v\Big{|}\leqslant\varepsilon(\lVert\eta\rVert_{\mathcal{H}^{0}}^{2}+\sigma^{2}\lVert\eta\rVert^{2}_{\mathcal{H}^{1}})+\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\eta\;h\Big{|}.$
(4.1.17)

On the other hand, thanks to Corollary A.5, we have that

$\Big{|}\int_{\mathbb{R}^{d}}[(\vartheta-\sigma^{2}\theta\Delta),\nabla]\eta\cdot
v\Big{|}=\Big{|}\int_{\mathbb{R}^{d}}\nabla\vartheta\eta\cdot
v-\sigma^{2}\nabla\theta\Delta\eta\cdot
v\Big{|}\lesssim\varepsilon(\lVert\eta\rVert_{\mathcal{H}^{0}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{2}})\lVert
v\rVert_{L^{2}}.$ (4.1.18)

Synthesizing these with (4.1.14) then yields (4.1.12).

Next, we turn to the proof of (4.1.13). We will prove this initially under the
stronger assumption that $v\in H^{3}(\mathbb{R}^{d};\mathbb{R}^{d})$ and
$\eta\in\mathcal{H}^{4}(\mathbb{R}^{d})$, and then we will conclude the proof
by showing how to reduce the general case to this special case.

We dot the second equation in (4.1.1) with $-\mu^{2}\Delta v$ and integrate to
see that

$-\int_{\mathbb{R}^{d}}(v-\mu^{2}\nabla\cdot\mathbb{S}v)\cdot\mu^{2}\Delta
v=\mu^{2}\int_{\mathbb{R}^{d}}-f\cdot\Delta v+(X\cdot\nabla)v\cdot\Delta
v+(\vartheta-\sigma^{2}\theta\Delta)\nabla\eta\cdot\Delta v.$ (4.1.19)

We will deal with the left and right sides separately. Integrating by parts
and using Korn’s inequality grants the coercive estimate for the left side:

$\mu^{2}\lVert\nabla v\rVert_{L^{2}}^{2}+\mu^{4}\lVert\nabla
v\rVert_{H^{1}}^{2}\lesssim-\int_{\mathbb{R}^{d}}(v-\mu^{2}\nabla\cdot\mathbb{S}v)\cdot\mu^{2}\Delta
v.$ (4.1.20)

Now we estimate the right side of (4.1.19). The $f$ term is easily handled
with Cauchy-Schwarz and an absorbing argument since $\lVert\Delta
v\rVert_{L^{2}}\lesssim\lVert\nabla v\rVert_{H^{1}}$. The $X\cdot\nabla v$
term controlled by integrating by parts:

$\Big{|}\mu^{2}\int_{\mathbb{R}^{d}}(X\cdot\nabla)v\cdot\Delta
v\Big{|}=\Big{|}-\mu^{2}\int_{\mathbb{R}^{d}}\frac{\nabla\cdot X}{2}|\nabla
v|^{2}+\mu^{2}\sum_{j=1}^{d}\int_{\mathbb{R}^{d}}(\partial_{j}X\cdot\nabla)v\cdot\partial_{j}v\Big{|}\lesssim\varepsilon\mu^{2}\lVert\nabla
v\rVert^{2}_{L^{2}}.$ (4.1.21)

The $\eta$-term in (4.1.19) is more involved. We have that

$-\mu^{2}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\nabla\eta\cdot\Delta
v=\mu^{2}\sum_{j=1}^{d}\int_{\mathbb{R}^{d}}(\partial_{j}\vartheta-\sigma^{2}\partial_{j}\theta\Delta)\nabla\eta\cdot\partial_{j}v\\\
-\mu^{2}\sum_{j=1}^{d}\int_{\mathbb{R}^{d}}\nabla\vartheta\partial_{j}\eta\cdot\partial_{j}v-\sigma^{2}\nabla\theta\Delta\partial_{j}\eta\cdot\partial_{j}v-\mu^{2}\sum_{j=1}^{d}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\partial_{j}\eta\;\nabla\cdot\partial_{j}v.$
(4.1.22)

The first two terms on the right here are straightforward to estimate:

$\mu^{2}\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\partial_{j}\vartheta-\sigma^{2}\partial_{j}\theta\Delta)\nabla\eta\cdot\partial_{j}v+\int_{\mathbb{R}^{d}}(\nabla\vartheta\partial_{j}\eta-\sigma^{2}\nabla\theta\Delta\eta)\cdot\partial_{j}v\Big{|}\lesssim\varepsilon|\mu|(\lVert\eta\rVert_{\mathcal{H}^{1}}+\sigma^{4}\lVert\eta\rVert_{\mathcal{H}^{3}})\cdot|\mu|\lVert\nabla
v\rVert_{L^{2}}.$ (4.1.23)

For the third term on the right side of (4.1.22), we use the identity
$\nabla\cdot\partial_{j}v+\nabla\cdot(X\partial_{j}\eta)=\partial_{j}h-\nabla\cdot(\partial_{j}X\eta)$
and appeal to Lemma 4.2 to bound

$\mu^{2}\Big{|}\sum_{j=1}^{d}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\partial_{j}\eta\;\nabla\cdot\partial_{j}v\Big{|}\lesssim\mu^{2}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\nabla\eta\cdot\nabla
h\Big{|}+\varepsilon\mu^{2}(\lVert\eta\rVert_{\mathcal{H}^{1}}^{2}+\sigma^{4}\lVert\eta\rVert^{2}_{\mathcal{H}^{3}}).$
(4.1.24)

Upon synthesizing these estimates for the various terms in (4.1.19), we arrive
at the bound

$\mu^{2}\lVert\nabla v\rVert_{L^{2}}^{2}+\mu^{4}\lVert\nabla
v\rVert_{H^{1}}^{2}\lesssim\lVert
f\rVert_{L^{2}}^{2}+\varepsilon\mu^{2}\lVert\nabla
v\rVert_{L^{2}}^{2}+\varepsilon|\mu|(\lVert\eta\rVert_{\mathcal{H}^{1}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{3}})\cdot|\mu|\lVert\nabla
v\rVert_{L^{2}}\\\
+\mu^{2}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\nabla\eta\cdot\nabla
h\Big{|}+\varepsilon\mu^{2}(\lVert\eta\rVert^{2}_{\mathcal{H}^{1}}+\sigma^{4}\lVert\eta\rVert_{\mathcal{H}^{3}}^{2}).$
(4.1.25)

Using an absorbing argument and taking $\rho_{\mathrm{AP0}}$ small enough, we
deduce from this that (4.1.13) holds in this special case of additional
smoothness assumptions.

To conclude, we now explain how to reduce the general case of (4.1.13) to the
special case. Suppose, then, that $v\in H^{2}(\mathbb{R}^{d};\mathbb{R}^{d})$
and $\eta\in\mathcal{H}^{3}(\mathbb{R}^{d})$ with
$0\not\in\mathrm{supp}\mathscr{F}[\eta]$. Given any $\delta\in\mathbb{R}^{+}$
there exists $v_{\delta}\in H^{3}(\mathbb{R}^{d};\mathbb{R}^{d})$ and
$\eta_{\delta}\in\mathcal{H}^{4}(\mathbb{R}^{d})$ with
$\mathrm{dist}(0,\mathrm{supp}\mathscr{F}[\eta_{\delta}])=\mathrm{dist}(0,\mathrm{supp}\mathscr{F}[\eta])$
and $\lVert v-v_{\delta}\rVert_{H^{2}}<\delta$,
$\lVert\eta-\eta_{\delta}\rVert_{\mathcal{H}^{3}}<\delta$. The equations
satisfied by $v_{\delta}$ and $\eta_{\delta}$ are (4.1.1) with right hand side
data

$h_{\delta}=h-\nabla\cdot(v-v_{\delta})-\nabla\cdot(X(\eta-\eta_{\delta}))$
(4.1.26)

and

$f_{\delta}=f-(1+X\cdot\nabla-\mu^{2}\nabla\cdot\mathbb{S})(v-v_{\delta})-(\vartheta-\sigma^{2}\theta\Delta)\nabla(\eta-\eta_{\delta}).$
(4.1.27)

We invoke the a priori estimate from the special case to deduce that

$|\mu|\lVert\nabla v_{\delta}\rVert_{L^{2}}^{2}+\mu^{2}\lVert\nabla
v_{\delta}\rVert_{H^{1}}\lesssim\lVert
f_{\delta}\rVert_{L^{2}}+\varepsilon^{1/2}(\lVert\eta_{\delta}\rVert_{\mathcal{H}^{0}}+\sigma^{2}\lVert\eta_{\delta}\rVert_{\mathcal{H}^{2}})\\\
+|\mu|\varepsilon^{1/2}(\lVert\eta_{\delta}\rVert_{\mathcal{H}^{1}}+\sigma^{2}\lVert\eta_{\delta}\rVert_{\mathcal{H}^{3}})+|\mu|\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\nabla\eta_{\delta}\cdot\nabla
h_{\delta}\Big{|}^{1/2}.$ (4.1.28)

We now send $\delta\to 0$ and observe that $f_{\delta}\to f$ in
$L^{2}(\mathbb{R}^{d};\mathbb{R}^{d})$, $h_{\delta}\to h$ in
$(\dot{H}^{-1}\cap H^{1})(\mathbb{R}^{d})$, $v_{\delta}\to v$ in
$H^{2}(\mathbb{R}^{d};\mathbb{R}^{d})$, and $\eta_{\delta}\to\eta$ in
$\mathcal{H}^{3}(\mathbb{R}^{d})$ (actually in $H^{3}(\mathbb{R}^{d})$). By
passing to the limit in (4.1.28), we thus acquire (4.1.13) in the general
case, which completes the proof. ∎

Our next result obtains closed (when $h=0$) a priori estimates at a base case
level of derivatives.

######  Theorem 4.4 (A priori estimates, II).

Suppose that $f\in H^{1}(\mathbb{R}^{d};\mathbb{R}^{d})$,
$h\in(\dot{H}^{-1}\cap H^{2})(\mathbb{R}^{d})$, $v\in
H^{3}(\mathbb{R}^{d};\mathbb{R}^{d})$, and
$\eta\in\mathcal{H}^{4}(\mathbb{R}^{d})$ with
$0\not\in\mathrm{supp}\mathscr{F}[\eta]$ solve system (4.1.1). Let
$\rho_{\mathrm{AP0}}\in\mathbb{R}^{+}$ be as in Proposition 4.3. Then there
exists $\mathbb{R}^{+}\ni\rho_{\mathrm{AP1}}\leqslant\rho_{\mathrm{AP0}}$ with
the property that for all
$\mathbb{R}^{+}\ni\varepsilon\leqslant\rho_{\mathrm{AP1}}$ and
$(v_{0},\eta_{0},\beta_{0})$ satisfying (4.1.3), we have the estimate

${{}_{\mu}}\lVert
v\rVert_{H^{1}}+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{1}}\lesssim\lVert
f\rVert_{H^{1}}+\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\eta\;h\Big{|}^{1/2}\\\
+\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\partial_{j}\eta\;\partial_{j}h\Big{|}^{1/2}+|\mu|\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\Delta)\nabla\partial_{j}\eta\cdot\nabla\partial_{j}h\Big{|}^{1/2}+|\sigma|\lVert\nabla
h\rVert_{L^{2}}.$ (4.1.29)

The implicit constant in (4.1.29) depends only on $\mathcal{Q}$ and $d$ from
Definition 4.1.

###### Proof.

For $j\in\\{1,\dotsc,d\\}$ we apply $\partial_{j}$ to (4.1.1) to obtain the
system

$\begin{cases}\nabla\cdot\partial_{j}v+\nabla\cdot(X\partial_{j}\eta)=\partial_{j}h-\nabla\cdot(\partial_{j}X\eta),\\\
\partial_{j}v+X\cdot\nabla\partial_{j}v-\mu^{2}\nabla\cdot\mathbb{S}\partial_{j}v+(\vartheta-\sigma^{2}\theta\Delta)\nabla\partial_{j}\eta=\partial_{j}f-\partial_{j}X\cdot\nabla
v-(\partial_{j}\vartheta-\sigma^{2}\partial_{j}\theta\Delta)\nabla\eta.\end{cases}$
(4.1.30)

To each of these we apply the a priori estimates of Proposition 4.3 and then
sum over $j$; we then sum this with the estimate for the non-differentiated
problem provided by the same proposition. This results in the bound

$\lVert v\rVert_{H^{1}}+\mu^{2}\lVert v\rVert_{H^{3}}\lesssim\lVert
f\rVert_{H^{1}}+\varepsilon^{1/2}(\lVert
v\rVert_{H^{1}}+\lVert\eta\rVert_{\mathcal{H}^{1}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{3}}+|\mu|(\lVert\eta\rVert_{\mathcal{H}^{2}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{4}}))\\\
+\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\partial_{j}\eta\;\nabla\cdot(\partial_{j}X\eta)\Big{|}^{1/2}+|\mu|\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\Delta)\nabla\partial_{j}\eta\cdot\nabla\nabla\cdot(\partial_{j}X\eta)\Big{|}^{1/2}+R_{\eta,h},$
(4.1.31)

where

$R_{\eta,h}=\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\eta\;h\Big{|}^{1/2}+\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\partial_{j}\eta\;\partial_{j}h\Big{|}^{1/2}+|\mu|\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\Delta)\nabla\partial_{j}\eta\cdot\nabla\partial_{j}h\Big{|}^{1/2}.$
(4.1.32)

It is straightforward to estimate

$\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\partial_{j}\eta\;\nabla\cdot(\partial_{j}X\eta)\Big{|}^{1/2}+|\mu|\sum_{j=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\Delta)\nabla\partial_{j}\eta\cdot\nabla\nabla\cdot(\partial_{j}X\eta)\Big{|}^{1/2}\\\
\lesssim\varepsilon^{1/2}(\lVert\eta\rVert_{\mathcal{H}^{1}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{3}})+\varepsilon^{1/2}|\mu|(\lVert\eta\rVert_{\mathcal{H}^{2}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{4}}).$
(4.1.33)

Thus, if $\rho_{\mathrm{AP1}}$ is small enough, this and (4.1.31) yield the
bound

$\lVert v\rVert_{H^{1}}+\mu^{2}\lVert v\rVert_{H^{3}}\lesssim\lVert
f\rVert_{H^{1}}+\varepsilon^{1/2}(\lVert\eta\rVert_{\mathcal{H}^{1}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{3}}+|\mu|(\lVert\eta\rVert_{\mathcal{H}^{2}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{4}}))+R_{\eta,h}.$
(4.1.34)

We now turn our attention to higher regularity estimates for $\eta$. We know
that $\nabla\eta$ satisfies the equation

$(\vartheta-\sigma^{2}\theta\Delta)\nabla\eta=f-v-X\cdot\nabla
v+\mu^{2}\nabla\cdot\mathbb{S}v,$ (4.1.35)

and, decreasing $\rho_{\mathrm{AP1}}$ if necessary, we may assume that the
operator $\vartheta-\sigma^{2}\theta\Delta$ is uniformly elliptic. Sobolev
interpolation allows us to bound

$\lVert X\cdot\nabla v\rVert_{L^{2}}+|\mu|\lVert X\cdot\nabla
v\rVert_{H^{1}}\lesssim\lVert v\rVert_{H^{1}}+|\mu|\lVert
v\rVert_{H^{2}}\lesssim\lVert v\rVert_{H^{1}}+|\mu|\lVert
v\rVert_{H^{1}}^{1/2}\lVert v\rVert_{H^{3}}^{1/2}\lesssim\lVert
v\rVert_{H^{1}}+\mu^{2}\left\lVert v\right\rVert_{H^{3}},$ (4.1.36)

and hence we may employ standard elliptic regularity on (4.1.35) to obtain the
bound

$\lVert\nabla\eta\rVert_{L^{2}}+\sigma^{2}\lVert\nabla\eta\rVert_{H^{2}}+|\mu|(\lVert\nabla\eta\rVert_{H^{1}}+\sigma^{2}\lVert\nabla\eta\rVert_{H^{3}})\lesssim\lVert
f\rVert_{H^{1}}+\lVert v\rVert_{H^{1}}+\mu^{2}\lVert v\rVert_{H^{3}}.$
(4.1.37)

On the other hand, from the identity $\nabla\cdot v+\nabla\cdot(X\eta)=0$, we
find that

$|\gamma|\lVert\mathcal{R}_{1}\eta\rVert_{L^{2}}\lesssim\lVert
v\rVert_{L^{2}}+\varepsilon\lVert\eta\rVert_{\mathcal{H}^{0}}.$ (4.1.38)

Combining these two estimates with the norm equivalence in the third item of
Proposition A.1 and the estimate (4.1.34) then shows that

$\lVert\eta\rVert_{\mathcal{H}^{1}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{3}}+|\mu|(\lVert\eta\rVert_{\mathcal{H}^{2}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{4}})\lesssim\varepsilon\lVert\eta\rVert_{\mathcal{H}^{0}}+\lVert
f\rVert_{H^{1}}+\lVert v\rVert_{H^{1}}+\mu^{2}\lVert v\rVert_{H^{3}}\\\
\lesssim\lVert
f\rVert_{H^{1}}+\varepsilon\lVert\eta\rVert_{\mathcal{H}^{0}}+\varepsilon^{1/2}(\lVert\eta\rVert_{\mathcal{H}^{1}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{3}}+|\mu|(\lVert\eta\rVert_{\mathcal{H}^{2}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{4}}))+R_{\eta,h}.$
(4.1.39)

Summing (4.1.34) and (4.1.39) and taking $\rho_{\mathrm{AP1}}$ sufficiently
small, we may absorb the $\eta$ terms on the right with those on the left,
resulting in the estimate

$\lVert v\rVert_{H^{1}}+\mu^{2}\lVert
v\rVert_{H^{3}}+\lVert\eta\rVert_{\mathcal{H}^{1}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{3}}+|\mu|(\lVert\eta\rVert_{\mathcal{H}^{2}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{4}})\lesssim\lVert
f\rVert_{H^{1}}+R_{\eta,h}.$ (4.1.40)

Thus, to finish the proof of (4.1.29) we only need to prove the estimate

$|\sigma|(\lVert\eta\rVert_{\mathcal{H}^{2}}+\sigma^{2}\lVert\eta\rVert_{\mathcal{H}^{4}})\lesssim\lVert
f\rVert_{H^{1}}+R_{\eta,h}+|\sigma|\lVert\nabla h\rVert_{L^{2}}.$ (4.1.41)

To this end, we take the divergence of the second equation in (4.1.1) and
substitute in the first equation to derive the identity

$(\vartheta-\sigma^{2}\theta\Delta)\Delta\eta=(X\cdot\nabla)\nabla\cdot(X\eta)+\nabla\cdot
f-\nabla X^{\mathrm{t}}:\nabla v-(1-3\mu^{2}\Delta)\nabla\cdot v-X\cdot\nabla
h\\\
-(\nabla\vartheta\cdot\nabla\eta-\sigma^{2}\nabla\theta\cdot\nabla\Delta\eta).$
(4.1.42)

Then we may again use elliptic regularity to bound

$|\sigma|(\lVert\nabla^{2}\eta\rVert_{L^{2}}+\sigma^{2}\lVert\nabla^{4}\eta\rVert_{L^{2}})\lesssim|\sigma|\lVert(X\cdot\nabla)\nabla\cdot(X\eta)+\nabla\cdot
f-\nabla X^{\mathrm{t}}:\nabla v-(1-3\mu^{2}\Delta)\nabla\cdot v-X\cdot\nabla
h\rVert_{L^{2}}\\\
+|\sigma|\lVert\nabla\vartheta\cdot\nabla\eta-\sigma^{2}\nabla\theta\cdot\nabla\Delta\eta\rVert_{L^{2}},$
(4.1.43)

and from here we can use Sobolev interpolation again (but now with powers of
$\sigma$) together with (4.1.40) to deduce (4.1.41). ∎

We can now use the previous result with a simple differentiation argument to
derive a priori estimates for higher derivatives of solutions to (4.1.1).

######  Theorem 4.5 (Principal part a priori estimates, omnisonic case).

Suppose that $\mathbb{N}\ni s\geqslant 2+\lfloor d/2\rfloor$, $f\in
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$, $h\in(\dot{H}^{-1}\cap
H^{1+s})(\mathbb{R}^{d})$, $v\in H^{2+s}(\mathbb{R}^{d};\mathbb{R}^{d})$, and
$\eta\in\mathcal{H}^{3+s}(\mathbb{R}^{d})$ solve system (4.1.1). There exists
$\mathbb{R}^{+}\ni\rho_{\mathrm{AP}s}\leqslant\rho_{\mathrm{WD}}$, depending
only on $\mathcal{Q}$, $\rho_{\mathrm{WD}}$, $r$ (from Definition 4.1) and
$s$, with the property that for all
$\mathbb{R}^{+}\ni\varepsilon\leqslant\rho_{\mathrm{AP}s}$ and
$(v_{0},\eta_{0},\beta_{0})$ satisfying (4.1.3), we have the estimate

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{1+s}\times H^{1+s}}\lVert
h,f\rVert_{\mathbb{Y}_{2+\lfloor d/2\rfloor}}$ (4.1.44)

The implicit constant in (4.1.44) depends only on $s$, $\mathcal{Q}$, and $d$
from Definition 4.1.

