# The inviscid limit for the $2d$ Navier-Stokes equations in bounded domains

Claude Bardos111Laboratoire J.-L. Lions, Sorbonne Université, BC 187, 4 place
Jussieu, 75252 Paris, Cedex 05, France. Email<EMAIL_ADDRESS>Trinh
T. Nguyen222Department of Mathematics, University of Southern California, LA,
CA 90089. Email<EMAIL_ADDRESS>TN is partly supported by the AMS-Simons
Travel Grant Award.  Toan T. Nguyen333Department of Mathematics, Penn State
University, State College, PA 16803. Email<EMAIL_ADDRESS>TN is partly
supported by the NSF under grant DMS-2054726. Edriss S. Titi444Department of
Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843-3368,
USA. Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, U.K. ALSO, Department of Computer Science and
Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel.
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We prove the inviscid limit for the incompressible Navier-Stokes equations for
data that are analytic only near the boundary in a general two-dimensional
bounded domain. Our proof is direct, using the vorticity formulation with a
nonlocal boundary condition, the explicit semigroup of the linear Stokes
problem near the flatten boundary, and the standard wellposedness theory of
Navier-Stokes equations in Sobolev spaces away from the boundary.

In memory of Robert T. Glassey

###### Contents

  1. 1 Introduction
    1. 1.1 Boundary vorticity formulation
    2. 1.2 Main results
    3. 1.3 Remarks
  2. 2 Navier-Stokes equations near the boundary
    1. 2.1 Global geodesic coordinates
    2. 2.2 Scaled coordinates
    3. 2.3 Vorticity equations near the boundary
    4. 2.4 Dirichlet-Neumann operator
  3. 3 Near boundary analytic spaces
    1. 3.1 Analytic norms
    2. 3.2 Elliptic estimates in the half-plane
    3. 3.3 Biot-Savart law in $\Omega$
    4. 3.4 Bilinear estimates
    5. 3.5 Semigroup estimates in the half-plane
    6. 3.6 Semigroup estimates near $\partial\Omega$
  4. 4 Nonlinear analysis
    1. 4.1 Global Sobolev-analytic norm
    2. 4.2 Analytic bounds near the boundary
    3. 4.3 Sobolev bounds away from the boundary
    4. 4.4 Proof of Theorem 1.1

##  1 Introduction

We are interested in the inviscid limit of solutions to the incompressible
Navier-Stokes equations (NSE)

$\displaystyle\partial_{t}u+u\cdot\nabla u+\nabla p$ $\displaystyle=\nu\Delta
u,$ (1.1) $\displaystyle\nabla\cdot u$ $\displaystyle=0,$

in a bounded domain $\Omega\subset\mathbb{R}^{2}$, with initial data
$u_{|_{t=0}}=u_{0}(x)$ and with the no-slip boundary condition

$u|_{\partial\Omega}=0.$ (1.2)

In the inviscid limit: $\nu\to 0$, one would intuitively expect that the
solutions $u_{\nu}$, of problem (1.1)-(1.2), converge to the corresponding
solutions of the Euler equations of ideal incompressible fluids

$\displaystyle\partial_{t}u+u\cdot\nabla u+\nabla p$
$\displaystyle=0,\quad\hbox{in}\quad\Omega,$ (1.3) $\displaystyle\nabla\cdot
u$ $\displaystyle=0,\quad\hbox{in}\quad\Omega,$ $\displaystyle u\cdot n$
$\displaystyle=0,\quad\hbox{on}\quad\partial\Omega,$

where $n$ denotes the unit normal vector to the boundary pointing inward.
However, the inviscid limit for problem (1.1)-(1.2) is strenuous and remains
open due to the appearance of boundary layers and strong shear near the
boundary that triggers the shedding of unbounded vorticity by the boundary. In
their celebrated work [22], Caflisch and Sammartino establish the boundary
layer expansion and the inviscid limit for analytic data on the half-plane.
Maekawa [20] proved a similar result that allows Sobolev data whose vorticity
is supported away from the boundary. The result and its proof was recently
simplified [21] and extended in [19, 18], which allow data that are only
analytic near the boundary.

