}({\mathbb{R}})$ we deduce that

$\displaystyle
II^{1,2}_{4}\lesssim\|u_{2}\|_{H^{s}}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}w\|_{L^{2}}+\|w\|_{L^{2}}\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}u_{2}\|_{H^{s}},$

with $s>\frac{3}{2}$. Consequently, the bound on $II_{4}$ is given by

$\displaystyle
II_{4}\lesssim_{\beta}\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2},$

which allows us to conclude that

$II\lesssim_{\beta}\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$

Control of $III$. By definition, we must estimate:

$\displaystyle III$
$\displaystyle=\big{(}u_{1}\partial_{x}\psi,\psi\big{)}_{L^{2}}+\big{(}(\mathcal{K}_{\mu}(D)+\eta_{1})\partial_{x}w,\psi\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}\partial_{x}\psi,(\mathcal{K}_{\mu}(D)+\eta_{1})w\big{)}_{L^{2}}+\big{(}u_{1}\partial_{x}w,(\mathcal{K}_{\mu}(D)+\eta_{1})w\big{)}_{L^{2}}$
$\displaystyle=A_{1}+A_{2}+A_{3}+A_{4}.$

The first term is handled by integration by parts and the Sobolev embedding

$\displaystyle A_{1}$
$\displaystyle\lesssim\|\partial_{x}u_{1}\|_{L^{\infty}}\|\psi\|_{L^{2}}^{2}\lesssim\|u_{1}\|_{H^{s}}\|\psi\|_{L^{2}}^{2}.$

Next, we observe a cancelation in the off-diagonal terms due to the symmetry.
Indeed, we see after integrating by parts that

$\displaystyle
A_{2}=-\big{(}(\partial_{x}\eta_{1})w,\psi\big{)}_{L^{2}}-A_{3}.$

Consequently, we observe after using Hölder’s inequality and the Sobolev
embedding that

$\displaystyle|A_{2}+A_{3}|\lesssim\|\partial_{x}\eta_{1}\|_{L^{\infty}}\|w\|_{L^{2}}\|\psi\|_{L^{2}}.$

The only term remaining is $A_{4}$, which contains the multiplier that will
need some more care. In particular, we write

$\displaystyle A_{4}$
$\displaystyle=\big{(}u_{1}\partial_{x}w,\eta_{1}w\big{)}_{L^{2}}+\big{(}u_{1}\partial_{x}w,\mathcal{K}_{\mu}(D)w\big{)}_{L^{2}}$
$\displaystyle=:A_{4}^{1}+A_{4}^{2}.$

The first term is again treated by integration by parts, and we obtain the
bound

$|A_{4}^{1}|\lesssim\|u_{1}\|_{H^{s}}\|\eta_{1}\|_{H^{s}}\|w\|_{L^{2}}^{2}.$

Lastly, to estimate $A_{4}^{2}$, we split the kernel $\mathcal{K}_{\mu}(D)$
into several pieces that are localized in low and high frequencies:

$\displaystyle\mathcal{K}_{\mu}(D)=((\chi^{(1)}_{\mu})^{2}\mathcal{K}_{\mu})(D)+((\chi_{\mu}^{(2)})^{2}(\sigma_{\mu,\frac{1}{2}})^{2})(D)-((\chi^{(2)}_{\mu})^{2}(\sigma_{\mu,0})^{2})(D),$
(4.11)

where $\sigma_{\mu,\frac{1}{2}}(D)$ is defined in (2.26), $\sigma_{\mu,0}(D)$
is defined in (2.29) and $\chi^{(i)}_{\mu}(D)$ with its porperties given by
Definition 2.10. Then, we get that

$\displaystyle\big{(}u_{1}\partial_{x}w,\mathcal{K}_{\mu}(D)w\big{)}_{L^{2}}$
$\displaystyle=\big{(}(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)(u_{1}\partial_{x}w),(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)w\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}(\chi^{(2)}_{\mu}\sigma_{\mu,\frac{1}{2}})(D)(u_{1}\partial_{x}w),(\chi^{(2)}_{\mu}\sigma_{\mu,\frac{1}{2}})(D)w\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt-\big{(}(\chi^{(2)}_{\mu}\sigma_{\mu,0})(D)(u_{1}\partial_{x}w),(\chi^{(2)}_{\mu}\sigma_{\mu,0})(D)w\big{)}_{L^{2}}$
$\displaystyle=:A^{2,1}_{4}+A^{2,2}_{4}-A^{2,3}_{4}.$

We treat each term individually using the commutator estimates in Lemma 2.11,
where the remaining part is symmetric and is treated by using integration by
parts and the Sobolev embedding in the usual way.

Control of $A^{2,1}_{4}$. Proceeding as explained above, we have that

$\displaystyle A^{2,1}_{4}$
$\displaystyle=\big{(}[(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D),u_{1}]\partial_{x}w,(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)w\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}u_{1}(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)\partial_{x}w,(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)w\big{)}_{L^{2}}$
$\displaystyle=A^{2,1,1}_{4}+A^{2,1,2}_{4}.$

For $A^{2,1,1}_{4}$ we use the Cauchy-Schwarz inequality, (2.24), and (2.25)
to obtain the bound

$\displaystyle A^{2,1,1}_{4}$
$\displaystyle\lesssim\|[(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D),u_{1}]\partial_{x}w\|_{L^{2}}\|(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)w\|_{L^{2}}$
$\displaystyle\lesssim\|u_{1}\|_{H^{s}}\|w\|_{L^{2}}^{2}.$

For the remaining term, we deduce from (2.24) that

$\displaystyle A^{2,1,2}_{4}$
$\displaystyle=-\frac{1}{2}\big{(}(\partial_{x}u_{1})(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)w,(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(D)w\big{)}_{L^{2}}$
$\displaystyle\lesssim\|\partial_{x}u_{1}\|_{L^{\infty}}\|w\|_{L^{2}}^{2}.$

Control of $A^{2,2}_{4}$. Similarly we get from the estimates (2.27), (2.28)
and the Sobolev embedding that

$\displaystyle A^{2,2}_{4}$
$\displaystyle=\big{(}[(\chi^{(2)}_{\mu}\sigma_{\mu,\frac{1}{2}})(D),u_{1}]\partial_{x}w,(\chi^{(2)}_{\mu}\sigma_{\mu,\frac{1}{2}})(D)w\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt-\frac{1}{2}\big{(}(\partial_{x}u_{1})(\chi^{(2)}_{\mu}\sigma_{\mu,\frac{1}{2}})(D)w,(\chi^{(2)}_{\mu}\sigma_{\mu,\frac{1}{2}})(D)w\big{)}_{L^{2}}$
$\displaystyle\lesssim\|u_{1}\|_{H^{s}}(\|w\|_{L^{2}}+\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}w\|_{L^{2}})^{2}.$

Control of $A^{2,3}_{4}$. By the same approach as above, combined with
estimates (2.30) and (2.31) leaves us with the bound

$\displaystyle A^{2,3}_{4}$
$\displaystyle=\big{(}[(\chi^{(2)}_{\mu}\sigma_{\mu,0})(D),u_{1}]\partial_{x}w,(\chi^{(2)}_{\mu}\sigma_{\mu,0})(D)w\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt-\frac{1}{2}\big{(}(\partial_{x}u_{1})(\chi^{(2)}_{\mu}\sigma_{\mu,0})(D)w,(\chi^{(2)}_{\mu}\sigma_{\mu,0})(D)w\big{)}_{L^{2}}$
$\displaystyle\lesssim\|u_{1}\|_{H^{s}}\|w\|_{L^{2}}^{2}+\|\partial_{x}u_{1}\|_{L^{\infty}}\|w\|_{L^{2}}^{2}.$