###### Proof.

We divide the proof into two stages. In the first of these we consider only
the special case in which $h=0$ and $0\not\in\mathrm{supp}\mathscr{F}[\eta]$.
An application of Theorem 4.4 yields the estimate

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{1}}\lesssim\lVert
f\rVert_{H^{1}}.$ (4.1.45)

In order to obtain higher-order estimates we will apply this same a priori
estimate to the derivatives of (4.1.1); it will suffice to consider only pure
derivatives of order $s-1$. To this end, we let $q=s-1$,
$j\in\\{{1,\dots,d}\\}$, and consider the equations satisfied by
$\partial_{j}^{q}v$ and $\partial_{j}^{q}\eta$:

$\begin{cases}\nabla\cdot\partial_{j}^{q}v+\nabla\cdot(X\partial_{j}^{q}\eta)=\nabla\cdot([X,\partial_{j}^{q}]\eta),\\\
\partial_{j}^{q}v+X\cdot\partial_{j}^{q}v-\mu^{2}\nabla\cdot\mathbb{S}\partial_{j}^{q}v+(\vartheta-\sigma^{2}\theta\Delta)\nabla\partial_{j}^{q}\eta=\partial_{j}^{q}f+[X,\partial_{j}^{q}]\cdot\nabla
v+([\vartheta,\partial_{j}^{q}]-\sigma^{2}[\theta,\partial_{j}^{q}]\Delta)\nabla\eta.\end{cases}$
(4.1.46)

Using Theorem 4.4 again, we acquire the estimate

${{}_{\mu}}\lVert\partial_{j}^{q}v\rVert_{H^{1}}+{{}_{\mu,\sigma}}\lVert\partial_{j}^{q}\eta\rVert_{\mathcal{H}^{1}}\lesssim\lVert\partial_{j}^{q}f+[X,\partial_{j}^{q}]\cdot\nabla
v+([\vartheta,\partial_{j}^{q}]-\sigma^{2}[\theta,\partial_{j}^{q}]\Delta)\nabla\eta\rVert_{H^{1}}\\\
+\sum_{k=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\partial_{k}\partial_{j}^{q}\eta\partial_{k}\nabla\cdot([X,\partial_{j}^{q}]\eta)\Big{|}^{1/2}+|\mu|\sum_{k=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\nabla\partial_{k}\partial_{j}^{q}\eta\cdot\nabla\partial_{k}\nabla\cdot([X,\partial_{j}^{q}]\eta)\Big{|}^{1/2}\\\
+|\sigma|\lVert\nabla\nabla\cdot([X,\partial_{j}^{q}]\eta)\rVert_{L^{2}}.$
(4.1.47)

The rest of our work in this stage consists of obtaining structured estimates
for the terms appearing on the right here.

For the first term on the right hand side of (4.1.47) we utilize the tame
estimates on commutators from Proposition A.24 to bound

$\lVert\partial_{j}^{q}f+[X,\partial_{j}^{q}]\cdot\nabla
v+([\vartheta,\partial_{j}^{q}]-\sigma^{2}[\theta,\partial_{j}^{q}]\Delta)\nabla\eta\rVert_{H^{1}}\lesssim\lVert
f\rVert_{H^{s}}+\varepsilon(\lVert
v\rVert_{H^{s}}+{{}_{\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}})\\\ +{\lVert
v_{0},\beta_{0},\eta_{0}\rVert_{H^{s}\times
H^{s}\times\mathcal{H}^{s}}}(\lVert v\rVert_{H^{2+\lfloor
d/2\rfloor}}+{{}_{\sigma}}\lVert\eta\rVert_{\mathcal{H}^{2+\lfloor
d/2\rfloor}}).$ (4.1.48)

For the remaining terms on the right hand side of (4.1.47), we first claim
that the following preliminary estimate holds:

$\lVert[X,\partial_{j}^{q}]\eta\rVert_{H^{2}}\lesssim\varepsilon\lVert\eta\rVert_{\mathcal{H}^{s}}+\lVert
v_{0}\rVert_{H^{s+1}}\lVert\eta\rVert_{\mathcal{H}^{1+\lfloor d/2\rfloor}}.$
(4.1.49)

To see this, we note that $[X,\partial_{j}^{q}]=[v_{0},\partial_{j}^{q}]$ and
employ the Fourier localization operators from (A.1.7) to bound

$\lVert[X,\partial_{j}^{q}]\eta\rVert_{H^{2}}\leqslant\lVert[v_{0},\partial_{j}^{q}]\Uppi^{1}_{\mathrm{H}}\eta\rVert_{H^{2}}+\lVert\partial_{j}^{q}(v_{0}\Uppi^{1}_{\mathrm{L}}\eta)\rVert_{H^{2}}+\lVert
v_{0}\partial_{j}^{q}\Uppi^{1}_{\mathrm{L}}\eta\rVert_{H^{2}}.$ (4.1.50)

Propositions A.3, A.23, and A.24, together with Corollary A.5 then allow us to
estimate

$\lVert[v_{0},\partial_{j}^{q}]\Uppi^{1}_{\mathrm{H}}\eta\rVert_{H^{2}}+\lVert\partial_{j}^{q}(v_{0}\Uppi^{1}_{\mathrm{L}}\eta)\rVert_{H^{2}}+\lVert
v_{0}\partial_{j}^{q}\Uppi^{1}_{\mathrm{L}}\eta\rVert_{H^{2}}\lesssim\varepsilon\lVert\eta\rVert_{\mathcal{H}^{s}}+\lVert
v_{0}\rVert_{H^{s+1}}\lVert\eta\rVert_{\mathcal{H}^{1+\lfloor d/2\rfloor}},$
(4.1.51)

and so we obtain (4.1.49) from this and (4.1.50), completing the proof of the
claim. With the claim in hand, we deduce that

$\sum_{k=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\partial_{k}\partial_{j}^{q}\eta\partial_{k}\nabla\cdot([X,\partial_{j}^{q}]\eta)\Big{|}^{1/2}+|\mu|\sum_{k=1}^{d}\Big{|}\int_{\mathbb{R}^{d}}(\vartheta-\sigma^{2}\theta\Delta)\nabla\partial_{k}\partial_{j}^{q}\eta\cdot\nabla\partial_{k}\nabla\cdot([X,\partial_{j}^{q}]\eta)\Big{|}^{1/2}\\\
+|\sigma|\lVert\nabla\nabla\cdot([X,\partial_{j}^{q}]\eta)\rVert_{L^{2}}\lesssim\;{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}}^{1/2}\lVert[X,\partial_{j}^{q}]\eta\rVert_{H^{2}}^{1/2}+|\sigma|\lVert[X,\partial_{j}^{q}]\eta\rVert_{H^{2}}.$
(4.1.52)

We may thus combine (4.1.45), (4.1.47), (4.1.48), (4.1.49), and (4.1.52) to
arrive at the bound

${{}_{\mu}}\lVert
v\rVert_{H^{s}}+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}}\asymp{{}_{\mu}}\lVert
v\rVert_{H^{1}}+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{1}}+\sum_{j=1}^{d}({{}_{\mu}}\lVert\partial_{j}^{s-1}v\rVert_{H^{1}}+{{}_{\mu,\sigma}}\lVert\partial_{j}^{s-1}\eta\rVert_{\mathcal{H}^{1}})\lesssim\lVert
f\rVert_{H^{s}}+\varepsilon(\lVert
v\rVert_{H^{s}}+{{}_{\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}})\\\ +{\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{s+1}\times H^{s+1}}}(\lVert
v\rVert_{H^{2+\lfloor
d/2\rfloor}}+{{}_{\sigma}}\lVert\eta\rVert_{\mathcal{H}^{2+\lfloor
d/2\rfloor}})+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}}^{1/2}(\varepsilon\lVert\eta\rVert_{\mathcal{H}^{s}}+\lVert
v_{0}\rVert_{H^{s+1}}\lVert\eta\rVert_{\mathcal{H}^{1+\lfloor
d/2\rfloor}})^{1/2}.$ (4.1.53)

Taking $\rho_{\mathrm{APs}}$ sufficiently small to absorb the $\varepsilon$
terms, (4.1.53) implies that

${{}_{\mu}}\lVert
v\rVert_{H^{s}}+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}}\lesssim\lVert
f\rVert_{H^{s}}+{\lVert v_{0},\beta_{0},\eta_{0}\rVert_{H^{s+1}\times
H^{s+1}\times\mathcal{H}^{s+1}}}(\lVert v\rVert_{H^{2+\lfloor
d/2\rfloor}}+{{}_{\sigma}}\lVert\eta\rVert_{\mathcal{H}^{2+\lfloor
d/2\rfloor}}).$ (4.1.54)

This is nearly (4.1.44); it remains to remove the $v$ and $\eta$ terms on the
right. For this we consider (4.1.54) in the case $s=s_{0}=2+\lfloor
d/2\rfloor$ and exploit (4.1.3) to make another absorption argument in order
to obtain the closed estimate

${{}_{\mu}}\lVert v\rVert_{H^{2+\lfloor
d/2\rfloor}}+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{2+\lfloor
d/2\rfloor}}\lesssim\lVert f\rVert_{H^{2+\lfloor d/2\rfloor}}.$ (4.1.55)

Thus, by combining (4.1.54) and (4.1.55), we find that for any $\mathbb{N}\ni
s\geqslant 2+\lfloor d/2\rfloor$ we have the bound

${{}_{\mu}}\lVert
v\rVert_{H^{s}}+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}}\lesssim\lVert
f\rVert_{H^{s}}+{\lVert v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{s+1}\times
H^{s+1}}}\lVert f\rVert_{H^{2+\lfloor d/2\rfloor}},$ (4.1.56)

which is (4.1.44). This completes the first stage of the proof.

In the second stage, the task is to add in the dependence of $h$ and to remove
the extra Fourier support hypotheses on $\eta$. Suppose first that the system
(4.1.1) is satisfied by $v$, $\eta$, and $f$ as in the previous stage, but now
with $h\in(\dot{H}^{-1}\cap H^{1+s})(\mathbb{R}^{d})$. As in Proposition 2.1,
we define $H\in H^{2+s}(\mathbb{R}^{d})$ via $H=\nabla\Delta^{-1}h$ and note
that $\left\lVert H\right\rVert_{H^{2+s}}\asymp\left\lVert
h\right\rVert_{\dot{H}^{-1}\cap H^{1+s}}$. We then set $w=v-H$ and observe
that $w$ and $\eta$ satisfy the system

$\begin{cases}\nabla\cdot w+\nabla\cdot(X\eta)=0,\\\ w+X\cdot\nabla
w-\mu^{2}\nabla\cdot\mathbb{S}w+(\vartheta-\sigma^{2}\theta\Delta)\nabla\eta=f-(I+X\cdot\nabla-\mu^{2}\nabla\cdot\mathbb{S})H,\end{cases}$
(4.1.57)

which means we are in a position to apply estimate (4.1.56) from the previous
stage. Doing so and handling the $X\cdot\nabla H$ term with Proposition A.23,
we obtain the bound

${{}_{\mu}}\lVert
w\rVert_{H^{s}}+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}}\lesssim\lVert
h,f\rVert_{\dot{H}^{-1}\cap H^{1+s}\times H^{s}}+{\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{s+1}\times H^{s+1}}}\lVert
h,f\rVert_{\dot{H}^{-1}\cap H^{3+\lfloor d/2\rfloor}\times H^{2+\lfloor
d/2\rfloor}}.$ (4.1.58)

Since ${{}_{\mu}}\lVert v\rVert_{H^{s}}\lesssim{{}_{\mu}}\lVert
w\rVert_{H^{s}}+\lVert h\rVert_{\dot{H}^{-1}\cap H^{1+s}}$, the estimate
(4.1.58) implies (4.1.44).

The final task in the proof is to remove the auxiliary Fourier support
assumption on $\eta$. This is straightforward, since we can frequency split
$\eta=\Uppi^{\kappa}_{\mathrm{H}}\eta+\Uppi^{\kappa}_{\mathrm{L}}\eta$ with
$\kappa\in(0,1)$ and consider the equations satisfied by $v$ and
$\Uppi^{\kappa}_{\mathrm{H}}\eta$:

$\begin{cases}\nabla\cdot
v+\nabla\cdot(X\Uppi^{\kappa}_{\mathrm{H}}\eta)=h-\nabla\cdot(X\Uppi^{\kappa}_{\mathrm{L}}\eta),\\\
v+X\cdot\nabla
v-\mu^{2}\nabla\cdot\mathbb{S}v+(\vartheta-\sigma^{2}\theta\Delta)\nabla\Uppi^{\kappa}_{\mathrm{H}}\eta=f-(\vartheta-\sigma^{2}\theta\Delta)\nabla\Uppi^{\kappa}_{\mathrm{L}}\eta.\end{cases}$
(4.1.59)

To this system we invoke the a priori estimates of the previous stage and send
$\kappa\to 0$. Since the right hand side of (4.1.59) converges in
$(\dot{H}^{-1}\cap H^{1+s})(\mathbb{R}^{d})\times H^{s}(\mathbb{R}^{d})$ to
$(h,f)$ as $\kappa\to 0$, we obtain the estimate (4.1.44) as a result of this
limiting procedure. ∎

For the final result of this subsection, we provide specialized a priori
estimates for (4.1.1) in the subsonic regime. The acronym $\mathrm{SAP}$
stands for ‘subsonic a priori’.

######  Theorem 4.6 (Principal part a priori estimates, subsonic case).

Assume that $\mathcal{Q}=I\times J\times K$ is subsonic in the sense that
$I\Subset(0,1)$ and suppose that $\mathbb{N}\ni s\geqslant 2+\lfloor
d/2\rfloor$. Further suppose that $f\in H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$,
$h\in(\dot{H}^{-1}\cap H^{1+s})(\mathbb{R}^{d})$, $v\in
H^{2+s}(\mathbb{R}^{d};\mathbb{R}^{d})$, and
$\eta\in\mathcal{H}^{3+s}(\mathbb{R}^{d})$ solve system (4.1.1). There exists
$\mathbb{R}^{+}\ni\rho_{\mathrm{SAP}s}\leqslant\rho_{\mathrm{WD}}$, depending
only on $\mathcal{Q}$, $\rho_{\mathrm{WD}}$, $r$ (from Definition 4.1) and
$s$, with the property that for all
$\mathbb{R}^{+}\ni\varepsilon\leqslant\rho_{\mathrm{SAP}s}$ and
$(v_{0},\eta_{0},\beta_{0})$ satisfying (4.1.3), we have the estimate

$\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{1+s}\times H^{1+s}}\lVert
h,f\rVert_{\mathbb{Y}_{2+\lfloor d/2\rfloor}}.$ (4.1.60)

Here the implicit constant in (4.1.44) depends only on $s$, $\mathcal{Q}$, and
$d$ from Definition 4.1.

###### Proof.

We shall seek $\rho_{\mathrm{SAP}s}\leqslant\rho_{\mathrm{AP}s}$, where the
latter parameter is the one granted by Theorem 4.1.1, so that estimate
(4.1.44) holds for our solution. To prove (4.1.60) we only need to promote the
regularity of $\eta$, as the required estimates on $v$ are already met; our
main tool in achieving this promotion is a more careful study of equation
(4.1.42). We can rewrite said equation as

$(\Delta-\gamma^{2}\partial_{1}^{2})\eta-\sigma^{2}\Delta^{2}\eta=-\eta_{0}(1-\sigma^{2}\Delta)\Delta\eta-\beta_{0}\Delta\eta+(X\otimes
X-\gamma^{2}e_{1}\otimes e_{1}):\nabla^{2}\eta+X\cdot\nabla(\eta\nabla\cdot
X)\\\ +\nabla\cdot f-\nabla X^{\mathrm{t}}:\nabla
v-(1-3\mu^{2}\Delta)\nabla\cdot v-X\cdot\nabla
h-(\nabla\vartheta\cdot\nabla\eta-\sigma^{2}\nabla\theta\cdot\nabla\Delta\eta)$
(4.1.61)

in order to make evident the utility of inclusion $\gamma\in I\Subset(0,1)$ in
the negative definiteness of the symbol of the operator on the left away from
zero. Note that this condition also implies the ellipticity of the operator on
the left when $\sigma=0$.

We shall take the norm of both sides of equation (4.1.61) in the space
$H^{s-1}(\mathbb{R}^{d})$. Due to ellipticity, we obtain the bound

$\lVert\nabla^{2}\eta\rVert_{{{}_{\sigma}}H^{s-1}}\lesssim\lVert-\eta_{0}(1-\sigma^{2}\Delta)\Delta\eta-\beta_{0}\Delta\eta+(X\otimes
X-\gamma^{2}e_{1}\otimes e_{1}):\nabla^{2}\eta\rVert_{H^{s-1}}\\\ +\lVert
X\cdot\nabla(\eta\nabla\cdot X)+\nabla\cdot f-\nabla X^{\mathrm{t}}:\nabla
v-(1-3\mu^{2}\Delta)\nabla\cdot v-X\cdot\nabla
h-(\nabla\vartheta\cdot\nabla\eta-\sigma^{2}\nabla\theta\cdot\nabla\Delta\eta)\rVert_{H^{s-1}}\\\
=\mathbf{I}+\mathbf{II},$ (4.1.62)

and we will estimate the two terms on the right separately. Thanks to
Corollary A.5 and Proposition A.23, we have the bounds

$\mathbf{I}\lesssim\varepsilon\lVert\nabla^{2}\eta\rVert_{{{}_{\sigma}}H^{s-1}}+\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{s-1}\times
H^{s-1}}\lVert\nabla^{2}\eta\rVert_{{{}_{\sigma}}H^{1+\lfloor d/2\rfloor}}$
(4.1.63)

and

$\mathbf{II}\lesssim\lVert
v,\eta,h,f\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}\times\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{s+1}\times H^{s+1}}\lVert
v,\eta,h,f\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{2+\lfloor
d/2\rfloor}\times\mathbb{Y}_{2+\lfloor d/2\rfloor}}.$ (4.1.64)

We combine (4.1.62), (4.1.63), and (4.1.64) with (4.1.44) to acquire the
estimate

$\lVert\nabla^{2}\eta\rVert_{{{}_{\sigma}}H^{s-1}}\lesssim\varepsilon\lVert\nabla^{2}\eta\rVert_{{{}_{\sigma}}H^{s-1}}+\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{s-1}\times
H^{s-1}}\lVert\nabla^{2}\eta\rVert_{{{}_{\sigma}}H^{1+\lfloor d/2\rfloor}}\\\
+\lVert h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{1+s}\times H^{1+s}}\lVert
h,f\rVert_{\mathbb{Y}_{2+\lfloor d/2\rfloor}}.$ (4.1.65)

By taking $s=2+\lfloor d/2\rfloor$ and
$\varepsilon\leqslant\rho_{\mathrm{SAP}s}$ sufficiently small in (4.1.65), we
may use an absorbing argument to learn that

$\lVert\nabla^{2}\eta\rVert_{{{}_{\sigma}}H^{1+\lfloor
d/2\rfloor}}\lesssim\lVert h,f\rVert_{\mathbb{Y}_{2+\lfloor d/2\rfloor}}.$
(4.1.66)

Inputting (4.1.66) into (4.1.65) for $s\geqslant 2+\lfloor d/2\rfloor$ and
taking $\varepsilon$ smaller, if necessary, we arrive at the bound

$\lVert\nabla^{2}\eta\rVert_{{{}_{\sigma}}H^{s-1}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{1+s}\times H^{1+s}}\lVert
h,f\rVert_{\mathbb{Y}_{2+\lfloor d/2\rfloor}}.$ (4.1.67)

Combining (4.1.67) with (4.1.44) then yields (4.1.60). ∎

###  4.2. Existence

The first goal of this subsection is to establish the existence of solutions
to system (4.1.1). The case of $\mu\cdot\sigma\neq 0$ is a direct application
of the method of continuity along with Theorems 2.3 and 4.5.

######  Proposition 4.7 (Preliminary existence result).

Suppose that $\mathbb{N}\ni s\geqslant 2+\lfloor d/2\rfloor$,
$\mathbb{R}^{+}\ni\varepsilon\leqslant\rho_{\mathrm{APs}}$, and
$(v_{0},\eta_{0},\beta_{0})$ satisfy (4.1.3). If
$(\mu,\sigma)\in(J\setminus\\{{0}\\})\times(K\setminus\\{{0}\\})$, and
$\gamma\in I\Subset\mathbb{R}$, then for every $(h,f)\in(\dot{H}^{-1}\cap
H^{1+s})(\mathbb{R}^{d})\times H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$ there
exists a unique $(v,\eta)\in
H^{2+s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{3+s}(\mathbb{R}^{d})$
satisfying system (4.1.1).