In this paper, we prove the inviscid limit of (1.1)-(1.2) for data that are
only analytic near the boundary of a general bounded analytic domain in
$\mathbb{R}^{2}$, thus further extending [22, 20, 21, 19] from the case of
half-plane to bounded domains with analytic boundaries. Precisely, we assume
that

  * •

$\Omega$ is a simply-connected bounded domain in $\mathbb{R}^{2}$ whose
boundary $\partial\Omega$ is an analytic curve, defined by an analytic map:
$\theta\in{\mathbb{T}}={\mathbb{R}}/({\mathbb{Z}}L)\mapsto
x(\theta)=(x_{1}(\theta),x_{2}(\theta))\in\partial\Omega\,.$

The analyticity of the boundary naturally extends to an analytic map which
maps the near-boundary part of the domain
$\\{x\in\Omega:\,\,d(x,\partial\Omega)<\delta\\}$ to the case of half-plane
$(z,\theta)\in(0,\delta)\times\mathbb{T}$, where $z$ is the distance function
from the boundary. Here, for sake of presentation, we have chosen to consider
the case of simply-connected domain $\Omega$. The results of this paper apply
to the general setting of multi-connected bounded domains whose boundaries
consist of closed analytic curves, i.e., including domains with holes. Our
analysis near each of the boundaries is close to that on the half-plane. A
crucial assumption, however, lies on the analyticity of initial data near the
boundary, which appears to be sharp.

The work is dedicated to the memory of Professor Robert T. Glassey, who was a
great mathematician, a close friend, and an inspiring teacher.

###  1.1 Boundary vorticity formulation

We shall work with the boundary vorticity formulation [1, 20, 21]. Precisely,
let $u=(u_{1},u_{2})$ be the velocity vector field and
$\omega=\nabla^{\bot}\cdot u=\partial_{x_{2}}u_{1}-\partial_{x_{1}}u_{2}$ be
the corresponding vorticity. Then, the vorticity equation reads

$\displaystyle\partial_{t}\omega+u\cdot\nabla\omega$
$\displaystyle=\nu\Delta\omega,$ (1.4) $\displaystyle u$
$\displaystyle=\nabla^{\perp}\Delta^{-1}\omega,\qquad\hbox{(the Biot-Savart
law).}$

Here and throughout the paper, $\Delta^{-1}$ denotes the inverse of the
Laplacian operator in $\Omega$ subject to the zero Dirichlet boundary
condition. Evidently, this, together with the Biot-Savart law, imply the
impermeability boundary condition $u\cdot n=0$ on $\partial\Omega$. To ensure
the full no-slip boundary condition, i.e., that $u\cdot\tau=0$ on the boundary
$\partial\Omega$, where $\tau$ in the unit tangent vector to the boundary, we
first require that the initial data satisfy the no-slip boundary condition
(1.2), and then we impose in addition that $\partial_{t}u\cdot\tau=0$ on the
boundary, $\partial\Omega$, for all positive time. This leads to the boundary
condition

$0=\tau\cdot\partial_{t}u=\tau\cdot\nabla^{\perp}\Delta^{-1}\partial_{t}\omega=\partial_{n}[\Delta^{-1}(\nu\Delta\omega-u\cdot\nabla\omega)]$
(1.5)

on the boundary. Introduce $\omega^{*}$ to be the solution of the
nonhomogeneous Dirichlet boundary-value problem

$\left\\{\begin{aligned} \Delta\omega^{*}&=0,\qquad\mbox{in}\quad\Omega\\\
{\omega^{*}}&=\omega,\qquad\mbox{on }\quad\partial\Omega.\end{aligned}\right.$
(1.6)

and define the Dirichlet-Neumann operator by

$DN\omega=-\partial_{n}\omega^{*},\qquad\mbox{on}\quad\partial\Omega,$ (1.7)

where $\omega^{*}$ solves (1.6). Observe that
$\partial_{n}[\Delta^{-1}\Delta\omega]=\partial_{n}[\Delta^{-1}\Delta(\omega-\omega^{*})]=(\partial_{n}+DN)\omega$.
Thus, by virtue of the boundary condition (1.5) the boundary condition on
vorticity reads

$\nu(\partial_{n}+DN)\omega_{|_{\partial\Omega}}=[\partial_{n}\Delta^{-1}(u\cdot\nabla\omega)]_{|_{\partial\Omega}},$
(1.8)

together with the Biot-Savart law (1.4).