Gathering all these estimates, we obtain the result

$\displaystyle A_{4}=A_{4}^{1}+A_{4}^{2}+A_{4}^{3}$
$\displaystyle\lesssim_{\beta}(\|u_{1}\|_{H^{s}}+\|\eta_{1}\|_{H^{s}})\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$

Adding $I+II+III$ concludes the proof. ∎

###  4.2. Estimates for system (1.6)

As in the prvious subsection, we let
$\mathbb{U}_{1}=(\eta_{1},u_{1})^{T}={\varepsilon}(\zeta_{1},v_{1})^{T}$ and
$\mathbb{U}_{2}=(\eta_{2},u_{2})^{T}={\varepsilon}(\zeta_{2},v_{2})^{T}$ be
two solutions of (1.6) and define the difference
$(\psi,w)=(\eta_{1}-\eta_{2},u_{1}-u_{2})$. Then $\mathbb{W}=(\psi,w)^{T}$
solves

$\partial_{t}\mathbb{W}+\mathcal{M}(\mathbb{U}_{1},D)\mathbb{W}=\mathbb{F},$
(4.12)

with $\mathcal{M}$ defined as in (3.11) and $\mathbb{F}$ will remain the same
as previously defined by (4.2). Then the energy associated to (4.12) is given
in terms of the symmetrizer (3.12):

$\tilde{\mathcal{E}}_{s}(\mathbb{W}):=\big{(}J_{\mu}^{\frac{1}{2}}J^{s}\mathbb{W},\mathcal{Q}(\mathbb{U}_{1},D)J_{\mu}^{\frac{1}{2}}J^{s}\mathbb{W}\big{)}_{L^{2}}.$
(4.13)

######  Proposition 4.3.

Take $s>\frac{3}{2}$ and ${\varepsilon},\mu\in(0,1)$. Let
$(\eta_{1},u_{1}),(\eta_{2},u_{2})\in
C([0,T_{0}]:V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ be two solutions of (1.6) on a
time interval $[0,T_{0}]$ for some $T_{0}>0$. Moreover, assume there exists
$h_{0}\in(0,1)$ and $h_{1}>0$ such that

$h_{0}-1\leq\eta_{1}(x,t),\quad\forall(x,t)\in\mathbb{R}\times[0,T_{0}]\quad\text{and}\quad\sup\limits_{t\in[0,T_{0}]}\|(\eta_{1},u_{1})\|_{H^{s}\times
H^{s}}\leq h_{1}.$ (4.14)

Define the difference to be
$\mathbb{W}=(\psi,w)=(\eta_{1}-\eta_{2},u_{1}-u_{2})$. Then, for the energy
defined by (4.13), there holds

$\frac{d}{dt}\tilde{\mathcal{E}}_{0}(\mathbb{W})\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2},$
(4.15)

and

$\|(\psi,w)\|^{2}_{V^{0}_{\sqrt{\mu}}}\lesssim\tilde{\mathcal{E}}_{0}(\mathbb{W})\lesssim\|(\psi,w)\|^{2}_{V^{0}_{\sqrt{\mu}}}.$
(4.16)

Furthermore, we have the following estimate at the $V^{s}_{\sqrt{\mu}}-$
level:

$\frac{d}{dt}\mathcal{\tilde{E}}_{s}(\mathbb{W})\lesssim|\big{(}J^{s}\mathbb{F},\mathcal{Q}(\mathbb{U}_{1},D)J^{s}\mathbb{W}\big{)}_{L^{2}}|+\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{s}_{\sqrt{\mu}}}^{2},$
(4.17)

and

$\|(\psi,w)\|^{2}_{V^{s}_{\sqrt{\mu}}}\lesssim\mathcal{\tilde{E}}_{s}(\mathbb{W})\lesssim\|(\psi,w)\|^{2}_{V^{s}_{\sqrt{\mu}}}.$
(4.18)

###### Proof.

The proofs of (4.16) and (4.18) are similar to the proof of (3.16).

Also, we only prove (4.15) since the control of (4.17) follows by the proof of
Proposition 3.3.

To prove (4.15), we use (4.12) and the self-adjointness of
$\mathcal{Q}(\mathbb{U}_{1},D)$ to write

$\displaystyle\frac{1}{2}\frac{d}{dt}\tilde{\mathcal{E}}_{0}(\mathbb{W})$
$\displaystyle=\frac{1}{2}\big{(}J_{\mu}^{\frac{1}{2}}\mathbb{W},(\partial_{t}\mathcal{Q}(\mathbb{U}_{1},D))J_{\mu}^{\frac{1}{2}}\mathbb{W}\big{)}_{L^{2}}+\big{(}J_{\mu}^{\frac{1}{2}}\mathbb{F},\mathcal{Q}(\mathbb{U}_{1},D)J_{\mu}^{\frac{1}{2}}\mathbb{W}\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt-\big{(}J_{\mu}^{\frac{1}{2}}\mathcal{M}(\mathbb{U}_{1},D)\mathbb{W},\mathcal{Q}(\mathbb{U}_{1},D)J_{\mu}^{\frac{1}{2}}\mathbb{W}\big{)}_{L^{2}}$
$\displaystyle=:\mathcal{I}-\mathcal{II}-\mathcal{III}.$

Control of $\mathcal{I}$. Using (4.12), (2.17), the Sobolev embedding and
(4.14) yields

$\displaystyle|\mathcal{I}|=\frac{1}{2}|\big{(}J_{\mu}^{\frac{1}{2}}w,(\partial_{t}\eta_{1})J_{\mu}^{\frac{1}{2}}w\big{)}_{L^{2}}|\lesssim\|(\eta_{1},u_{1})\|_{V^{s}_{\sqrt{\mu}}}\|J_{\mu}^{\frac{1}{2}}w\|_{L^{2}}^{2},$

since $s>\frac{3}{2}$.

Control of $\mathcal{II}$. The contribution of the source term is given by

$\displaystyle\mathcal{II}$
$\displaystyle=\big{(}J_{\mu}^{\frac{1}{2}}(w\partial_{x}\eta_{2}),\mathcal{T}_{\mu}(D)J_{\mu}^{\frac{1}{2}}\psi\big{)}_{L^{2}}+\big{(}J_{\mu}^{\frac{1}{2}}(\psi\partial_{x}u_{2}),\mathcal{T}_{\mu}(D)J_{\mu}^{\frac{1}{2}}\psi\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}J_{\mu}^{\frac{1}{2}}(w\partial_{x}u_{2}),J_{\mu}^{\frac{1}{2}}w\big{)}_{L^{2}}+\big{(}J_{\mu}^{\frac{1}{2}}(w\partial_{x}u_{2}),\eta_{1}J_{\mu}^{\frac{1}{2}}w\big{)}_{L^{2}}$
$\displaystyle=:\mathcal{II}_{1}+\mathcal{II}_{2}+\mathcal{II}_{3}+\mathcal{II}_{4}.$