###### Proof.

We consider the affine homotopy of bounded linear operators

$L_{t}:H^{2+s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{3+s}(\mathbb{R}^{d})\to(\dot{H}^{-1}\cap
H^{1+s})(\mathbb{R}^{d})\times H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})\text{ for
}t\in[0,1]$ (4.2.1)

defined via

$L_{t}(v,\eta)=\begin{pmatrix}\nabla\cdot
w-\gamma\partial_{1}\eta+t\nabla\cdot(v_{0}\eta)\\\
v-\gamma\partial_{1}v-\mu^{2}\nabla\cdot\mathbb{S}v+(1-\sigma^{2}\Delta)\nabla\eta+t(v_{0}\nabla\cdot
v+(\eta_{0}+\beta_{0}-\sigma^{2}\eta_{0}\Delta)\nabla\eta)\end{pmatrix}.$
(4.2.2)

Theorem 4.5 provides a constant $C>0$ such that

${{}_{\mu}}\lVert
v\rVert_{H^{s}}+{{}_{\mu,\sigma}}\lVert\eta\rVert_{\mathcal{H}^{s}}\leqslant
C\lVert L_{t}(v,\eta)\rVert_{\dot{H}^{-1}\cap H^{1+s}\times H^{s}}\text{ for
all }t\in[0,1].$ (4.2.3)

On the other hand, Theorem 4.5 shows that $L_{0}$ is a linear isomorphism.
Hence, the method of continuity (see, for instance, Theorem 5.2 in Gilbarg and
Trudinger [24]) guarantees that the map $L_{1}$ is also an isomorphism, and
$L_{1}$ corresponds to the operator in system (4.1.1). ∎

Our main existence theorems are the following two results.

######  Theorem 4.8 (Omnisonic and subsonic existence result).

Suppose that $\mathbb{N}\ni s\geqslant\max\\{{3,2+\lfloor d/2\rfloor}\\}$. The
following hold.

  1. (1)

If $\mathbb{R}^{+}\ni\varepsilon\leqslant\rho_{\mathrm{APs}}$, where the
latter parameter is from Theorem 4.5, $(v_{0},\eta_{0},\beta_{0})$ satisfy
(4.1.3), $(\mu,\sigma)\in J\times K$, and $\gamma\in I\Subset\mathbb{R}^{+}$,
then for every $(h,f)\in\mathbb{Y}_{s}$ there exists a unique
$(v,\eta)\in\mathbb{X}_{s}$ satisfying system (4.1.1). Moreover, the solution
obeys estimate (4.1.44).

  2. (2)

If $\mathbb{R}^{+}\ni\varepsilon\leqslant\rho_{\mathrm{SAPs}}$, where the
latter parameter is from Theorem 4.6, $(v_{0},\eta_{0},\beta_{0})$ satisfy
(4.1.3), $(\mu,\sigma)\in J\times K$, and $\gamma\in I\Subset(0,1)$, then for
every $(h,f)\in\mathbb{Y}_{s}$ there exists a unique
$(v,\eta)\in\mathbb{X}^{\mathrm{sub}}_{s}$ satisfying system (4.1.1).
Moreover, the solution obeys estimate (4.1.60).

###### Proof.

We only prove the first item, as the second item follows from a nearly
identical argument. We begin by justifying the uniqueness assertion. By
shifting to the system (4.1.57) as in the proof of Theorem 4.5, we see that it
is sufficient to prove uniqueness in the case that $h=0$. Uniqueness in this
case is a consequence of estimate (4.1.29) from Theorem 4.4.

We now prove existence. Let
$\\{{(\mu_{n},\sigma_{n})}\\}_{n\in\mathbb{N}}\subset(J\setminus\\{{0}\\})\times(K\setminus\\{{0}\\})$
be any sequence satisfying $(\mu_{n},\sigma_{n})\to(\mu,\sigma)$ as
$n\to\infty$. Thanks to Proposition 4.7, we obtain a corresponding sequence
$\\{{(v_{n},\eta_{n})}\\}_{n\in\mathbb{N}}\subset
H^{2+s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{3+s}(\mathbb{R}^{d})$
such that

$\begin{cases}\nabla\cdot v_{n}+\nabla\cdot(X\eta_{n})=h,\\\
v_{n}+X\cdot\nabla
v_{n}-\mu_{n}^{2}\nabla\cdot\mathbb{S}v_{n}+(\vartheta-\sigma_{n}^{2}\theta\Delta)\nabla\eta_{n}=f.\end{cases}$
(4.2.4)

Theorem 4.5, applied along this sequence, yields the uniform bounds

$\lVert
v_{n},\eta_{n}\rVert_{{{}_{\mu_{n},\sigma_{n}}}\mathbb{X}_{s}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{1+s}\times H^{1+s}}\lVert
h,f\rVert_{\mathbb{Y}_{2+\lfloor d/2\rfloor}}.$ (4.2.5)

Thus, by the reflexivity of the container space, we may extract a subsequence
that converges weakly to $(v,\eta)\in
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{s}(\mathbb{R}^{d})$ in
the weak
$H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{s}(\mathbb{R}^{d})$
topology. Taking weak limits along the subsequence in (4.2.4), we deduce that
$(v,\eta)$ is the desired solution. Sequential weak lower semicontinuity and
(4.2.5) then provide the bound (4.1.44). ∎

###  4.3. Synthesis

The goal of this subsection is to combine the principal part linear analysis
of Sections 4.1 and 4.2 with the splitting analysis of Section 3.3 to complete
the satisfaction of the linear hypotheses of the Nash-Moser theorem.

The main emphasis of this subsection is to prove the results in full detail in
the omnisonic regime. We then, at the end, consider the completely analogous
results in the subsonic regime. The proofs for these are then greatly
abbreviated, as the are very similar to the omnisonic case except for a few
minor distinctions.

We begin with the following definition of maps and function spaces, which are
crucially utilized in the subsequent proofs. Throughout the rest of this
subsection we will employ the notation set in Proposition 3.8.

######  Definition 4.9 (Operators and adapted domains, omnisonic case).

Given $\gamma$, $\mu$, $\sigma$, $v_{0}$, $\eta_{0}$, and $\beta_{0}$ as in
Definition 4.1, we define the following.

  1. (1)

For $s\in\mathbb{N}$ the map
$L^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}:H^{2+s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{3+s}(\mathbb{R}^{d})\to(\dot{H}^{-1}\cap
H^{1+s})(\mathbb{R}^{d})\times H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$ is given
by

$L^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)=P^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]+(0,\beta_{0}\nabla\eta).$
(4.3.1)

  2. (2)

For $\mathbb{N}\ni s\geqslant 3$ we define the space
${{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}_{s}=\\{{(v,\eta)\in{{}_{\mu,\sigma}}\mathbb{X}_{s}\;:L_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}(v,\eta)\in\mathbb{Y}_{s}}\\}$
and equip it with the graph norm

$\lVert
v,\eta\rVert_{{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}_{s}}=\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}\mathbb{X}_{s}}}+\lVert
L^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)\rVert_{\mathbb{Y}_{s}}.$
(4.3.2)

We now come to the first part of the synthesis. The subscript $\mathrm{S}$
stands for synthesis and the $s$ refers to the data belonging to
$\mathbb{Y}_{s}$.

######  Proposition 4.10 (Synthesis I, omnisonic case).

Let $(\gamma,\mu,\sigma)\in\mathcal{Q}$ with $\mathcal{Q}$ as in Definition
4.1. For each $\mathbb{N}\ni s\geqslant\max\\{{3,2+\lfloor d/2\rfloor}\\}$
there exists
$\mathbb{R}^{+}\ni\widetilde{\rho}_{\mathrm{S}s}\leqslant\rho_{\mathrm{AP}s}$,
depending only on $s$ and $\mathcal{Q}$, such that if
$\mathbb{R}^{+}\ni\varepsilon\in(0,\widetilde{\rho}_{\mathrm{S}s}]$,
$(v_{0},\eta_{0},\beta_{0})$ satisfy (4.1.3), $\upvarphi_{0}\in
B_{H^{r}}(0,\varepsilon)\cap H^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$, and
$\uptau_{0}\in B_{H^{r}}(0,\varepsilon)\cap
H^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d\times d})$, then the following hold.

  1. (1)

The following linear map is well-defined and bounded:

${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\mathbb{X}_{s}\ni(v,\eta)\mapsto
L^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)+(0,\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta)+(0,R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta])\in\mathbb{Y}_{s}.$
(4.3.3)

Moreover, this map is an isomorphism.

  2. (2)

For all $(h,f)\in\mathbb{Y}_{s}$ the unique
$(v,\eta)\in{{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}_{s}}$
satisfying the system

$L^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)+(0,\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta)+(0,R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta])=(h,f)$
(4.3.4)

obeys the tame estimate

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0},\upvarphi_{0}\rVert_{\mathbb{X}_{3+s}\times
H^{3+s}\times H^{3+s}\times H^{3+s}}\lVert
h,f\rVert_{\mathbb{Y}_{\max\\{{3,2+\lfloor d/2\rfloor}\\}}},$ (4.3.5)

where the implicit constant depends only on $s$, $d$, and $\mathcal{Q}$.

###### Proof.

We begin by verifying the map (4.3.3) is well-defined and bounded. Due to the
definition of the norm on the space
${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\mathbb{X}_{s}$, which is
provided in Definition 4.9, and the embedding
${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\mathbb{X}_{s}\hookrightarrow{{}_{\mu,\sigma}}\mathbb{X}_{s}$,
it is sufficient to verify that the map

${{}_{\mu,\sigma}}\mathbb{X}_{s}\ni(v,\eta)\mapsto\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta+R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]\in
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$ (4.3.6)

is well-defined and bounded. This is a consequence of the first item of
Proposition 3.9, Corollary A.5, and the definition of the norm on
${{}_{\mu,\sigma}}\mathbb{X}_{s}$ from (3.1.13).

It remains to show that the map (4.3.3) is an isomorphism that obeys estimate
(4.3.5). We will prove this in two steps. For the first step we shall use the
following pair of definitions.

  * •

For $s\in\mathbb{N}$ the map
$M_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}:H^{2+s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{3+s}(\mathbb{R}^{d})\to(\dot{H}^{-1}\cap
H^{1+s})(\mathbb{R}^{d})\times H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$ is given
via

$M_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}(v,\eta)=Q^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]+(0,\beta_{0}\nabla\eta).$
(4.3.7)

  * •

For $\mathbb{N}\ni s\geqslant 3$ we define the space
${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\widetilde{\mathbb{X}}_{s}=\\{{(v,\eta)\in{{}_{\mu,\sigma}}\mathbb{X}_{s}\;:\;M^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)\in\mathbb{Y}_{s}}\\}$
and equip it with the graph norm

$\lVert
v,\eta\rVert_{{{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\widetilde{\mathbb{X}}_{s}}=\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}\mathbb{X}_{s}}}+\lVert
M^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)\rVert_{\mathbb{Y}_{s}}.$
(4.3.8)

The claim of the first step is then as follows. Let $b=\max\\{{3,2+\lfloor
d/2\rfloor}\\}$. For each $\mathbb{N}\ni s\geqslant b$ there exists
$\mathbb{R}^{+}\ni\rho_{\mathrm{XS}s}\leqslant\rho_{\mathrm{AP}s}$ with the
property that for all $\mathbb{R}^{+}\ni\varepsilon\in(0,\rho_{\mathrm{XS}s}]$
and $(h,f)\in\mathbb{Y}_{s}$ there exists a unique
$(v,\eta)\in{{{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\widetilde{\mathbb{X}}_{s}}$
satisfying

$M^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)+(0,\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta)+(0,S^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v])=(h,f)$
(4.3.9)

as well as the estimate

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0},\upvarphi_{0}\rVert_{\mathbb{X}_{2+s}\times
H^{2+s}\times H^{2+s}\times H^{2+s}}\lVert h,f\rVert_{\mathbb{Y}_{b}}$
(4.3.10)

for an implicit constant depending only on $s$, $d$, and $\mathcal{Q}$.

We begin the proof of the claim by considering the homotopy of operators
$M_{t}:\;{{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\widetilde{\mathbb{X}}_{s}\to\mathbb{Y}_{s}$
defined for $t\in[0,1]$ via

$M_{t}(v,\eta)=M^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)+t(0,\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta)+t(0,S^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v]).$
(4.3.11)

Thanks to the embedding
${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\widetilde{\mathbb{X}}_{s}\hookrightarrow\;{{}_{\mu,\sigma}}\mathbb{X}_{s}$
and the remainder estimate of the second item of Proposition 3.9, we readily
deduce that each $M_{t}$ is a well-defined and bounded linear map. Our goal is
to use the method of continuity on the family $\\{M_{t}\\}_{t\in[0,1]}$, but
to do so we must first establish the requisite estimates.

Provided that $\rho_{\mathrm{XS}s}\leqslant\rho_{\mathrm{AP}s}$, the estimates
of Theorem 4.8 grant us the bound

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}+\lVert\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta+S^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v]\rVert_{H^{s}}\\\
+\lVert v_{0},\eta_{0},\beta_{0}\rVert_{\mathbb{X}_{1+s}\times H^{1+s}}(\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{2+\lfloor
d/2\rfloor}}+\lVert\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta+S^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v]\rVert_{H^{2+\lfloor
d/2\rfloor}}).$ (4.3.12)

We next take $s=b$ in (4.3.12) and estimate

$\lVert\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta\rVert_{H^{b}}\lesssim\varepsilon\lVert\eta\rVert_{{{}_{\mu,\sigma}}\mathcal{H}^{b}}\text{
and }\lVert
S^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v]\rVert_{H^{b}}\lesssim\varepsilon\cdot{{}_{\mu}}\lVert
v\rVert_{H^{b}},$ (4.3.13)

by employing Corollary A.5 for the former and the second item of Proposition
3.9 for the latter. Combining these with (4.3.12) shows that

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{b}}\lesssim\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}}+\varepsilon\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{b}},$ (4.3.14)

and thus if $\varepsilon\leqslant\rho_{\mathrm{XS}b}$ is taken sufficiently
small we may use an absorbing argument to obtain the estimate

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{b}}\lesssim\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}}.$ (4.3.15)

Armed with (4.3.15), we return to (4.3.12) for $s\geqslant b$, but this time
we estimate

$\lVert\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta\rVert_{H^{s}}\lesssim\varepsilon\lVert\eta\rVert_{{{}_{\mu,\sigma}}\mathcal{H}^{s}}+\lVert\upvarphi_{0},\uptau_{0}\rVert_{H^{s}\times
H^{s}}\lVert\eta\rVert_{{{}_{\mu,\sigma}}\mathcal{H}^{b}}\\\ \text{and }\lVert
S^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v]\rVert_{H^{s}}\lesssim\varepsilon\cdot{{}_{\mu}}\lVert
v\rVert_{H^{s}}+\lVert v_{0},\eta_{0}\rVert_{\mathbb{X}_{2+s}}\lVert
v\rVert_{H^{b}}$ (4.3.16)

in order to deduce from (4.3.12), (4.3.15), and (4.3.16) that

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0},\upvarphi_{0}\rVert_{\mathbb{X}_{2+s}\times
H^{2+s}\times H^{2+s}\times H^{2+s}}\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}}+\varepsilon\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}.$ (4.3.17)

Further decreasing
$\varepsilon\leqslant\rho_{\mathrm{XS}s}\leqslant\rho_{\mathrm{XS}b}$ if
necessary, we can again use an absorbing argument to see that

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0}\upvarphi_{0}\rVert_{\mathbb{X}_{2+s}\times
H^{2+s}\times H^{2+s}\times H^{2+s}}\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}},$ (4.3.18)

for an implicit constant depending only on $s$ and $\mathcal{Q}$.

Estimate (4.3.18) is not quite what we need for the method of continuity, so
we continue to work. To control the full norm on
${{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\widetilde{\mathbb{X}}_{s}$
it remains to estimate $\lVert M_{0}(v,\eta)\rVert_{\mathbb{Y}_{s}}$. To this
end, we first use (4.3.11) along with estimate (4.3.16):

$\lVert M_{0}(v,\eta)\rVert_{\mathbb{Y}_{s}}\leqslant\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}+\lVert(M_{0}-M_{t})(v,\eta)\rVert_{\mathbb{Y}_{s}}\lesssim\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}\\\ +\varepsilon\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}+\lVert
v_{0},\eta_{0},\upvarphi_{0},\uptau_{0}\rVert_{\mathbb{X}_{2+s}\times
H^{s}\times H^{s}}\lVert v,\eta\rVert_{\mathbb{X}_{b}}.$ (4.3.19)

Next, we use (4.3.15) and (4.3.18) to handle the $(v,\eta)$ terms on the right
hand side of (4.3.19), which results in the bound

$\lVert M_{0}(v,\eta)\rVert_{\mathbb{Y}_{s}}\lesssim\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0}\upvarphi_{0}\rVert_{\mathbb{X}_{2+s}\times
H^{2+s}\times H^{2+s}\times H^{2+s}}\lVert
M_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}}.$ (4.3.20)

Therefore, by combining (4.3.18) and (4.3.20) we obtain a constant
$C\in\mathbb{R}^{+}$, depending on $s$, the tuple
$(v_{0},\eta_{0},\beta_{0},\uptau_{0},\upvarphi_{0})$, $d$, and $\mathcal{Q}$
but independent of $(v,\eta)$, such that

$\lVert
v,\eta\rVert_{{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\widetilde{\mathbb{X}}_{s}}\leqslant
C\lVert M_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}\text{ for all }t\in[0,1].$
(4.3.21)

This, finally, is the estimate needed for the method of continuity. Theorem
4.8 shows that the operator $M_{0}$ is an isomorphism, and so the bounds
(4.3.21) and the the method of continuity establish that the operator $M_{1}$
is also an isomorphism. Thus, the existence and uniqueness of solutions to
(4.3.9) is established. The claimed estimate (4.3.10) then follows from
(4.3.18), and this concludes the first step of the proof.

In the second step we aim to build on the result of the first step to complete
the proof, and so we will actually seek
$\widetilde{\rho}_{\mathrm{S}s}\leqslant\rho_{\mathrm{XS}s}$ in order to
justify use of the first step. Once more we will proceed via the method of
continuity by considering the operator homotopy
$L_{t}:{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}_{s}\to\mathbb{Y}_{s}$
defined for $t\in[0,1]$ via

$L_{t}(v,\eta)=M_{1}((1+\eta_{0})v,\eta)+t(0,R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]).$
(4.3.22)

Thanks to the embedding
${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}}\mathbb{X}_{s}\hookrightarrow\;{{}_{\mu,\sigma}}\mathbb{X}_{s}$,
the remainder estimate in the first item of Proposition 3.9, and the identity

$M_{1}((1+\eta_{0})v,\eta)=P^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]+(0,\beta_{0}\nabla\eta+\upvarphi_{0}\eta)=L^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)+(0,\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta),$
(4.3.23)

which is a consequence of the second item of Proposition 3.8, we have that the
$L_{t}$ are well-defined and bounded linear maps. As above, we now turn to the
derivation of the estimates needed for the method of continuity.

The condition
$\varepsilon\leqslant\widetilde{\rho}_{\mathrm{S}s}\leqslant\rho_{\mathrm{XS}s}$
allows us to invoke estimate (4.3.18) to acquire the following bound for
$s\geqslant b=\max\\{{3,2+\lfloor d/2\rfloor}\\}$:

$\lVert(1+\eta_{0})v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
L_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}+\lVert
R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]\rVert_{H^{s}}\\\ +\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0},\upvarphi_{0}\rVert_{\mathbb{X}_{2+s}\times
H^{2+s}\times H^{2+s}\times H^{2+s}}(\lVert
L_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}}+\lVert
R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]\rVert_{H^{b}}).$ (4.3.24)

By taking $s=b$ in (4.3.24), using
$\lVert\eta_{0}v\rVert_{{{}_{\mu}}H^{b}}\lesssim\varepsilon\lVert
v\rVert_{{{}_{\mu}}H^{b}}$, and invoking the remainder estimate (3.3.11), we
find that $\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{b}}\lesssim\lVert
L_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}}+\varepsilon\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{b}}$; and hence, if
$\varepsilon\leqslant\rho_{\mathrm{S}b}$ is sufficiently small we have the
bound

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{b}}\lesssim\lVert
L_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}}.$ (4.3.25)

We now return to (4.3.24) with the base estimate (4.3.25) in hand; again
invoking the remainder estimate (3.3.11) and using the bound
$\lVert\eta_{0}v\rVert_{{{}_{\mu}}H^{s}}\lesssim\varepsilon\lVert
v\rVert_{H^{s}}+\lVert\eta_{0}\rVert_{\mathcal{H}^{2+s}}\lVert
v\rVert_{H^{b}}$, together with (4.3.25), we find that if
$\varepsilon\leqslant\widetilde{\rho}_{\mathrm{S}s}\leqslant\rho_{\mathrm{S}b}$
is sufficiently small then

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
L_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0},\upvarphi_{0}\rVert_{\mathbb{X}_{3+s}\times
H^{3+s}\times H^{3+s}\times H^{3+s}}\lVert
L_{t}(v,\eta)\rVert_{\mathbb{Y}_{b}}.$ (4.3.26)

As in the proof of the first step, estimate (4.3.26) will lead us to the
estimates we need for the operator homotopy $\\{L_{t}\\}_{t\in[0,1]}$ with a
bit more work. The remaining piece of the norm on
${{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}_{s}$ is $\lVert
L_{0}(v,\eta)-(0,\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta)\rVert_{\mathbb{Y}_{s}}$.
By the same strategy used in (4.3.19), we derive the estimate

$\lVert
L_{0}(v,\eta)-(0,\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta)\rVert_{\mathbb{Y}_{s}}\lesssim\lVert
L_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}+\varepsilon\lVert
v,\eta\rVert_{{}_{\mu,\sigma}\mathbb{X}_{s}}\\\ +\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0},\upvarphi_{0}\rVert_{\mathbb{X}_{3+s}\times
H^{3+s}\times H^{3+s}\times H^{3+s}}\lVert v,\eta\rVert_{\mathbb{X}_{b}}.$
(4.3.27)

Combining (4.3.26) and (4.3.27) then provides a constant
$C^{\prime}\in\mathbb{R}^{+}$, depending only on $s$, $d$, $\mathcal{Q}$, and
$(v_{0},\eta_{0},\beta_{0},\upvarphi_{0})$, such that

$\lVert
v,\eta\rVert_{{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}_{s}}\leqslant
C^{\prime}\lVert L_{t}(v,\eta)\rVert_{\mathbb{Y}_{s}}\text{ for all
}t\in[0,1].$ (4.3.28)

This and the method of continuity show that in order to show that $L_{1}$ is
an isomorphism it suffices to prove that the map $L_{0}$ is an isomorphism.

