same way by choosing $k_{\beta}^{1}\geq\frac{C_{1}}{\beta}\geq\frac{c}{\beta
h_{\beta}}$ for $C_{1}>0$ large enough.

The hypotheses of Proposition 3.1 are now verified, leaving us (3.6) and (3.7)
at our disposal. With this at hand, we observe that

$E_{s}(\mathbb{U}^{\delta})(t)\leq
E_{s}(\mathbb{U}^{\delta})(0)+c^{2}_{\beta}\int_{0}^{t}\big{(}E_{s}(\mathbb{U}^{\delta})(s^{\prime})\big{)}^{\frac{3}{2}}\>ds^{\prime}:=\psi(t).$

By the above inequality, we then have $\psi^{\prime}(t)\leq
c^{2}_{\beta}\>\big{(}E_{s}(\mathbb{U}^{\delta})(t)\big{)}^{\frac{3}{2}}\leq
c^{2}_{\beta}\>(\psi(t))^{\frac{3}{2}}$. We solve the differential inequality
and use (3.7) to relate the energy with the $V^{s}_{\sqrt{\mu}}-$norm of the
solution and deduce that

$\displaystyle
c_{\beta}^{1}\|(\eta^{\delta},u^{\delta})(t)\|_{V^{s}_{\sqrt{\mu}}}\leq\frac{c_{\beta}^{2}\|(\eta_{0}^{\delta},u_{0}^{\delta})\|_{V^{s}_{\sqrt{\mu}}}}{1-\frac{(c_{\beta}^{2})^{2}}{2}t\|(\eta_{0}^{\delta},u_{0}^{\delta})\|_{V^{s}_{\sqrt{\mu}}}},$
(5.14)

for all $t\in[0,\tilde{T}_{\delta}]$. Finally, if $C_{1},C_{2}>0$ is large
enough then since $\tilde{T}_{\delta}\leq T_{0}$ we have that

$\displaystyle\|(\eta^{\delta},u^{\delta})(t)\|_{V^{s}_{\sqrt{\mu}}}$
$\displaystyle\leq
2k^{2}_{\beta}\|(\eta_{0}^{\delta},u_{0}^{\delta})\|_{V^{s}_{\sqrt{\mu}}}.$

Though, by continuity of the solution in time $t\in[0,T_{\delta}^{\star})$,
there exists $\tau>0$ such that
$\|(\eta^{\delta},u^{\delta})(\tau)\|_{V^{s}_{\sqrt{\mu}}}\leq
3k^{2}_{\beta}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}$ for
$\tilde{T}_{\delta}<\tau<T_{\delta}^{\star}$. This contradicts the definition
of $\tilde{T}_{\delta}$. Thus, we may conclude $T_{0}<\tilde{T}_{\delta}$ for
all $\delta>0$ and that $T_{0}$ is independent from $\delta$ by its definition
in (5.9).

∎

######  Remark 5.3.

For $0<\beta<\frac{1}{3}$ we observe that $k^{1}_{\beta}\sim
k^{2}_{\beta}\sim\frac{1}{\beta}$ and is due to the appearance of
$c_{\beta}^{1}$ in the coercivity estimate (3.7). This will impact the size of
the time interval when $\beta$ is small (see Remark 1.9). On the other hand,
for system (1.7) the coercivity estimate (3.26) is independent from $\beta$
and therefore gives a longer time of existence, as noted in Remark 1.15.

Step $4$: Uniqueness. Given a solution of (1.5), then we claim that it must be
unique.

We consider two solutions

${\varepsilon}(\zeta_{1},v_{1})=(\eta_{1},u_{1})\>\text{and}\>{\varepsilon}(\zeta_{1},v_{1})=(\eta_{1},u_{1})\>\text{in}\>C([0,T_{0}];V^{s}_{\sqrt{\mu}}({\mathbb{R}})),$

with the same initial data. Then define
$\mathbb{W}=(\eta_{1}-\eta_{2},u_{1}-u_{2})^{T}$, which is associated to the
initial datum $\mathbb{W}(0)=\mathbb{0}$. Since $(\eta_{1},u_{1})\in
H^{s}({\mathbb{R}})$, there exist a number $h_{1}>0$ such that
$\|(\eta_{1},u_{1})\|_{H^{s}\times H^{s}}\leq h_{1}$. Moreover, $\eta_{1}$
satisfy the non-cavitation condition by the Fundamental Theorem of Calculus
and the argument made in the proof of Lemma 5.2. Thus, we may use Proposition
4.1 to deduce

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

Then Grönwall’s lemma and (4.7) implies that
$\|(\eta_{1}-\eta_{2},u_{1}-u_{2})(t)\|_{V_{\sqrt{\mu}}^{s}}=0$ for all
$t\in[0,T_{0}]$. We therefore conclude the proof of the uniqueness.

Step 5: Existence of solutions. We claim that for all $0\leq s^{\prime}<s$
there exists a solution $(\zeta,v)={\varepsilon}^{-1}(\eta,u)\in
C([0,T_{0}];V_{\sqrt{\mu}}^{s^{\prime}}(\mathbb{R}))\cap
L^{\infty}([0,T_{0}];V_{\sqrt{\mu}}^{s}(\mathbb{R}))$ of (1.5) with
$T_{0}=\mathcal{O}(\frac{1}{{\varepsilon}})$ defined by (5.9).

Using the change of variable $(\zeta,v)={\varepsilon}^{-1}(\eta,u)$, we see
that the claim in Step $5$ is equivalent to proving that
$(\eta^{\delta},u^{\delta})$ solving (5.2) will satisfy system (3.1) in the
limit $\delta\searrow 0$ on $[0,T_{0}]$. In fact, the main idea is to prove
the convergence of $\\{(\eta^{\delta},u^{\delta})\\}$ as $\delta\searrow 0$ by
considering the difference between two solutions

$\mathbb{W}^{\delta}=(\psi^{\delta},w^{\delta}):=(\eta^{\delta_{1}}-\eta^{\delta_{2}},u^{\delta_{1}}-u^{\delta_{2}}).$

Here we let $0<\delta_{1}<\delta_{2}<\delta\leq 1$ and
$(\eta^{\delta_{i}},u^{\delta_{i}})$ be the solution of (5.2), obtained in
Step $1$. Moreover, let
$\varphi_{\delta}(D)=(\varphi_{\delta_{1}}-\varphi_{\delta_{2}})(D)$, then
$(\psi^{\delta},w^{\delta})$ satisfies a regularized version of (4.1):

$\partial_{t}\mathbb{W}^{\delta}+M(\mathbb{U}_{1}^{\delta},D)\cdot\varphi_{\delta}(D)\mathbb{W}^{\delta}=\mathbb{F}^{\delta},$

with

$\mathbb{F}^{\delta}=-\begin{pmatrix}w^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)\eta^{\delta_{2}}+\psi^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)u^{\delta_{2}}\\\
w^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)u^{\delta_{2}}\end{pmatrix},$
(5.15)

and initial data

$(\psi^{\delta}(0),w^{\delta}(0))=(\varphi_{\delta}(D)\eta_{0},\varphi_{\delta}(D)u_{0}).$
(5.16)

The system also satisfies the estimates of Propositions 3.6 and 4.1 by the
arguments used in the proof of Lemma 5.2.

Step 5.1: Convergence in $C([0,T_{0}];V^{0}_{\sqrt{\mu}}(\mathbb{R}))$. Define
the difference $(\psi^{\delta},w^{\delta})$ as above, then use (4.6) and
(4.7), combined with Grönwall’s inequality and (5.10) to find the estimate

$\sup\limits_{t\in[0,T_{0}]}\|(\psi^{\delta},w^{\delta})(t)\|_{V^{0}_{\sqrt{\mu}}}\leq
e^{c_{\beta}^{2}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}T_{0}}\|(\psi^{\delta}(0),w^{\delta}(0))\|_{V^{0}_{\sqrt{\mu}}},$

with $\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}T_{0}\lesssim_{\beta}1$ by
defintion (5.9). As a result, we use (5.16), the triangle inequality and
(2.40) to deduce that

$\displaystyle\sup\limits_{t\in[0,T_{0}]}\|(\psi^{\delta},w^{\delta})(t)\|_{V^{0}_{\sqrt{\mu}}}\lesssim_{\beta}\delta^{s}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}\underset{\delta\rightarrow
0}{\longrightarrow}0.$ (5.17)

Consequently, $\\{(\eta^{\delta},u^{\delta})\\}_{0<\delta\leq 1}$ defines a
Cauchy sequence in $C([0,T_{0}];V^{0}_{\sqrt{\mu}}(\mathbb{R}))$ and we
conclude that there exists a limit $(\eta,u)\in
C([0,T_{0}];V^{0}_{\sqrt{\mu}}(\mathbb{R}))$ by completeness.

Step $5.2$: Solution in
$C([0,T_{0}];V^{s^{\prime}}_{\sqrt{\mu}}(\mathbb{R}))\cap
L^{\infty}([0,T_{0}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ for
$s^{\prime}\in[0,s)$. As a direct consequence of (5.10) and (5.17), we deduce
by interpolation

$\displaystyle\|(\psi^{\delta},w^{\delta})\|_{L^{\infty}_{T_{0}}V^{s^{\prime}}_{\sqrt{\mu}}}$
$\displaystyle\lesssim_{\beta}\|(\psi^{\delta},w^{\delta})\|_{L^{\infty}_{T_{0}}V^{s}_{\sqrt{\mu}}}^{\frac{s^{\prime}}{s}}\|(\psi^{\delta},w^{\delta})\|_{L^{\infty}_{T_{0}}V^{0}_{\sqrt{\mu}}}^{1-\frac{s^{\prime}}{s}}$
$\displaystyle\lesssim_{\beta}\delta^{s-s^{\prime}}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}\underset{\delta\rightarrow
0}{\longrightarrow}0.$ (5.18)

Step 6: Presistence of the solution. We claim that there exists a unique
solution $(\zeta,v)={\varepsilon}^{-1}(\eta,u)\in
C([0,T_{0}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ of (1.5).