Throughout this paper, we shall deal with the Navier-Stokes solutions that
solve (1.4)-(1.7), or equivalently (1.4) and (1.8). Such a solution will be
constructed via the Duhamel’s integral representation, treating the
nonlinearity as a source term. As we observed earlier the boundary condition
$u\cdot n=0$ on $\partial\Omega$ follows from the Biot-Savart law and the
definition of $\Delta^{-1}$ with the zero Dirichlet boundary condition.

###  1.2 Main results

Our main result reads as follows.

######  Theorem 1.1.

Let $u_{0}\in H^{5}(\Omega)$ be an initial data that vanishes on the boundary.
We assume that the initial vorticity $\omega_{0}$ is analytic near the
boundary $\partial\Omega$ (see Section 3). Then, there is a positive time $T$,
independent of $\nu$, so that the unique solution $u_{\nu}(t)$ to the Navier-
Stokes problem (1.1)-(1.2), for every $\nu>0$, with initial data $u_{0}$,
exists on $[0,T]$ and has vorticity $\omega_{\nu}=\nabla^{\perp}\cdot u_{\nu}$
that remains analytic near the boundary, and satisfies

$\lim_{\nu\rightarrow
0}\sqrt{\nu}\|\omega_{\nu}\|_{L^{\infty}([0,T]\times\partial\Omega))}<\infty.$
(1.9)

Moreover, in the inviscid limit as $\nu\to 0$, $u_{\nu}$ converges strongly in
$L^{\infty}([0,T];L^{p}(\Omega))$, for any $2\leq p<\infty$, to the
corresponding solution $u$ of the Euler equations (1.3) with the same initial
data $u_{0}$.

The fact that Euler solutions remain analytic near the boundary is a classical
result [3, 17], which is a direct consequence of the main theorem. The main
difficulty in establishing the inviscid limit is to control the vorticity on
the boundary and derive uniform estimates such as (1.9), which is the main
contribution of this paper. The inviscid limit then follows easily. In fact, a
much weaker bound than (1.9) is sufficient to guarantee the convergence of
solutions to the Navier-Stokes to a corresponding solution of the Euler
equations. Precisely, we have the following simple Kato’s type theorem.

######  Theorem 1.2.

Let $T>0$ and $u$ be a weak solution to the Euler equations (1.3) in
$[0,T]\times\Omega$ satisfying $\|\nabla
u\|_{L^{\infty}([0,T]\times\Omega)}<\infty$. Suppose that, for every $\nu>0$,
$u_{\nu}$ are Leray weak solutions to the Navier-Stokes problem (1.1)-(1.2) on
$[0,T]\times\Omega$, satisfying

$\sup_{0<t<T}\|u_{\nu}(t)\|^{2}_{L^{2}(\Omega)}+\nu\int_{0}^{T}\|\nabla_{x}u_{\nu}(t)\|_{L^{2}(\Omega)}dt\leq
C_{0},$ (1.10)

uniformly in $\nu\to 0$. Assume that the vorticity
$\omega_{\nu}=\nabla^{\bot}\cdot u_{\nu}$ satisfies

$\limsup_{\nu\to
0}\Big{(}-\int_{0}^{T}\int_{\partial\Omega}\nu\omega_{\nu}(t,\sigma)u(t,\sigma)\cdot\tau(\sigma)d\sigma
dt\Big{)}=0,$ (1.11)

then any $\overline{u}_{\nu}$, which is a weak$-*$ limit in
$L^{\infty}([0,T];L^{2}(\Omega))$ of a subsequence $u_{\nu_{j}}$ of the Leray
weak solutions, as $\nu_{j}\to 0$, satisfies the stability estimate:

$\|\overline{u_{\nu}(t)}-u(t)\|_{L^{2}(\Omega)}^{2}\leq e^{2t\|\nabla
u\|_{L^{\infty}([0,T]\times\Omega)}}\|\overline{u_{\nu}(0)}-u(0)\|_{L^{2}(\Omega)}^{2}.$
(1.12)