Estimate (4.3.28) proves the injectivity of $L_{0}$. For surjectivity we let
$(h,f)\in\mathbb{Y}_{s}$. The first step of the proof show that the map
$M_{1}$ is an isomorphism, so there exists
$(w,\eta)\in{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\widetilde{\mathbb{X}}_{s}$
such that $M_{1}(w,\eta)=(h,f)$. We set $v=w/(1+\eta_{0})$ and note that Lemma
3.7 and Corollary A.5 guarantee the inclusion
$v\in{{}_{\mu}}H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$. Then

$L^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)=(h,f)-(0,\upvarphi_{0}\eta+(\mu+\sigma)\uptau_{0}\nabla\eta)\in\mathbb{Y}_{s},$
(4.3.29)

and hence
$(v,\eta)\in{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}_{s}$
and $L_{0}(v,\eta)=(h,f)$. This shows that $L_{0}$ is surjective, and so
$L_{1}$ is an isomorphism. This establishes that the map (4.3.3) is an
isomorphism; the estimate (4.3.5) follows from (4.3.26) with $t=1$. ∎

######  Corollary 4.11 (Synthesis II, omnisonic case).

Let $(\gamma,\mu,\sigma)\in\mathcal{Q}$ with $\mathcal{Q}$ as in Definition
4.1. For each $\mathbb{N}\ni s\geqslant\max\\{{3,2+\lfloor d/2\rfloor}\\}$
there exists
$\mathbb{R}^{+}\ni\rho_{\mathrm{S}s}\leqslant\widetilde{\rho}_{\mathrm{S}s}$,
depending only on $s$ and $\mathcal{Q}$, such that if
$\varepsilon\in(0,\rho_{\mathrm{S}s}]$ and

$(v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell})\in
B_{\mathbb{X}_{r}\times\mathbb{W}_{r}}(0,\varepsilon)\cap(\mathbb{X}_{\infty}\times\mathbb{W}_{\infty})\text{
for }r=\max\\{{6,5+\lfloor d/2\rfloor}\\},$ (4.3.30)

then the following hold.

  1. (1)

Upon setting

$\uptau_{0}=\sum_{i=0}^{\ell}\eta_{0}^{i}\tau_{0i},\quad\beta_{0}=\mathrm{Tr}\uptau_{0},\text{
and
}\upvarphi_{0}=\sum_{i=1}^{\ell}i\eta_{0}^{i-1}(\tau_{0i}\nabla\eta_{0}+\varphi_{0i}),$
(4.3.31)

the linear map (4.3.3) is well-defined, bounded, and an isomorphism.

  2. (2)

For all $(h,f)\in\mathbb{Y}_{s}$ the unique
$(v,\eta)\in{{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}_{s}}$
satisfying the system (4.3.4) obeys the tame estimate

$\lVert v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}_{s}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{X}_{3+s}\times\mathbb{W}_{3+s}}\lVert
h,f\rVert_{\mathbb{Y}_{\max\\{{3,2+\lfloor d/2\rfloor}\\}}},$ (4.3.32)

where the implicit constant depends only on $s$, $d$, and $\mathcal{Q}$.

###### Proof.

Thanks to Corollary A.5 and Proposition A.23, we make take
$0<\varepsilon\leqslant\rho_{\mathrm{S}s}$ sufficiently small to guarantee
that $\uptau_{0}\in B_{H^{r}}(0,\widetilde{\rho}_{\mathrm{S}s})\cap
H^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d\times d})$, $\beta_{0}\in
B_{H^{r}}(0,\widetilde{\rho}_{\mathrm{S}s})\cap H^{\infty}(\mathbb{R}^{d})$,
and $\upvarphi_{0}\in B_{H^{r}}(0,\widetilde{\rho}_{\mathrm{S}s})\cap
H^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$, where the parameter
$\widetilde{\rho}_{\mathrm{S}s}\in\mathbb{R}^{+}$ is granted by Proposition
4.10. Therefore, by invoking this latter result, we obtain the first item and
estimate (4.3.5). To swap to the claimed estimate (4.3.32), one simply need
note that another application of Corollary A.5 and Proposition A.23 permits
one the bound

$\lVert
v_{0},\eta_{0},\beta_{0},\uptau_{0},\upvarphi_{0}\rVert_{\mathbb{X}_{3+s}\times
H^{3+s}\times H^{3+s}\times H^{3+s}}\lesssim\lVert
v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{X}_{3+s}\times\mathbb{W}_{3+s}}.$
(4.3.33)

∎

######  Remark 4.12.

We are free to assume that the sequence
$\\{{\rho_{\mathrm{S}s}}\\}_{s=3+\lfloor
d/2\rfloor}^{\infty}\subset(0,\rho_{\mathrm{WD}}]$ generated by by Corollary
4.11 is nonincreasing.

We are now ready for the first of our main theorems of our linear analysis.

######  Theorem 4.13 (Synthesis III, omnisonic case).

Set $b=\max\\{{3,2+\lfloor d/2\rfloor}\\}$, and let
$\overline{\Uppsi}:\mathbb{E}_{b}\to\mathbb{F}_{b-3}$ be the map given in
(3.2.4), which is well-defined and smooth in light of the first item of
Theorem 3.6. Let $\mathbb{N}\ni s\geqslant b$ and
$\rho_{\mathrm{S}s}\in\mathbb{R}^{+}$ be given by Corollary 4.11. Suppose that
$0<\varepsilon\leqslant\min\\{\rho_{\mathrm{S}s},\rho_{\mathrm{WD}}\\}$
(recall that the latter is defined in Lemma 3.7), $I\Subset\mathbb{R}^{+}$ and
$J,K\Subset\mathbb{R}$ are bounded open intervals, and

$\mathbf{p}_{0}=(\gamma_{0},\mu_{0},\sigma_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell},w_{0},\xi_{0},v_{0},\eta_{0})\in[I\times
J\times K\times B_{\mathbb{W}_{r}}(0,\varepsilon)\times
B_{\mathbb{X}_{r}\times\mathbb{X}_{r}}(0,\varepsilon)]\cap\mathbb{E}_{\infty},$
(4.3.34)

where $r\in\mathbb{N}$ is as in Definition 4.1. We set
$\beta_{0}=\mathrm{Tr}\sum_{i=0}^{\ell}\eta_{0}^{i}\tau_{0i}\in
H^{\infty}(\mathbb{R}^{d})$. The following hold.

  1. (1)

The restriction of $D\overline{\Uppsi}(\mathbf{p}_{0})$ to
$\mathbb{R}^{3}\times\mathbb{W}_{s}\times\mathbb{X}_{s}\times[{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}]$
is well-defined and takes values in $\mathbb{F}_{s}$, and the induced linear
map

$D\overline{\Uppsi}(\mathbf{p}_{0}):\mathbb{R}^{3}\times\mathbb{W}_{s}\times\mathbb{X}_{s}\times[{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}]\to\mathbb{F}_{s}$
(4.3.35)

is an isomorphism.

  2. (2)

Assume that $\mathbf{f}\in\mathbb{F}_{s}$ and
$\mathbf{e}\in\mathbb{R}^{3}\times\mathbb{W}_{s}\times\mathbb{X}_{s}\times[{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}]$
is defined via
$\mathbf{e}=(D\overline{\Uppsi}(\mathbf{p}_{0}))^{-1}\mathbf{f}$. We then have
the tame estimate

$\lVert\mathbf{e}\rVert_{\mathbb{E}_{s}}\lesssim\lVert\mathbf{f}\rVert_{\mathbb{F}_{s}}+\langle\lVert\mathbf{p}_{0}\rVert_{\mathbb{E}_{s+3}}\rangle\lVert\mathbf{f}\rVert_{\mathbb{F}_{b}}$
(4.3.36)

for an implicit constant depending only on $s$, $d$, $I$, $J$, and $K$.

###### Proof.

We break the proof into two steps. In the first step we use (3.2.12) to view
$\Upupsilon:\mathbb{R}^{3}\times[\mathbb{X}_{b}\times\mathbb{W}_{b}]\to\mathbb{Y}_{b-3}\times\mathbb{W}_{b-3}$,
defined in (3.2.3), as a smooth map with a domain comprised of two factors
(namely, $\mathbb{R}^{3}$ and $\mathbb{X}_{b}\times\mathbb{W}_{b}$) and study
the derivative with respect to each factor evaluated at the point
$\mathbf{q}_{0}=(\gamma_{0},\mu_{0},\sigma_{0},v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell})$.
We begin with
$D_{1}\Upupsilon(\mathbf{q}_{0}):\mathbb{R}^{3}\to\mathbb{Y}_{b-3}\times\mathbb{W}_{b-3}$.
The inclusion (3.2.12) and the definition of strong $3-$tame smoothness (see
Definition A.9) requires that

$\lVert
D_{1}\Upupsilon(\mathbf{q}_{0})(\gamma,\mu,\sigma)\rVert_{\mathbb{Y}_{t-3}\times\mathbb{W}_{t-3}}=\lVert
D\Upupsilon(\mathbf{q}_{0})(\gamma,\mu,\sigma,0,0,0,0,0)\rVert_{\mathbb{Y}_{t-3}\times\mathbb{W}_{t-3}}\lesssim\langle\lVert\mathbf{q}_{0}\rVert_{\mathbb{R}^{3}\times\mathbb{W}_{t}\times\mathbb{X}_{t}}\rangle\left\lVert\gamma,\mu,\sigma\right\rVert_{\mathbb{R}^{3}}$
(4.3.37)

for every $t\geqslant b$, and so we actually have that

$D_{1}\Upupsilon(\mathbf{q}_{0})(\gamma,\mu,\sigma)\in\mathbb{Y}_{\infty}\times\mathbb{W}_{\infty}$
(4.3.38)

for all $(\gamma,\mu,\sigma)\in\mathbb{R}^{3}$.

Next, we turn our attention to
$D_{2}\Upupsilon(\mathbf{q}_{0}):\mathbb{X}_{b}\times\mathbb{W}_{b}\to\mathbb{Y}_{b-3}\times\mathbb{W}_{b-3}$.
By assumption, we have that $s\geqslant b$, and so
${{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}\hookrightarrow\mathbb{X}_{b}$,
which means that the space
${{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}\times\mathbb{W}_{s}$
lies within the domain of $D_{2}\Upupsilon(\mathbf{q}_{0})$. We claim that for
$s\geqslant b$ and $\varepsilon\leqslant\rho_{\mathrm{S}s}$ we have the
inclusion
$D_{2}\Upupsilon(\mathbf{q}_{0})({{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}\times\mathbb{W}_{s})\subseteq\mathbb{Y}_{s}\times\mathbb{W}_{s}$
and that the induced map

$D_{2}\Upupsilon(\mathbf{q}_{0}):{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}\times\mathbb{W}_{s}\to\mathbb{Y}_{s}\times\mathbb{W}_{s}.$
(4.3.39)

is a bounded isomorphism obeying the tame bound

$\lVert
v,\eta,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell}\rVert_{{{}_{\mu_{0},\sigma_{0}}}\mathbb{X}_{s}\times\mathbb{W}_{s}}\lesssim\lVert
D_{2}\Upupsilon(\mathbf{q}_{0})[v,\eta,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell}]\rVert_{\mathbb{Y}_{s}\times\mathbb{W}_{s}}\\\
+\lVert
v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{X}_{3+s}\times\mathbb{W}_{3+s}}\lVert
D_{2}\Upupsilon(\mathbf{q}_{0})[v,\eta,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell}]\rVert_{\mathbb{Y}_{b}\times\mathbb{W}_{b}}.$
(4.3.40)

We begin the proof of the claim by establishing the stated inclusion. Suppose
that

$D_{2}\Upupsilon(\mathbf{q}_{0})[v,\eta,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell}]=(h,f,\\{{\widetilde{\tau}_{i}}\\}_{i=0}^{\ell},\\{{\widetilde{\varphi}_{i}}\\}_{i=0}^{\ell})$
(4.3.41)

for
$(v,\eta,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}\times\mathbb{W}_{s}$.
Since $\Upupsilon$ acts as the identity map on the $\mathbb{W}_{s}$ factor, we
have that

$(\\{{\widetilde{\tau}_{i}}\\}_{i=0}^{\ell},\\{{\widetilde{\varphi}_{i}}\\}_{i=0}^{\ell})=(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathbb{W}_{s}.$
(4.3.42)

The remaining piece of (4.3.41) manifests as the equation

$D_{2}\Uppsi(\gamma_{0},\mu_{0},\sigma_{0},v_{0},\eta_{0})[v,\eta]+\sum_{i=0}^{\ell}(0,\eta_{0}^{i}(\mathrm{Tr}\tau_{0i}I+(\mu_{0}+\sigma_{0})\tau_{0i})\nabla\eta+i\eta_{0}^{i-1}(\tau_{0i}\nabla\eta_{0}+\varphi_{0i})\eta)\\\
=(h,f)-\sum_{i=0}^{\ell}\eta_{0}^{i}(0,(\mathrm{Tr}\tau_{i}I+(\mu_{0}+\sigma_{0})\tau_{i})\nabla\eta_{0}+\varphi_{i}),$
(4.3.43)

which, after defining $\uptau_{0}$, $\beta_{0}$, and $\upvarphi_{0}$ as in
(4.3.31) is equivalent to

$L^{\gamma,\mu,\sigma}_{v_{0},\eta_{0},\beta_{0}}(v,\eta)+(0,\upvarphi_{0}\eta+(\mu_{0}+\sigma_{0})\uptau_{0}\nabla\eta)+(0,R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta])\\\
=(h,f)-\sum_{i=0}^{\ell}\eta_{0}^{i}(0,(\mathrm{Tr}\tau_{i}I+(\mu_{0}+\sigma_{0})\tau_{i})\nabla\eta_{0}+\varphi_{i}),$
(4.3.44)

thanks to the the first item of Proposition 3.8 and Definition 4.9. Corollary
4.11 then guarantees that
$(h,f)-\sum_{i=0}^{\ell}\eta_{0}^{i}(0,(\mathrm{Tr}\tau_{i}I+(\mu_{0}+\sigma_{0})\tau_{i})\nabla\eta_{0}+\varphi_{i})\in\mathbb{Y}_{s}$,
from which it immediately follows that $(h,f)\in\mathbb{Y}_{s}$. Hence,
$D_{2}\Upupsilon(\mathbf{q}_{0})({{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}\times\mathbb{W}_{s})\subseteq\mathbb{Y}_{s}\times\mathbb{W}_{s}$.

To complete the proof of the claim, it remains to show that (4.3.39) is an
isomorphism obeying the estimate (4.3.40). Once more we consider the equation
(4.3.41), but now with the aim of solving it for given data
$(h,f,\\{{\widetilde{\tau}_{i}}\\}_{i=0}^{\ell},\\{{\widetilde{\varphi}_{i}}\\}_{i=0}^{\ell})\in\mathbb{Y}_{s}\times\mathbb{W}_{s}$.
The identity (4.3.42) remains true, and so again we reduce to (4.3.44).
Applying Corollary 4.11, we obtain the existence and uniqueness of
$(v,\eta)\in{{}^{\gamma_{0},\mu_{0},\sigma_{0}}_{v_{0},\eta_{0},\beta_{0}}}\mathbb{X}_{s}$
satisfying (4.3.44). Furthermore, the proposition guarantees that the solution
obeys the bound

$\lVert
v,\eta\rVert_{{{}_{\mu_{0},\sigma_{0}}}\mathbb{X}_{s}}\lesssim\Big{\lVert}(h,f)-\sum_{i=0}^{\ell}\eta_{0}^{i}(0,(\mathrm{Tr}\tau_{i}I+(\mu_{0}+\sigma_{0})\tau_{i})\nabla\eta_{0}+\varphi_{i})\Big{\rVert}_{\mathbb{Y}_{s}}\\\
+\lVert
v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{X}_{s+3}\times\mathbb{W}_{s+3}}\Big{\lVert}(h,f)-\sum_{i=0}^{\ell}\eta_{0}^{i}(0,(\mathrm{Tr}\tau_{i}I+(\mu_{0}+\sigma_{0})\tau_{i})\nabla\eta_{0}+\varphi_{i})\Big{\rVert}_{\mathbb{Y}_{b}},$
(4.3.45)

but thanks to Corollary A.5 and Proposition A.23, estimate (4.3.45)
immediately provides the bound

$\lVert v,\eta\rVert_{{{}_{\mu_{0},\sigma_{0}}}\mathbb{X}_{s}}\lesssim\lVert
h,f,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{Y}_{s}\times\mathbb{W}_{s}}\\\
+\lVert
v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{X}_{s+3}\times\mathbb{W}_{s+3}}\lVert
h,f,\tau,\\{{\varphi_{i}}\\}_{i=0}^{\ell},\\{{\psi_{i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{Y}_{b}\times\mathbb{W}_{b}}.$
(4.3.46)

As usual, the implicit constants here depend only on $s$, $d$, $I$, $J$, and
$K$. Upon combining the above existence and uniqueness results, (4.3.42), and
estimate (4.3.46) we then readily deduce that (4.3.39) is an isomorphism
obeying the estimate (4.3.40), which completes the proof of the claim and
concludes the first step of the proof.

We now move on to the second step of the proof, in which the goal is to attain
the theorem statement by building on the results of the first step. Let
$\mathbf{p}_{0}$ be as in (4.3.34) and let $\mathbf{q}_{0}$ be as in the first
step. The system

$D\overline{\Uppsi}(\mathbf{p}_{0})(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},w,\xi,v,\eta)=(\widetilde{\gamma},\widetilde{\mu},\widetilde{\sigma},h,f,\\{{\widetilde{\tau}_{i}}\\}_{i=0}^{\ell},\\{{\widetilde{\varphi}_{i}}\\}_{i=0}^{\ell},u,\zeta)$
(4.3.47)

is equivalent to the following system of four equations:

$(\gamma,\mu,\sigma)=(\widetilde{\gamma},\widetilde{\mu},\widetilde{\sigma}),$
(4.3.48)
$D_{1}\Upupsilon(\mathbf{q}_{0})(\gamma,\mu,\sigma)+D_{2}\Upupsilon(\mathbf{q}_{0})(v,\eta,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})=(h,f,\\{{\widetilde{\tau}_{i}}\\}_{i=0}^{\ell},\\{{\widetilde{\varphi}_{i}}\\}_{i=0}^{\ell}),$
(4.3.49) $w=\langle\mu_{0}\nabla\rangle^{2}v-u+2\mu\mu_{0}|\nabla|^{2}v_{0},$
(4.3.50)

and

$\xi=(1+(\mu_{0}+\sigma_{0})|\nabla|)\langle\sigma_{0}\nabla\rangle^{2}\eta-\zeta+(\mu+\sigma)|\nabla|\langle\sigma_{0}\nabla\rangle^{2}\eta_{0}+(1+(\mu_{0}+\sigma_{0})|\nabla|)\sigma\sigma_{0}|\nabla|^{2}\eta_{0}.$
(4.3.51)

For brevity we denote in what follows

$\mathbf{e}=(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},w,\xi,v,\eta)\text{
and
}\mathbf{f}=(\widetilde{\gamma},\widetilde{\mu},\widetilde{\sigma},h,f,\\{{\widetilde{\tau}_{i}}\\}_{i=0}^{\ell},\\{{\widetilde{\varphi}_{i}}\\}_{i=0}^{\ell},u,\zeta).$
(4.3.52)

If
$\mathbf{e}\in\mathbb{R}^{3}\times\mathbb{W}_{s}\times\mathbb{X}_{s}\times[{{}_{v_{0},\beta_{0},\eta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}]$,
then we use the first step on (4.3.49) and trivial estimates on (4.3.48),
(4.3.50), and (4.3.51) to see that $\mathbf{f}\in\mathbb{F}_{s}$ and

$\left\lVert\mathbf{f}\right\rVert_{\mathbb{F}_{s}}\lesssim\left\lVert\mathbf{e}\right\rVert_{\mathbb{W}_{s}\times\mathbb{X}_{s}\times[{{}_{v_{0},\beta_{0},\eta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}]}.$
(4.3.53)

This shows that the linear map (4.3.35) is well-defined and bounded.