We again use the scaled variables $(\eta^{\delta},u^{\delta})$, where we will
apply estimate (4.8), following the Bona-Smith argument [7]. But first we must
control $\mathbb{F}^{\delta}$ in $V_{\sqrt{\mu}}^{s}(\mathbb{R})$. The
elements of $\mathbb{F}^{\delta}$ are given in (4.2), and we must therefore
control the terms given by:

$\displaystyle\big{(}J^{s}\mathbb{F}^{\delta},Q(\mathbb{U}^{\delta_{1}},D)J^{s}\mathbb{W}^{\delta}\big{)}_{L^{2}}$
$\displaystyle=\big{(}J^{s}(w^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)\eta^{\delta_{2}}),J^{s}\psi^{\delta}\big{)}_{L^{2}}+\big{(}J^{s}(\psi^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)\eta^{\delta_{2}}),J^{s}\psi^{\delta}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}J^{s}(w^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)u^{\delta_{2}}),\eta^{\delta_{1}}J^{s}w^{\delta}\big{)}_{L^{2}}+\big{(}J^{s}(w^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)u^{\delta_{2}}),\mathcal{K}_{\mu}(D)J^{s}w^{\delta}\big{)}_{L^{2}}$
$\displaystyle=:A_{1}+A_{2}+A_{3}+A_{4}.$

The terms $A_{1},A_{2}$ and $A_{3}$ are treated similarly. For instance, take
$A_{1}$. Then we observe that

$\displaystyle
A_{1}\leq\|J^{s}(w^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)\eta^{\delta_{2}})\|_{L^{2}}\|J^{s}\psi^{\delta}\|_{L^{2}}.$

Furthermore, using (2.22) and the Sobolev embedding we obtain that

$\displaystyle\|J^{s}(w^{\delta}\partial_{x}\varphi_{\delta_{2}}(D)\eta^{\delta_{2}})\|_{L^{2}}$
$\displaystyle\lesssim\|w^{\delta}\|_{L^{\infty}}\|J^{s}\partial_{x}\varphi_{\delta}(D)\eta^{\delta_{2}}\|_{L^{2}}+\|J^{s}w^{\delta}\|_{L^{2}}\|\eta^{\delta_{2}}\|_{H^{s}}.$
(5.19)

Using the triangle inequality, (2.39), and (5.10) we observe that

$\|J^{s}\partial_{x}\varphi_{\delta}(D)\eta^{\delta_{2}}\|_{L^{2}}\leq\delta^{-1}\|\eta^{\delta_{2}}\|_{H^{s}},$
(5.20)

which needs to be compensated to close the estimate. With this in mind, we use
the Sobolev embedding $H^{\frac{1}{2}^{+}}({\mathbb{R}})\hookrightarrow
L^{\infty}({\mathbb{R}})$ and (5.18) to deduce

$\displaystyle\|w^{\delta}\|_{L^{\infty}_{T_{0}}L^{\infty}}\lesssim\|(\psi^{\delta},w^{\delta})\|_{L^{\infty}_{T_{0}}V_{\sqrt{\mu}}^{\frac{1}{2}^{+}}}\lesssim\delta^{s-\frac{1}{2}^{+}}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}.$
(5.21)

Thus, combining (5.19) with (5.20) and (5.21) we get that

$\displaystyle|A_{1}|\lesssim\sup\limits_{t\in[0,T_{0}]}\big{(}\|w^{\delta}\|_{H^{s}}\|\eta^{\delta_{2}}\|_{H^{s}}+\delta^{s-\frac{3}{2}^{+}}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}\big{)}\|\psi^{\delta}\|_{H^{s}},$

as $\delta\searrow 0$. Arguing similarly, and using estimate (5.10), we deduce
that

$\displaystyle|A_{1}|+|A_{2}|+|A_{3}|\lesssim\sup\limits_{t\in[0,T_{0}]}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}\big{(}\|(\psi^{\delta},w^{\delta})\|_{V^{s}_{\sqrt{\mu}}}^{2}+\delta^{s-\frac{3}{2}^{+}}\|(\psi^{\delta},w^{\delta})\|_{V^{s}_{\sqrt{\mu}}}\big{)}.$

For $A_{4}$, we write

$\displaystyle A_{4}$
$\displaystyle=\big{(}[J^{s}\sqrt{\mathcal{K}_{\mu}}(D),w^{\delta}]\partial_{x}u^{\delta_{2}},J^{s}\sqrt{\mathcal{K}_{\mu}}(D)w^{\delta}\big{)}_{L^{2}}$
$\displaystyle\hskip
11.38092pt+\big{(}w^{\delta}J^{s}\sqrt{\mathcal{K}_{\mu}}(D)\partial_{x}u^{\delta_{2}},J^{s}\sqrt{\mathcal{K}_{\mu}}(D)w^{\delta}\big{)}_{L^{2}}.$

The commutator is treated by (2.7). While in the second term, we use (2.7) and
argue as above, giving the estimate

$\displaystyle|A_{4}|\lesssim_{\beta}\sup\limits_{t\in[0,T_{0}]}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}\big{(}\|(\psi^{\delta},w^{\delta})\|_{V^{s}_{\sqrt{\mu}}}^{2}+\delta^{s-\frac{3}{2}^{+}}\|(\psi^{\delta},w^{\delta})\|_{V^{s}_{\sqrt{\mu}}}\big{)}.$

We may therefore conclude by (4.8):

$\displaystyle\frac{d}{dt}\tilde{E}_{s}(\mathbb{W}^{\delta})$
$\displaystyle\lesssim_{\beta}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}\big{(}\tilde{E}_{s}(\mathbb{W}^{\delta})+\delta^{s-\frac{3}{2}^{+}}\tilde{E}_{s}(\mathbb{W}^{\delta})^{\frac{1}{2}}\big{)}.$

Then Grönwall’s inequality and (4.9) implies

$\|(\psi^{\delta},w^{\delta})\|_{L^{\infty}_{T_{0}}V^{s}_{\sqrt{\mu}}}\lesssim_{\beta}\delta^{s-\frac{3}{2}^{+}}\|(\eta_{0},u_{0})\|_{V^{s}_{\sqrt{\mu}}}\underset{\delta\rightarrow
0}{\longrightarrow}0.$

Thus, $(\eta^{\delta},u^{\delta})$ is a Cauchy sequence in
$C([0,T_{0}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ and we conclude by the
uniqueness of the limit that the solution $(\eta,u)\in
C([0,T_{0}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ .

Step 7: The solution is bounded by the initial data. We claim that the
solution obtained in Step $6$ satisfy (1.13).

Indeed, using the notation from previous step, we deduce by (5.10) that

$\\{(\eta^{\delta},u^{\delta})\\}_{0<\delta\leq 1}\subset
C([0,T_{0}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$

is a bounded sequence in a reflexive Banach space. As a result, we have by
Eberlein-S̆umulian’s Theorem that
$(\eta^{\delta},u^{\delta})\underset{\delta\rightarrow
0}{\rightharpoonup}(\eta,u)$ weakly in $V^{s}_{\sqrt{\mu}}(\mathbb{R})$ for
all $t \in[0,T_{0}]$ and implies

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

Step 8: Continuous dependence of the flow map data solution. Consider two sets
of initial data $(\zeta_{1},v_{1})(0),(\zeta_{2},v_{2})(0)\in
V^{s}_{\sqrt{\mu}}({\mathbb{R}})$. Then we claim that for all $\lambda>0$,
there exists $\gamma>0$ such that having

$\|(\zeta_{1}-\zeta_{2},v_{1}-v_{2})(0)\|_{V^{s}_{\sqrt{\mu}}}<\gamma,$

implies

$\|(\zeta_{1}-\zeta_{2},v_{1}-v_{2})\|_{L^{\infty}_{\frac{T_{0}}{2}}V^{s}_{\sqrt{\mu}}}<\lambda.$

Equivalently, we will prove that for
${\varepsilon}(\zeta_{1},\zeta_{2},v_{1},v_{2})=(\eta_{1},\eta_{2},u_{1},u_{2})$
such that

$\|(\eta_{1}-\eta_{2},u_{1}-u_{2})(0)\|_{V^{s}_{\sqrt{\mu}}}<{\varepsilon}\gamma,$

implies

$\|(\eta_{1}-\eta_{2},u_{1}-u_{2})\|_{L^{\infty}_{\frac{T_{0}}{2}}V^{s}_{\sqrt{\mu}}}<{\varepsilon}\lambda.$

Using the notation in Step $6$, we let $0<\delta<1$ to be fixed, and
$(\eta_{1}^{\delta},u_{1}^{\delta}),(\eta_{2}^{\delta},u_{2}^{\delta})\in
C([0,\frac{T_{0}}{2}];V^{s}_{\sqrt{\mu}}(\mathbb{R}))$ be two solutions of
(5.2) on large time with corresponding initial data
$(\varphi_{\delta}(D)\eta_{1},\varphi_{\delta}(D)u_{1})(0)$ and
$(\varphi_{\delta}(D)\eta_{2},\varphi_{\delta}(D)u_{2})(0)$. Then observe

$\displaystyle\|(\eta_{1}-\eta_{2},u_{1}-u_{2})\|_{V^{s}_{\sqrt{\mu}}}$
$\displaystyle\leq\|(\eta_{1}-\eta_{1}^{\delta},u_{1}-u_{1}^{\delta})\|_{V_{\sqrt{\mu}}^{s}}+\|(\eta_{2}-\eta_{2}^{\delta},u_{2}-u_{2}^{\delta})\|_{V_{\sqrt{\mu}}^{s}}$
$\displaystyle\hskip
11.38092pt+\|(\eta_{1}^{\delta}-\eta_{2}^{\delta},u_{1}^{\delta}-u_{2}^{\delta})\|_{V^{s}_{\sqrt{\mu}}}$
$\displaystyle=:B_{1}+B_{2}+B_{3}.$ (5.22)

For the first two terms we use that
${\varepsilon}^{-1}(\eta^{\delta},u^{\delta})=(\zeta^{\delta},v^{\delta})\rightarrow(\zeta,v)={\varepsilon}^{-1}(\eta,u)$
as $\delta\searrow 0$ by Step $6$. Therefore it follows that $B_{1}$ and
$B_{2}$ must at least satisfy the estimate,

$\sup\limits_{t\in[0,\frac{T_{0}}{2}]}(B_{1}+B_{2})(t)\lesssim_{\beta}{\varepsilon}o_{\delta}(1).$
(5.23)

While for $B_{3}$, we need the continuity of the flow map of the regularized
system (5.2) on a long time (see Remark 5.1).

We let
$\mathbb{\tilde{W}}^{\delta}=(\tilde{\psi}^{\delta},\tilde{w}^{\delta})=(\eta_{1}^{\delta}-\eta_{2}^{\delta},u_{1}^{\delta}-u_{2}^{\delta})$.
Then staying consistent with previous notation, we have that the difference of
two regularized solutions will satisfy the equation:

$\partial_{t}\mathbb{\tilde{W}}^{\delta}+M(\mathbb{U}_{1}^{\delta},D)\varphi_{\delta}(D)\mathbb{\tilde{W}}^{\delta}=\mathbb{\tilde{F}}^{\delta},$
(5.24)

with

$\mathbb{\tilde{F}}^{\delta}=-\begin{pmatrix}\tilde{w}^{\delta}\partial_{x}\varphi_{\delta}(D)\eta_{2}^{\delta}+\tilde{\psi}^{\delta}\partial_{x}\varphi_{\delta}(D)u_{2}^{\delta}\\\
\tilde{w}^{\delta}\partial_{x}\varphi_{\delta}(D)u_{2}^{\delta}\end{pmatrix},$

and initial data

$(\tilde{\psi}^{\delta},\tilde{w}^{\delta})(0)=(\varphi_{\delta}(D)\eta_{1}-\varphi_{\delta}(D)\eta_{2},\varphi_{\delta}(D)u_{1}-\varphi_{\delta}(D)u_{2})(0).$

We will use this information to estimate $B_{3}$ by suitable energy estimates
at $V^{0}_{\sqrt{\mu}}({\mathbb{R}})$ and
$V^{s}_{\sqrt{\mu}}({\mathbb{R}})-$level.