Control of $\mathcal{II}_{1}+\mathcal{II}_{2}$. The estimate of
$\mathcal{II}_{1}$ is a direct consequence of the Cauchy-Schwarz inequality,
(2.16) and the Sobolev embedding. Indeed, since $s>\frac{3}{2}$, we get

$\displaystyle|\mathcal{II}_{1}|\leq\|\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}}(w\partial_{x}\eta_{2})\|_{L^{2}}\|\sqrt{\mathcal{T}_{\mu}}(D)J^{\frac{1}{2}}_{\mu}\psi\|_{L^{2}}\lesssim\|\eta_{2}\|_{H^{s}}\|w\|_{L^{2}}\|\psi\|_{L^{2}}.$

Next, the control of $\mathcal{II}_{2}$ follows by the same estimates and
gives

$\displaystyle|\mathcal{II}_{2}|\lesssim\|\eta_{2}\|_{H^{s}}\|\psi\|_{H^{1}}^{2}.$

Control of $\mathcal{II}_{3}+\mathcal{II}_{4}$. We first deduce from (2.13)
that

$\displaystyle\|J^{\frac{1}{2}}_{\mu}(w\partial_{x}u_{2})\|_{L^{2}}$
$\displaystyle\leq\|w\partial_{x}u_{2}\|_{L^{2}}+\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}(w\partial_{x}u_{2})\|_{L^{2}}.$

The first term is estimated by the Sobolev embedding, while the second term is
equal to the term $II^{1,2}_{4}$ in the proof of Proposition 4.1. Since the
terms $w$ and $u_{2}$ in $II^{1,2}_{4}$ belong to the same function space, we
can apply the same estimates. Thus, there holds for $s>\frac{3}{2}$ that

$\|J^{\frac{1}{2}}_{\mu}(w\partial_{x}u_{2})\|_{L^{2}}\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$
(4.19)

Therefore, by using the Cauchy-Schwarz inequality, (4.19), (2.17), (4.14), and
the Sobolev embedding implies

$\displaystyle|\mathcal{II}_{3}|+|\mathcal{II}_{4}|\lesssim(1+\|\eta_{1}\|_{L^{\infty}})\|J_{\mu}^{\frac{1}{2}}(w\partial_{x}u_{2})\|_{L^{2}}\|J_{\mu}^{\frac{1}{2}}w\|_{L^{2}}\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$

Control of $\mathcal{III}$. Lastly, the symmetrized term reads:

$\displaystyle\mathcal{III}$
$\displaystyle=\big{(}J_{\mu}^{\frac{1}{2}}(u_{1}\partial_{x}\psi),\mathcal{T}_{\mu}(D)J_{\mu}^{\frac{1}{2}}\psi\big{)}_{L^{2}}+\big{(}J_{\mu}^{\frac{1}{2}}\big{(}(1+\eta_{1})\partial_{x}w\big{)},\mathcal{T}_{\mu}(D)J_{\mu}^{\frac{1}{2}}\psi\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}\mathcal{T}_{\mu}(D)J_{\mu}^{\frac{1}{2}}\partial_{x}\psi,(1+\eta_{1})J_{\mu}^{\frac{1}{2}}w\big{)}_{L^{2}}+\big{(}J_{\mu}^{\frac{1}{2}}(u_{1}\partial_{x}w),(1+\eta_{1})J_{\mu}^{\frac{1}{2}}w\big{)}_{L^{2}}$
$\displaystyle=\mathcal{A}_{11}+\mathcal{A}_{12}+\mathcal{A}_{21}+\mathcal{A}_{22}.$

Each term is treated by using integration by parts and suitable commutator
estimates.

Control of $\mathcal{A}_{11}$. For $\mathcal{A}_{11}$, we use integration by
parts to find that

$\displaystyle\mathcal{A}_{11}=\big{(}[\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}},u_{1}]\partial_{x}\psi,\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}}\psi\big{)}_{L^{2}}-\frac{1}{2}\big{(}(\partial_{x}u_{1})\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}}\psi,\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}}\psi\big{)}_{L^{2}}.$

Thus, it follows from the commutator estimate (2.32) with $s>\frac{3}{2}$ and
estimate (2.16) that

$|\mathcal{A}_{11}|\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|\psi\|^{2}_{L^{2}}.$

Control of $\mathcal{A}_{12}+\mathcal{A}_{21}$. Treating the off-diagonal
terms we first observe,

$\displaystyle\mathcal{A}_{12}$
$\displaystyle=\big{(}[\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}},\eta_{1}]\partial_{x}w,\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}}\psi\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}(1+\eta_{1})\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}}\partial_{x}w,\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}}\psi\big{)}_{L^{2}}$
$\displaystyle=\mathcal{A}_{12}^{1}+\mathcal{A}_{12}^{2}.$

The commutator estimate (2.32) and estimate (2.16) deals with the first term.
Indeed, we get the bound

$\displaystyle|\mathcal{A}_{12}^{1}|\leq\|[\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}},\eta_{1}]\partial_{x}w\|_{L^{2}}\|\sqrt{\mathcal{T}_{\mu}}(D)J_{\mu}^{\frac{1}{2}}\psi\|_{L^{2}}\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$

Next, we integrate $\mathcal{A}_{12}^{2}$ by parts to obtain two new terms

$\displaystyle\mathcal{A}_{12}^{2}$
$\displaystyle=-\big{(}(\partial_{x}\eta_{1})J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)w,J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\psi\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-\big{(}(1+\eta_{1})J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)w,J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}\psi\big{)}_{L^{2}}$
$\displaystyle=\mathcal{A}_{12}^{2,1}+\mathcal{A}_{12}^{2,2}.$

Arguing as above, we find that

$\displaystyle|\mathcal{A}_{12}^{2,1}|\leq\|\partial_{x}\eta_{1}\|_{L^{\infty}}\|J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)w\|_{L^{2}}\|J^{\frac{1}{2}}_{\mu}\sqrt{\mathcal{T}_{\mu}}(D)\psi\|_{L^{2}}\lesssim\|\eta_{1}\|_{H^{s}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2},$

for $s>\frac{3}{2}$. On the other hand, the term $\mathcal{A}_{12}^{2,2}$, is
absorbed by $\mathcal{A}_{21}$. Indeed,

$\displaystyle\mathcal{A}_{21}$
$\displaystyle=-\big{(}\sqrt{\mathcal{T}_{\mu}}(D)J^{\frac{1}{2}}_{\mu}\psi,\partial_{x}[\sqrt{\mathcal{T}_{\mu}}(D),\eta_{1}]J^{\frac{1}{2}}_{\mu}w\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}\sqrt{\mathcal{T}_{\mu}}(D)J^{\frac{1}{2}}_{\mu}\partial_{x}\psi,(1+\eta_{1})\sqrt{\mathcal{T}_{\mu}}(D)J^{\frac{1}{2}}_{\mu}w\big{)}_{L^{2}}$
$\displaystyle=\mathcal{A}_{21}^{1}+\mathcal{A}_{21}^{2},$

with $\mathcal{A}_{21}^{2}=-\mathcal{A}_{12}^{2,2}$. We estimate
$\mathcal{A}_{21}^{1}$ by using the Cauchy-Schwarz inequality, (2.16), (2.35),
and (2.17) to get

$\displaystyle|\mathcal{A}_{21}^{1}|\leq\|\sqrt{\mathcal{T}_{\mu}}(D)J^{\frac{1}{2}}_{\mu}\psi\|_{L^{2}}\|\partial_{x}[\sqrt{\mathcal{T}_{\mu}}(D),\eta_{1}]J^{\frac{1}{2}}_{\mu}w\|_{L^{2}}\lesssim\|\eta_{1}\|_{H^{s}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$