Now suppose that $\mathbf{f}\in\mathbb{F}_{s}$. We seek to find a unique
$\mathbf{e}\in\mathbb{W}_{s}\times\mathbb{X}_{s}\times[{{}_{v_{0},\beta_{0},\eta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}]$
solving (4.3.47). The identity (4.3.48) uniquely selects
$(\gamma,\mu,\sigma)$, and then (4.3.49) and the work from the previous step
uniquely selects
$(v,\eta,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in{{}_{v_{0},\eta_{0},\beta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}_{s}\times\mathbb{W}_{s}$.
From here it’s a simple matter to use (4.3.50) and (4.3.51) to uniquely select
$(w,\xi)\in\mathbb{X}_{s}$. This process yields the desired $\mathbf{e}$ and
completes the proof of the first item. For the second item we employ (4.3.40)
to see that the pair $(v,\eta)$ determined by this process obeys the bound

$\lVert
v,\eta\rVert_{{{}_{\mu_{0},\sigma_{0}}}\mathbb{X}_{s}}\lesssim\lVert\gamma,\mu,\sigma,h,f,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{R}^{3}\times\mathbb{Y}_{s}\times\mathbb{W}_{s}}\\\
+\langle\lVert\mathbf{p}_{0}\rVert_{\mathbb{E}_{s+3}}\rangle\lVert\gamma,\mu,\sigma,h,f,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{R}^{3}\times\mathbb{Y}_{b}\times\mathbb{W}_{b}},$
(4.3.54)

which, together with (4.3.50) and (4.3.51), then provides the estimate

$\lVert
w,\xi\rVert_{\mathbb{X}_{s}}\lesssim\lVert\mathbf{f}\rVert_{\mathbb{F}_{s}}+\langle\lVert\mathbf{p}_{0}\rVert_{\mathbb{E}_{3+s}}\rangle\lVert\mathbf{f}\rVert_{\mathbb{F}_{b}}.$
(4.3.55)

Combining these two estimates with the trivial bounds obtained from (4.3.48)
then completes the proof of the second item. ∎

The remainder of this subsection is meant to provide subsonic analogs of the
previous results. In the following definition, we recall that the space
${{}_{\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}$ is defined in (3.1.13) and
the operator $P^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}$ is given in the first
item of Proposition 3.8.

######  Definition 4.14 (Operators and adapted domains, subsonic case).

Given $\gamma$, $\mu$, $\sigma$, $v_{0}$, and $\eta_{0}$ as in Definition 4.1
and $\mathbb{N}\ni s\geqslant 3$ we define the space
${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}}\mathbb{X}^{\mathrm{sub}}_{s}=\\{{(v,\eta)\in{{{}_{\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}}\;:\;P^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]\in\mathbb{Y}_{s}}\\}$
and equip it with the graph norm

$\lVert
v,\eta\rVert_{{{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}}\mathbb{X}^{\mathrm{sub}}_{s}}=\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}}+\lVert
P^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]\rVert_{\mathbb{Y}_{s}}.$
(4.3.56)

The following result is the subsonic refinement of Proposition 4.10, as such
the subscript $\mathrm{SS}$ is an acronym meaning ‘subsonic synthesis’.

######  Proposition 4.15 (Synthesis I, subsonic case).

Let $(\gamma,\mu,\sigma)\in\mathcal{Q}$ with $\mathcal{Q}=I\times J\times K$,
as in Definition 4.1, but suppose additionally that $I\Subset(0,1)$. For each
$\mathbb{N}\ni s\geqslant\max\\{{3,2+\lfloor d/2\rfloor}\\}$ there exists
$\mathbb{R}^{+}\ni\widetilde{\rho}_{\mathrm{SS}s}\leqslant\rho_{\mathrm{SAP}s}$,
depending only on $s$ and $\mathcal{Q}$, such that if
$\mathbb{R}^{+}\ni\varepsilon\in(0,\widetilde{\rho}_{\mathrm{SS}s}]$,

$(v_{0},\eta_{0},\uptau_{0},\upvarphi_{0})\in
B_{\mathbb{X}^{\mathrm{sub}}_{r}\times H^{r}\times
H^{r}}(0,\varepsilon)\cap(\mathbb{X}_{\infty}\times
H^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d\times d})\times
H^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})),$ (4.3.57)

with $r=\max\\{{6,5+\lfloor d/2\rfloor}\\}$, then the following hold.

  1. (1)

The following linear map is well-defined and bounded:

${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}}\mathbb{X}^{\mathrm{sub}}_{s}\ni(v,\eta)\mapsto
P^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]+(0,\upvarphi_{0}\eta+\uptau_{0}\nabla\eta)+(0,R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta])\in\mathbb{Y}_{s}.$
(4.3.58)

Moreover, this map is an isomorphism.

  2. (2)

For all $(h,f)\in\mathbb{Y}_{s}$ the unique
$(v,\eta)\in{{{}_{v_{0},\eta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}}$
satisfying the system

$P^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]+(0,\upvarphi_{0}\eta+\uptau_{0}\nabla\eta)+(0,R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta])=(h,f)$
(4.3.59)

obeys the tame estimate

$\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\uptau_{0},\upvarphi_{0}\rVert_{\mathbb{X}^{\mathrm{sub}}_{2+s}\times
H^{2+s}\times H^{2+s}}\lVert h,f\rVert_{\mathbb{Y}_{\max\\{{3,2+\lfloor
d/2\rfloor}\\}}},$ (4.3.60)

where the implicit constant depends only on $s$, $d$, and $\mathcal{Q}$.

###### Proof.

The proof exactly follows the strategy we employed to prove Proposition 4.10,
namely a double use of the method of continuity. As such, we will only provide
a sketch of the argument here.

It is routine to verify that the map (4.3.58) is well-defined and bounded. In
order to see that this map is an isomorphism obeying estimate (4.3.60), we
again work in two steps. In the first step we apply the method continuity to
an auxiliary homotopy of operators, which are defined as follows. For
$\mathbb{N}\ni s\geqslant 3$ we define the space
${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}}\widetilde{\mathbb{X}^{\mathrm{sub}}_{s}}=\\{{(v,\eta)\in{{}_{\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}\;:\;Q^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]\in\mathbb{Y}_{s}}\\}$,
where $Q^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}$ is the operator from the second
item of Proposition 3.8, and equip it with the graph norm

$\lVert
v,\eta\rVert_{{{}_{v_{0},\eta_{0}}^{\gamma,\mu,\sigma}}\widetilde{\mathbb{X}^{\mathrm{sub}}_{s}}}=\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}}+\lVert
Q^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]\rVert_{\mathbb{Y}_{s}}.$
(4.3.61)

The auxiliary mappings are then defined, for $t\in[0,1]$, to be

${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}}\widetilde{\mathbb{X}^{\mathrm{sub}}_{s}}\ni(v,\eta)\mapsto
Q^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]+t(0,\upvarphi_{0}\eta+\uptau_{0}\nabla\eta)+t(0,S^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v])\in\mathbb{Y}_{s}.$
(4.3.62)

By combining the a priori estimates of Theorem 4.6, the existence of Theorem
4.8, and the remainder estimates of Proposition 3.9 we verify that the
mappings (4.3.62) are well-defined and bounded and find
$\widetilde{\rho}_{\mathrm{SS}s}\in\mathbb{R}^{+}$ as in the proposition
statement such that for $\varepsilon\leqslant\widetilde{\rho}_{\mathrm{SS}s}$
the maps (4.3.62) obey the requisite tame closed range estimates uniformly in
$t\in[0,1]$. The method of continuity then applies, and we may indeed argue as
in the first step of the proof of Proposition 4.10.

The second step of the proof builds on the previous step by considering a
different operator homotopy, namely

${{}^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}}\mathbb{X}^{\mathrm{sub}}_{s}\ni(v,\eta)\mapsto
P^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta]+(0,\upvarphi_{0}\eta+\uptau_{0}\nabla\eta)+t(0,R^{\gamma,\mu,\sigma}_{v_{0},\eta_{0}}[v,\eta])\in\mathbb{Y}_{s},\quad
t\in[0,1].$ (4.3.63)

Supplementing the previous step with more remainder estimates from Proposition
3.9 allows us to take $\uprho_{\mathrm{SS}s}$ smaller, if necessary, and
deduce the claimed bounds (4.3.60) and also invoke the method of continuity on
(4.3.63), leading us to a complete proof of the first and second items. ∎

######  Corollary 4.16 (Synthesis II, subsonic case).

Let $(\gamma,\mu,\sigma)\in\mathcal{Q}$ with $\mathcal{Q}=I\times J\times K$,
as in Definition 4.1, but suppose additionally that $I\Subset(0,1)$. For each
$\mathbb{N}\ni s\geqslant\max\\{{3,2+\lfloor d/2\rfloor}\\}$ there exists
$\mathbb{R}^{+}\ni\rho_{\mathrm{SS}s}\leqslant\widetilde{\rho}_{\mathrm{SS}s}$,
depending only on $s$ and $\mathcal{Q}$, such that if
$\mathbb{R}^{+}\ni\varepsilon\in(0,\rho_{\mathrm{SS}s}]$ and
$(v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell})$
satisfy (4.3.30) then the following hold.

  1. (1)

Upon defining $\uptau_{0}$, $\beta_{0}$, and $\upvarphi_{0}$ as in (4.3.31),
we have that the linear map (4.3.58) is well-defined, bounded, and an
isomorphism.

  2. (2)

For all $(h,f)\in\mathbb{Y}_{s}$ the unique
$(v,\eta)\in{{{}_{v_{0},\eta_{0}}^{\gamma,\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}}$
satisfying (4.3.59) obeys the tame estimate

$\lVert
v,\eta\rVert_{{{}_{\mu,\sigma}}\mathbb{X}^{\mathrm{sub}}_{s}}\lesssim\lVert
h,f\rVert_{\mathbb{Y}_{s}}+\lVert
v_{0},\eta_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell}\rVert_{\mathbb{X}^{\mathrm{sub}}_{2+s}\times\mathbb{W}_{2+s}}\lVert
h,f\rVert_{\mathbb{Y}_{\max\\{{3,2+\lfloor d/2\rfloor}\\}}},$ (4.3.64)

where the implicit constant depends only on $s$, $d$, and $\mathcal{Q}$.

###### Proof.

We derive these claims from Proposition (4.15) by arguing as in the proof of
Corollary 4.11. ∎

######  Theorem 4.17 (Synthesis III, subsonic case).

Set $b=\max\\{{3,2+\lfloor d/2\rfloor}\\}$, and let
$\overline{\Uppsi}^{\mathrm{sub}}:\mathbb{E}_{b}\to\mathbb{F}_{b-2}$ be the
map given in (3.2.7), which is well-defined and smooth in light of the second
item of Theorem 3.6. Let $\mathbb{N}\ni s\geqslant b$ and
$\rho_{\mathrm{SS}s}\in\mathbb{R}^{+}$ be given by Corollary 4.16. Suppose
that $0<\varepsilon\leqslant\min\\{\rho_{\mathrm{SS}s},\rho_{\mathrm{WD}}\\}$
(recall that the latter is defined in Lemma 3.7), $I\Subset(0,1)$ and
$J,K\Subset\mathbb{R}$ are bounded open intervals, and

$\mathbf{p}_{0}=(\gamma_{0},\mu_{0},\sigma_{0},\\{{\tau_{0i}}\\}_{i=0}^{\ell},\\{{\varphi_{0i}}\\}_{i=0}^{\ell},w_{0},\xi_{0},v_{0},\eta_{0})\in[I\times
J\times K\times B_{\mathbb{W}_{r}}(0,\varepsilon)\times
B_{\mathbb{X}^{\mathrm{sub}}_{r}\times\mathbb{X}^{\mathrm{sub}}_{r}}(0,\varepsilon)]\cap\mathbb{E}^{\mathrm{sub}}_{\infty},$
(4.3.65)

where $r=\max\\{{6,5+\lfloor d/2\rfloor}\\}$. The following hold.

  1. (1)

The restriction of $D\overline{\Uppsi}^{\mathrm{sub}}(\mathbf{p}_{0})$ to
$\mathbb{R}^{3}\times\mathbb{W}_{s}\times\mathbb{X}^{\mathrm{sub}}_{s}\times[{{}_{v_{0},\eta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}^{\mathrm{sub}}_{s}]$
is well-defined and takes values in $\mathbb{F}^{\mathrm{sub}}_{s}$, and the
induced linear map

$D\overline{\Uppsi}^{\mathrm{sub}}(\mathbf{p}_{0}):\mathbb{R}^{3}\times\mathbb{W}_{s}\times\mathbb{X}^{\mathrm{sub}}_{s}\times[{{}_{v_{0},\eta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}^{\mathrm{sub}}_{s}]\to\mathbb{F}^{\mathrm{sub}}_{s}$
(4.3.66)

is an isomorphism.

  2. (2)

Assume that $\mathbf{f}\in\mathbb{F}^{\mathrm{sub}}_{s}$ and
$\mathbf{e}\in\mathbb{R}^{3}\times\mathbb{W}_{s}\times\mathbb{X}^{\mathrm{sub}}_{s}\times[{{}_{v_{0},\eta_{0}}^{\gamma_{0},\mu_{0},\sigma_{0}}}\mathbb{X}^{\mathrm{sub}}_{s}]$
is defined via
$\mathbf{e}=(D\overline{\Uppsi}^{\mathrm{sub}}(\mathbf{p}_{0}))^{-1}\mathbf{f}$.
We then have the tame estimate

$\lVert\mathbf{e}\rVert_{\mathbb{E}^{\mathrm{sub}}_{s}}\lesssim\lVert\mathbf{f}\rVert_{\mathbb{F}^{\mathrm{sub}}_{s}}+\langle\lVert\mathbf{p}_{0}\rVert_{\mathbb{E}^{\mathrm{sub}}_{s+2}}\rangle\lVert\mathbf{f}\rVert_{\mathbb{F}^{\mathrm{sub}}_{b}}$
(4.3.67)

for an implicit constant depending only on $s$, $d$, $I$, $J$, and $K$.

###### Proof.

The same strategy used in the proof of Theorem 4.13 applies here, with minor
modifications; for instance, instead of the omnisonic results such as
Corollary 4.11, we instead invoke the corresponding subsonic result from
Corollary 4.16. In the interest of brevity, we omit further details. ∎

##  5\. Conclusion

In this section we piece together the nonlinear analysis of Section 3 and the
linear analysis of Section 4 along with the Nash-Moser inverse function
theorem of Appendix A.3 to prove Theorems 3 and 4 along with Corollaries 5.3
and 5.4.

###  5.1. Abstract construction

In the following first formulations of the main theorem for the omnisonic and
subsonic cases, we will utilize the following variations on the main nonlinear
operators $\overline{\Uppsi}$ and $\overline{\Uppsi}^{\mathrm{sub}}$, which we
recall are defined in (3.2.4) and (3.2.7), respectively. For
$b=\max\\{{3,2+\lfloor d/2\rfloor}\\}\in\mathbb{N}$ and any fixed
$\mathfrak{q}=(\overline{\gamma},\overline{\mu},\overline{\sigma})\in\mathbb{R}^{+}\times\mathbb{R}^{2}$
we define the operators
$\overline{\Uppsi}_{\mathfrak{q}}:\mathbb{E}_{b}\to\mathbb{F}_{b-3}$ and
$\overline{\Uppsi}_{\mathfrak{q}}^{\mathrm{sub}}:\mathbb{E}^{\mathrm{sub}}_{b}\to\mathbb{F}^{\mathrm{sub}}_{b-2}$
via the assignments

$\overline{\Uppsi}_{\mathfrak{q}}(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},w,\xi,v,\eta)=\overline{\Uppsi}(\gamma+\overline{\gamma},\mu+\overline{\mu},\sigma+\overline{\sigma},\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},w,\xi,v,\eta)-(\overline{\gamma},\overline{\mu},\overline{\sigma},0,0,0)$
(5.1.1)

and

$\overline{\Uppsi}^{\mathrm{sub}}_{\mathfrak{q}}(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},w,\xi,v,\eta)=\overline{\Uppsi}^{\mathrm{sub}}(\gamma+\overline{\gamma},\mu+\overline{\mu},\sigma+\overline{\sigma},\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},w,\xi,v,\eta)-(\overline{\gamma},\overline{\mu},\overline{\sigma},0,0,0)$
(5.1.2)

The utility of these definitions is that
$\overline{\Uppsi}_{\mathfrak{q}}(0)=\overline{\Uppsi}^{\mathrm{sub}}_{\mathfrak{q}}(0)=0$.

######  Theorem 5.1 (Abstract formulation, omnisonic case).

Let $\varsigma=19+2\lfloor d/2\rfloor$. For each
$\mathfrak{q}=(\overline{\gamma},\overline{\mu},\overline{\sigma})\in\mathbb{R}^{+}\times\mathbb{R}^{2}$,
there exists a nonincreasing sequence
$\\{{\uprho_{s}(\mathfrak{q})}\\}_{s=0}^{\infty}\subset(0,\min\\{{\rho_{\mathrm{WD}},\overline{\gamma}/2}\\}]$
and $\upkappa(\mathfrak{q})\in\mathbb{R}^{+}$ with the following properties.

  1. (1)

_Existence and uniqueness:_ Given $\mathbf{f}\in
B_{\mathbb{F}_{\varsigma}}(0,\uprho_{0}(\mathfrak{q}))$ there exists a unique
$\mathbf{e}\in
B_{\mathbb{E}_{\varsigma}}(0,\upkappa(\mathfrak{q})\uprho_{0}(\mathfrak{q}))$
such that $\overline{\Uppsi}_{\mathfrak{q}}(\mathbf{e})=\mathbf{f}$. This
induces the local inverse map

$\overline{\Uppsi}^{-1}_{\mathfrak{q}}:B_{\mathbb{F}_{\varsigma}}(0,\uprho_{0}(\mathfrak{q}))\to\overline{\Uppsi}^{-1}_{\mathfrak{q}}(B_{\mathbb{F}_{\varsigma}}(0,\uprho_{0}(\mathfrak{q})))\subseteq
B_{\mathbb{E}_{\varsigma}}(0,\upkappa(\mathfrak{q})\uprho_{0}(\mathfrak{q})).$
(5.1.3)

  2. (2)

_Higher regularity, given low norm smallness:_ If $s\in\mathbb{N}$ and
$\mathbf{f}\in\mathbb{F}_{s+\varsigma}\cap
B_{\mathbb{F}_{\varsigma}}(0,\uprho_{s}(\mathfrak{q}))$, then we have the
inclusion
$\overline{\Uppsi}_{\mathfrak{q}}^{-1}(\mathbf{f})\in\mathbb{E}_{s+\varsigma}$
and the tame estimate

$\lVert\overline{\Uppsi}_{\mathfrak{q}}^{-1}(\mathbf{f})\rVert_{\mathbb{E}_{\varsigma+s}}\lesssim\lVert\mathbf{f}\rVert_{\mathbb{F}_{\varsigma+s}},$
(5.1.4)

for an implicit constant depending only on $s$, the dimension, and
$\mathfrak{q}$.

  3. (3)

_Continuous dependence:_ For every $s\in\mathbb{N}$, the restricted map

$\overline{\Uppsi}_{\mathfrak{q}}^{-1}:\mathbb{F}_{s+\varsigma}\cap
B_{\mathbb{F}_{\varsigma}}(0,\uprho_{s}(\mathfrak{q}))\to\mathbb{E}_{\varsigma+s}\cap
B_{\mathbb{E}_{\varsigma}}(0,\upkappa(\mathfrak{q})\uprho_{0}(\mathfrak{q}))$
(5.1.5)

is continuous with respect to the norms on $\mathbb{F}_{s+\varsigma}$ and
$\mathbb{E}_{s+\varsigma}$.

###### Proof.

Our aim is to show that the hypotheses of Theorem A.19 are satisfied by the
map $\overline{\Uppsi}_{\mathfrak{q}}$, defined in (5.1.1), acting between the
Banach scales $\boldsymbol{\mathbb{E}}$ and $\boldsymbol{\mathbb{F}}$, which
we recall are defined in (3.1.7). Evidently, these Banach scales consist of
reflexive (in fact, Hilbert) spaces and, thanks to Lemma 3.1, we know that
$\boldsymbol{\mathbb{E}}$ and $\boldsymbol{\mathbb{F}}$ are LP-smoothable in
the sense of Definition A.15.