In particular, if $u_{\nu}(0)\rightarrow u(0)$ in $L^{2}(\Omega)$, as $\nu\to
0$, then $u_{\nu}$ converges strongly to $u$ in
$L^{\infty}([0,T];L^{2}(\Omega)).$

###### Proof.

An elementary manipulation (e.g., [4]) yields the following energy inequality

$\displaystyle\|u_{\nu}(t)-u(t)\|_{L^{2}(\Omega)}^{2}+\nu\int_{0}^{t}\|\nabla
u_{\nu}(s)\|^{2}_{L^{2}(\Omega)}ds$ (1.13)
$\displaystyle\leq\|u_{\nu}(0)-u(0)\|_{L^{2}(\Omega)}^{2}+\nu\int_{0}^{t}\|\nabla
u(s)\|^{2}_{L^{2}(\Omega)}ds-\int_{0}^{t}\int_{\partial\Omega}\nu(\partial_{n}u_{\nu}(s,\sigma))\cdot
u(s,\sigma)\;d\sigma ds$
$\displaystyle+\int_{0}^{t}\int_{\Omega}|\Big{(}(\nabla
u+\nabla^{\bot}u)(u_{\nu}-u)\Big{)}\cdot(u_{\nu}-u))|dxds$
$\displaystyle\leq\|u_{\nu}(0)-u(0)\|_{L^{2}(\Omega)}^{2}+\nu\int_{0}^{t}\|\nabla
u(s)\|^{2}_{L^{2}(\Omega)}ds-\int_{0}^{t}\int_{\partial\Omega}\nu\omega_{\nu}(s,\sigma)(u(s,\sigma)\cdot\tau(\sigma))\;d\sigma
ds$ $\displaystyle+2\|\nabla
u\|_{L^{\infty}([0,T]\times\Omega)}\int_{0}^{t}\|u_{\nu}(s)-u(s)\|_{L^{2}(\Omega)}^{2}\,ds,$

where in the third term in the right-hand side of the last inequality we used
the fact that $(\partial_{n}u_{\nu})\cdot u=\omega_{\nu}(u\cdot\tau)$ on the
boundary. Let $u_{\nu_{j}}$ be a subsequence which converges weak$-*$ in
$L^{\infty}([0,T];L^{2}(\Omega))$, as $\nu_{j}\to 0$. We apply the above
energy inequality to $u_{\nu_{j}}$ and invoke Gronwall’s Lemma. Observe that
since the Leray weak solutions belong to $C([0,T];L^{2}(\Omega))$ then
$\|u_{\nu}(0)\|^{2}_{L^{2}(\Omega)}\leq C_{0}$ by virtue of (1.10). Thanks to
the Banach-Alaoglu Theorem and assumption (1.11) we conclude (1.12). The last
part of the theorem is an immediate consquence of (1.12). ∎

###  1.3 Remarks

As mentioned in the introduction, our main results extend the previous works
[22, 20, 21, 19] from the case of the half-plane to bounded domains. The
analyticity near the boundary is required to control the unbounded vorticity
in the inviscid limit. It may be possible to extend the present analysis to
include the propagation of boundary layers and the classical Prandtl’s
boundary layer expansions, whose validity near general boundary layers again
requires analyticity.

The first such a result was due to the celebrated work by Asano [2] and
Sammartino-Caflisch [22], where the boundary layer expansion was established
for data on the half-plane that are analytic in both horizontal and vertical
variables. When constructing solutions to the Prandtl equation, the
analyticity in the vertical variable can be dropped [15]. It is not known
however if such an assumption can be dropped at the level of Navier-Stokes
equations. Maekawa [20] established the Prandtl’s expansion for data whose
vorticity is compactly supported away from the boundary, while recently
Kukavica, Nguyen, Vicol and Wang [18] extended the result to include data that
are analytic only near the boundary, building upon the vorticity formulation
revived by Maekawa [20], the direct proof of the inviscid limit for analytic
data developed in Nguyen and Nguyen [21], and the Sobolev-analytic norm
developed in Kukavica, Vicol and Wang [19]. All these aforementioned works are
on the half-plane. We mention a recent result [23], which to the best of our
knowledge was the first to establish a Prandtl asymptotic expansion in a
curved domain.