Thus, we deduce by gathering all these estimates that

$|\mathcal{A}_{12}+\mathcal{A}_{21}|\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$

Control of $\mathcal{A}_{22}$. Lastly, the term $\mathcal{A}_{22}$ is
estimated by (2.36) for $s>\frac{3}{2}$, (4.14), and integration by parts

$\displaystyle|\mathcal{A}_{22}^{2}|$
$\displaystyle\leq|\big{(}[J_{\mu}^{\frac{1}{2}},u_{1}]\partial_{x}w,(1+\eta_{1})J_{\mu}^{\frac{1}{2}}w\big{)}_{L^{2}}|+|\big{(}u_{1}J_{\mu}^{\frac{1}{2}}\partial_{x}w,(1+\eta_{1})J_{\mu}^{\frac{1}{2}}w\big{)}_{L^{2}}|$
$\displaystyle\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2}.$

Therefore, we deduce that

$\displaystyle\frac{d}{dt}\tilde{\mathcal{E}}_{0}(\mathbb{W})\lesssim|\mathcal{I}|+|\mathcal{II}|+|\mathcal{III}|\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{V^{0}_{\sqrt{\mu}}}^{2},$

which concludes the proof of Proposition 4.3.

∎

###  4.3. Estimates for system (1.7)

Again, we let
$\mathbb{U}_{1}=(\eta_{1},u_{1})^{T}={\varepsilon}(\zeta_{1},v_{1})^{T}$ and
$\mathbb{U}_{2}=(\eta_{2},u_{2})^{T}={\varepsilon}(\zeta_{2},v_{2})^{T}$ be
two solutions of (1.7) and define the difference
$(\psi,w)=(\eta_{1}-\eta_{2},u_{1}-u_{2})$. Then $\mathbb{W}=(\psi,w)^{T}$
solves

$\partial_{t}\mathbb{W}+\mathscr{M}(\mathbb{U}_{1},D)\mathbb{W}=\mathbb{F},$
(4.20)

with $\mathscr{M}$ defined as in (3.21) and $\mathbb{F}$ is defined by

$\mathbb{F}=-\begin{pmatrix}\mathcal{T}_{\mu}(D)(w\partial_{x}\eta_{2})+\mathcal{T}_{\mu}(D)(\psi\partial_{x}u_{2})\\\
\mathcal{T}_{\mu}(D)(w\partial_{x}u_{2})\end{pmatrix}.$ (4.21)

The energy associated to (4.20) is given in terms of the symmetrizer (3.22) by

$\tilde{\mathscr{E}}_{s}(\mathbb{W}):=\big{(}J^{s}\mathbb{W},\mathscr{Q}(\mathbb{U}_{1},D)J^{s}\mathbb{W}\big{)}_{L^{2}}.$
(4.22)

######  Proposition 4.4.

Take $s>\frac{3}{2}$, ${\varepsilon},\mu\in(0,1)$ and $\beta>0$. Let
$(\eta_{1},u_{1}),(\eta_{2},u_{2})\in
C([0,T_{0}]:X^{s}_{\beta,\mu}(\mathbb{R}))$ be two solutions of (1.7) on a
time interval $[0,T_{0}]$ for some $T_{0}>0$. Moreover, assume there exist
$h_{0}\in(0,1)$ and $h_{1}>0$ such that

$h_{0}-1\leq\eta_{1}(x,t),\quad\forall(x,t)\in\mathbb{R}\times[0,T_{0}]\quad\text{and}\quad\sup\limits_{t\in[0,T_{0}]}\|(\eta_{1},u_{1})\|_{H^{s}\times
H^{s}}\leq h_{1}.$ (4.23)

Define the difference to be
$\mathbb{W}=(\psi,w)=(\eta_{1}-\eta_{2},u_{1}-u_{2})$. Then, for the energy
defined by (4.22), there holds

$\frac{d}{dt}\tilde{\mathscr{E}}_{0}(\mathbb{W})\lesssim_{\beta}\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{X^{0}_{\beta,\mu}}^{2},$
(4.24)

and

$\|(\psi,w)\|^{2}_{X^{0}_{\beta,\mu}}\lesssim\tilde{\mathscr{E}}_{0}(\mathbb{W})\lesssim\|(\psi,w)\|^{2}_{X^{0}_{\beta,\mu}}.$
(4.25)

Furthermore, we have the following estimate at the $X^{s}_{\beta,\mu}-$ level:

$\frac{d}{dt}\mathscr{\tilde{E}}_{s}(\mathbb{W})\lesssim_{\beta}|\big{(}J^{s}\mathbb{F},\mathscr{Q}(\mathbb{U}_{1},D)J^{s}\mathbb{W}\big{)}_{L^{2}}|+\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{X^{s}_{\beta,\mu}}^{2},$
(4.26)

and

$\|(\psi,w)\|^{2}_{X^{s}_{\beta,\mu}}\lesssim\mathscr{\tilde{E}}_{s}(\mathbb{W})\lesssim\|(\psi,w)\|^{2}_{X^{s}_{\beta,\mu}}.$
(4.27)

###### Proof.

By previous arguments, we note that the proofs of (4.25) and (4.27) are
similar to the proof of (3.26).

Moreover, we will only prove (4.24) since the control of (4.26) follows by the
proof of Proposition 3.4.

We will now prove (4.24). Then we first use (4.20) and the self-adjointness of
$\mathscr{Q}(\mathbb{U}_{1},D)$ to write

$\displaystyle\frac{1}{2}\frac{d}{dt}\tilde{\mathscr{E}}_{0}(\mathbb{W})$
$\displaystyle=\frac{1}{2}\big{(}\mathbb{W},(\partial_{t}\mathscr{Q}(\mathbb{U}_{1},D))\mathbb{W}\big{)}_{L^{2}}+\big{(}\mathbb{F},\mathscr{Q}(\mathbb{U}_{1},D)\mathbb{W}\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt-\big{(}\mathscr{M}(\mathbb{U}_{1},D)\mathbb{W},\mathscr{Q}(\mathbb{U}_{1},D)\mathbb{W}\big{)}_{L^{2}}$
$\displaystyle=:\mathscr{I}-\mathscr{II}-\mathscr{III}.$

Control of $\mathscr{I}$. By (4.20), Hölder’s inequality, the Sobolev
embedding and (4.23) we deduce

$\displaystyle|\mathscr{I}|=\frac{1}{2}|\big{(}w,(\partial_{t}\eta_{1})w\big{)}_{L^{2}}|\lesssim\|u_{1}\|_{H^{s}}(1+\|\eta_{1}\|_{H^{s}})\|w\|_{L^{2}}^{2}\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{X^{0}_{\beta,\mu}}^{2}.$

Control of $\mathscr{II}$. The contribution from the source term is given by,

$\displaystyle\mathscr{II}$
$\displaystyle=\big{(}w\partial_{x}\eta_{2},\mathcal{K}_{\mu}(D)\psi\big{)}_{L^{2}}+\big{(}\psi\partial_{x}\eta_{2},\mathcal{K}_{\mu}(D)\psi\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}w\partial_{x}u_{2},w\big{)}_{L^{2}}+\big{(}\mathcal{T}_{\mu}(D)(w\partial_{x}u_{2}),\eta_{1}w\big{)}_{L^{2}}$
$\displaystyle=:\mathscr{II}_{1}+\mathscr{II}_{2}+\mathscr{II}_{3}+\mathscr{II}_{4}.$