We now assess how the triple
$(\boldsymbol{\mathbb{E}},\boldsymbol{\mathbb{F}},\overline{\Uppsi}_{\mathfrak{q}})$
satisfies the LRI mapping hypotheses of Definition A.18 with parameters

$(\mu,r,R)=(3,6+\lfloor d/2\rfloor,33+3\lfloor d/2\rfloor),$ (5.1.6)

which obey the inequalities $1\leqslant\mu\leqslant r<R<\infty$ and
$2(r+\mu)+1<(r+R)/2$ as well as the identity $\varsigma=2(r+\mu)+1$. We then
set $\delta_{r}=\min\\{{\rho_{\mathrm{WD}},\overline{\gamma}/2}\\}$. By
heeding to these definitions, Theorem 3.6 and Corollary A.12, we see that the
first and second items of Definition A.18 are satisfied. The satisfaction of
the third and final item of Definition A.18 is a consequence of Theorem 4.13
and Remark 4.12 with $\delta_{R}=\min\\{{\delta_{r},\rho_{\mathrm{S}R}}\\}$
(the parameter $\rho_{\mathrm{S}R}$ is granted by the theorem) and
$I=(\overline{\gamma}/2,2\overline{\gamma})$,
$J=(-2|\overline{\mu}|,2|\overline{\mu}|)$, and
$K=(-2|\overline{\sigma}|,2|\overline{\sigma}|)$.

Thus, the hypotheses of Theorem A.19 are satisfied and we therefore obtain
$\varepsilon=\uprho_{0}(\mathfrak{q})$ such that the first item holds, the map
(5.1.3) is continuous, and we have the estimate
$\lVert\overline{\Uppsi}_{\mathfrak{q}}^{-1}(\mathbf{f})\rVert_{\mathbb{E}_{\varsigma}}\lesssim\lVert\mathbf{f}\rVert_{\mathbb{F}_{\varsigma}}$
for $\mathbf{f}\in B_{\mathbb{F}_{\varsigma}}(0,\uprho_{0}(\mathfrak{q}))$.

Finally, suppose that $s\in\mathbb{N}$. We again apply the Nash-Moser theorem,
but this time with the parameter triple $(\mu,r,\widetilde{R})=(\mu,r,R+s)$
and
$\delta_{\widetilde{R}}=\min\\{{\delta_{R},\rho_{\mathrm{S}\widetilde{R}}}\\}$
and let $\uprho_{s}(\mathfrak{q})\in(0,\rho_{\mathrm{WD}}]$ denote the
smallness parameter $\varepsilon$ provided by this application of Theorem
A.19. By local uniqueness, we are thus granted the second and third items. ∎

We are now interested in the subsonic sharpening of Theorem 5.1, which we
state and prove below.

######  Theorem 5.2 (Abstract formulation, subsonic case).

Let $\varpi=17+2\lfloor d/2\rfloor$. For each
$\mathfrak{q}=(\overline{\gamma},\overline{\mu},\overline{\sigma})\in(0,1)\times\mathbb{R}^{2}$,
there exists nonincreasing
$\\{{\uprho^{\mathrm{sub}}_{s}(\mathfrak{q})}\\}_{s=0}^{\infty}\subset(0,\min\\{{\rho_{\mathrm{WD}},\overline{\gamma}/2,(1-\overline{\gamma})2}\\}]$
and $\upkappa^{\mathrm{sub}}(\mathfrak{q})\in\mathbb{R}^{+}$ with the
following properties.

  1. (1)

_Existence and uniqueness:_ For $\mathbf{f}\in
B_{\mathbb{F}^{\mathrm{sub}}_{\varpi}}(0,\uprho^{\mathrm{sub}}_{0}(\mathfrak{q}))$
there exists a unique $\mathbf{e}\in
B_{\mathbb{E}^{\mathrm{sub}}_{\varpi}}(0,\upkappa^{\mathrm{sub}}(\mathfrak{q})\uprho^{\mathrm{sub}}_{0}(\mathfrak{q}))$
such that
$\overline{\Uppsi}^{\mathrm{sub}}_{\mathfrak{q}}(\mathbf{e})=\mathbf{f}$. This
induces the local inverse map

$[\overline{\Uppsi}^{\mathrm{sub}}_{\mathfrak{q}}]^{-1}:B_{\mathbb{F}^{\mathrm{sub}}_{\varpi}}(0,\uprho^{\mathrm{sub}}_{0}(\mathfrak{q}))\to[\overline{\Uppsi}^{\mathrm{sub}}_{\mathfrak{q}}]^{-1}(B_{\mathbb{F}^{\mathrm{sub}}_{\varpi}}(0,\uprho^{\mathrm{sub}}_{0}(\mathfrak{q})))\subseteq
B_{\mathbb{E}^{\mathrm{sub}}_{\varpi}}(0,\upkappa^{\mathrm{sub}}(\mathfrak{q})\uprho^{\mathrm{sub}}_{0}(\mathfrak{q})).$
(5.1.7)

  2. (2)

_Higher regularity, given low norm smallness:_ If $s\in\mathbb{N}$ and
$\mathbf{f}\in\mathbb{F}^{\mathrm{sub}}_{s+\varpi}\cap
B_{\mathbb{F}^{\mathrm{sub}}_{\varpi}}(0,\uprho^{\mathrm{sub}}_{s}(\mathfrak{q}))$,
then we have the inclusion
$[\overline{\Uppsi}^{\mathrm{sub}}_{\mathfrak{q}}]^{-1}(\mathbf{f})\in\mathbb{E}^{\mathrm{sub}}_{s+\varpi}$
and the tame estimate

$\lVert[\overline{\Uppsi}^{\mathrm{sub}}_{\mathfrak{q}}]^{-1}(\mathbf{f})\rVert_{\mathbb{E}^{\mathrm{sub}}_{\varpi+s}}\lesssim\lVert\mathbf{f}\rVert_{\mathbb{F}^{\mathrm{sub}}_{\varpi+s}},$
(5.1.8)

for an implicit constant depending only on $s$, the dimension, and
$\mathfrak{q}$.

  3. (3)

_Continuous dependence:_ For every $s\in\mathbb{N}$, the restricted map

$[\overline{\Uppsi}_{\mathfrak{q}}^{\mathrm{sub}}]^{-1}:\mathbb{F}^{\mathrm{sub}}_{s+\varpi}\cap
B_{\mathbb{F}^{\mathrm{sub}}_{\varpi}}(0,\uprho^{\mathrm{sub}}_{s}(\mathfrak{q}))\to\mathbb{E}^{\mathrm{sub}}_{\varpi+s}\cap
B_{\mathbb{E}^{\mathrm{sub}}_{\varpi}}(0,\upkappa^{\mathrm{sub}}(\mathfrak{q})\uprho^{\mathrm{sub}}_{0}(\mathfrak{q}))$
(5.1.9)

is continuous with respect to the norms on
$\mathbb{F}^{\mathrm{sub}}_{s+\varpi}$ and
$\mathbb{E}^{\mathrm{sub}}_{s+\varpi}$.

###### Proof.

The proof is much the same as that of Theorem 5.1; the main differences are
that we use the second item of Theorem 3.6 in place of the first and Theorem
4.17 instead of Theorem 4.13. The reason for the difference between
$\varsigma$ and $\varpi$ is because the map $\overline{\Uppsi}$ is $3$-tame,
while $\overline{\Uppsi}^{\mathrm{sub}}$ is only $2$-tame. We omit further
details. ∎

###  5.2. PDE construction

The goal of this subsection is to take the abstract formulations of Theorems
5.1 and 5.2 and transform them into Theorems 3 and 4 along with a few
corollaries.

###### Proof of Theorem 3.

For each
$\mathfrak{q}=(\overline{\gamma},\overline{\mu},\overline{\sigma})\in\mathbb{R}^{+}\times[0,\infty)^{2}$
we may invoke Theorem 5.1 and acquire a nonincreasing sequence
$\\{{\uprho_{\mathfrak{s}}(\mathfrak{q})}\\}_{\mathfrak{s}=0}^{\infty}\subset(0,\min\\{{\rho_{\mathrm{WD}},\overline{\gamma}/2}\\}]$
and $\upkappa(\mathfrak{q})\in\mathbb{R}^{+}$ such that the various
conclusions of the theorem hold. We then define the open and relatively open
sets

$\mathcal{V}(\mathfrak{q})=B_{\mathbb{X}_{\varsigma}}(0,\upkappa(\mathfrak{q})\uprho_{0}(\mathfrak{q}))\subset\mathbb{X}_{\varsigma}.$
(5.2.1)

and, for $\mathbb{N}\ni s\geqslant\varsigma$,

$\mathcal{U}_{s}=\Big{\\{}{(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\bigcup_{\mathfrak{q}\in\mathbb{R}^{+}\times[0,\infty)^{2}}B_{\mathbb{R}^{3}\times\mathbb{W}_{\varsigma}}((\mathfrak{q},0),\uprho_{s-\varsigma}(\mathfrak{q}))\;:\mu\geqslant
0,\;\sigma\geqslant
0}\Big{\\}}\subset\mathbb{R}^{3}\times\mathbb{W}_{\varsigma}.$ (5.2.2)

The first properties of $\mathcal{U}_{s}$ and $\mathcal{V}(\mathfrak{q})$ in
(1.3.9) and (1.3.10) are now clear.

For the first item, we apply the first conclusion of Theorem 5.1. Indeed,
given
$(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathcal{U}_{\varsigma}$
we define $(v,\eta)\in\mathcal{V}(\gamma,\mu,\sigma)$ to be the last two
components of the tuple
$\overline{\Uppsi}_{(\gamma,\mu,\sigma)}^{-1}(0,0,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},0,0)$.
This grants us existence and uniqueness, and hence the second item is proved.

It remains to prove the second and third items, which we do in tandem. It is
clear from Theorem 5.1 that, given
$(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathcal{U}_{s}\cap(\mathbb{R}^{3}\times\mathbb{W}_{s})$,
the solution $(v,\eta)$ belongs also to $\mathbb{X}_{s}$ and the map

$\mathcal{U}_{s}\cap(\mathbb{R}^{3}\times\mathbb{W}_{s})\ni(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\mapsto(v,\eta)\in\mathbb{X}_{s}$
(5.2.3)

is continuous. It is at this point where we crucially utilize the last two
components of the mapping $\overline{\Uppsi}$ from (3.2.4). Indeed, we have
that
$\overline{\Uppsi}(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},w,\xi,v,\eta)=(\gamma,\mu,\sigma,0,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell},0,0)$
and, by arguments similar to the above, the map

$\mathcal{U}_{s}\cap(\mathbb{R}^{3}\times\mathbb{W}_{s})\ni(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\mapsto(w,\xi)\in\mathbb{X}_{s}$
(5.2.4)

is also continuous; however, we have the identities

$w=\langle\mu\nabla\rangle^{2}v\quad\text{and}\quad\xi=(1+(\mu+\sigma)|\nabla|)\langle\sigma\nabla\rangle^{2}\eta$
(5.2.5)

which, in turn, lead to the inclusion
$(v,\eta)\in{{}_{\mu,\sigma}}\mathbb{X}_{s}$ (which implies (1.3.12)) and the
continuity of the maps (1.3.13). ∎

###### Proof of Theorem 4.

The proof is very similar to the proof of Theorem 3; the main difference is
that we use Theorem 5.2 in place of Theorem 5.1. We omit further details for
the sake of brevity. ∎

We now shall enumerate some important consequences of Theorems 3 and 4.

######  Corollary 5.3 (Well-posedness for fixed physical parameters).

Let $(\gamma,\mu,\sigma)\in\mathbb{R}^{+}\times[0,\infty)^{2}$. The following
hold.

  1. (1)

There exists a nonincreasing sequence of open sets

$\\{{0}\\}^{\ell+1}\times\\{{0}\\}^{\ell+1}\subset\mathcal{U}_{s}(\gamma,\mu,\sigma)\subset(H^{\varsigma}(\mathbb{R}^{d};\mathbb{R}^{d\times
d}))^{\ell+1}\times(H^{\varsigma}(\mathbb{R}^{d};\mathbb{R}^{d}))^{\ell+1},\quad\mathbb{N}\ni
s\geqslant\varsigma$ (5.2.6)

such that the following hold.

    1. (a)

For all
$(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathcal{U}_{\varsigma}(\gamma,\mu,\sigma)$
there exists a unique $(v,\eta)\in\mathcal{V}(\gamma,\mu,\sigma)$ such that
system (1.1.9) is classically satisfied by the latter tuple with physical
parameters $(\gamma,\mu,\sigma)$ and data $\mathcal{F}$ determined from
$\eta$, $\mu+\sigma$, and
$(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})$ via the
omnisonic case of (1.1.12).

    2. (b)

If $\mathbb{N}\ni s\geqslant\varsigma$ and
$(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathcal{U}_{s}\cap((H^{s}(\mathbb{R}^{d};\mathbb{R}^{d\times
d}))^{\ell+1}\times(H^{s}(\mathbb{R}^{d};\mathbb{R}^{d}))^{\ell+1})$ then the
corresponding solution $(v,\eta)\in\mathcal{V}(\gamma,\mu,\sigma)$ produced by
the previous item in fact obeys the inclusions
$\langle\mu\nabla\rangle^{2}v\in H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$ and
$(1+(\mu+\sigma)|\nabla|)\langle\sigma\nabla\rangle^{2}\eta\in\mathcal{H}^{s}(\mathbb{R}^{d})$;
moreover the mapping from the data to the solution is continuous for the
spaces determined via this inclusion.

  2. (2)

If we assume additionally that $\gamma<1$, then there exists another
nonincreasing sequence of open sets

$\\{{0}\\}^{\ell+1}\times\\{{0}\\}^{\ell+1}\subset\mathcal{U}^{\mathrm{sub}}_{s}(\gamma,\mu,\sigma)\subset(H^{\varpi}(\mathbb{R}^{d};\mathbb{R}^{d\times
d}))^{\ell+1}\times(H^{\varpi}(\mathbb{R}^{d};\mathbb{R}^{d}))^{\ell+1},\quad\mathbb{N}\ni
s\geqslant\varpi$ (5.2.7)

such that the following hold.

    1. (a)

For all
$(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathcal{U}^{\mathrm{sub}}_{\varpi}$
there exists a unique
$(v,\eta)\in\mathcal{V}^{\mathrm{sub}}(\gamma,\mu,\sigma)$ such that system
(1.1.9) is classically satisfied by the latter tuple with physical parameters
$(\gamma,\mu,\sigma)$ and data $\mathcal{F}$ determined from $\eta$ and
$(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})$ via the
subsonic case of (1.1.12).

    2. (b)

If $\mathbb{N}\ni s\geqslant\varpi$ and
$(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathcal{U}^{\mathrm{sub}}_{s}\cap((H^{s}(\mathbb{R}^{d};\mathbb{R}^{d\times
d}))^{\ell+1}\times(H^{s}(\mathbb{R}^{d};\mathbb{R}^{d}))^{\ell+1})$ then the
corresponding solution
$(v,\eta)\in\mathcal{V}^{\mathrm{sub}}(\gamma,\mu,\sigma)$ produced by the
previous item in fact obeys the inclusions $\langle\mu\nabla\rangle^{2}v\in
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$ and
$\langle\sigma\nabla\rangle^{2}\eta\in\mathcal{H}^{1+s}(\mathbb{R}^{d})$;
moreover the mapping from the data to the solution is continuous for the
spaces determined via this inclusion.

###### Proof.

Throughout the proof we shall let $\mathrm{int}$ denote the topological
interior. We begin by defining the open sets of (5.2.6) and (5.2.7); we set:

$\mathcal{U}_{s}(\gamma,\mu,\sigma)=\mathrm{int}\\{{(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\;:\;(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathcal{U}_{s}}\\},$
(5.2.8)

and, if $\gamma<1$, similarly define

$\mathcal{U}^{\mathrm{sub}}_{s}(\gamma,\mu,\sigma)=\mathrm{int}\\{{(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\;:\;(\gamma,\mu,\sigma,\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\mathcal{U}^{\mathrm{sub}}_{s}}\\}.$
(5.2.9)

The sets in (5.2.8) and (5.2.9) are manifestly open and nonempty since due to
the fact that $\mathcal{U}_{s}$ and $\mathcal{U}_{s}^{\mathrm{sub}}$ are are
relatively open and satisfy the nondegeneracy conditions of Theorems 3 and 4,
respectively. With these open sets in hand, we see that the first and second
items of above are just restrictions of the conclusions of Theorems 3 and 4 to
the slices defined above. ∎

######  Corollary 5.4 (On the limits of vanishing viscosity and surface
tension).

Let $\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}}\in\mathbb{R}^{+}$. The
following hold.

  1. (1)

There exists a nonincreasing sequence of open sets

$\\{{0}\\}^{\ell+1}\times\\{{0}\\}^{\ell+1}\subset\widetilde{\mathcal{U}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})\subset(H^{\varsigma}(\mathbb{R}^{d};\mathbb{R}^{d\times
d}))^{\ell+1}\times(H^{\varsigma}(\mathbb{R}^{d};\mathbb{R}^{d}))^{\ell+1},\quad\mathbb{N}\ni
s\geqslant\varsigma$ (5.2.10)

such that the following hold.

    1. (a)

We have the inclusion
$\widetilde{\mathcal{U}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})\subseteq\bigcap_{0\leqslant\mu\leqslant
R_{\mathrm{visc}}}\bigcap_{0\leqslant\sigma\leqslant
R_{\mathrm{surf}}}\mathcal{U}_{s}(\gamma,\mu,\sigma)$.

    2. (b)

For each fixed
$(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\widetilde{\mathcal{U}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})\cap(H^{s}(\mathbb{R}^{d};\mathbb{R}^{d\times
d}))^{\ell+1}\times(H^{s}(\mathbb{R}^{d};\mathbb{R}^{d}))^{\ell+1})$, where
$\mathbb{N}\ni s\geqslant\varsigma$, there exists a continuous mapping

$[0,R_{\mathrm{visc}}]\times[0,R_{\mathrm{surf}}]\ni(\mu,\sigma)\mapsto(v,\eta)\in
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{s}(\mathbb{R}^{d};\mathbb{R}^{d})$
(5.2.11)

where $(v,\eta)$ solve (1.1.9) as in Theorem 3.

    3. (c)

The following derivatives of each mapping in (5.2.11) are well-defined and
continuous between the stated spaces:

$[0,R_{\mathrm{visc}}]\times[0,R_{\mathrm{surf}}]\ni(\mu,\sigma)\mapsto(\mu\nabla
v,\mu^{2}\nabla^{2}v)\in H^{s}(\mathbb{R}^{d};\mathbb{R}^{d^{2}})\times
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d^{3}})$ (5.2.12)

and

$[0,R_{\mathrm{visc}}]\times[0,R_{\mathrm{surf}}]\ni(\mu,\sigma)\mapsto(\mu+\sigma)(\nabla\eta,\sigma\nabla^{2}\eta,\sigma^{2}\nabla^{3}\eta)\in
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})\times
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d^{2}})\times
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d^{3}}).$ (5.2.13)

  2. (2)

If we assume additionally that $\gamma<1$, then there exists an additional
nonincreasing sequence of open sets

$\\{{0}\\}^{\ell+1}\times\\{{0}\\}^{\ell+1}\subset\widetilde{\mathcal{U}}^{\mathrm{sub}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})\subset(H^{\varpi}(\mathbb{R}^{d};\mathbb{R}^{d\times
d}))^{\ell+1}\times(H^{\varpi}(\mathbb{R}^{d};\mathbb{R}^{d}))^{\ell+1},\quad\mathbb{N}\ni
s\geqslant\varpi$ (5.2.14)

such that the following hold.

    1. (a)

We have the inclusion
$\widetilde{\mathcal{U}}^{\mathrm{sub}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})\subseteq\bigcap_{0\leqslant\mu\leqslant
R_{\mathrm{visc}}}\bigcap_{0\leqslant\sigma\leqslant
R_{\mathrm{surf}}}\mathcal{U}^{\mathrm{sub}}_{s}(\gamma,\mu,\sigma)$.

    2. (b)

For each
$(\\{{\tau_{i}}\\}_{i=0}^{\ell},\\{{\varphi_{i}}\\}_{i=0}^{\ell})\in\widetilde{\mathcal{U}}^{\mathrm{sub}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})\cap(H^{s}(\mathbb{R}^{d};\mathbb{R}^{d\times
d}))^{\ell+1}\times(H^{s}(\mathbb{R}^{d};\mathbb{R}^{d}))^{\ell+1})$, where
$\mathbb{N}\ni s\geqslant\varpi$, there exists a continuous mapping

$[0,R_{\mathrm{visc}}]\times[0,R_{\mathrm{surf}}]\ni(\mu,\sigma)\mapsto(v,\eta)\in
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d})\times\mathcal{H}^{1+s}(\mathbb{R}^{d};\mathbb{R}^{d})$
(5.2.15)

where $(v,\eta)$ solve (1.1.9) as in Theorem 4.