Similar to Step $5.1$, we first obtain the estimate in
$V^{0}_{\sqrt{\mu}}({\mathbb{R}})$ by using (4.7) and (4.9). Indeed, there
holds

$\frac{d}{dt}\tilde{E}_{0}(\mathbb{\tilde{W}}^{{\delta}})\lesssim_{\beta}\max\limits_{i=1,2}\|(\eta_{i}^{\delta},u_{i}^{\delta})\|_{V^{s}_{\sqrt{\mu}}}\tilde{E}_{0}(\mathbb{\tilde{W}}^{{\delta}}).$
(5.25)

For simplicity we let
$\|(\eta_{1},u_{1})(0)\|_{V^{s}_{\sqrt{\mu}}}={\varepsilon}K$. Moreover, we
observe that if
${\varepsilon}\gamma<\frac{1}{2}\|(\eta_{1},u_{1})(0)\|_{V^{s}_{\sqrt{\mu}}}$,
then we have by (5.10) that

$\displaystyle\|(\eta_{1}^{\delta},u_{1}^{\delta})\|_{L^{\infty}_{\frac{T_{0}}{2}}V^{s}_{\sqrt{\mu}}}+\|(\eta_{2}^{\delta},u_{2}^{\delta})||_{L^{\infty}_{\frac{T_{0}}{2}}V^{s}_{\sqrt{\mu}}}$
$\displaystyle\lesssim_{\beta}\|(\eta_{1},u_{1})(0)\|_{V^{s}_{\sqrt{\mu}}}+\|(\eta_{2},u_{2})(0)\|_{V^{s}_{\sqrt{\mu}}}$
$\displaystyle\lesssim_{\beta}{\varepsilon}K.$ (5.26)

As a result, we have an estimate on the difference in
$V^{0}_{\sqrt{\mu}}({\mathbb{R}})$. Indeed, by Grönwall’s inequality, (5.25),
(5.26), the triangle inequality, and (2.40) implies

$\displaystyle\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})\|_{V^{0}_{\sqrt{\mu}}}\lesssim_{\beta}\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})(0)\|_{V^{0}_{\sqrt{\mu}}}\lesssim_{\beta}{\varepsilon}K(\delta^{s}+K^{-1}\gamma).$
(5.27)

We will now use this decay estimate to deal with (4.8), which is at the
$V^{s}_{\sqrt{\mu}}({\mathbb{R}})-$level. Similar to Step $6$, we decompose
the source term (4.2) in four pieces

$\displaystyle\tilde{A}$
$\displaystyle:=\big{(}J^{s}\mathbb{\tilde{F}}^{\delta},Q(\mathbb{U}_{1}^{\delta},D)J^{s}\mathbb{\tilde{W}}^{\delta}\big{)}_{L^{2}}$
$\displaystyle=\big{(}J^{s}(\tilde{w}^{\delta}\partial_{x}\varphi_{\delta}(D)\eta_{2}^{\delta}),J^{s}\tilde{\psi}^{\delta}\big{)}_{L^{2}}+\big{(}J^{s}(\tilde{\psi}^{\delta}\partial_{x}\varphi_{\delta}(D)\eta_{2}^{\delta}),J^{s}\tilde{\psi}^{\delta}\big{)}_{L^{2}}$
$\displaystyle\hskip
14.22636pt+\big{(}J^{s}(\tilde{w}^{\delta}\partial_{x}\varphi_{\delta}(D)u_{2}^{\delta}),\eta_{2}^{\delta}J^{s}\tilde{w}^{\delta}\big{)}_{L^{2}}+\big{(}J^{s}(\tilde{w}^{\delta}\partial_{x}\varphi_{\delta}(D)u_{2}^{\delta}),\mathcal{K}_{\mu}(D)J^{s}\tilde{w}^{\delta}\big{)}_{L^{2}}$
$\displaystyle=:\tilde{A}_{1}+\tilde{A}_{2}+\tilde{A}_{3}+\tilde{A}_{4}.$

To estimate $\tilde{A}_{1}$, we first obtain a bound similar to (5.18).
Indeed, using the Sobolev emebedding, interpolation, (5.26), and (5.27) yields

$\displaystyle\sup\limits_{t\in[0,T_{0}]}\|\tilde{w}^{\delta}\|_{L^{\infty}}$
$\displaystyle\lesssim\sup\limits_{t\in[0,T_{0}]}\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})\|_{V_{\sqrt{\mu}}^{\frac{1}{2}^{+}}}$
$\displaystyle\lesssim\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})\|_{L^{\infty}_{T_{0}}V^{s}_{\sqrt{\mu}}}^{\frac{1}{2s}^{+}}\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})\|_{L^{\infty}_{T_{0}}V^{0}_{\sqrt{\mu}}}^{1-\frac{1}{2s}^{+}}$
$\displaystyle\lesssim{\varepsilon}K(\delta^{s-\frac{1}{2}^{+}}+(K^{-1}\gamma)^{1-\frac{1}{2s}^{+}}),$

where $1-\frac{1}{2s}^{+}>0$ for $s>\frac{1}{2}^{+}$. Then arguing as we did
for $A_{1}$ in Step $6$, we obtain that

$\displaystyle|\tilde{A}_{1}|$
$\displaystyle\lesssim\sup\limits_{t\in[0,T_{0}]}\big{(}\|\tilde{w}^{\delta}\|_{H^{s}}\|\eta_{2}^{\delta}\|_{H^{s}}+\|\tilde{w}^{\delta}\|_{L^{\infty}}\delta^{-1}\|\eta^{\delta}_{2}\|_{H^{s}}\big{)}\|\tilde{\psi}^{\delta}\|_{H^{s}}$
$\displaystyle\lesssim{\varepsilon}K\sup\limits_{t\in[0,T_{0}]}\big{(}\|\tilde{w}^{\delta}\|_{H^{s}}+\delta^{s-\frac{3}{2}^{+}}+\delta^{-1}(K^{-1}\gamma)^{1-\frac{1}{2s}^{+}}\big{)}\|\tilde{\psi}^{\delta}\|_{H^{s}}.$

Moreover, for the remaining terms, we can use similar estimates, recalling
that for $\tilde{A}_{4}$ we also need to deal with the non-local operator
$\mathcal{K}_{\mu}(D)$ (see step $6$ for details). Indeed,

$\displaystyle\tilde{A}$
$\displaystyle\lesssim_{\beta}{\varepsilon}K\sup\limits_{t\in[0,T_{0}]}\big{(}\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})\|_{V_{\sqrt{\mu}}^{s}}+\delta^{s-\frac{3}{2}^{+}}+\delta^{-1}(K^{-1}\gamma)^{1-\frac{1}{2s}^{+}}\big{)}\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})\|_{V_{\sqrt{\mu}}^{s}}.$
(5.28)

Consequently, combining estimates (4.8) and (4.9) with (5.28) yields

$\displaystyle\frac{d}{dt}\tilde{E}_{s}(\tilde{\mathbb{W}}^{\delta})\lesssim_{\beta}{\varepsilon}K\big{(}\tilde{E}_{s}(\tilde{\mathbb{W}}^{\delta})+(\delta^{s-\frac{3}{2}^{+}}+\delta^{-1}(K^{-1}\gamma)^{1-\frac{1}{2s}^{+}})\tilde{E}_{s}(\tilde{\mathbb{W}}^{\delta})^{\frac{1}{2}}\big{)}.$

Thus, we have an estimate on $B_{3}$ by the energy estimate (4.9), Grönwall’s
inequality and (2.41). Indeed, there holds

$\displaystyle
B_{3}=\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})\|_{V_{\sqrt{\mu}}^{s}}$
$\displaystyle\lesssim_{\beta}\|(\tilde{\psi}^{\delta},\tilde{w}^{\delta})(0)\|_{V_{\sqrt{\mu}}^{s}}+{\varepsilon}K(\delta^{s-\frac{3}{2}^{+}}+\delta^{-1}(K^{-1}\gamma)^{1-\frac{1}{2s}^{+}})$
$\displaystyle\lesssim_{\beta}{\varepsilon}o_{\delta}(1)+{\varepsilon}\gamma+{\varepsilon}K(\delta^{s-\frac{3}{2}^{+}}+\delta^{-1}(K^{-1}\gamma)^{1-\frac{1}{2s}^{+}}).$
(5.29)

Returning to (5), we may conclude the proof of the continuous dependence. We
first fix $0<\delta<1$ to be small enough and satisfying

$o_{\delta}(1)+K\delta^{s-\frac{3}{2}^{+}}<\frac{\lambda}{2c_{\beta}},$

for some constant $c_{\beta}$ depending on $\beta$. Then let $\gamma$ verify
the restriction:

${\varepsilon}\gamma<\frac{1}{2}\|(\eta_{1},u_{1})(0)\|_{V^{s}_{\sqrt{\mu}}},$

such that
$\gamma+K\delta^{-1}(K^{-1}\gamma)^{1-\frac{1}{2s}^{+}}<\frac{\lambda}{2c_{\beta}}$.
Consequently, we have by (5), (5.23) and (5.29) that

$\displaystyle\sup\limits_{t\in[0,\frac{T_{0}}{2}]}\|(\eta_{1}-\eta_{2},u_{1}-u_{2})(t)\|_{V^{s}_{\sqrt{\mu}}}$
$\displaystyle\leq{\varepsilon}c_{\beta}(o_{\delta}(1)+\gamma+K(\delta^{s-\frac{3}{2}^{+}}+\delta^{-1}(K^{-1}\gamma)^{1-\frac{1}{2s}^{+}}))$
$\displaystyle<{\varepsilon}\lambda.$

As a result, we have demonstrated that the solution of (1.5) depends
continuously on the initial data and thus completes the proof of Theorem 1.6.

∎

##  6\. Final comments

The proof of Theorem 1.11 and Theorem 1.14 are similar and is therefore
omitted.

## Appendix A

###  A.1. Pointwise estimates for $\sqrt{K_{\mu}(\xi)}$ and
$\sqrt{T_{\mu}(\xi)}$

Before turning to the proof of the pointwise estimates in Lemma 2.1 and Lemma
2.5, we make an important observation. Let $\sqrt{\mathcal{T}_{\mu}}(D)$ be
the Fourier multiplier associated with the symbol

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

Then the operator is regularizing for $\mu>0$ and acts similar to the Bessel
potential $J^{-\frac{1}{2}}$ in the $L^{2}-$norm. While $\sqrt{K_{\mu}(\xi)}$
has a similar behaviour in low frequency for $\beta<\frac{1}{3}$, but acts
like $J^{\frac{1}{2}}$ in high frequencies.

Figure 3. The multiplier $\sqrt{K_{1}(\xi)}$ in the cases
$\beta\geq\frac{1}{3}$ (line) and $\beta<\frac{1}{3}$ (dash-dots). The red
curve is a plot of $\sqrt{T_{1}(\xi)}$ (dash).

######  Lemma A.1.

Let $\mu>0$ and take any $n\in{\mathbb{N}}$.

  * •

Then $T_{\mu}(\xi)$ satisfies

$\Big{|}\frac{d^{n}}{d\xi^{n}}\sqrt{T_{\mu}(\xi)}\Big{|}\lesssim\mu^{\frac{n}{2}}\langle\sqrt{\mu}\xi\rangle^{-\frac{1}{2}-n}.$
(A.1)

  * •

Similarly, $K_{\mu}(\xi)$ satisfies

$\Big{|}\frac{d^{n}}{d\xi^{n}}\sqrt{K_{\mu}(\xi)}\Big{|}\lesssim_{\beta}\mu^{\frac{n}{2}}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}-n}.$
(A.2)

###### Proof.

The proof is a generalization of Lemma $8$ in [16]. Following their arguments,
we observe that since $T_{1}(\sqrt{\mu}r)=T_{\mu}(r)$ for $r>0$, it is
sufficient to show that

$\Big{|}\frac{d^{n}}{dr^{n}}\sqrt{T_{1}(r)}\Big{|}\lesssim\langle
r\rangle^{-\frac{1}{2}-n}.$

We divide the proof into two steps.