When background boundary layers have no inflection point, the analyticity can
be relaxed to include perturbations in Gevrey-$\frac{3}{2}$ spaces [7, 8],
which is sharp in view of the Kelvin-Helmholtz type of instability of generic
boundary layers and shear flows [10, 11]. When Sobolev data is allowed, the
Prandtl’s asymptotic expansion is false due to counter-examples given in [9,
12, 13], where the failure of the convergence from Navier-Stokes to Euler
solutions, plus a Prandtl corrector, is due to an emergence of viscous
boundary sublayers that reach to order one, independent of viscosity, in
$L^{\infty}$ norm for velocity [12].

##  2 Navier-Stokes equations near the boundary

###  2.1 Global geodesic coordinates

Following a construction done in [5] we introduce a well adapted
representation of $\partial\Omega\,,$

$\theta\in{\mathbb{T}}={\mathbb{R}}/({\mathbb{Z}}L)\mapsto
x(\theta)=(x_{1}(\theta),x_{2}(\theta))\in\partial\Omega$

which, being global, preserves the analyticity hypothesis. Let
$\vec{\tau}(\theta)$ and $\vec{n}(\theta)$ be the unit tangent and interior
normal vectors at the boundary:

$\displaystyle\vec{\tau}(\theta)=\vec{\tau}(x(\theta)=(x^{\prime}_{1}(\theta),x^{\prime}_{2}(\theta)),\quad\hbox{
and}\quad\vec{n}(\theta)=\vec{n}(x(\theta))=(-x^{\prime}_{2}(\theta),x^{\prime}_{1}(\theta))$
(2.1) $\displaystyle\hbox{
with}\quad|x^{\prime}(\theta)|^{2}=(x^{\prime}_{1}(\theta))^{2}+(x^{\prime}_{1}(\theta))^{2}=1.$

Let $d(x,\partial\Omega)$ denotes the distance of any point
$x\in{\mathbb{R}}^{2}$ to $\partial\Omega$ . Then we have the following
classical result.

######  Proposition 2.1.

There exists a $\delta>0$ such that for each $x$ on the open set

$V_{\delta}=\\{x\in{\mathbb{R}}^{2}\quad\hbox{with}\quad
d(x,\partial\Omega)<{\delta}\\}$ (2.2)

there is a unique point $\hat{x}(\theta)\in\partial\Omega$ with
$d(x,\partial\Omega)=|x-\hat{x}(\theta)|.$ The mapping
$x\mapsto\hat{x}(\theta)$ is an analytic map from $V_{\delta}$ with value in
$\partial\Omega$. In addition, for $x\in V_{\delta}\,,$ one has the formula

$\nabla_{x}d(x,\partial\Omega)=\vec{n}(x(\theta)).$ (2.3)

When no confusion is possible, for $x\in V_{\delta}$ the notations
$\vec{n}(x)$ and $\vec{\tau}(x)$ will be used for $\vec{n}(x(\theta))$ and
$\vec{\tau}(x(\theta))$ respectively. Observe that

$\vec{\tau}^{\prime}(\theta)\wedge\vec{n}(\theta)=x^{\prime}_{1}(\theta)x_{1}^{\prime\prime}(\theta)+x^{\prime}_{2}(\theta)x_{2}^{\prime\prime}(\theta)=\frac{d}{d\theta}|x^{\prime}(\theta)|^{2}=0\,,$
(2.4)

which implies the relation

$\vec{n}^{\prime}(\theta)=\gamma(\theta)\vec{\tau}(\theta)\quad\hbox{and}\quad\vec{\tau}^{\prime}(\theta)=\gamma(\theta)\vec{n}(\theta)\,,$
(2.5)

with

$\gamma(\theta)=x_{1}^{\prime\prime}(\theta)x_{2}^{\prime}(\theta)-x^{\prime}_{1}(\theta)x_{2}^{\prime\prime}(\theta),$
(2.6)

being the curvature of the boundary $\partial\Omega\,.$ Therefore the mapping:

$(z,\theta)\mapsto X(z,\theta)=x(\theta)+z\vec{n}(x(\theta)),$ (2.7)

defines au global $C^{2}$ diffeomorphisme of
$[-\delta,\delta]\times({\mathbb{R}}/(L{\mathbb{Z}})) $ on
$\overline{V_{\delta}}\,.$ Moreover, for any vector field
$x\in\overline{\Omega}\mapsto v(x)\,,$ as soon as
$x\in\overline{V}_{\delta}\,,$ using the above notations, one has:

$v(x)=(v(x)\cdot\vec{\tau}(x))\vec{\tau}(x)+(v(x)\cdot\vec{n}(x))\vec{n}(x)\,.$
(2.8)

Below, for sake of clarity, the symbol $X$ is used for any $x=X(z,\theta)$.
There hold

$\displaystyle\partial_{z}X(z,\theta)=\vec{n}(\theta)\,,\quad\partial_{\theta}X(z,\theta)=J(z,\theta)\vec{\tau}(\theta)\,,$
(2.9) $\displaystyle\hbox{and}\quad J(z,\theta)=1+z\gamma(\theta)>0
\quad\hbox{for}\quad|z|<\delta\,,$

provided $\delta>0$ is chosen to be small enough. From the relation

$\begin{pmatrix}\partial_{z}X_{1}&\partial_{\theta}X_{1}\\\
\partial_{z}X_{2}&\partial_{\theta}X_{2}\end{pmatrix}\begin{pmatrix}\partial_{X_{1}}z&\partial_{X_{2}}z\\\
\partial_{X_{1}}\theta&\partial_{X_{2}}\theta\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\,,$ (2.10)

one deduces the formula:

$\nabla_{X}\theta=\frac{\vec{\tau}(z,\theta))}{J(z,\theta)}\quad\hbox{and}\quad\nabla_{X}z=\vec{n}(\theta)\,.$
(2.11)

We collect the following useful relations whose derivations are classical. For
any vector field $u$, we have

$\displaystyle\nabla\cdot u$
$\displaystyle=\frac{1}{J}(\partial_{z}(J(u\cdot\vec{n}))+\partial_{\theta}(u\cdot\vec{\tau}))=\partial_{z}(u\cdot\vec{n})+\frac{1}{J}\partial_{\theta}(u\cdot\vec{\tau}))+\frac{\gamma}{J}u\cdot\vec{n}\,,$
(2.12) $\displaystyle\nabla\wedge u$
$\displaystyle=\frac{1}{J}(\partial_{z}(Ju\cdot\vec{\tau})-\partial_{\theta}(u\cdot\vec{\tau}))=\partial_{z}(u\cdot\vec{\tau})-\frac{1}{J}\partial_{\theta}(u\cdot\vec{\tau}))+\frac{\gamma}{J}(u\cdot\vec{\tau})\,.$

For any scalar function $\Psi$, we have

$\nabla\wedge\Psi=\frac{1}{J}\Bigg{(}\begin{aligned} &\partial_{z}(J\Psi)\\\
&-\partial_{\theta}\Psi\end{aligned}\Bigg{)}=\Bigg{(}\begin{aligned}
&\partial_{z}\Psi\\\
&-\frac{1}{J}\partial_{\theta}\Psi\end{aligned}\Bigg{)}+\Bigg{(}\begin{aligned}
&\frac{\gamma}{J}\Psi\\\ &0\end{aligned}\Bigg{)}\,,$ (2.13)

and

$\displaystyle\Delta\Psi$
$\displaystyle=\frac{1}{J}\partial_{z}(J\partial_{z}\Psi)+\frac{1}{J}\partial_{\theta}(\frac{1}{J}\partial_{\theta}\Psi)=\Delta_{z,\theta}\Psi+R_{\Delta}\Psi\,,$
(2.14)

in which we denote

$\Delta_{z,\theta}=(\partial^{2}_{z}+\partial^{2}_{\theta}),\quad
R_{\Delta}=m(z,\theta)\partial^{2}_{\theta}+\frac{\gamma}{1+z\gamma}\partial_{z}-\frac{z\gamma^{\prime}}{(1+z\gamma)^{3}}\partial_{\theta}\quad\hbox{and}\quad
m(z,\theta)=-\frac{2z\gamma+(z\gamma)^{2}}{(1+z\gamma)^{2}}.$