Control of $\mathscr{II}_{1}+\mathscr{II}_{2}$. We first apply the Cauchy-
Schwarz inequality, (2.1), and the Sobolev embedding to deduce that for
$s>\frac{3}{2}$

$\displaystyle|\mathscr{II}_{1}|+|\mathscr{II}_{2}|$
$\displaystyle\lesssim(\|w\|_{L^{2}}+\|\psi\|_{L^{2}})\|\eta_{2}\|_{H^{s}}\|\mathcal{K}_{\mu}(D)\psi\|_{L^{2}}$
$\displaystyle\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{X^{0}_{\beta,\mu}}^{2}.$

Control of $\mathscr{II}_{3}+\mathscr{II}_{4}$. Both terms are treated with
the Cauchy-Schwatz inequality, (2.14) and the Sobolev embedding. Consequently,
for $s>\frac{3}{2}$ there holds

$\displaystyle|\mathscr{II}_{3}|+|\mathscr{II}_{4}|\lesssim(1+\|\eta_{1}\|_{H^{s}})\|u_{2}\|_{H^{s}}\|w\|_{L^{2}}^{2}\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{X^{0}_{\beta,\mu}}^{2}.$

Gathering all these estimates yields

$|\mathscr{II}|\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{X^{0}_{\beta,\mu}}^{2}.$

Control of $\mathscr{III}$. The symmetrized term $\mathscr{III}$ reads:

$\displaystyle\mathscr{III}=$
$\displaystyle\big{(}u_{1}\partial_{x}\psi,\mathcal{K}_{\mu}(D)\psi\big{)}_{L^{2}}+\big{(}(1+\mathcal{T}_{\mu}(D)\eta_{1})\partial_{x}w,\mathcal{T}^{-1}_{\mu}(D)\mathcal{K}_{\mu}(D)\psi\big{)}_{L^{2}}$
$\displaystyle+\big{(}\mathcal{K}_{\mu}(D)\partial_{x}\psi,(\mathcal{T}_{\mu}^{-1}(D)+\eta_{1})w\big{)}_{L^{2}}+\big{(}\mathcal{T}_{\mu}(D)(u_{1}\partial_{x}w),(\mathcal{T}_{\mu}^{-1}(D)+\eta_{1})w\big{)}_{L^{2}}$
$\displaystyle=\mathscr{A}_{11}+\mathscr{A}_{12}+\mathscr{A}_{21}+\mathscr{A}_{22}.$

Control of $\mathscr{A}_{11}$. We decompose $\mathscr{A}_{11}$ as

$\displaystyle\mathscr{A}_{11}$
$\displaystyle=\big{(}u_{1}\partial_{x}\psi,((\chi^{(1)}_{\mu})^{2}\mathcal{K}_{\mu})(D)\psi\big{)}_{L^{2}}+\big{(}u_{1}\partial_{x}\psi,((\chi_{\mu}^{(2)})^{2}(\sigma_{\mu,\frac{1}{2}})^{2})(D)\psi\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt-\big{(}u_{1}\partial_{x}\psi,((\chi^{(2)}_{\mu})^{2}(\sigma_{\mu,0})^{2})(D)\psi\big{)}_{L^{2}},$

where we have divided the multiplier $\mathcal{K}_{\mu}(D)$ into three pieces
in the same way as we did in (4.11). We may therefore apply the same estimates
as for $A_{4}^{3}$ in the proof of Proposition 4.1, where we change the role
of $\psi$ and $w$ to obtain

$\displaystyle|\mathscr{A}_{11}|=\big{(}u_{1}\partial_{x}\psi,\mathcal{K}_{\mu}(D)\psi\big{)}_{L^{2}}\lesssim_{\beta}\|u_{1}\|_{H^{s}}(\|\psi\|_{L^{2}}^{2}+\sqrt{\beta}\mu^{\frac{1}{4}}\|\psi\|_{H^{\frac{1}{2}}}^{2}).$

Then use inequality (3.27) to conclude that

$|\mathscr{A}_{11}|\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{X^{0}_{\beta,\mu}}^{2}.$

Control of $\mathscr{A}_{12}+\mathscr{A}_{21}$. Treating the off-diagonal
terms we first observe by integrating by parts that

$\displaystyle\mathscr{A}_{12}$
$\displaystyle=-\big{(}(\partial_{x}\eta_{1})w,\mathcal{K}_{\mu}(D)\psi\big{)}_{L^{2}}-\mathscr{A}_{21}.$

Therefore, we may apply Hölder’s inequality and the Sobolev embedding to
deduce

$|\mathscr{A}_{12}+\mathscr{A}_{21}|\lesssim\|\eta_{1}\|_{H^{s}}\|(\psi,w)\|_{X^{0}_{\beta,\mu}}.$

Control of $\mathscr{A}_{22}$. We decompose $\mathscr{A}_{22}$ into two terms

$\displaystyle\mathscr{A}_{22}$
$\displaystyle=\big{(}u_{1}\partial_{x}w,w\big{)}_{L^{2}}+\big{(}\mathcal{T}_{\mu}(D)(u_{1}\partial_{x}w),\eta_{1}w\big{)}_{L^{2}}$
$\displaystyle=\mathscr{A}_{22}^{1}+\mathscr{A}_{22}^{2}.$

We see that $\mathscr{A}_{22}^{1}$ is easily treated by the Cauchy-Schwarz
inequality, integration by parts, the Sobolev embedding, and (4.23). Indeed,
there holds

$|\mathscr{A}_{22}^{1}|\lesssim\|u_{1}\|_{H^{s}}\|w\|_{L^{2}}^{2}.$

Next, we decompose $\mathscr{A}_{22}^{2}$ into three parts

$\displaystyle\mathscr{A}_{22}^{2}$
$\displaystyle=\big{(}[\sqrt{\mathcal{T}_{\mu}}(D),u_{1}]\partial_{x}w,\sqrt{\mathcal{T}_{\mu}}(D)(\eta_{1}w)\big{)}_{L^{2}}+\big{(}u_{1}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}w,[\sqrt{\mathcal{T}_{\mu}}(D),\eta_{1}]w\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}u_{1}\sqrt{\mathcal{T}_{\mu}}(D)\partial_{x}w,\eta_{1}\sqrt{\mathcal{T}_{\mu}}(D)w\big{)}_{L^{2}}.$
$\displaystyle=\mathscr{A}_{22}^{2,1}+\mathscr{A}_{22}^{2,2}+\mathscr{A}_{22}^{2,3}.$

For $\mathscr{A}_{22}^{2,1}$, we simply apply Hölder’s inequality, (2.34),
(2.14), the Sobolev embedding to find that

$\displaystyle|\mathscr{A}_{22}^{2,1}|\leq\|[\sqrt{\mathcal{T}_{\mu}}(D),u_{1}]\partial_{x}w\|_{L^{2}}\|\sqrt{\mathcal{T}_{\mu}}(D)(\eta_{1}w)\|_{L^{2}}\lesssim\|w\|_{L^{2}}^{2}.$