    3. (c)

The following derivatives of each mapping in (5.2.15) are well-defined and
continuous between the stated spaces:

$[0,R_{\mathrm{visc}}]\times[0,R_{\mathrm{surf}}]\ni(\mu,\sigma)\mapsto(\mu\nabla
v,\mu^{2}\nabla^{2}v)\in H^{s}(\mathbb{R}^{d};\mathbb{R}^{d^{2}})\times
H^{s}(\mathbb{R}^{d};\mathbb{R}^{d^{3}})$ (5.2.16)

and

$[0,R_{\mathrm{visc}}]\times[0,R_{\mathrm{surf}}]\ni(\mu,\sigma)\mapsto(\sigma\nabla\eta,\sigma^{2}\nabla^{2}\eta)\in
H^{1+s}(\mathbb{R}^{d};\mathbb{R}^{d})\times
H^{1+s}(\mathbb{R}^{d};\mathbb{R}^{d^{2}}).$ (5.2.17)

###### Proof.

Given $\gamma$, $R_{\mathrm{visc}}$, and $R_{\mathrm{surf}}$ as in the
statement of the corollary, we define

$\widetilde{\mathcal{U}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})=\mathrm{int}\bigcap_{0\leqslant\mu\leqslant
R_{\mathrm{visc}}}\bigcap_{0\leqslant\sigma\leqslant
R_{\mathrm{surf}}}\mathcal{U}_{s}(\gamma,\mu,\sigma)$ (5.2.18)

and, provided $\gamma<1$,

$\widetilde{\mathcal{U}}^{\mathrm{sub}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})=\mathrm{int}\bigcap_{0\leqslant\mu\leqslant
R_{\mathrm{visc}}}\bigcap_{0\leqslant\sigma\leqslant
R_{\mathrm{surf}}}\mathcal{U}^{\mathrm{sub}}_{s}(\gamma,\mu,\sigma),$ (5.2.19)

where again we use $\mathrm{int}$ to denote the topological interior. We note
that the set
$\widetilde{\mathcal{U}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})$ is
indeed non-empty since inspection of the definition of the set
$\mathcal{U}_{s}$ in (5.2.2) allows one to conclude that

$\mathrm{dist}\Big{(}\\{{\gamma}\\}\times[0,R_{\mathrm{visc}}]\times[0,R_{\mathrm{surf}}]\times\\{{0}\\},\partial\Big{(}\bigcup_{\mathfrak{q}\in\mathbb{R}^{+}\times[0,\infty)^{2}}B_{\mathbb{R}^{3}\times\mathbb{W}_{\varsigma}}((\mathfrak{q},0),\uprho_{s-\varsigma}(\mathfrak{q}))\Big{)}\Big{)}>0.$
(5.2.20)

Hence
$\widetilde{\mathcal{U}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})$
contains balls of positive radii of length at most the above distance. A
similar argument verifies that
$\widetilde{\mathcal{U}}^{\mathrm{sub}}_{s}(\gamma,R_{\mathrm{visc}},R_{\mathrm{surf}})$
is also non-empty. Armed with the open sets of (5.2.18) and (5.2.19), we find
that the conclusions stated above are immediate consequences of Theorems 3 and
4. ∎

##  Appendix A Analysis background

###  A.1. On the anisotropic Sobolev spaces

For $\mathbb{R}\ni s\geqslant 0$ and $d\in\mathbb{N}^{+}$ we define the
anisotropic Sobolev space

$\mathcal{H}^{s}(\mathbb{R}^{d})=\\{{f\in\mathscr{S}^{\ast}(\mathbb{R}^{d};\mathbb{R})\;:\;\mathscr{F}[f]\in
L^{1}_{\mathrm{loc}}(\mathbb{R}^{d};\mathbb{C}),\;\lVert
f\rVert_{\mathcal{H}^{s}}<\infty}\\},$ (A.1.1)

equipped with the norm

$\lVert
f\rVert_{\mathcal{H}^{s}}=\Big{(}\int_{\mathbb{R}^{d}}\big{(}|\xi|^{-2}(\xi_{1}^{2}+|\xi|^{4})\mathds{1}_{B(0,1)}(\xi)+\langle\xi\rangle^{2s}\mathds{1}_{\mathbb{R}^{d}\setminus
B(0,1)}(\xi)\big{)}|\mathscr{F}[f](\xi)|^{2}\;\mathrm{d}\xi\Big{)}^{1/2}.$
(A.1.2)

These spaces were introduced in Leoni and Tice [38], where it was shown, in
Proposition 5.3 and Theorem 5.6, that $\mathcal{H}^{s}(\mathbb{R}^{d})$ is a
Hilbert space and
$H^{s}(\mathbb{R}^{d})\hookrightarrow\mathcal{H}^{s}(\mathbb{R}^{d})\hookrightarrow
H^{s}(\mathbb{R}^{d})+C^{\infty}_{0}(\mathbb{R}^{d})$, with equality in the
first embedding if and only if $d=1$.

We shall find the following characterizations useful.

######  Proposition A.1 (Characterizations of the anisotropic Sobolev spaces).

The following hold for $\mathbb{R}\ni s\geqslant 0$.

  1. (1)

If $f\in\mathscr{S}^{\ast}(\mathbb{R}^{d};\mathbb{R})$ satisfies $\nabla f\in
H^{s-1}(\mathbb{R}^{d};\mathbb{R}^{d})$ and
$\partial_{1}f\in\dot{H}^{-1}(\mathbb{R}^{d})$, then there exists a constant
$c\in\mathbb{R}$ such that $f-c\in\mathcal{H}^{s}(\mathbb{R}^{d})$.

  2. (2)

Assume $d\geqslant 2$ and write $p_{d}=\frac{2(d+1)}{d-1}>2$. Then we have the
equality

$\mathcal{H}^{s}(\mathbb{R}^{d})=\Big{\\{}{f\in
L^{2}(\mathbb{R}^{d})+\bigcup_{p_{d}<p<\infty}L^{p}(\mathbb{R}^{d})\;:\;\nabla
f\in H^{s-1}(\mathbb{R}^{d};\mathbb{R}^{d})\text{ and
}\partial_{1}f\in\dot{H}^{-1}(\mathbb{R}^{d})}\Big{\\}},$ (A.1.3)

with the equivalence of norms

$\lVert f\rVert_{\mathcal{H}^{s}}\asymp\sqrt{\lVert\nabla
f\rVert_{H^{s-1}}^{2}+[{\partial_{1}f}]_{\dot{H}^{-1}}^{2}},$ (A.1.4)

where the implied constants depend only on $d$ and $s$.

  3. (3)

With $d$ and $p_{d}$ as in the previous item, we have the equality

$\mathcal{H}^{s}(\mathbb{R}^{d})=\Big{\\{}{f\in
L^{2}(\mathbb{R}^{d})+\bigcup_{p_{d}<p<\infty}L^{p}(\mathbb{R}^{d})\;:\;\nabla
f\in H^{s-1}(\mathbb{R}^{d};\mathbb{R}^{d})\text{ and }\mathcal{R}_{1}f\in
L^{2}(\mathbb{R}^{d})}\Big{\\}}$ (A.1.5)

with the equivalence of norms

$\lVert f\rVert_{\mathcal{H}^{s}}\asymp\sqrt{\lVert\nabla
f\rVert_{H^{s-1}}^{2}+\lVert\mathcal{R}_{1}f\rVert_{L^{2}}^{2}},$ (A.1.6)

where $\mathcal{R}_{1}$ is the Riesz transform in the $e_{1}$-direction.

###### Proof.

The first and second items are the content of Proposition B.4 in Stevenson and
Tice [57]. That the third item holds follows from the fact that
$\partial_{1}f\in\dot{H}^{-1}(\mathbb{R}^{d})$ is equivalent to
$\mathcal{R}_{1}f\in L^{2}(\mathbb{R}^{d})$ for any
$f\in(L^{2}+L^{q})(\mathbb{R}^{d})$ for $p_{d}<q<\infty$. ∎

The following result is Lemma B.6 in Stevenson and Tice [57].

######  Proposition A.2 (LP-smoothability).

The Banach scale $\\{{\mathcal{H}^{s}(\mathbb{R}^{d})}\\}_{s\in\mathbb{N}}$ is
LP-smoothable in the sense of Definition A.15.

We next discuss a high-low decomposition for which the following notational
convention is set. For $\kappa\in\mathbb{R}^{+}$, we define the linear
operators $\Uppi^{\kappa}_{\mathrm{L}}$ and $\Uppi^{\kappa}_{\mathrm{H}}$ on
the subspace of $f\in\mathscr{S}^{\ast}(\mathbb{R}^{d};\mathbb{C})$ such that
$\mathscr{F}[f]$ is locally integrable via

$\Uppi^{\kappa}_{\mathrm{L}}f=\mathscr{F}^{-1}[\mathds{1}_{B(0,\kappa)}\mathscr{F}[f]]\text{
and }\Uppi^{\kappa}_{\mathrm{H}}f=(I-\Uppi^{\kappa}_{\mathrm{L}})f.$ (A.1.7)

######  Proposition A.3 (Frequency splitting for anisotropic Sobolev spaces).

The following hold for $0\leqslant s\in\mathbb{R}$,
$f\in\mathcal{H}^{s}(\mathbb{R}^{d})$, and $\kappa\in\mathbb{R}^{+}$.

  1. (1)

We have the equivalence $\lVert
f\rVert_{\mathcal{H}^{s}}\asymp_{s,\kappa}\sqrt{\lVert\Uppi_{\mathrm{L}}^{\kappa}f\rVert^{2}_{\mathcal{H}^{0}}+\lVert\Uppi_{\mathrm{H}}^{\kappa}f\rVert^{2}_{H^{s}}}$.

  2. (2)

We have that $\Uppi_{\mathrm{L}}^{\kappa}f\in C^{\infty}_{0}(\mathbb{R}^{d})$
with the estimates
$\lVert\Uppi_{\mathrm{L}}^{\kappa}f\rVert_{W^{k,\infty}}\lesssim_{k,\kappa}\lVert\Uppi_{\mathrm{L}}^{\kappa}f\rVert_{\mathcal{H}^{0}}$
for every $k\in\mathbb{N}$.

###### Proof.

This is Theorem 5.5 in Leoni and Tice [38]. ∎

Our next result, which is Proposition B.2 in Stevenson and Tice [57],
enumerates important algebra properties of the anisotropic Sobolev spaces. In
what follows, we denote

$\mathcal{H}^{0}_{(\kappa)}(\mathbb{R}^{d})=\\{{f\in\mathcal{H}^{0}(\mathbb{R}^{d}),\;\mathrm{supp}\mathscr{F}[f]\subseteq\overline{B(0,\kappa)}}\\}.$
(A.1.8)

######  Proposition A.4 (Algebra properties of anisotropic Sobolev spaces).

Suppose that $f_{1},f_{2}\in\mathcal{H}^{0}_{(\kappa)}(\mathbb{R}^{d})$ for
some $\kappa\in\mathbb{R}^{+}$. The following hold.

  1. (1)

The product $f_{1}f_{2}$ belongs to
$\mathcal{H}_{(2\kappa)}^{0}(\mathbb{R}^{d})$ and satisfies the estimate
$\lVert f_{1}f_{2}\rVert_{\mathcal{H}^{0}}\lesssim_{\kappa}\lVert
f_{1}\rVert_{\mathcal{H}^{0}}\lVert f_{2}\rVert_{\mathcal{H}^{0}}$.

  2. (2)

Set

$r_{d}=\begin{cases}1+\big{\lfloor}\frac{d+1}{d-1}\big{\rfloor},&\text{if
}d>1,\\\ 1&\text{if }d=1.\end{cases}$ (A.1.9)

Assume additionally that
$f_{3},\dots,f_{r_{d}}\in\mathcal{H}_{(\kappa)}^{0}(\mathbb{R}^{d})$. Then the
pointwise product $\prod_{j=1}^{r_{d}}f_{j}$ belongs to
$(L^{2}\cap\mathcal{H}^{0}_{(r_{d}\cdot\kappa)})(\mathbb{R}^{d})$ and
satisfies

$\Big{\lVert}\prod_{j=1}^{r_{d}}f_{j}\Big{\rVert}_{L^{2}}\lesssim_{\kappa,d}\prod_{j=1}^{r_{d}}\lVert
f_{j}\rVert_{\mathcal{H}^{0}}.$ (A.1.10)

  3. (3)

If $g\in\mathcal{H}^{s}(\mathbb{R}^{d})$, then the pointwise product $f_{1}g$
belongs to $\mathcal{H}^{s}(\mathbb{R}^{d})$ and satisfies

$\lVert f_{1}g\rVert_{\mathcal{H}^{s}}\lesssim_{\kappa,s}\lVert
f_{1}\rVert_{\mathcal{H}^{0}}\lVert g\rVert_{\mathcal{H}^{s}}.$ (A.1.11)

######  Corollary A.5 (More algebra properties).

The following hold for $s\geqslant 1+\lfloor d/2\rfloor$, $t\leqslant s$, and
$g\in\mathcal{H}^{s}(\mathbb{R}^{d})$.

  1. (1)

If $f\in H^{t}(\mathbb{R}^{d})$, then the pointwise product $fg$ belongs to
$H^{t}(\mathbb{R}^{d})$ and obeys the estimate

$\lVert fg\rVert_{H^{t}}\lesssim\lVert f\rVert_{H^{t}}\lVert
g\rVert_{\mathcal{H}^{1+\lfloor d/2\rfloor}}+\begin{cases}0&\text{if
}t\leqslant 1+\lfloor d/2\rfloor,\\\ \lVert f\rVert_{H^{1+\lfloor
d/2\rfloor}}\lVert g\rVert_{\mathcal{H}^{t}}&\text{if }t>1+\lfloor
d/2\rfloor.\end{cases}.$ (A.1.12)

  2. (2)

If $h\in\mathcal{H}^{t}(\mathbb{R}^{d})$, then the pointwise product $gh$
belongs to $\mathcal{H}^{t}(\mathbb{R}^{d})$ and obeys the estimate

$\lVert hg\rVert_{\mathcal{H}^{t}}\lesssim\lVert
h\rVert_{\mathcal{H}^{t}}\lVert g\rVert_{\mathcal{H}^{1+\lfloor
d/2\rfloor}}+\begin{cases}0&\text{if }t\leqslant 1+\lfloor d/2\rfloor,\\\
\lVert h\rVert_{\mathcal{H}^{1+\lfloor d/2\rfloor}}\lVert
g\rVert_{\mathcal{H}^{t}}&\text{if }t>1+\lfloor d/2\rfloor\end{cases}$
(A.1.13)

  3. (3)

If $g\in\mathcal{H}^{s}(\mathbb{R}^{d})$, then $g^{r_{d}}$, for $r_{d}$ as in
(A.1.9), belongs to $H^{s}(\mathbb{R}^{d})$ and obeys the estimate

$\lVert g^{r_{d}}\rVert_{H^{s}}\lesssim\lVert g\rVert_{\mathcal{H}^{s}}\lVert
g\rVert_{\mathcal{H}^{1+\lfloor d/2\rfloor}}^{r_{d}-1}.$ (A.1.14)

###### Proof.

For the first item we decompose
$g=\Uppi^{1}_{\mathrm{L}}g+\Uppi^{1}_{\mathrm{H}}g$. For the low piece
$f\Uppi^{1}_{\mathrm{L}}g$, we use the second item of Proposition A.3 along
with the fourth item of Proposition A.23. For the high piece
$f\Uppi^{1}_{\mathrm{H}}g$, we instead use the first and second items of the
two aforementioned propositions, respectively.

For the second item, we decompose both $g$ and $h$ into high and low pieces
via $\Uppi^{1}_{\mathrm{L}}$ and $\Uppi^{1}_{\mathrm{H}}$ and inspect the four
terms in the expanded product individually. For either of the mixed high-low
terms, we invoke the third item of Proposition A.4. For the low-low term, we
instead use the first item. Finally, for the high-high term, we use the first
item of Proposition A.3 along with the second item of Proposition A.23.

For the third item, we use the binomial theorem to write

$g^{r_{d}}=\sum_{\ell=0}^{r_{d}}\binom{r_{d}}{\ell}(\Uppi^{1}_{\mathrm{L}}g)^{\ell}(\Uppi^{1}_{\mathrm{H}}g)^{r_{d}-\ell}.$
(A.1.15)

The terms indexed with $\ell<r_{d}$ are handled via the first and second items
of this result. The $\ell=r_{d}$ term is handled via the second item of
Proposition A.4. ∎

###  A.2. Tame structure abstraction

The goal of this sub-appendix is to review notions of Banach scales and
tameness of mappings between them. This is a rapid treatment of the thorough
development of Section 2.1 in Stevenson and Tice [57].

######  Definition A.6.

Given $N\in\mathbb{N}\cup\\{{\infty}\\}$ we define $\llbracket
N\rrbracket\subseteq\mathbb{N}$ to be the set $\llbracket
N\rrbracket=\\{{n\in\mathbb{N}\;:\;n\leqslant N}\\}$.

######  Definition A.7 (Banach scales).

Let $N\in\mathbb{N}^{+}\cup\\{{\infty}\\}$ and let
$\mathbf{E}=\\{E^{s}\\}_{s\in\llbracket N\rrbracket}$ be a collection of
Banach spaces over a common field, either $\mathbb{R}$ or $\mathbb{C}$.

  1. (1)

We say that $\mathbf{E}$ is a Banach scale if for each $s\in\llbracket
N-1\rrbracket$ we have the non-expansive inclusion $E^{s+1}\hookrightarrow
E^{s}$, i.e. $\lVert\cdot\rVert_{E^{s}}\leqslant\lVert\cdot\rVert_{E^{s+1}}$.

  2. (2)

If $\mathbf{E}$ is a Banach scale, then we define the scale’s terminal space
to be $E^{N}=\bigcap_{s\in\llbracket N\rrbracket}E^{s}$ and endow it with the
Fréchet topology induced by the collection of norms
$\\{\left\lVert\cdot\right\rVert_{E^{s}}\\}_{s\in\llbracket N\rrbracket}$.
Note that if $N<\infty$, then $E^{N}$ has the standard Banach topology from
its norm.

  3. (3)

If $\mathbf{E}$ is a Banach scale, then we write $B_{E^{r}}(u,\delta)\subseteq
E^{r}$ for the $E^{r}$-open ball of radius $\delta>0$, centered at $u\in
E^{r}$.

  4. (4)

If $\mathbf{E}$ is a Banach scale and $E^{N}$ is dense in $E^{s}$ for each
$s\in\llbracket N\rrbracket$, then we say that $\mathbf{E}$ is terminally
dense.

######  Remark A.8 (Products of Banach scales).

Suppose that $\mathbf{E}_{i}=\\{E^{s}_{i}\\}_{s\in\llbracket N\rrbracket}$ is
a Banach scale for $i\in\\{1,\dotsc,n\\}$, each over the same field. Then
$\mathbf{F}=\prod_{i=1}^{n}\mathbf{E}_{i}=\\{\prod_{i=1}^{n}E^{s}_{i}\\}_{s\in\llbracket
N\rrbracket}$ is a Banach scale when each $\prod_{i=1}^{n}E^{s}_{i}$ is
endowed with the norm

$\left\lVert
u_{1},\dotsc,u_{n}\right\rVert_{\prod_{i=1}^{n}E^{s}_{i}}=\Big{(}\sum_{i=1}^{n}\left\lVert
u_{i}\right\rVert_{E^{s}_{i}}^{2}\Big{)}^{1/2}.$ (A.2.1)

######  Definition A.9 (Tame maps).

Let $\mathbf{E}=\\{{E^{s}}\\}_{s\in\llbracket N\rrbracket}$ and
$\mathbf{F}=\\{{F^{s}}\\}_{s\in\llbracket N\rrbracket}$ be Banach scales over
the same field, $\mu,r\in\llbracket N\rrbracket$ with $\mu\leqslant r$,
$U\subseteq E^{r}$ be an open set, and $P:U\to F^{0}$.

  1. (1)

We say that $P$ satisfies tame estimates of order $\mu$ and base $r$ (with
respect to the Banach scales $\mathbf{E}$ and $\mathbf{F}$) if for all
$\llbracket N\rrbracket\ni s\geqslant r$, there exists a constant
$C_{s}\in\mathbb{R}^{+}$ such that for all $f\in U\cap E^{s}$ we have the
inclusion $P(f)\in F^{s-\mu}$ as well as the estimate

$\lVert P(f)\rVert_{F^{s-\mu}}\leqslant C_{s}\langle\lVert
f\rVert_{E^{s}}\rangle.$ (A.2.2)

  2. (2)

We say that $P$ is $\mu$-tame with base $r$ if for each $f\in U$, there exists
an $E^{r}$-open set $V\subseteq U$ such that $f\in V$ and the restricted map
$P\restriction V:V\to F^{0}$ satisfies tame estimates of order $\mu$ and base
$r$.

  3. (3)

We say that $P$ is strongly $\mu$-tame with base $r$ if on every $E^{r}$-open
and bounded subset $V\subseteq U$, the restricted map $P\restriction V$
satisfies tame estimates of order $\mu$ and base $r$.

  4. (4)

We say that $P$ is (strongly) $\mu$-tamely $C^{0}$ with base $r$ if $P$ is
(strongly) $\mu$-tame with base $r$ and if for every $\llbracket
N\rrbracket\ni s\geqslant r$ we have that $P:U\cap E^{s}\to F^{s-\mu}$ is
continuous as a map from $E^{s}$ to $F^{s-\mu}$. The collections of such maps
will be denoted by $T^{0}_{\mu,r}(U,\mathbf{E};\mathbf{F})$ and
${{}_{\mathrm{s}}}T^{0}_{\mu,r}(U,\mathbf{E};\mathbf{F})$, respectively.