###  2.2 Scaled coordinates

In view of (2.14), we observe that the Laplacian $\Delta$ is nearly the flat
Laplacian $\Delta_{z,\theta}$, in the $(z,\theta)$ coordinates, near the
boundary. To make use of this fact, we introduce the following scaled
variables

$(\widetilde{z},\widetilde{\theta})=(\lambda z,\lambda\theta)$ (2.15)

for sufficiently small $\lambda\in(0,1)$. By construction, we compute

$\Delta=\lambda^{2}\Big{(}\Delta_{\widetilde{z},\widetilde{\theta}}+\lambda^{2}\widetilde{R}_{\Delta}\Big{)}\,,$
(2.16)

in which
$\Delta_{\widetilde{z},\widetilde{\theta}}=(\partial^{2}_{\widetilde{z}}+\partial^{2}_{\widetilde{\theta}})$
and

$\widetilde{R}_{\Delta}=\widetilde{m}(\widetilde{z},\widetilde{\theta})\partial^{2}_{\widetilde{\theta}}+\frac{\widetilde{\gamma}}{1+\lambda^{2}\widetilde{z}\widetilde{\gamma}}\partial_{\widetilde{z}}-\frac{\widetilde{z}\widetilde{\gamma}^{\prime}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{3}}\partial_{\widetilde{\theta}}\quad\hbox{and}\quad\widetilde{m}(\widetilde{z},\widetilde{\theta})=-\frac{2\widetilde{z}\widetilde{\gamma}+\lambda^{2}(\widetilde{z}\widetilde{\gamma})^{2}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{2}}\,,$
(2.17)

where $\gamma=\lambda^{3}\widetilde{\gamma}(\widetilde{\theta})$. In the
analysis, $\lambda$ will be taken sufficiently small, and so $\Delta$ is
indeed approximated by $\lambda^{2}\Delta_{\widetilde{z},\widetilde{\theta}}$,
treating $\lambda^{2}\widetilde{R}_{\Delta}$ as a perturbation.

###  2.3 Vorticity equations near the boundary

In this section, we derive vorticity equations in the geodesic coordinates
near the boundary in the region $V_{\delta}$ defined as in Proposition 2.1.
Introduce a smooth cutoff function $\phi^{b}(x)$ so that

$\phi^{b}(x)=\left\\{\begin{aligned} 1,\qquad&\mbox{if}\quad\lambda
d(x,\partial\Omega)\leq\delta_{0}+\rho_{0}\\\ 0,\qquad&\mbox{if}\quad\lambda
d(x,\partial\Omega)\geq\delta_{0}+2\rho_{0}\end{aligned}\right.$ (2.18)

for small positive constants $\delta_{0},\rho_{0}$ so that
$\delta_{0}+2\rho_{0}<\lambda\delta$ to guarantee that
$\hbox{supp}(\phi^{b})\subset V_{\delta}$ as in Proposition 2.1. Define

${\omega}^{b}=\phi^{b}(x){\omega}(t,x).$ (2.19)

It follows from (1.4) that

$\partial_{t}{\omega}^{b}-\nu\Delta{\omega}^{b}=N^{b},$ (2.20)

where

$N^{b}:=-u\cdot\nabla{\omega}^{b}+(u\cdot\nabla\phi^{b}){\omega}-\nu(\Delta\phi^{b}){\omega}-2\nu\nabla\phi^{b}\cdot\nabla{\omega}.$

Observe that $N^{b}(u,\omega)=0$ on $\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}+2\rho_{0}\\}$ where the cutoff function
$\phi^{b}$ vanishes. We then introduce the following scaled vorticity

$\omega^{b}(t,x)=\widetilde{\omega}(\lambda^{2}t,\lambda\theta,\lambda
z),\qquad(\widetilde{t},\widetilde{z},\widetilde{\theta})=(\lambda^{2}t,\lambda
z,\lambda\theta),$ (2.21)

for small $\lambda>0$. Using (2.16), we rewrite the vorticity equation as

$\left(\partial_{\widetilde{t}}-\nu\Delta_{\widetilde{z},\widetilde{\theta}}\right)\widetilde{\omega}=-\nu\lambda^{2}\widetilde{R}_{\Delta}\widetilde{\omega}+\lambda^{-2}N^{b}.$
(2.22)