First, let $0<r<\frac{1}{2}$ and prove that any derivative of
$\sqrt{T_{1}(r)}$ is bounded. We have that $\sqrt{T_{1}(r)}$ is bounded in
this region. By direct computation, we observe

$\displaystyle\frac{d}{dr}\sqrt{T_{1}(r)}=\frac{sech^{2}(r)}{(2r)^{2}\sqrt{T_{1}(r)}}\bigg{(}2r+\frac{e^{-2r}-e^{2r}}{2}\bigg{)}=:\frac{sech^{2}(r)}{\sqrt{T_{1}(r)}}G(r),$
(A.3)

where $G(r)$ can be written as a series by expanding the exponentials. Indeed,
we have that

$\displaystyle
G(r)=\frac{1}{(2r)^{2}}\bigg{(}2r+\frac{e^{-2r}-e^{2r}}{2}\bigg{)}=-\sum\limits_{k=0}^{\infty}\frac{(2r)^{2k+1}}{(2k+3)!}.$

The series is uniformly convergent for $r\geq 0$. Moreover, $G(r)$ and its
derivatives are bounded for $0<r<\frac{1}{2}$. By extension, since for all
$n\geq 0$ there holds $\frac{d^{n}}{dr^{n}}sech^{2}(r)\lesssim e^{-2r}$ we
have that

$\Big{|}\frac{d^{n}}{dr^{n}}\sqrt{T_{1}(r)}\Big{|}\lesssim 1.$

Now, we let $r\geq\frac{1}{2}$ and prove the necessary decay estimate. We use
the identity

$tanh(r)=1-\frac{2}{e^{2r}+1},$ (A.4)

and deduce by the chain rule that

$\displaystyle\Big{|}\frac{d^{n}}{dr^{n}}\sqrt{T_{1}(r)}\Big{|}\lesssim\sum_{k=0}^{n}\Big{(}\frac{1}{r}\big{(}1-\frac{2}{e^{2r}+1}\big{)}\Big{)}^{\frac{1}{2}-k}r^{-k-n}\lesssim\langle
r\rangle^{-\frac{1}{2}-n}.$

Lastly, we have that (A.2) follows by the Leibniz rule. Indeed, we observe

$\displaystyle\Big{|}\frac{d^{n}}{dr^{n}}\sqrt{K_{1}(r)}\Big{|}$
$\displaystyle=\Big{|}\frac{d^{n}}{dr^{n}}\sqrt{T_{1}(r)(1+\beta
r^{2})}\Big{|}$
$\displaystyle\lesssim\sum\limits_{k=0}^{n}\Big{|}\frac{d^{n-k}}{dr^{n-k}}\sqrt{T_{1}(r)}\Big{|}\Big{|}\frac{d^{k}}{dr^{k}}\sqrt{1+\beta
r^{2}}\Big{|}$ $\displaystyle\lesssim_{\beta}\langle
r\rangle^{\frac{1}{2}-n},$

which concludes the proof of Lemma A.1. ∎

###  A.2. Proof of Lemmas 2.1 and 2.5

###### Proof of Lemma 2.1.

First, we again make the observation that $K_{1}(\sqrt{\mu}\xi)=K_{\mu}(\xi)$.
Therefore, we simply let $r>0$ and consider $K_{1}(r)$. To establish the upper
bound given in (2.1), we note that for $r<1$ we have

$\displaystyle K_{1}(r)\leq 1+\beta.$

This is because $\frac{tanh(r)}{r}\leq 1$. On the other hand, when $r\geq 1$
then $tanh(r)<1$ and it follows that

$\displaystyle K_{1}(r)\leq 1+\beta r.$

Consequently, for all $r>0$ there holds $K_{1}(r)\leq c_{\beta}^{2}(1+r)$ with
$c_{\beta}^{2}$ as defined in (1.15).

Next, we prove the lower bound given by (2.2) with $\beta\geq\frac{1}{3}$. We
will again split $r$ into two intervals, where we aim to prove

$K_{1}(r)=\frac{tanh(r)}{r}(1+\beta r^{2})\geq(1-\frac{h_{0}}{2})+cr,$ (A.5)

for some positive constant $c>0$ and any $h_{0}\in(0,1)$. We prove (A.5) by
considering two cases for $r$. When $0\leq r\leq\frac{h_{0}}{4}$ we use that

$tanh(r)=\int_{0}^{r}(1-tanh^{2}(x))\>dx\geq r-\frac{r^{3}}{3},$ (A.6)

since $tanh^{2}(x)\leq x^{2}$ by the mean value theorem. Therefore, we have
that

$\displaystyle
K_{1}(r)\geq\Big{(}1-\frac{r^{2}}{3}\Big{)}\Big{(}1+\frac{r^{2}}{3}\Big{)}\geq\Big{(}1-\frac{h_{0}}{2}\Big{)}+\Big{(}\frac{h_{0}}{4}-\frac{r^{4}}{9}\Big{)}+r,$

which implies (A.5) since $0\leq r\leq\frac{h_{0}}{4}$.

For the remaining part, we use the identity (A.4) and show that (A.5) holds
for $r\geq\frac{h_{0}}{4}$ if:

$\displaystyle\hskip
28.45274pttanh(r)\bigg{(}1+\frac{r^{2}}{3}\bigg{)}-r(1+cr)>0$
$\displaystyle\iff\bigg{(}1+\frac{r^{2}}{3}-r(1+cr)\bigg{)}-\frac{2}{e^{2r}+1}\bigg{(}1+\frac{r^{2}}{3}\bigg{)}>0$
$\displaystyle\iff\bigg{(}1+\frac{r^{2}}{3}-r(1+cr)\bigg{)}\big{(}e^{2r}+1\big{)}-2\bigg{(}1+\frac{r^{2}}{3}\bigg{)}:=G(r)>0.$

But this holds since

$\displaystyle G^{\prime\prime\prime}(r)$
$\displaystyle=\frac{4}{3}e^{2r}\big{(}2r^{2}-3c(2r^{2}+6r+3)\big{)}$
$\displaystyle\geq\frac{4}{3}e^{2r}\Big{(}r^{2}(1-6c)+r(\frac{h_{0}}{8}-18c)+(\frac{h_{0}^{2}}{32}-9c)\Big{)},$

and is positive for $0<c<10^{-3}h_{0}^{2}$ with $r\geq\frac{h_{0}}{4}$.
Indeed, as a consequence we have the following chain of implications

$\displaystyle 0<G^{\prime\prime}\Big{(}\frac{h_{0}}{2}\Big{)}\leq
G^{\prime\prime}(r)=\frac{2}{3}\big{(}e^{2r}(1-2r+2r^{2}-3c(2r^{2}+4r+1))-3c-1\big{)}$
$\displaystyle\implies$ $\displaystyle
0<G^{\prime}\Big{(}\frac{h_{0}}{2}\Big{)}\leq
G^{\prime}(r)=\frac{1}{3}\big{(}e^{2r}(3-4r+2r^{2}-6cr(r+1))-2r-3-6cr\big{)}$
$\displaystyle\implies$ $\displaystyle 0<G\Big{(}\frac{h_{0}}{2}\Big{)}\leq
G(r).$

We have therefore verified (A.5) for all $r\geq 0$ and we conclude that (2.2)
holds true.

Similarly, for $0<\beta<\frac{1}{3}$, we have that (2.3) is a consequence of
the inequality

$\frac{tanh(r)}{r}(1+\beta r^{2})\geq\beta+cr.$

One should note that we do not require sharp estimates. In fact, we simply
need to obtain the estimate

$\bigg{(}1+\beta
r^{2}-r(\beta+cr)\bigg{)}\big{(}e^{2r}+1\big{)}-2\bigg{(}1+\beta
r^{2}\bigg{)}=:H(r)\geq 0,$

for $r\geq 0$. On the other hand, we observe that

$H^{\prime\prime\prime}(r)=4e^{2r}\Big{(}2+2\beta r(2+r)-3c+2cr(3+r)\Big{)}>0$

for all $r\geq 0$ if

$\displaystyle(2-3c)+2r(2\beta-3c)+2r^{2}(\beta-c)>0$

and is ensured for $0<c\leq 10^{-3}\beta$. Consequently,

$\displaystyle 0<H^{\prime\prime}(0)\leq
H^{\prime\prime}(r)=2e^{2r}\Big{(}2+\beta(2r^{2}+2r-1)-c(2r^{2}+4r+1)\Big{)}-2(\beta+c)$
$\displaystyle 0<H^{\prime}(0)\leq
H^{\prime}(r)=e^{2r}\Big{(}2+\beta(2r^{2}-1)-2c(r^{2}+2r)\Big{)}-\beta(2r+1)-2cr$
$\displaystyle 0<H(0)\leq H(r),$

and we argue as above to conclude.

The proof of estimate (2.4) is a direct consequence of Lemma A.1 and (A.2)
with $n=1$ if we trace the dependence in $\beta$:

$\displaystyle\big{|}\frac{d}{dr}\sqrt{K_{1}(r)}\big{|}\lesssim\langle
r\rangle^{-\frac{1}{2}-1}(1+\beta r^{2})^{\frac{1}{2}}+\langle
r\rangle^{-\frac{1}{2}}\frac{\beta r}{(1+\beta
r^{2})^{\frac{1}{2}}}\lesssim\langle r\rangle^{-1}+\sqrt{\beta}\langle
r\rangle^{-\frac{1}{2}}.$

Estimate (2.5) concerns the following bound on the difference:

$\displaystyle\Big{|}\sqrt{K_{\mu}(\xi)}-\sqrt{\beta}\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\Big{|}=\Big{|}\Big{(}\frac{tanh(\sqrt{\mu}|\xi|)}{\sqrt{\mu}|\xi|}(1+\beta\mu\xi^{2})\Big{)}^{\frac{1}{2}}-\sqrt{\beta}\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\Big{|}.$

For $\beta\sqrt{\mu}|\xi|\leq 1$ there holds trivially by using the triangle
inequality that

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

While for $\beta\sqrt{\mu}|\xi|>1$ we observe by direct calculations that

$\displaystyle\Big{|}\sqrt{K_{\mu}(\xi)}-\sqrt{\beta}\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\Big{|}$
$\displaystyle=\sqrt{\beta}\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\Big{|}\Big{(}\frac{tanh(\sqrt{\mu}|\xi|)}{\beta\mu\xi^{2}}+tanh(\sqrt{\mu}|\xi|)\Big{)}^{\frac{1}{2}}-1\Big{|}$
$\displaystyle\lesssim\sqrt{\beta}\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\Big{|}\frac{tanh(\sqrt{\mu}|\xi|)}{\beta\mu\xi^{2}}+(tanh(\sqrt{\mu}|\xi|)-1)\Big{|}$
$\displaystyle\lesssim\frac{1}{(\beta\sqrt{\mu}|\xi|)^{\frac{1}{2}}}\frac{1}{\sqrt{\mu}|\xi|}+\sqrt{\beta}(\sqrt{\mu}|\xi|)^{\frac{1}{2}}e^{-2\sqrt{\mu}|\xi|}$
$\displaystyle\lesssim\beta+\sqrt{\beta},$

where we used the triangle inequality and that $\beta\sqrt{\mu}|\xi|>1$.