For $\mathscr{A}_{22}^{2,2}$, we first remark that

$\|[\sqrt{\mathcal{T}_{\mu}}(D),\eta_{1}]w\|_{L^{2}}\lesssim\|\eta_{1}\|_{L^{\infty}}\|w\|_{L^{2}},$
(4.28)

simply by using Hölder’s inequality and (2.14). Then after integrating by
parts, we use Hölder’s inequality, the Sobolev embedding, (2.35), (4.23), and
(4.28) to deduce that

$\displaystyle|\mathscr{A}_{22}^{2,2}|$
$\displaystyle\leq\|\partial_{x}u_{1}\|_{L^{\infty}}\|\sqrt{\mathcal{T}_{\mu}}(D)w\|_{L^{2}}\|[\sqrt{\mathcal{T}_{\mu}}(D),\eta_{1}]w\|_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\|u_{1}\|_{L^{\infty}}\|\sqrt{\mathcal{T}_{\mu}}(D)w\|_{L^{2}}\|\partial_{x}[\sqrt{\mathcal{T}_{\mu}}(D),\eta_{1}]w\|_{L^{2}}$
$\displaystyle\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|w\|_{L^{2}}^{2}.$

Lastly, we use integration by parts, then apply Hölder’s inequality, (2.14),
the Sobolev embedding, and (4.23) to obtain that

$\displaystyle|\mathscr{A}_{22}^{2,3}|\leq\frac{1}{2}\|\partial_{x}(u_{1}\eta_{1})\|_{L^{\infty}}\|\sqrt{\mathcal{T}_{\mu}}(D)w\|^{2}_{L^{2}}\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|w\|_{L^{2}}^{2}.$

We may now gather these estimates to conclude that

$|\mathscr{A}_{22}|\lesssim\max\limits_{i=1,2}\|(\eta_{i},u_{i})\|_{V^{s}_{\sqrt{\mu}}}\|(\psi,w)\|_{X^{0}_{\beta,\mu}}^{2},$

and as a result the proof of Proposition 4.4 is now complete. ∎

##  5\. Proof of Theorem 1.6

###### Proof.

The proof is divided into eigth steps, utilizing the results above.

Step $1$: Existence of solutions for a regularized system. Let
$s>\frac{1}{2}$, $\delta>0$, and $\varphi_{\delta}(D)$ be as given in
Definition 2.15. Then, for initial data on the form
$\varphi_{\delta}(D)\mathbb{U}_{0}:=(\varphi_{\delta}(D)\eta_{0},\varphi_{\delta}(D)u_{0})$,
we claim that there exist $c_{\beta}>0$, and a time

$0<T_{\delta}:=T_{\delta}\big{(}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}\big{)}=\frac{c_{\beta}\delta}{1+\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}}$
(5.1)

such that $\mathbb{U}^{\delta}:=(\eta^{\delta},u^{\delta})^{T}\in
C([0,T_{\delta}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ is a unique solution of the
regularised Cauchy problem:

$\begin{cases}\partial_{t}\eta^{\delta}+u^{\delta}\partial_{x}\varphi_{\delta}(D)\eta^{\delta}+(\mathcal{K}_{\mu}(D)+\eta^{\delta})\partial_{x}\varphi_{\delta}(D)u^{\delta}=0\\\
\partial_{t}u^{\delta}+\partial_{x}\varphi_{\delta}(D)\eta^{\delta}+u^{\delta}\partial_{x}\varphi_{\delta}(D)u^{\delta}=0\end{cases}$
(5.2)

The proof of the existence of a unique solution is a consequence of the
contraction mapping principle. First, define the notation
$\varphi_{\delta}(D)\mathbb{U}^{\delta}(t):=(\varphi_{\delta}(D)\eta^{\delta},\varphi_{\delta}(D)u^{\delta})^{T}(t)$.
Then we shall show for $M$ defined by (3.2) that

$\displaystyle\mathbb{U}^{\delta}(t)=\varphi_{\delta}(D)\mathbb{U}_{0}-\int_{0}^{t}M(\mathbb{U}^{\delta}(s),D)\cdot\varphi_{\delta}(D)\mathbb{U}^{\delta}(s)\>ds=:\Phi_{\mathbb{U}_{0}}(\mathbb{U}^{\delta})(t)$
(5.3)

defines a contraction on the compete metric space

$\displaystyle Y^{s}(T,a):=\big{\\{}(f^{1},f^{2})\in
C([0,T];V_{\sqrt{\mu}}^{s}(\mathbb{R}))\>:\>\|(f^{1},f^{2})\|_{L^{\infty}_{T}V_{\sqrt{\mu}}^{s}}\leq
a\big{\\}}.$

Step $1.1$: $\Phi$ is well-defined on $Y^{s}(T,a)$. Indeed, let
$\mathbb{U}^{\delta}=(\eta^{\delta},u^{\delta})\in Y^{s}(T,a)$ and let
$\Phi:=(\Phi^{1},\Phi^{2})^{T}$. Then for $\Phi^{1}$ we have by (2.1), (2.22),
the Sobolev embedding and (2.39) that

$\displaystyle\|\Phi^{1}(\mathbb{U}^{\delta})\|_{L^{\infty}_{T}H^{s}}$
$\displaystyle\leq\|\eta_{0}\|_{H^{s}}+T\big{(}\delta^{-1}\|u^{\delta}\|_{L^{\infty}_{T}H^{s}}\|\eta^{\delta}\|_{L^{\infty}_{T}H^{s}}$
$\displaystyle\hskip
14.22636pt+\delta^{-\frac{3}{2}}\|\mathcal{K}_{\mu}(D)u^{\delta}\|_{L^{\infty}_{T}H^{s-\frac{1}{2}}}+\delta^{-1}\|\eta^{\delta}\|_{L^{\infty}_{T}H^{s}}\|u^{\delta}\|_{L^{\infty}_{T}H^{s}}\big{)}$
$\displaystyle\leq\|\eta_{0}\|_{H^{s}}+T(2a^{2}\delta^{-1}+c_{\beta}a\delta^{-\frac{3}{2}}).$

Similarly, for the second coordinate there holds

$\displaystyle\|\Phi^{2}(\mathbb{U}^{\delta})\|_{H^{s}}+\mu^{\frac{1}{4}}\|D^{\frac{1}{2}}\Phi^{2}(\mathbb{U}^{\delta})\|_{H^{s}}\leq\|u_{0}\|_{H^{s}}+\mu^{\frac{1}{4}}\|D^{\frac{1}{4}}u_{0}\|_{H^{s}}+Tc_{\beta}(a+a^{2})(\delta^{-1}+\delta^{-\frac{3}{2}}).$

Gathering these estimates, with
$\|(\eta_{0},u_{0})||_{V_{\sqrt{\mu}}^{s}}\leq\frac{a}{2}$ and
$T\leq\frac{\tilde{c}_{\beta}\delta}{2(a+1)(1+\delta^{-\frac{1}{2}})}$ implies

$\displaystyle\|(\Phi^{1}(\mathbb{U}^{\delta}),\Phi^{2}(\mathbb{U}^{\delta}))\|_{V_{\sqrt{\mu}}^{s}}\leq
a.$

Step $1.2$: $\Phi$ defines a contraction map on $Y^{s}(T,a)$. Let
$\mathbb{U}^{\delta}_{1}=(\eta_{1}^{\delta},u_{1}^{\delta})^{T},\mathbb{U}^{\delta}_{2}=(\eta_{2}^{\delta},u_{2}^{\delta})^{T}\in
Y^{s}(T,a)$. Then if we define
$\mathbb{W}^{\delta}=(\psi^{\delta},w^{\delta})^{T}=\mathbb{U}^{\delta}_{1}-\mathbb{U}^{\delta}_{2}$
we have that