  5. (5)

For $k\in\mathbb{N}^{+}$ we say that $P$ is (strongly) $\mu$-tamely $C^{k}$
with base $r$ if for all $\llbracket N\rrbracket\ni s\geqslant r$ the map
$P:U\cap E^{s}\to F^{s-\mu}$ is $C^{k}$ and for all $j\in\\{{0,1,\dots,k}\\}$
the $j^{\mathrm{th}}$ derivative map, thought of as mapping
$D^{j}P:U\times\prod_{\ell=1}^{j}E^{r}\to F^{0}$, is (strongly) $\mu$-tame
with base $r$ with respect to the Banach scales $\mathbf{E}^{1+j}$ and
$\mathbf{F}$. In other words, $D^{j}P\in
T^{0}_{\mu,r}(U\times\prod_{\ell=1}^{j}E^{r},\mathbf{E}^{1+j};\mathbf{F})$ or,
in the strong case,
$D^{j}P\in{{}_{\mathrm{s}}}T^{0}_{\mu,r}(U\times\prod_{\ell=1}^{j}E^{r},\mathbf{E}^{1+j};\mathbf{F})$.
The collections of such maps will be denoted by
$T^{k}_{\mu,r}(U,\mathbf{E};\mathbf{F})$ and
${{}_{\mathrm{s}}}T^{k}_{\mu,r}(U,\mathbf{E};\mathbf{F})$, respectively.

  6. (6)

We denote the collections of (strongly) $\mu$-tamely smooth with base $r$ maps
via
$T^{\infty}_{\mu,r}(U,\mathbf{E};\mathbf{F})=\bigcap_{k\in\mathbb{N}}T^{k}_{\mu,r}(U,\mathbf{E};\mathbf{F})$
and
${{}_{\mathrm{s}}}T^{\infty}_{\mu,r}(U,\mathbf{E};\mathbf{F})=\bigcap_{k\in\mathbb{N}}{{}_{\mathrm{s}}}T^{k}_{\mu,r}(U,\mathbf{E};\mathbf{F})$.

  7. (7)

When $U=E^{r}$ we will often use the abbreviated notation
$T^{k}_{\mu,r}(\mathbf{E},\mathbf{F})$ and
${{}_{\mathrm{s}}}T^{k}_{\mu,r}(\mathbf{E};\mathbf{F})$ in place of
$T^{k}_{\mu,r}(E^{r},\mathbf{E};\mathbf{F})$ and
${{}_{\mathrm{s}}}T^{k}_{\mu,r}(U,\mathbf{E};\mathbf{F})$, respectively.

The following result, which is proved in Lemma 2.5 in Stevenson and Tice [57],
is a useful characterization of tameness when there is multilinear structure
present in the map.

######  Lemma A.10 (Multilinearity and tame maps).

Let $k\in\mathbb{N}^{+}$ and $r,\mu\in\llbracket N\rrbracket$ with
$\mu\leqslant r$. Let $\mathbf{F}=\\{{F^{s}}\\}_{s\in\llbracket N\rrbracket}$,
$\mathbf{G}=\\{{G^{s}}\\}_{s\in\llbracket N\rrbracket}$, and
$\mathbf{E}_{j}=\\{{E^{s}_{j}}\\}_{s\in\llbracket N\rrbracket}$, for
$j\in\\{{1,\dots,k}\\}$, be Banach scales over the same field. Let $U\subseteq
G^{r}$ be an open set. Then the following are equivalent for all maps
$P:U\times\prod_{j=1}^{k}E^{r}_{j}\to F^{0}$ such that $P(g,\cdot)$ is
$k$-multilinear for all $g\in U$.

  1. (1)

$P\in
T^{0}_{\mu,r}(U\times\prod_{j=1}^{k}E^{r}_{j};\mathbf{G}\times\prod_{j=1}^{k}\mathbf{E}_{j};\mathbf{F})$.

  2. (2)

The restriction of $P$ to $(U\cap G^{s})\times\prod_{j=1}^{k}E^{s}_{j}$ is
continuous as a map into $F^{s-\mu}$ for all $\llbracket N\rrbracket\ni
s\geqslant r$, and for all $g\in U$ there exists a $G^{r}$-open subset
$V\subseteq U$ such that $g\in V$ and whenever $\llbracket N\rrbracket\ni
s\geqslant r$, $f\in V\cap G^{s}$, and $h_{i}\in E_{i}^{s}$ for
$i\in\\{{1,\dots,k}\\}$ we have the estimate

$\lVert P(f,h_{1},\dots,h_{k})\rVert_{F^{s-\mu}}\lesssim\langle\lVert
f\rVert_{G^{s}}\rangle\prod_{i=1}^{k}\lVert
h_{i}\rVert_{E_{i}^{r}}+\sum_{j=1}^{k}\lVert
h_{j}\rVert_{E_{j}^{s}}\prod_{i\neq j}\lVert h_{i}\rVert_{E_{i}^{r}}.$ (A.2.3)

A similar equivalence holds for maps
$P\in{{}_{\mathrm{s}}}T^{0}_{\mu,r}(U\times\prod_{j=1}^{k}E^{r}_{j};\mathbf{G}\times\prod_{j=1}^{k}\mathbf{E}_{j};\mathbf{F})$
if we change the space in the first item to the space of strongly tame maps
and we change the second item’s quantification of $V$ to ‘for all bounded
$G^{r}$-open sets $V\subseteq U$.’

######  Remark A.11.

If $P\in T^{0}_{\mu,r}(\prod_{j=1}^{k}\mathbf{E}_{j};\mathbf{F})$ is a
$k$-multilinear mapping, then a simple modification of Lemma A.10 shows that
$P$ is actually strongly tame. Moreover, by multilinearity, we have that $P$
is automatically a smooth function whose derivatives are also multilinear
maps. This combines with the previous fact to show that actually
$P\in{{}_{\mathrm{s}}}T^{\infty}_{\mu,r}(\prod_{j=1}^{k}\mathbf{E}_{j};\mathbf{F})$.

######  Corollary A.12 (Derivative estimates on tame maps).

Let $\mathbf{E}=\\{{E^{s}}\\}_{s\in\llbracket N\rrbracket}$ and
$\mathbf{F}=\\{{F^{s}}\\}_{s\in\llbracket N\rrbracket}$ be Banach scales over
a common field. Let $r,\mu\in\llbracket N\rrbracket$ with $\mu\leqslant r$,
$U\subseteq E^{r}$ be an open set, and $P\in
T^{k}_{\mu,r}(U,\mathbf{E};\mathbf{F})$. Then for each $f_{0}\in U$, there
exists an open set $f_{0}\in V\subseteq U$ with the property that for every
$\llbracket N\rrbracket\ni s\geqslant r$ there exists a constant
$C_{s,V}\in\mathbb{R}^{+}$, depending on $s$ and $V$, such that for all $f\in
V\cap E^{s}$ and all $g_{1},\dots,g_{k}\in E^{s}$ we have the estimate

$\lVert D^{k}P(f)[g_{1},\dots,g_{k}]\rVert_{F^{s-\mu}}\leqslant
C_{s,V}\sum_{\ell=1}^{k}\lVert g_{\ell}\rVert_{E^{s}}\prod_{j\neq\ell}\lVert
g_{j}\rVert_{E^{r}}+C_{s,V}\langle\lVert
f\rVert_{E^{s}}\rangle\prod_{j=1}^{k}\lVert g_{j}\rVert_{E^{r}}.$ (A.2.4)

Moreover, if $P\in{{}_{\mathrm{s}}}T^{k}_{\mu,r}(U,\mathbf{E};\mathbf{F})$,
then a similar assertion holds for every bounded and open subset $V\subseteq
U$.

###### Proof.

This is a direct application of Lemma A.10. ∎

######  Proposition A.13 (Composition of tame maps).

Let $\mathbf{E}=\\{{E^{s}}\\}_{s\in\llbracket N\rrbracket}$,
$\mathbf{F}=\\{{F^{s}}\\}_{s\in\llbracket N\rrbracket}$, and
$\mathbf{G}=\\{{G^{s}}\\}_{s\in\llbracket N\rrbracket}$ be a triple of Banach
scales over a common field ($\mathbb{R}$ or $\mathbb{C}$), and let
$k\in\mathbb{N}$ and $\mu,\mu^{\prime},r,r^{\prime}\in\llbracket N\rrbracket$
be such that $\mu\leqslant r$, $\mu^{\prime}\leqslant r^{\prime}$,
$r^{\prime}+\mu\in\llbracket N\rrbracket$. Suppose that $U\subseteq E^{r}$,
$U^{\prime}\subseteq F^{r^{\prime}}$ are open sets, $P\in
T^{k}_{\mu,r}(U,\mathbf{E};\mathbf{F})$, and $Q\in
T^{k}_{\mu^{\prime},r^{\prime}}(U^{\prime},\mathbf{F};\mathbf{G})$. Set
$V=U\cap E^{\max\\{{r,r^{\prime}+\mu}\\}}$. If $P(V)\subseteq U^{\prime}$,
then $Q\circ P\in
T^{k}_{\mu+\mu^{\prime},\max\\{{r,\mu+r^{\prime}}\\}}(V,\mathbf{E};\mathbf{G})$.
A similar assertion holds for the composition of strongly tame maps.

######  Corollary A.14 (Tame products of tame $C^{k}$ maps).

Let $\ell,k\in\mathbb{N}$ with $\ell\geqslant 1$. For
$j\in\\{{1,\dots,\ell}\\}$ let
$\mathbf{E}_{j}=\\{{E^{s}_{j}}\\}_{s\in\llbracket N\rrbracket}$ and
$\mathbf{F}_{j}=\\{{E^{s}_{j}}\\}_{s\in\llbracket N\rrbracket}$ be Banach
scales over a common field, and let $\mathbf{G}=\\{{G^{s}}\\}_{s\in\llbracket
N\rrbracket}$ be a Banach scale over the same field. Let
$r_{1},\dots,r_{\ell}\in\llbracket N\rrbracket$, $\mu_{j}\in\llbracket
r_{j}\rrbracket$ for $j\in\\{{1,\dots,\ell}\\}$, $r^{\prime}\in\llbracket
N\rrbracket$, and $\mu^{\prime}\in\llbracket r^{\prime}\rrbracket$. Finally,
suppose that $U_{j}\subseteq E^{r_{j}}_{j}$ is open, $P_{j}\in
T^{k}_{\mu_{j},r_{j}}(U_{j},\mathbf{E}_{j};\mathbf{F}_{j})$, and $B\in
T^{0}_{\mu^{\prime},r^{\prime}}(\prod_{j=1}^{\ell}\mathbf{F}_{j};\mathbf{G})$
is $k$-multilinear. If we set

$\mu=\mu^{\prime}+\max\\{{\mu_{1},\dots,\mu_{\ell}}\\}\text{ and
}r=\max\\{{r_{1},\dots,r_{\ell},\mu_{1}+r^{\prime},\dots,\mu_{\ell}+r^{\prime}}\\}$
(A.2.5)

and assume that $r\in\llbracket N\rrbracket$, then the product map satisfies
the inclusion

$B(P_{1},\dots,P_{\ell})\in T^{k}_{\mu,r}\Big{(}\prod_{j=1}^{\ell}U_{j}\cap
E^{r}_{j},\prod_{j=1}^{\ell}\mathbf{E}_{j};\mathbf{G}\Big{)}.$ (A.2.6)

A similar assertion holds for products of strongly tame maps.

######  Definition A.15 (Smoothable and LP-smoothable Banach scales).

Let $\mathbf{E}=\\{E^{s}\\}_{s\in\llbracket N\rrbracket}$ be a Banach scale.

  1. (1)

We say that $\mathbf{E}$ is smoothable if for each $j\in\mathbb{N}^{+}$ there
exists a linear map $S_{j}:E^{0}\to E^{N}$ satisfying the following smoothing
conditions for every $u\in E^{0}$:

$\left\lVert S_{j}u\right\rVert_{E^{s}}\lesssim\left\lVert
u\right\rVert_{E^{s}}\text{ for all }s\in\llbracket N\rrbracket,$ (A.2.7)
$\left\lVert S_{j}u\right\rVert_{E^{t}}\lesssim 2^{j(t-s)}\left\lVert
S_{j}u\right\rVert_{E^{s}}\text{ for all }s,t\in\llbracket N\rrbracket\text{
with }s<t,$ (A.2.8) $\left\lVert(I-S_{j})u\right\rVert_{E^{s}}\lesssim
2^{-j(t-s)}\left\lVert(I-S_{j})u\right\rVert_{E^{t}}\text{ for all
}s,t\in\llbracket N\rrbracket\text{ with }s<t,$ (A.2.9)
$\left\lVert(S_{j+1}-S_{j})u\right\rVert_{E^{t}}\lesssim
2^{j(t-s)}\left\lVert(S_{j+1}-S_{j})u\right\rVert_{E^{s}}\text{ for all
}s,t\in\llbracket N\rrbracket,$ (A.2.10)

where the implicit constants are independent of $j$ and are increasing in $s$
and $t$.

  2. (2)

We say that $\mathbf{E}$ is LP-smoothable if $\mathbf{E}$ is smoothable and
the smoothing operators satisfy the following Littlewood-Paley condition: for
$s\in\llbracket N\rrbracket$ and $u\in E^{s}$ we have that

$\lim_{j\to\infty}\lVert(I-S_{j})u\rVert_{E^{s}}=0,$ (A.2.11)

and there exists a constant $A>0$, possibly depending on $s$, such that

$A^{-1}\left\lVert
u\right\rVert_{E^{s}}\leqslant\Big{(}\sum_{j=0}^{\infty}\left\lVert\Updelta_{j}u\right\rVert_{E^{s}}^{2}\Big{)}^{1/2}\leqslant
A\left\lVert u\right\rVert_{E^{s}},$ (A.2.12)

where the operators $\\{\Updelta_{j}\\}_{j=0}^{\infty}$ are defined via
$\Updelta_{j}=S_{j+1}-S_{j}$ with the convention that $S_{0}=0$.

######  Example A.16 (A fixed Banach space).

Suppose that $X$ is a Banach space and $N\in\mathbb{N}\cup\\{\infty\\}$. Then
the scale $\\{{X}\\}_{s\in\llbracket N\rrbracket}$, generated by $X$ alone, is
LP-smoothable in the sense of Definition A.15, provided we take $S_{j}=I$ for
all $j\in\mathbb{N}^{+}$.

######  Example A.17 (Sobolev spaces on $\mathbb{R}^{d}$).

Let $V$ be a finite dimensional real vector space and
$N\in\mathbb{N}\cup\\{\infty\\}$. The real Banach scales of Sobolev spaces
$\\{{H^{s}(\mathbb{R}^{d};V)}\\}_{s\in\llbracket N\rrbracket}$ and
$\\{{(\dot{H}^{-1}\cap H^{s})(\mathbb{R}^{d};V)}\\}_{s\in\llbracket
N\rrbracket}$ are LP-smoothable for the smoothing operators
$\\{{S_{j}}\\}_{j=0}^{\infty}$ given by
$S_{j}=\mathds{1}_{B(0,2^{j})}(\nabla/2\pi\mathrm{i})$.

###  A.3. A variation on the Nash-Moser inverse function theorem

The following definition gives us a sense of the mapping hypotheses one needs
to verify to invoke a Nash-Moser inverse function theorem.

######  Definition A.18 (Mapping hypotheses).

We say that a triple $(\mathbf{E},\mathbf{F},\Psi)$ satisfies the RI mapping
hypotheses with parameters $(\mu,r,R)\in\llbracket N\rrbracket^{3}$ if
$\mathbf{E}=\\{E^{s}\\}_{s\in\llbracket N\rrbracket}$ and
$\mathbf{F}=\\{F^{s}\\}_{s\in\llbracket N\rrbracket}$ are Banach scales over a
common field, $1\leqslant\mu\leqslant r<R<\infty$, $R+\mu\in\llbracket
N\rrbracket$, and there exists $0<\delta_{r}\in\mathbb{R}$ such that
$\Psi:B_{E^{r}}(0,\delta_{r})\to F^{r-\mu}$ is a map satisfying the following.

  1. (1)

$\Psi(0)=0$.

  2. (2)

$\mu$-tamely $C^{2}$: For every $r-\mu\leqslant s\in\llbracket
N-\mu\rrbracket$ we have that $\Psi:B_{E^{r}}(0,\delta_{r})\cap E^{s+\mu}\to
F^{s}$ is $C^{2}$, and for every $u_{0}\in B_{E^{r}}(0,\delta_{r})\cap
E^{s+\mu}$ we have the tame estimate

$\left\lVert D^{2}\Psi(u_{0})[v,w]\right\rVert_{F^{s}}\leqslant
C_{1}(s)(\left\lVert v\right\rVert_{E^{s+\mu}}\left\lVert
w\right\rVert_{E^{r}}+\left\lVert v\right\rVert_{E^{r}}\left\lVert
w\right\rVert_{E^{s+\mu}}+\left\langle\left\lVert
u_{0}\right\rVert_{E^{s+\mu}}\right\rangle\left\lVert
v\right\rVert_{E^{r}}\left\lVert w\right\rVert_{E^{r}}).$ (A.3.1)

Here the constant $C_{1}(s)$ is increasing in $s$. In other words
$\Psi\in{{}_{\mathrm{s}}}T^{2}_{\mu,r}(B_{E^{r}}(0,\delta_{r}),\mathbf{E};\mathbf{F})$
according to the notation from Definition A.9.

  3. (3)

Derivative inversion: There exists $\delta_{R}\in\mathbb{R}$ satisfying
$0<\delta_{R}\leqslant\delta_{r}$ such that for every $u_{0}\in
B_{E^{r}}(0,\delta_{R})\cap E^{N}$ there exists a bounded linear operator
$L(u_{0}):F^{r}\to E^{r}$ satisfying the following three conditions.

    1. (a)

For every $s\in\mathbb{N}\cap[r,R]$ we have that the restriction of $L(u_{0})$
to $F^{s}$ defines a bounded linear operator with values in $E^{s}$, i.e.
$L(u_{0})\in\mathcal{L}(F^{s};E^{s})$.

    2. (b)

$D\Psi(u_{0})L(u_{0})f=f$ for every $f\in F^{r}$.

    3. (c)

We have the tame estimate

$\left\lVert L(u_{0})f\right\rVert_{E^{s}}\leqslant C_{2}(s)(\left\lVert
f\right\rVert_{F^{s}}+\left\langle\left\lVert
u_{0}\right\rVert_{E^{s+\mu}}\right\rangle\left\lVert f\right\rVert_{F^{r}})$
(A.3.2)

for every $f\in F^{s}$ and $r\leqslant s\leqslant R$, where again the constant
$C_{2}(s)$ is increasing in $s$.

Here the use of the prefix RI- is meant to indicate that the maps $L(u_{0})$
are only required to be right inverses. We say that the triple
$(\mathbf{E},\mathbf{F},\Psi)$ satisfies the LRI mapping hypotheses if
condition $(b)$ in the third item is augmented by the left-inverse condition
$(b^{\prime}):$

$L(u_{0})D\Psi(u_{0})v=v\text{ for every }v\in E^{r+\mu}.$ (A.3.3)

We now state a slightly specified version of the Nash-Moser inverse function
theorem. For a proof and a more general formulation, the reader is referred to
Section 2 of Stevenson and Tice [57].

######  Theorem A.19 (Inverse function theorem).

Let $\mathbf{E}=\\{E^{s}\\}_{s\in\llbracket N\rrbracket}$ and
$\mathbf{F}=\\{F^{s}\\}_{s\in\llbracket N\rrbracket}$ be Banach scales over
the same field consisting of reflexive spaces, and assume that $\mathbf{E}$
and $\mathbf{F}$ are LP-smoothable (see Definition A.15). Assume the triple
$(\mathbf{E},\mathbf{F},\Psi)$ satisfies the LRI mapping hypotheses of
Definition A.18 with parameters $(\mu,r,R)$ satisfying $2(r+\mu)+1<(r+R)/2$,
and set $\beta=2(r+\mu)+1\in\llbracket N\rrbracket$. Then there exist
$\varepsilon,\kappa_{1},\kappa_{2}>0$ such that the following hold.

  1. (1)

Existence of local inverse: For every $g\in B_{F^{\beta}}(0,\varepsilon)$
there exists a unique $u\in B_{E^{\beta}}(0,\kappa_{1}\varepsilon)$ such that
$\Psi(u)=g$.

  2. (2)

Estimates of local inverse: The induced bijection
$\Psi^{-1}:B_{F^{\beta}}(0,\varepsilon)\to\Psi^{-1}(B_{F^{\beta}}(0,\varepsilon))\cap
B_{E^{\beta}}(0,\kappa_{1}\varepsilon)$ obeys the estimate

$\lVert\Psi^{-1}(g)\rVert_{E^{\beta}}\leqslant\kappa_{1}\lVert
g\rVert_{F^{\beta}}\text{ for all }g\in B_{F^{\beta}}(0,\varepsilon).$ (A.3.4)

Moreover, if $\mathbb{N}\ni\nu\leqslant R+r-2\beta$, then we have that