Lastly, we prove (2.6) by using (2.5):

$\displaystyle\sqrt{K_{\mu}(\xi)}\langle\xi\rangle^{s-1}|\xi|$
$\displaystyle=\bigg{(}\sqrt{K_{\mu}(\xi)}-\sqrt{\beta}\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\bigg{)}\langle\xi\rangle^{s-1}|\xi|+\sqrt{\beta}\mu^{\frac{1}{4}}\langle\xi\rangle^{s-1}|\xi|^{\frac{3}{2}}$
$\displaystyle\lesssim(\beta+\sqrt{\beta})\langle\xi\rangle^{s}+\sqrt{\beta}\mu^{\frac{1}{4}}\langle\xi\rangle^{s}|\xi|^{\frac{1}{2}}.$

∎

###### Proof of Lemma 2.5.

To prove (2.9), since $T_{1}(\sqrt{\mu}\xi)=T_{\mu}(\xi)$, we only need to
establish the following inequality:

$1-\frac{h_{0}}{2}+cr\leq\frac{r}{tanh(r)}\lesssim 1+r,$ (A.7)

for all $r>0$ and some $c>0$. We also note that the upper bound is trivial, so
we only prove the lower bound. Let $h_{0}\in(0,1)$. By the mean value theorem
we find that $tanh(r)\leq r$ and observe

$\frac{r}{tanh(r)}=(1-\frac{h_{0}}{2})\frac{r}{tanh(r)}+\frac{h_{0}}{2}\frac{r}{tanh(r)}\geq
1-\frac{h_{0}}{2}+\frac{h_{0}}{2}r.$

Next, we consider (2.10). For $\sqrt{\mu}|\xi|\leq 1$ we have that
$T_{\mu}(\xi)\sim 1$ and $\langle\sqrt{\mu}\xi\rangle\sim 1$. On the other
hand, when $\sqrt{\mu}|\xi|\geq 1$ then
$T_{\mu}(\xi)\sim\frac{1}{\sqrt{\mu}|\xi|}$ and
$\langle\sqrt{\mu}\xi\rangle\sim\sqrt{\mu}|\xi|$. Multiplying the two
functions, we obtain the desired result.

We estimate the derivative (2.11) directly and using that $\mu\in(0,1)$:

$\displaystyle\Big{|}\frac{d}{d\xi}\langle\xi\rangle^{s}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}\Big{|}\lesssim\langle\xi\rangle^{s-1}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}+\langle\xi\rangle^{s}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}\sqrt{\mu}\langle\sqrt{\mu}\xi\rangle^{-1}\lesssim\langle\xi\rangle^{s-1}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}},$

since $\sqrt{\mu}\langle\sqrt{\mu}\xi\rangle^{-1}\leq\langle\xi\rangle^{-1}$.

Similarly, we have that (2.12) follows by the same argument after using (A.1)
and (2.11):

$\displaystyle\Big{|}\frac{d}{d\xi}\sqrt{T_{\mu}(\xi)}\langle\xi\rangle^{s}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}\Big{|}$
$\displaystyle\lesssim\sqrt{\mu}\langle\xi\rangle^{-\frac{3}{2}}\langle\xi\rangle^{s}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}+\langle\sqrt{\mu}\xi\rangle^{-\frac{1}{2}}\langle\xi\rangle^{s-1}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}$
$\displaystyle\lesssim\langle\xi\rangle^{s-1}.$

For estimate (2.13), we observe that

$\displaystyle\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}-\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}$
$\displaystyle=\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}\Big{(}(\frac{1}{\mu\xi^{2}}+1)^{\frac{1}{4}}-1\Big{)}\lesssim
1.$

∎

###  A.3. Proof of Lemmas 2.11 and 2.12

For the proof of Lemma 2.11 and Lemma 2.12, we need a “generalized” version of
the Kato-Ponce commutator estimate which holds for symbols defined by:

######  Definition A.2 (Symbol class [33] Def. $B.7$).

We say that a symbol $\sigma(D)$ is a member of the symbol class
$\mathcal{S}^{s}$ with $s\in\mathbb{R}$, if
$\xi\mapsto\sigma(\xi)\in\mathbb{C}$ is smooth and satisfies

$\forall\alpha\in\mathbb{N},\quad\sup\limits_{\xi\in\mathbb{R}}\langle\xi\rangle^{\alpha-s}\Big{|}\frac{d^{\alpha}}{d\xi^{\alpha}}\sigma(\xi)\Big{|}<\infty.$

One also associates the following seminorm on $\mathcal{S}^{s}:$

$\mathcal{N}^{s}(\sigma)=\sup\limits_{\alpha\in\mathbb{N},\>\>\alpha\leq
4}\>\>\>\sup\limits_{\xi\in\mathbb{R}}\langle\xi\rangle^{\alpha-s}\Big{|}\frac{d^{\alpha}}{d\xi^{\alpha}}\sigma(\xi)\Big{|}.$
(A.8)

The following result is found in Appendix B of [33].

######  Lemma A.3.

Let $t_{0}>1/2$, $s\geq 0$ and $\sigma\in\mathcal{S}^{s}$. If $f\in H^{s}\cap
H^{t_{0}+1}(\mathbb{R})$, then for all $g\in H^{s-1}(\mathbb{R})$,

$\|[\sigma(D),f]g\|_{L^{2}}\lesssim\mathcal{N}^{s}(\sigma)\|f\|_{H^{\max\\{t_{0}+1,s\\}}}\|g\|_{H^{s-1}}$
(A.9)

With this at hand, we may give the proof.

###### Proof of Lemma 2.11.

To prove (2.24) and (2.25), it suffices to verify for all $n\in{\mathbb{N}}$
that

$\sup\limits_{\xi\in\mathbb{R}}\>\langle\xi\rangle^{n}\>\Big{|}\frac{d^{n}}{d\xi^{n}}(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})(\xi)\Big{|}\lesssim_{\beta}1,$
(A.10)

for any $0<\mu<1$. Indeed, in agreement with Definition A.2, then
$(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})\in\mathcal{S}^{0}$ and (2.25)
holds true due to Lemma A.3. Moreover, using Plancherel and (A.10) with $n=0$
we have

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

proving (2.24). Now, let us prove (A.10). We observe that

$\mu^{\frac{k}{2}}\langle\xi\rangle^{k}\Big{|}(\frac{d^{k}}{d\xi^{k}}\chi^{(1)})(\sqrt{\mu}\xi)\Big{|}\lesssim
1,\quad k\geq 0,$ (A.11)

since $\sqrt{\mu}|\xi|\lesssim 1$ on the support of $\chi^{(1)}_{\mu}(\xi)$.
Moreover, we observe by Lemma A.1 and $\mu\in(0,1)$ that

$\displaystyle\chi^{(1)}_{\mu}(\xi)\>\Big{|}\frac{d^{k}}{d\xi^{k}}\sqrt{K_{\mu}(\xi)}\Big{|}\lesssim_{\beta}\chi^{(1)}_{\mu}(\xi)\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}\mu^{\frac{k}{2}}\langle\sqrt{\mu}\xi\rangle^{-k}\lesssim_{\beta}\langle\xi\rangle^{-k}.$

Combining these estimates with the Leibniz rule yields

$\displaystyle\langle\xi\rangle^{n}\Big{|}\frac{d^{n}}{d\xi^{n}}\big{(}\chi^{(1)}_{\mu}(\xi)\sqrt{K_{\mu}(\xi)}\big{)}\Big{|}$
$\displaystyle\lesssim\langle\xi\rangle^{n}\sum_{k=0}^{n}\Big{|}\frac{d^{n-k}}{d\xi^{n-k}}\big{(}\chi^{(1)}(\sqrt{\mu}\xi)\big{)}\>\frac{d^{k}}{d\xi^{k}}\big{(}\sqrt{K_{\mu}(\xi)}\big{)}\Big{|}$
$\displaystyle\lesssim_{\beta}\langle\xi\rangle^{n}\sum_{k=0}^{n}\mu^{\frac{n-k}{2}}\big{|}(\frac{d^{n-k}}{d\xi^{n-k}}\chi^{(1)})(\sqrt{\mu}\xi)\big{|}\langle\xi\rangle^{-k}$
$\displaystyle\lesssim_{\beta}\sum_{k=0}^{n}\mu^{\frac{n-k}{2}}\langle\xi\rangle^{n-k}\big{|}(\frac{d^{n-k}}{d\xi^{n-k}}\chi^{(1)})(\sqrt{\mu}\xi)\big{|}\lesssim_{\beta}1.$

Hence, $(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})\in\mathcal{S}^{0}$ and
$\mathcal{N}^{0}(\chi^{(1)}_{\mu}\sqrt{\mathcal{K}_{\mu}})\lesssim_{\beta}1$
independently from $\mu$, proves (A.10).

Next, we consider estimates (2.27) and (2.28). Recalling (2.26) we define

$\displaystyle\tilde{\sigma}_{\mu,\frac{1}{2}}(\xi)=\mu^{-\frac{1}{4}}\sigma_{\mu,\frac{1}{2}}(\xi)=\frac{1}{\mu^{\frac{1}{4}}}\>\frac{1}{\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}}\>(1+\mu\beta\xi^{2})^{\frac{1}{2}}.$
(A.12)

Then, it suffices to prove that

$\sup\limits_{\xi\in\mathbb{R}}\>\langle\xi\rangle^{n-\frac{1}{2}}\Big{|}\frac{d^{n}}{d\xi^{n}}(\chi^{(2)}_{\mu}\tilde{\sigma}_{\mu,\frac{1}{2}})(\xi)\Big{|}\lesssim_{\beta}1,$
(A.13)

for all $n\in{\mathbb{N}}$ and any $\mu\in(0,1)$. Indeed, if we assume (A.13)
and take $n=0$ we deduce from Plancherel’s identity that

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

which proves (2.27). Moreover, (A.13) also implies that
$(\chi^{(2)}_{\mu}\tilde{\sigma}_{\mu,\frac{1}{2}})\in\mathcal{S}^{\frac{1}{2}}$
with
$\mathcal{N}^{\frac{1}{2}}(\chi^{(2)}_{\mu}\tilde{\sigma}_{\mu,\frac{1}{2}})\lesssim_{\beta}1$
so that

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

by Lemma A.3. Now we prove (A.13). First, we consider the functions,

$\displaystyle
a_{\mu}(\xi)=\frac{1}{\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}}\quad\text{and}\quad
b_{\mu}(\xi)=(1+\mu\beta\xi^{2})^{\frac{1}{2}}.$

Then, since $|\xi|>\sqrt{\mu}|\xi|>1$ on the support of $\chi^{(2)}_{\mu}$, we
observe that

$\displaystyle\chi^{(2)}_{\mu}(\xi)\Big{|}\frac{d^{k}}{d\xi^{k}}a_{\mu}(\xi)\Big{|}\lesssim\mu^{\frac{1}{4}}\chi^{(2)}_{\mu}(\xi)\frac{1}{\sqrt{\mu}|\xi|}\frac{1}{|\xi|^{k-\frac{1}{2}}}\lesssim\mu^{\frac{1}{4}}\langle\xi\rangle^{\frac{1}{2}-k}\langle\sqrt{\mu}\xi\rangle^{-1}.$
(A.14)

While $b_{\mu}(\xi)\lesssim_{\beta}\langle\sqrt{\mu}\xi\rangle$ and its
deriatives satisfy the bound,

$\displaystyle\chi^{(2)}_{\mu}(\xi)\Big{|}\frac{d^{k}}{d\xi^{k}}b_{\mu}(\xi)\Big{|}\lesssim_{\beta}\chi^{(2)}_{\mu}(\xi)\mu^{\frac{k}{2}}\langle\sqrt{\mu}\xi\rangle^{1-k}\lesssim_{\beta}\langle\sqrt{\mu}\xi\rangle\langle\xi\rangle^{-k}.$
(A.15)

Thus, if all derivatives falls on $\tilde{\sigma}_{\mu,\frac{1}{2}}$, the
Leibniz rule, (A.14) and (A.15) imply

$\displaystyle\chi^{(2)}_{\mu}(\xi)\Big{|}\frac{d^{k}}{d\xi^{k}}\tilde{\sigma}_{\mu,\frac{1}{2}}(\xi)\Big{|}\lesssim\mu^{-\frac{1}{4}}\chi^{(2)}_{\mu}(\xi)\sum_{j=0}^{k}\Big{|}\frac{d^{k-j}}{d\xi^{k-j}}a_{\mu}(\xi)\Big{|}\>\Big{|}\frac{d^{j}}{d\xi^{j}}b_{\mu}(\xi)\Big{|}\lesssim_{\beta}\langle\xi\rangle^{\frac{1}{2}-k}.$