$\Phi(\mathbb{U}^{\delta}_{1})-\Phi(\mathbb{U}^{\delta}_{2})=\int_{0}^{t}\big{(}M(\mathbb{U}_{1}^{\delta},D)\cdot\varphi_{\delta}(D)\mathbb{W}^{\delta}+(M(\mathbb{U}_{2}^{\delta},D)-M(\mathbb{U}_{1}^{\delta},D))\cdot\varphi_{\delta}(D)\mathbb{U}^{\delta}_{2}\big{)}\>ds.$

Reapplying the same estimates as in the previous step, we conclude that

$\displaystyle\|(\Phi^{1}(\mathbb{U}_{1}^{\delta})-\Phi^{1}(\mathbb{U}_{2}^{\delta}),\Phi^{2}(\mathbb{U}_{1}^{\delta})-\Phi^{2}(\mathbb{U}_{2}^{\delta}))\|_{V_{\sqrt{\mu}}^{s}}<\|(\psi^{\delta},w^{\delta})\|_{V^{s}_{\sqrt{\mu}}}.$

by choosing $T\lesssim T_{\delta}(a)$.

We conclude by the Fixed Point Theorem that there exist a unique solution of
(5.2) in $C([0,T_{\delta}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$.

######  Remark 5.1.

A consequence of Step $1$, is the continuity of the flow map associated to
(5.2). But this is only for a time $T_{\delta}\lesssim_{\beta}\delta$, and is
therefore not usefull for the limit equation.

Step 2: The blow-up alternative. We define the maximal time of existence to be

$T^{\star}_{\delta}=\sup\Big{\\{}T_{\delta}>0\>:\>\exists!\>\mathbb{U}^{\delta}=(\eta^{\delta},u^{\delta})^{T}\>\text{solution
of}\>\eqref{regularised Cauchy
problem}\>\text{in}\>C([0,T_{\delta}];V^{s}_{\sqrt{\mu}}({\mathbb{R}}))\Big{\\}}.$

Then we claim that the solution of (5.2) satisfy the blow-up alternative:

$\text{If}\quad
T_{\delta}^{\star}<\infty,\quad\text{then}\quad\lim\limits_{t\nearrow
T_{\delta}^{\star}}\|(\eta^{\delta},u^{\delta})(t)\|_{V^{s}_{\sqrt{\mu}}}=\infty.$
(5.4)

First, we argue by contradiction that $T^{\star}_{\delta}<\infty$ and there
exist $A\in{\mathbb{R}}^{+}$ such that

$\sup\limits_{t\in[0,T^{\star}_{\delta})}\|(\eta^{\delta},u^{\delta})(t)\|_{V^{s}_{\sqrt{\mu}}}=A.$
(5.5)

We use (5.1) to define
$\tau_{\delta,A}=T^{\star}_{\delta}-\frac{T_{\delta}(A)}{2}$. Then we have
that

$a:=\|(\eta^{\delta},u^{\delta})(\tau_{\delta,A})\|_{V^{s}_{\sqrt{\mu}}}\leq
A.$

Therefore, if we let
$\mathbb{V}^{\delta}_{0}=(\eta^{\delta},u^{\delta})^{T}(\tau_{\delta,A})$
serve as initial data, then (5.2) has a unique solution given by

$\displaystyle\mathbb{V}^{\delta}(t)=\mathbb{V}^{\delta}_{0}-\int_{0}^{t}M(\mathbb{V}^{\delta}(s),D)\cdot\varphi_{\delta}(D)\mathbb{V}^{\delta}(s)\>ds,$
(5.6)

with $\mathbb{V}^{\delta}\in
C([0,T_{\delta}(a)];V^{s}_{\sqrt{\mu}}({\mathbb{R}}))$. Here, $T_{\delta}(a)$
is given by (5.1) due to Step $1$. Moreover, we observe that
$T_{\delta}(a)\geq T_{\delta}(A)$ by definition (5.1), and implies
$\tau_{\delta,A}+T_{\delta}(a)\geq
T^{\star}_{\delta}+\frac{T_{\delta}(A)}{2}$. Thus, we define the extension of
$\mathbb{U}^{\delta}=(\eta^{\delta},u^{\delta})^{T}$ by the function

$\mathbb{Z}^{\delta}(t)=\begin{cases}\mathbb{U}^{\delta}(t),\quad&\text{if}\quad
0\leq t<\tau_{\delta,A}\\\
\mathbb{V}^{\delta}(t-\tau_{\delta,A}),\quad&\text{if}\quad\tau_{\delta,A}\leq
t\leq\tau_{\delta,A}+T_{\delta}(a),\end{cases}$

and one can verify that it is a solution of (5.2) for all
$t\in[0,T_{\delta}^{\star}+\frac{T_{\delta}(A)}{2}]\subset[0,\tau_{\delta,A}+T_{\delta}(a)]$.
This contradicts the definition of $T^{\star}_{\delta}$. Thus, we conclude
that if $T^{\star}_{\delta}<\infty$, then necessarily $A=\infty$ in (5.5), and
implies

$\limsup\limits_{t\nearrow
T^{\star}_{\delta}}\|(\eta^{\delta},u^{\delta}(t))\|_{V^{s}_{\sqrt{\mu}}}=\infty.$
(5.7)

To conlcude the proof of the claim, we use (5.7) to verify that for all $R>0$
there exists an open interval $(t_{R},T^{\star}_{\delta})$ such that
$\|(\eta^{\delta},u^{\delta}(t))\|_{V^{s}_{\sqrt{\mu}}}>R$, for all
$t\in(t_{R},T^{\star}_{\delta})$. Indeed, we argue by contradiction that there
exists $R\in{\mathbb{R}}^{+}$ such that for all $0<t_{R}<T^{\star}$, we have

$\|(\eta^{\delta},u^{\delta})(t)\|_{V^{s}_{\sqrt{\mu}}}\leq R,$ (5.8)

for some $t\in(t_{R},T^{\star}_{\delta})$. By (5.7) there is a time such that
$\tau_{R,0}>T^{\star}_{\delta}-\frac{T_{\delta}(R)}{2}$ and satisfying

$\|(\eta^{\delta},u^{\delta})(\tau_{R,0})\|_{V^{s}_{\sqrt{\mu}}}>R.$

On the other hand, by assumption (5.8) we can take $t_{R}=\tau_{R,0}$ and use
the fact that there is a time $\tau_{R,1}\in(t_{R},T^{\star}_{\delta})$ such
that

$\|(\eta^{\delta},u^{\delta})(\tau_{R,1})\|_{V^{s}_{\sqrt{\mu}}}\leq R.$

Thus, by the same argument as above we can take
$(\eta^{\delta},u^{\delta})(\tau_{R,1})$ as initial data of (5.2) to find an
extended solution defined on
$[0,T^{\star}_{\delta}+\frac{T_{\delta}(R)}{2}]\subset[0,\tau_{R,1}+T_{\delta}(R)]$.
This contradicts the definition of $T_{\delta}^{\star}$. As a result, we
conclude that (5.4) holds true.