On the other hand, when derivatives fall the cut-off function, we observe

$\mu^{\frac{k}{2}}\langle\xi\rangle^{k}\Big{|}(\frac{d^{k}}{d\xi^{k}}\chi^{(2)})(\sqrt{\mu}\xi)\Big{|}\lesssim
1,\quad k\geq 1,$ (A.16)

since the support of $\frac{d^{k}}{d\xi^{k}}\chi^{(2)}_{\mu}$ is contained in
the support of $\chi^{(1)}_{\mu}$. As a result, there holds

$\displaystyle\langle\xi\rangle^{n-\frac{1}{2}}\Big{|}\frac{d^{n}}{d\xi^{n}}\big{(}\chi^{(2)}_{\mu}(\xi)\tilde{\sigma}_{\mu,\frac{1}{2}}(\xi)\big{)}\Big{|}$
$\displaystyle\lesssim\langle\xi\rangle^{n-\frac{1}{2}}\sum_{k=0}^{n}\Big{|}\frac{d^{n-k}}{d\xi^{n-k}}\big{(}\chi^{(2)}(\sqrt{\mu}\xi)\big{)}\Big{|}\>\Big{|}\frac{d^{k}}{d\xi^{k}}\big{(}\tilde{\sigma}_{\mu,\frac{1}{2}}(\xi)\big{)}\Big{|}$
$\displaystyle\lesssim_{\beta}\langle\xi\rangle^{n-\frac{1}{2}}\sum_{k=0}^{n}\mu^{\frac{n-k}{2}}\big{|}(\frac{d^{n-k}}{d\xi^{n-k}}\chi^{(2)})(\sqrt{\mu}\xi)\big{|}\>\langle\xi\rangle^{\frac{1}{2}-k}$
$\displaystyle\lesssim_{\beta}\sum_{k=0}^{n}\mu^{\frac{n-k}{2}}\langle\xi\rangle^{n-k}\big{|}(\frac{d^{n-k}}{d\xi^{n-k}}\chi^{(2)})(\sqrt{\mu}\xi)\big{|}$
$\displaystyle\lesssim_{\beta}1.$

The estimate is uniform in $\mu\in(0,1)$, and (A.13) is proved, which provides
the desired result.

Lastly, we prove (2.30) and (2.31) arguing in the same vein. First, we write:

$\displaystyle\chi^{(2)}_{\mu}(\xi)\sigma_{\mu,0}(\xi)$
$\displaystyle=\chi^{(2)}_{\mu}(\xi)\cdot\>\frac{1}{\mu^{\frac{1}{4}}|\xi|^{\frac{1}{2}}}\>\cdot\Big{(}1+\mu\beta\xi^{2}\Big{)}^{\frac{1}{2}}\cdot\Big{(}\frac{2}{e^{2\sqrt{\mu}|\xi|}+1}\Big{)}^{\frac{1}{2}}$
$\displaystyle=:\chi^{(2)}_{\mu}(\xi)\>a_{\mu}(\xi)\>b_{\mu}(\xi)\>c_{\mu}(\xi),$

making use of the identity (A.4). Then we observe for all $N\in{\mathbb{N}}$
that

$\displaystyle\langle\xi\rangle^{k}\langle\sqrt{\mu}\xi\rangle^{N}\Big{|}\frac{d^{k}}{d\xi^{k}}c_{\mu}(\xi)\Big{|}$
$\displaystyle\lesssim\mu^{\frac{k}{2}}\langle\xi\rangle^{k}\langle\sqrt{\mu}\xi\rangle^{N}e^{-\sqrt{\mu}|\xi|}\lesssim
1.$ (A.17)

As a result, we deduce by (A.15) and (A.17) with $N=1$ that

$\displaystyle\chi^{(2)}_{\mu}(\xi)\Big{|}\frac{d^{k}}{d\xi^{k}}(b_{\mu}(\xi)\>c_{\mu}(\xi))\Big{|}$
$\displaystyle\lesssim\chi^{(2)}_{\mu}(\xi)\sum_{j=0}^{k}\big{|}\frac{d^{k-j}}{d\xi^{k-j}}\big{(}b_{\mu}(\xi)\big{)}\big{|}\>\big{|}\frac{d^{j}}{d\xi^{j}}\big{(}c_{\mu}(\xi)\big{)}\big{|}$
$\displaystyle\lesssim_{\beta}\sum_{j=0}^{k}\langle\xi\rangle^{-(k-j)}\langle\sqrt{\mu}\xi\rangle\>\langle\xi\rangle^{-j}\langle\sqrt{\mu}\xi\rangle^{-1}$
$\displaystyle\lesssim_{\beta}\langle\xi\rangle^{-k}.$

Moreover, we use (A.14) to deduce

$\displaystyle\chi^{(2)}_{\mu}(\xi)\Big{|}\frac{d^{k}}{d\xi^{k}}\sigma_{\mu,0}(\xi)\Big{|}\lesssim\chi^{(2)}_{\mu}(\xi)\sum_{j=0}^{k}\Big{|}\frac{d^{k-j}}{d\xi^{k-j}}\big{(}a_{\mu}(\xi)\big{)}\Big{|}\>\Big{|}\frac{d^{j}}{d\xi^{j}}\big{(}b_{\mu,\beta}(\xi)\>c_{\mu}(\xi)\big{)}\Big{|}\lesssim_{\beta}\langle\xi\rangle^{-k},$

from which we find

$\displaystyle\langle\xi\rangle^{n}\Big{|}\frac{d^{n}}{d\xi^{n}}\big{(}\chi^{(2)}_{\mu}(\xi)\sigma_{\mu,0}(\xi)\big{)}\Big{|}\lesssim\langle\xi\rangle^{n}\sum_{k=0}^{n}\Big{|}\frac{d^{n-k}}{d\xi^{n-k}}\big{(}\chi^{(2)}_{\mu}(\xi)\big{)}\Big{|}\>\Big{|}\frac{d^{k}}{d\xi^{k}}\big{(}\sigma_{\mu,0}(\xi)\big{)}\Big{|}\lesssim_{\beta}1,$

by (A.16). Arguing as above, we may conclude that the estimates (2.30) and
(2.31) holds. ∎

###### Proof of Lemma 2.12.

In order to prove (2.32), we simply verify that
$\sqrt{\mathcal{T}_{\mu}}J^{\frac{1}{2}}_{\mu}\in\mathcal{S}^{0}$ and
$\mathcal{N}^{0}(\sqrt{\mathcal{T}_{\mu}}J^{\frac{1}{2}}_{\mu})\lesssim 1$
uniformly in $\mu\in(0,1)$. But this is a direct consequence of Lemma A.1 and
the Leibniz rule:

$\displaystyle\langle\xi\rangle^{n}\Big{|}\frac{d^{n}}{d\xi^{n}}\sqrt{T_{\mu}(\xi)}J^{\frac{1}{2}}_{\mu}\Big{|}$
$\displaystyle\lesssim\langle\xi\rangle^{n}\sum_{k=0}^{n}\Big{|}\frac{d^{n-k}}{d\xi^{n-k}}\sqrt{T_{\mu}(\xi)}\Big{|}\>\Big{|}\frac{d^{k}}{d\xi^{k}}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}\Big{|}$
$\displaystyle\lesssim\langle\xi\rangle^{n}\sum_{k=0}^{n}\mu^{\frac{n-k}{2}}\langle\sqrt{\mu}\xi\rangle^{-\frac{1}{2}-(n-k)}\mu^{\frac{k}{2}}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}-k}$
$\displaystyle\lesssim\langle\xi\rangle^{n}\mu^{\frac{n}{2}}\langle\sqrt{\mu}\xi\rangle^{-n},$

and is bounded by a constant independent from $\mu\in(0,1)$. Hence, we may
conclude by Lemma A.3 that (2.32) holds true.

A similar approach is used for the proof of (2.33). Indeed, we observe that

$\displaystyle\langle\xi\rangle^{n-s}\Big{|}\frac{d^{n}}{d\xi^{n}}\sqrt{T_{\mu}(\xi)}J^{s}\Big{|}\lesssim\langle\xi\rangle^{n-s}\sum_{k=0}^{n}\mu^{\frac{n-k}{2}}\langle\sqrt{\mu}\xi\rangle^{-\frac{1}{2}-(n-k)}\langle\xi\rangle^{s-k}\lesssim
1.$

Hence, $\sqrt{\mathcal{T}_{\mu}}J^{s}\in\mathcal{S}^{s}$ and
$\mathcal{N}^{s}(\sqrt{\mathcal{T}_{\mu}}J^{s})\lesssim 1$ uniformly in
$\mu\in(0,1)$, allowing us to conclude by Lemma A.3.

The proof of (2.34) is the same, by a direct application of (A.1) we deduce
that $\sqrt{\mathcal{T}_{\mu}}\in\mathcal{S}^{0}$ uniformly in $\mu\in(0,1)$.

Next, we consider (2.35). Define the bilinear form:
$a_{1}(D)(f,g)=\partial_{x}[\sqrt{\mathcal{T}_{\mu}}(D),f]g$. Then we may use
Plancherel to write

$\displaystyle|\hat{a}_{1}(\xi)(f,g)|$
$\displaystyle\leq\int_{{\mathbb{R}}}|\xi|\Big{|}\sqrt{T_{\mu}(\xi)}-\sqrt{T_{\mu}(\rho)}\Big{|}|\hat{f}(\xi-\rho)|\>|\hat{g}(\rho)|\>d\rho.$

Clearly, if we can prove that

$b_{1}(\xi,\rho):=|\xi|\Big{|}\sqrt{T_{\mu}(\xi)}-\sqrt{T_{\mu}(\rho)}\Big{|}\lesssim
1+|\xi-\rho|,$ (A.18)

then we can conclude as we did for the proof of Lemma 2.7. Indeed, assuming
the claim (A.18), then there holds

$\displaystyle\|\partial_{x}[\sqrt{\mathcal{T}_{\mu}}(D),f]g\|_{L^{2}}=\|\hat{a}_{1}(\xi)(f,g)\|_{L^{2}}\lesssim(\|f\|_{H^{t_{0}}}+\|\partial_{x}f\|_{H^{t_{0}}})\|g\|_{L^{2}}.$

Now, in order to estimate $b_{1}(\xi,\rho)$ we consider three cases. First, if
$|\rho|\leq 1$ then we have by the triangle inequality,

$b_{1}(\xi,\rho)\leq(1+|\xi-\rho|)\Big{(}\sqrt{T_{\mu}(\xi)}+\sqrt{T_{\mu}(\rho)}\Big{)}\lesssim
1+|\xi-\rho|,$

since $\xi\mapsto\sqrt{T_{\mu}(\xi)}$ is bounded by one. Secondly, consider
the region where $|\rho|>1$ and $|\xi|\geq|\rho|$. Then since $\xi\mapsto
tanh(\sqrt{\mu}|\xi|)$ is increasing and $\xi\mapsto T_{\mu}(\xi)$ is
decreasing, we have that

$\frac{|\rho|}{|\xi|}\leq\bigg{(}\frac{|\rho|}{|\xi|}\bigg{)}^{\frac{1}{2}}\leq\bigg{(}\frac{T_{\mu}(\xi)}{T_{\mu}(\rho)}\bigg{)}^{\frac{1}{2}}\leq
1.$

Thus, there holds

$\displaystyle
b_{1}(\xi,\rho)=|\xi|\Big{(}1-\bigg{(}\frac{T_{\mu}(\xi)}{T_{\mu}(\rho)}\bigg{)}^{\frac{1}{2}}\Big{)}\sqrt{T_{\mu}(\rho)}\leq|\xi|-|\rho|\leq|\xi-\rho|.$

For $|\rho|>1$ and $|\xi|<|\rho|$ we use a similar argument to find,

$\displaystyle
b_{1}(\xi,\rho)=|\xi|\Big{(}1-\bigg{(}\frac{T_{\mu}(\rho)}{T_{\mu}(\xi)}\bigg{)}^{\frac{1}{2}}\Big{)}\sqrt{T_{\mu}(\xi)}\leq\frac{|\xi|}{|\rho|}(|\rho|-|\xi|)\leq|\xi-\rho|.$

Finally, we estimate (2.36) using a similar approach. We define the bilinear
form $a_{2}(D)(f,g)=[J_{\mu}^{\frac{1}{2}},f]\partial_{x}g$ and look in
frequency:

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

Then by the same argument as above, we only need to prove that

$b_{2}(\xi,\eta)=\Big{|}\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}-\langle\sqrt{\mu}\rho\rangle^{\frac{1}{2}}\Big{|}\frac{|\rho|}{\langle\sqrt{\mu}\rho\rangle^{\frac{1}{2}}}\lesssim
1+|\xi-\rho|.$ (A.19)

We consider three cases. If $|\rho|\leq 1$ then since $\mu\in(0,1)$, there
holds by the triangle inequality:

$\displaystyle b_{2}(\xi,\rho)\lesssim
1+\langle\xi-\rho\rangle^{\frac{1}{2}}.$

In the case $|\xi|\geq|\rho|>1$, observe that

$\frac{1+\mu\rho^{2}}{1+\mu\xi^{2}}\leq\frac{(1+\mu\rho^{2})^{\frac{1}{4}}}{(1+\mu\xi^{2})^{\frac{1}{4}}}\leq
1,$ (A.20)

and we have

$\frac{\xi^{2}-\rho^{2}}{|\xi-\rho|\>|\xi|}\leq\frac{|\xi|+|\rho|}{|\xi|}\lesssim
1.$ (A.21)

As a consequence, recalling $\mu\in(0,1)$, we have that

$\displaystyle b_{2}(\xi,\rho)$
$\displaystyle\leq\Big{(}1-\frac{1+\mu\rho^{2}}{1+\mu\xi^{2}}\Big{)}\frac{\langle\sqrt{\mu}\xi\rangle^{\frac{1}{2}}}{\langle\sqrt{\mu}\rho\rangle^{\frac{1}{2}}}|\rho|$
$\displaystyle\leq\frac{\mu(\xi^{2}-\rho^{2})}{\mu^{\frac{1}{4}}(1+\mu\xi^{2})^{\frac{3}{4}+\frac{1}{4}}}\frac{(1+\mu\xi^{2})^{\frac{1}{4}}}{\langle\rho\rangle^{\frac{1}{2}}}|\rho|$
$\displaystyle\leq\frac{\xi^{2}-\rho^{2}}{|\xi|}\frac{|\rho|}{|\xi|^{\frac{1}{2}}\langle\rho\rangle^{\frac{1}{2}}}$
$\displaystyle\lesssim|\xi-\rho|.$

Lastly, in the case $|\rho|>1$ and $|\xi|<|\rho|$, we can simply change the
role of $\xi$ and $\rho$ in (A.20) and (A.21). As result, we get

$\displaystyle b_{2}(\xi,\rho)$
$\displaystyle\leq\Big{(}1-\frac{1+\mu\xi^{2}}{1+\mu\rho^{2}}\Big{)}|\rho|\leq\frac{\rho^{2}-\xi^{2}}{|\rho|}\lesssim|\xi-\rho|.$

We may therefore conclude that (A.19) holds and the estimate (2.36) follows. ∎

## Acknowledgements

This research was supported by a Trond Mohn Foundation grant. I also thank my
advisor, Didier Pilod, for many long and helpful mathematical discussions,
Henrik Kalisch for providing references and Vincent Duchêne for some important
comments on the introduction.

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Oriol Nieto, Zeyu Jin, Franck Dernoncourt, Justin Salamon

# Efficient Spoken Language Recognition via Multilabel Classification

###### Abstract

Spoken language recognition (SLR) is the task of automatically identifying the
language present in a speech signal. Existing SLR models are either too
computationally expensive or too large to run effectively on devices with
limited resources. For real-world deployment, a model should also gracefully
handle unseen languages outside of the target language set, yet prior work has
focused on closed-set classification where all input languages are known
a-priori. In this paper we address these two limitations: we explore efficient
model architectures for SLR based on convolutional networks, and propose a
multilabel training strategy to handle non-target languages at inference time.
Using the VoxLingua107 dataset, we show that our models obtain competitive
results while being orders of magnitude smaller and faster than current state-
of-the-art methods, and that our multilabel strategy is more robust to unseen
non-target languages compared to multiclass classification.

Index Terms: spoken language recognition, efficient architectures, multilabel
classification

##  1 Introduction

Automatic speech recognition (ASR) techniques have achieved near human
accuracy for various languages [1, 2], and enable applications such as text-
based audio/video editing [3, 4], voice translation [5], and virtual
assistants [6]. When the language is not known in advance, Spoken Language
Recognition (SLR) is often a necessary first step before running ASR, as most
ASR systems require this information (the target language) to correctly
transcribe the speech signal. Past research in SLR has been organized around
challenges such as NIST Language Recognition Evaluations [7], focusing on
improving accuracy with large neural networks and extensive data. Recent
advances in large-scale and self-supervised models have achieved impressive
generalization across hundreds of languages with near-perfect accuracy [8, 9,
10].

Despite the significant progress, there are two important limitations that
make real-world deployment of such models challenging in resource-constrained
scenarios: (1) models are too large or resource intensive for devices with
limited compute power, (2) they operate on a closed set of target languages,
i.e., they are not designed to handle the scenario where an unknown language
is presented to the model at inference time. Most modern SLR systems are based
on deep architectures, all of which contain millions if not tens or hundreds
of millions of parameters. One of the smallest models is based on X-vectors
[11], which were initially introduced for the task of speaker recognition
[12]. Such vectors are computed using blocks of Time-Delayed Neural Networks
(TDNNs) [13] and an aggregation layer, allowing for a large receptive field
with small kernels, yielding good performance with a reduced amount of
parameters. A larger and more powerful version of this model was recently
introduced for the same task of speaker recognition, and it uses higher
capacity TDNNs with an emphasized channel attention, propagation, and
aggregation (ECAPA) set of mechanisms [14]. Such a model was later used
successfully for SLR [15]. ECAPA-TDNNs have around four times the number of
parameters as the original X-vector model. More recent and larger systems are
capable of jointly recognizing a given language and solving additional tasks
in a single pass. For example, Whisper [8] can perform multi-lingual speech to
text transcription, having an implicit SLR system embedded in the model. This
architecture is an encoder-decoder Transformer [16] trained on 680k hours of
speech. The smallest version of this model has almost 2x the parameters of
ECAPA-TDNN, and its medium version is over an order of magnitude larger than
its smallest counterpart. Finally, XLS-R [9] is another self-supervised model
whose smallest version has around 300 million parameters. It is based on
Wav2Vec 2.0 [17] and is pre-trained with over half a million hours of speech
covering 128 languages. When fine-tuned on the SLR task, it achieves state-of-
the-art results on most test datasets. While self-supervised models lead the
charts in terms of accuracy, they require significant compute power to
operate.

There is a growing need for robust SLR models that can run efficiently
embedded on-device. SLR on-device eliminates computation and networking costs
involved with running SLR in the cloud, and can prevent tracking and other
potential threats to user privacy. However, large-scale models such as those
discussed above are impractical to use on, e.g., a mobile device, due to size
and runtime constraints. There are use cases where a user only requires on-
device SLR for a limited number of languages (e.g., the languages they speak),
not hundreds, providing an opportunity to trade-off the language set size for
model efficiency. Furthermore, the models in the studies discussed above are
incapable of detecting unsupported languages: they assume the input is always
one of the target languages, and would incorrectly classify a non-target
language as one of the target languages the model was trained on. For real-
world deployment, a model should be able to identify this scenario and ``fail
gracefully'' by classifying the input as ``unknown'' or ``other''.

In this paper we address these two aforementioned limitations. We investigate
small-footprint models for SLR that can run effectively on-device. We propose
novel variants of widely-used architectures, including TC-ResNets [18] and
ECAPA-TDNNs [13], and compare them to top-performing large models using
VoxLingua107 [19], a sizeable speech dataset recently introduced for SLR
research [9]. We show that our proposed lightweight architectures achieve
competitive error rates with models that are two orders of magnitude larger in
terms of parameters. To handle non-target (unseen) languages, we propose a
multilabel classification approach that is novel in the context of SLR, and
show that it produces models that are more robust compared to modeling all
non-target languages via a single ``Other'' class in the commonly used
multiclass setup. To the best of our knowledge, this is the first work to
study SLR from an efficiency viewpoint and address the problem of non-target
languages at inference time.

##  2 Models for Efficient SLR

We explore two families of architectures for efficient SLR: Temporal
Convolution ResNets (TC-ResNets) [18], and ECAPA-TDNN [15]. For the latter, we
propose a modification that makes it significantly lighter, which we call
LECAPAT. For all the proposed models, our input is a log-melspectrogram with
64 mel-frequency bins computed from an input audio signal sampled at 16 kHz.
The mel-spectrogram is computed using a 25 ms Hann window, FFT size of 64 ms,
and a 10 ms hop size.

###  2.1 Temporal Convolution Residual Networks

(a) TC-ResNet10

(b) ResNet Block

Figure 1: Proposed TC-ResNet10 architecture, where $T$ is time frames, $c$
number of channels, $s$ stride, $L$ number of languages, and $i$ and $o$ input
and output to the ResNet Block, respectively.

Temporal convolution networks apply a 2D convolution to the input spectrogram
with a kernel whose height matches the number of frequency bins, compacting
the frequency dimension into a set of 1D time representations (one per
channel). Then 1D convolutions can be applied in the subsequent layers of the
architecture, making it highly efficient in both number of parameters and
inference time. These networks were employed for early approaches to music
recommendation using deep learning [20], and more recently added residual
connections [21] for efficient keyword spotting [18].

We propose two different flavors of TC-ResNets for efficient SLR: TC-ResNet10
is a slightly larger version of the model introduced in [18], but with fewer
layers (10 as opposed to 14), depicted in Figure 1. The two ResNet blocks are
repeated (i.e., four ResNet blocks in total), where their size changes based
on the repetition number $\lambda\in\\{0,1\\}$. This model has 200k
parameters. The second model is TC-ResNet14, introduced by Choi et al. [18],
with 100k parameters.

###  2.2 LECAPAT: Light ECAPA Time-Delayed NNs

(a) LECAPAT

(b) SEResNet Block

Figure 2: Proposed LECAPAT architecture. $d$ is dilation size, BN is Batch
Normalization [22], and _scale_ is the scale for the Res2Net model [23]. All
convolution strides are set to $s=1$.

TDNNs [13] were introduced to efficiently model long temporal contexts. They
use dilated convolutions to increase the receptive field, and are thus capable
of modeling long-term structures without adding extra complexity in terms of
computation or model size. ECAPA-TDNNs [14] added the following set of
mechanisms: (i) per-channel attention, (ii) squeeze-excitation (SE) residual
blocks that contain an SE mechanism [24] and a Res2Net model [23], and (iii)
an aggregation technique that attends to several levels of feature statistics
across the architecture. While more computationally expensive, ECAPA-TDNNs
achieve better performance on speaker recognition, and have been shown to be
highly effective for SLR [15].