Step $3$: The existence time is independent of $\delta>0$. We claim that there
exists

$T=\frac{1}{k_{\beta}^{1}\|(\zeta_{0},v_{0})\|_{V^{s}_{\sqrt{\mu}}}},$

as in (1.12), such that the regularized solution
${\varepsilon}(\zeta^{\delta},v^{\delta})=(\eta^{\delta},u^{\delta})$ exists
on the interval $[0,\frac{T}{{\varepsilon}}]$.

The proof relies on a bootstrap argument similar to the proof of Lemma $5.1$
in [28]. In fact, the long time existence is a direct consequence of the
following lemma.

######  Lemma 5.2.

Let $s>2$ and ${\varepsilon}$ be as in (1.11). Let
$(\eta^{\delta},u^{\delta})={\varepsilon}(\zeta^{\delta},v^{\delta})\in
C([0,T_{\delta}^{\star});V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ be a solution of
(5.2) with initial data ${\varepsilon}(\zeta_{0},v_{0})=(\eta_{0},u_{0})\in
V^{s}_{\sqrt{\mu}}(\mathbb{R})$, defined on its maximal time of existence and
satisfying the blow-up alternative (5.4). Moreover, let
$\eta_{0}={\varepsilon}\zeta_{0}$ satisfy either the non-cavitation condition
(1.9) or the $\beta-$dependent surface condition (1.10), depending on whether
$\beta\geq\frac{1}{3}$ or $0<\beta<\frac{1}{3}$, respectivly. Then there
exists a time

$T_{0}=\frac{1}{k_{\beta}^{1}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}},$
(5.9)

such that $T^{\star}_{\delta}>T_{0}$ and

$\sup\limits_{t\in[0,T_{0}]}\|(\eta^{\delta},u^{\delta})(t)\|_{V^{s}_{\sqrt{\mu}}}\leq
4k^{2}_{\beta}\|(\eta_{0}^{\delta},u_{0}^{\delta})\|_{V^{s}_{\sqrt{\mu}}}.$
(5.10)

The constants are on the form

$k^{2}_{\beta}=\frac{c_{\beta}^{2}}{c_{\beta}^{1}}\quad\text{and}\quad
k_{\beta}^{1}=\begin{cases}\frac{C_{1}}{\beta}\>\>\quad\quad\text{for}\>\>0<\beta<\frac{1}{3}\\\
C_{2}\beta^{2}\>\>\>\quad\text{for}\>\>\beta\geq\frac{1}{3}\end{cases}$

where $C_{1}$ and $C_{2}$ are two positive constants to be fixed in the proof.

###### Proof.

We define the set

$\tilde{T}_{\delta}=\sup\limits\Big{\\{}T_{\delta}\in(0,T_{\delta}^{\star})\>:\>\sup\limits_{t\in[0,T_{\delta}]}\|(\eta^{\delta},u^{\delta})(t)\|_{V^{s}_{\sqrt{\mu}}}\leq
4k^{2}_{\beta}\|(\eta_{0}^{\delta},u^{\delta}_{0})\|_{V^{s}_{\sqrt{\mu}}}\Big{\\}}.$
(5.11)

Then we first note that $\tilde{T}_{\delta}<T^{\star}_{\delta}$, or else it
would contradict the blow-up alternative (5.4). For the proof we argue by
contradiction that $\tilde{T}_{\delta}\leq T_{0}$.

The main idea is to improve the estimate given in (5.11). First, we verify
that the solution $(\eta^{\delta},u^{\delta})$ satisfy (3.4). Indeed,
recalling assumption (1.11):

$0<{\varepsilon}\leq\frac{1}{k^{2}_{\beta}\|(\zeta_{0},v_{0})\|_{V^{s}_{\sqrt{\mu}}}},$

implies

$\displaystyle\|(\eta^{\delta},u^{\delta})\|_{H^{s}}\leq
k^{2}_{\beta}\|(\eta_{0}^{\delta},u_{0}^{\delta})\|_{V_{\sqrt{\mu}}^{s}}=4{\varepsilon}k^{2}_{\beta}\|(\zeta_{0},v_{0})\|_{V^{s}_{\sqrt{\mu}}}\leq
4,$ (5.12)

for all $t\in[0,\tilde{T}_{\delta}]$. Next, the solution
$(\eta^{\delta},u^{\delta})$ satisfy the non-cavitation condition. We will
prove this as a consequence of two claims.

Claim 1: If $\eta_{0}$ satisfies the non-cavitation condition, then the same
is true for $\eta^{\delta}_{0}:=\varphi_{\delta}(D)\eta_{0}$. Indeed, by the
Sobolev embedding $H^{\frac{1}{2}^{+}}(\mathbb{R})\hookrightarrow
L^{\infty}(\mathbb{R})$ and (2.41) to deduce that

$\|\eta_{0}^{\delta}-\eta_{0}\|_{L^{\infty}}\lesssim\|\varphi_{\delta}(D)\eta_{0}-\eta_{0}\|_{H^{\frac{1}{2}^{+}}}\underset{\delta\rightarrow
0}{=}o(\delta),$

since $s-1>\frac{1}{2}$.

Claim 2: We have the following bound on $\partial_{t}\eta^{\delta}$:

$\sup\limits_{\tau\in[0,\tilde{T}_{\delta}]}|\partial_{t}\eta^{\delta}(\tau)|\leq
k_{\beta}^{2}\beta^{2}.$

Indeed, by similar argument as for (3.8), we use (5.2), (5.12), and
$H^{\frac{1}{2}^{+}}(\mathbb{R})\hookrightarrow L^{\infty}(\mathbb{R})$ to
find

$\displaystyle\|\partial_{t}\eta^{\delta}\|_{L^{\infty}}\leq
c_{\beta}^{2}\|(\eta^{\delta},u^{\delta})\|_{V^{s}_{\sqrt{\mu}}}+\|\eta^{\delta}\|_{H^{s}}\|u^{\delta}\|_{H^{s}}\leq
4k^{2}_{\beta}c_{\beta}^{2}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}.$

Now, let $\delta$ small such that $0<o(\delta)<\frac{h_{0}}{4}$. Then, by
Claim $1$ and Claim $2$, we use the Fundamental Theorem of Calculus to obtain

$\displaystyle
1+\eta^{\delta}(x,t)=1+\eta_{0}^{\delta}+\int^{t}_{0}\partial_{t}\eta^{\delta}(x,s)\>ds\geq\frac{3}{4}h_{0}-k_{\beta}^{2}\beta^{2}\tilde{T}_{\delta},$
(5.13)

for all $t\in[0,\tilde{T}_{\delta}]$. On the one hand, if
$\beta\geq\frac{1}{3}$, then

$k^{2}_{\beta}=\frac{c^{2}_{\beta}}{c^{1}_{\beta}}=c\beta.$

Thus, for $C_{2}>0$ large enough, we get that $k^{1}_{\beta}\geq
C_{2}\beta^{2}\geq\frac{c\beta^{2}}{h_{0}}$. Moreover, by the assumption
$\tilde{T}_{\delta}\leq T_{0}$, we conclude from (5.13) that

$1+\eta^{\delta}(x,t)\geq\frac{3}{4}h_{0}-k_{\beta}^{2}\beta^{2}T_{0}\geq\frac{h_{0}}{2},$

for all $t\in[0,\tilde{T}_{\delta}]$. On the other hand, in the case when
$\beta\in(0,\frac{1}{3})$ we need to verify (3.5). But this can be done the