We propose a more efficient and lightweight version of ECAPA-TDNN, LECAPAT
(for Light ECAPA-TDNN). Our main intuition is that, while the original model
was designed to recognize thousands or millions of speakers, SLR typically
involves recognizing dozens or at most hundreds of languages, suggesting a
model with reduced capacity may still perform well on this task. We reduce the
complexity of ECAPA-TDNN making it two orders of magnitude smaller: from 21
million parameters down to 600k. LECAPAT employs a single SE residual block
instead of three, and we reduce the number of parameters in each of the main
blocks, as depicted in Figure 2.

###  2.3 Multilabel Classification

ASR applications typically only support a subset of languages. Thus, an SLR
model should recognize when a non-target language, i.e., a language that is
not one of the languages the model is trained to recognize, is provided as
input, to avoid sending it to be transcribed by ASR. A common approach in
multiclass tasks for handling non-target classes is to add a single ``Other''
class. Our early experiments revealed that this approach, i.e., adding a
single class for all non-target languages, resulted in poor performance for
SLR. The ``Other'' class must capture many different languages, including some
that may be similar to a target language. Our conjecture is that the model
struggles to simultaneously group all non-target languages _and_ separate them
from the target languages. For example, in our experiments Spanish and German
are target languages, while Catalan and Yiddish are not. The multiclass setup
has to group Catalan and Yiddish and at the same time separate Catalan from
the relatively close Spanish and Yiddish from the relatively close German, a
challenging proposition.

Instead, we propose to train a _multilabel_ model for SLR, a novel approach in
this context. Unlike the multiclass setup which is forced to always return a
single positive class, multilabel models allow for (1) several classes to be
positive at the same time and (2) no positive classes for a given input. Thus,
the model can focus on representing the target languages and separating them
from non-target languages, without explicitly modeling all possible non-target
languages. Implementation-wise, we swap the final softmax layer in our models
with sigmoids, such that zero positives are allowed. We return the language
with the highest output activation as the model's prediction, unless all
activations are below a threshold (0.5) in which case the ``Other'' class is
detected. Note that this adds no extra complexity to the models during
inference.

##  3 Experiments

###  3.1 Data

We use the VoxLingua107 dataset [19] for all of our experiments. The dataset
consists of a training set and a ``development'' set for model evaluation. The
training set comprises 6,628 hours of speech from 107 languages, with an
average of 62 hours per language, though it is heavily imbalanced. The
development set contains 1,609 manually validated speech segments from 33
languages.

Our focus is on real-world scenarios where the target language set is limited,
as is often the case for speech applications, for example due to localization
constraints. This common scenario is one for which efficient models are
particularly promising as, we hypothesize, the reduced language set suggests
we should be able to train performant models with significantly lower capacity
compared to models targeting hundreds of languages. To this end, we construct
a new test set taking a subset of $L=11$ languages from the VoxLingua107
development (evaluation) set, which are the languages for which speech-to-text
(ASR) is supported in a commercial video editing tool. We call this test set
VoxLingua11, with the distribution of seconds per language depicted in Figure
3. We maintain the class imbalance for these languages from the VoxLingua107
development set, to make VoxLingua11 easy to reproduce.

We use VoxLingua11 to evaluate performance of our models and baselines under
the multiclass setup where only target languages are included in the test set.
Since our final goal is to evaluate our models when non-target languages are
present in the test set, we also create VoxLingua11+O, which expands
VoxLingua11 with an ``Other'' class containing 128 samples with at least one
sample from each of the non-target languages in the VoxLingua107 development
set. VoxLingua11+O can be used to evaluate models under both the multiclass
and multilabel setups, where in the former models have 12 output neurons (11
target languages + ``Other''), and in the latter they have 11 output neurons
and ``Other'' is predicted as described in Section 2.3.

To make the models robust to real-world recordings, we augment the training
set with noise, reverb, and random equalization following the work of Su et
al. [25]. Since the training set is imbalanced, we use the augmentation to
balance our training data, such that each training epoch contains a balanced
sampling of the target languages. The ``Other'' class is treated as a 12th
language by randomly sampling across the non-target languages during training.
All the models evaluated in this work take log-melspectrograms of 10-second
audio clips as input. If the input recording is shorter than 10 s, we center
it and zero pad to the left and right to obtain a 10 s clip. If it is longer,
we take a random 10 s subclip during training. At test time, for longer
recordings we slide a 10 s window over the input with a 5 s hop size and
average the predictions over time.

Figure 3: Seconds per language in the VoxLingua11 test set.

###  3.2 Baselines

We use the SLR models reviewed in Section 1 as baselines. The open source
implementations of the X-Vector111https://github.com/KrishnaDN/x-vector-
pytorch and ECAPA-TDNN [26] are trained from scratch on the VoxLingua107
training set. We verify that we obtain similar results to those reported in
the original publications. For the larger baselines, XLS-R and Whisper, we
rely on their published weights, as these higher capacity models were trained
on a diverse conglomerate of datasets–including VoxLingua107–that make up much
larger amounts of data. We use the 300M parameter (i.e., smallest) version of
XLS-R fine-tuned on
VoxLingua107222https://huggingface.co/TalTechNLP/voxlingua107-xls-r-300m-wav2vec
and the ``Medium" and ``Tiny" versions of Whisper [8].

###  3.3 Metrics and Optimization

We report the models' error rate $err$, the standard for SLR evaluation, where
$err=100(1-acc)$ and $acc$ is the multiclass classification accuracy. We
report the inference runtime of each model on VoxLingua11 using either a CPU
(Intel Cascade Lake on a g4dn.4xlarge G4 AWS instance) or a V100 GPU as a
real-time factor (rtf) coefficient, i.e., the total test set audio duration
divided by the total model inference time (larger rtf = faster).

Our proposed models are trained on the augmented VoxLingua107 training set
using a V100 GPU, the Adam optimizer ($\beta_{1}{=}0.9$ and
$\beta_{2}{=}0.999$), early stopping, and hyperparameter Bayesian optimization
to determine learning rates and batch sizes, which fluctuate between
$[10^{-3},10^{-5}]$ and $[32,128]$, respectively. We minimize the categorical
and binary cross-entropy loss for multiclass and multilabel models,
respectively.

##  4 Results

###  4.1 Accuracy versus size and runtime

Figure 4:  Model size versus error rate, multiclass classification on
VoxLingua11. Proposed models in bold. * indicates original weights used, all
other models were trained from scratch.

We start by examining the trade-off between model capacity and error rate,
depicted in Figure 4, with our proposed models in bold. Results are shown for
multiclass classification on VoxLingua11 so that we can compare to all
baselines. The models trained from scratch only see the 11 selected languages
from VoxLingua107 during training. The error rates are also reported in Table
1. Note the log scale of the parameters (y) axis. Expectedly, we see a
correlation between model size and error rate in both the baselines and our
proposed models. For the baselines, we see Whisper (Tiny) and ECAPA-TDNN
provide an order-of-magnitude (OOM) reduction in model size compared to
Whisper (Medium) and XLS-R (20-40M vs 300-800M) with a relatively low drop in
error rate of 4-5 points. X-Vector is a further OOM smaller but its error rate
is significant.

Most interestingly, we see that our proposed LECAPAT model performs very
closely to the Whisper (Tiny) and ECAPA-TDNN baselines with an error rate of
7.5, but is a whole two OOM smaller (and four OOM smaller than the largest
baselines), with just 0.6M parameters. The TC-ResNet models, though smaller
still, are notably less accurate, thus providing a less desirable tradeoff.
LECAPAT (and the ResNets) are also significantly faster to run inference with,
as shown in Table 1, running over 800 times faster than real-time on GPU,
which is 12 times faster than Whisper (Tiny) and 1.5 times faster than ECAPA-
TDNN. It is interesting to note that despite their similar size and accuracy,
the latter is around 8 times faster than the former, which shows that model
size is not a good proxy for model runtime and highlights the importance of
reporting runtime metrics when considering models for deployment. It is also
noteworthy that LECAPAT remains fast on CPU, whereas ECAPA-TDNN becomes 6
times slower. This makes our proposed model a strong candidate for deployment
scenarios where a GPU may not be available. While our best proposed model is
outperformed by the strongest baselines in terms of accuracy, it trades off 6
accuracy points for a dramatic four OOM reduction in size and almost two OOM
increase in speed.

Model |  $err$ $\downarrow$ |  rtf (CPU) $\uparrow$ |  rtf (GPU) $\uparrow$
---|---|---|---
XLS-R (Smallest) | 0.7 | 11.02 | 13.03
Whisper (Medium) | 1.0 | 3.43 | 28.31
Whisper (Tiny) | 5.3 | 39.54 | 65.56
ECAPA-TDNN | 6.7 | 83.95 | 522.40
X-Vector | 19.5 | 524.2 | 656.54
LECAPAT (Ours) | 7.5 | 678.42 | 810.93
TC-ResNet10 (Ours) | 12.8 | 966.91 | 821.56
TC-ResNet14 (Ours) | 21.9 | 932.12 | 824.26
Table 1: Multiclass error rate and rtf on VoxLingua11.

###  4.2 Dealing with non-target languages

Figure 5: Normalized confusion matrices for LECAPAT on VoxLingua11+O for multiclass (left) and multilabel (right). Model | Multiclass | Multilabel
---|---|---
LECAPAT | 22.56 | 14.85
TC-ResNet10 | 29.02 | 23.93
TC-ResNet14 | 32.07 | 31.01
Table 2: Error rates for our proposed models using the multiclass and
multilabel training strategies on VoxLingua11+O.

Finally, we examine how our proposed multilabel training strategy for handling
non-target languages at inference time (cf. Section 2.3) compares to a
multiclass setup which explicitly models them via an additional ``Other''
class. For this experiment, we train both multiclass and multilabel models
with the full VoxLingua107 training set. The results for our proposed models
on VoxLingua11+O using both strategies are reported in Table 2. We note the
error rates are generally higher than those for VoxLingua11. This is expected,
as the classification problem is more challenging now given the presence of
the additional, highly heterogeneous, class ``Other''. It also shows that
studies which only evaluate SLR against a known set of target languages are
most likely not representative of how models would perform under deployment
conditions where non-target languages may be presented to the model. We see
that our multilabel strategy significantly outperforms the multiclass setup
for all three models, with the improvement being most notable for our top-
performing LECAPAT model. We hypothesize that the multiclass model struggles
to learn an embedding space that can model a large number of non-target
languages as a single ``Other'' class while keeping it disjoint from the
target languages. To explore this, we plot the normalized confusion matrices
for LECAPAT trained with each of the two strategies in Figure 5. We see the
multiclass model (left) is able to better classify non-target languages as
``Other,'' but suffers from notable leakage of target classes into the
``Other'' class. The multilabel model (right) alleviates this considerably,
e.g., improving the accuracy for Spanish, Italian, Russian, and Norwegian by
over 10 percentage points each.

##  5 Conclusion

In this paper we addressed the problem of efficient Spoken Language
Recognition (SLR) in the presence of non-target languages. To the best of our
knowledge this is the first study on efficient SLR, and the first to propose
multilabel training to handle non-target languages at inference time. Our
experiments show that our top-performing proposed model, LECAPAT, performs
almost on par with models that are two OOM larger, and only 6 accuracy points
lower than the largest models while being four OOM smaller and almost two OOM
faster. Additionally, we show that our proposed multilabel training strategy
outperforms the multiclass setting by a considerable margin when non-target
languages are present at inference time. We hope this study will stimulate
further work on the topic of efficient SLR under real-world conditions.

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