Equation (2.22) is defined on
$(\widetilde{z},\widetilde{\theta})\in\mathbb{R}_{+}\times{\mathbb{T}}$ (in
fact, the equation vanishes for $\widetilde{z}\geq\delta_{0}+2\rho_{0}$). We
shall solve (2.22) together with the boundary condition (1.8), which now reads

$\nu(\partial_{\widetilde{z}}+\widetilde{DN})\widetilde{\omega}_{|_{\widetilde{z}=0}}=\lambda^{-1}[\partial_{n}\Delta^{-1}(u\cdot\nabla\omega)]_{|_{\partial\Omega}}.$
(2.23)

System (2.22)-(2.23) will be our main equation for the scaled vorticity near
the boundary. Away from the boundary, we construct vorticity using the
original system as derived in Section 1.1.

###  2.4 Dirichlet-Neumann operator

Let us precise the Dirichlet-Neumann operator defined as in (1.6)-(1.7).

######  Lemma 2.2.

For $\omega\in H^{1/2}(\partial\Omega)$, let $DN\omega$ be the Dirichlet-
Neumann operator defined as in (1.6)-(1.7). In the scaled variables, there
holds

$\widetilde{DN}\widetilde{\omega}=|\partial_{\widetilde{\theta}}|\widetilde{\omega}+\widetilde{B}\widetilde{\omega}$
(2.24)

for some linear bounded operator $\widetilde{B}$ from $L^{2}(\partial\Omega)$
to itself: namely,

$\|\widetilde{B}\widetilde{\omega}\|_{L^{2}(\partial\Omega)}\leq
C_{0}\|\widetilde{\omega}\|_{L^{2}(\partial\Omega)}$

for some positive constant $C_{0}$.

###### Proof.

Let $\phi^{b}$ be the cutoff function defined as in (2.18), and set
$\omega^{*b}=\phi^{b}\omega^{*}$, where $\phi^{*}$ solves (1.6). It follows
that

$\left\\{\begin{aligned}
\Delta\omega^{*b}&=(\Delta\phi^{b}){\omega}^{*}-2\nabla\phi^{b}\cdot\nabla{\omega}^{*},\qquad\mbox{in}\quad\Omega\\\
{\omega^{*b}}&=\omega,\qquad\mbox{on
}\quad\partial\Omega.\end{aligned}\right.$ (2.25)

Since $\phi^{p}$ vanishes away from the boundary, we can work in the scaled
variables, which reads
$\widetilde{DN}\widetilde{\omega}=-\partial_{\widetilde{z}}\widetilde{\omega}^{*}_{|_{\widetilde{z}=0}}$.
Recalling (2.16), the scaled function
$\widetilde{\omega}^{*}(\widetilde{t},\widetilde{z},\widetilde{\theta})$ of
$\omega^{*b}$ solves

$\Delta_{\widetilde{z},\widetilde{\theta}}\widetilde{\omega}^{*}=-\lambda^{2}\widetilde{R}_{\Delta}\widetilde{\omega}^{*}+\lambda^{-2}[(\Delta\phi^{b}){\omega}^{*}-2\nabla\phi^{b}\cdot\nabla{\omega}^{*}],\qquad\widetilde{\omega}^{*}|_{\widetilde{z}=0}=\widetilde{\omega}|_{\widetilde{z}=0}\,,$

on $\mathbb{R}_{+}\times{\mathbb{T}}$, which can be solved explicitly. Indeed,
let $\widetilde{\omega}_{\alpha}$ be the Fourier coefficient of
$\widetilde{\omega}(\widetilde{z},\widetilde{\theta})$ in variable
$\widetilde{\theta}$. Note that $\widetilde{\omega}_{\alpha}$ vanishes for
$\alpha=0$, and thus we focus on the case when $\alpha\not=0$. Let
$K_{\alpha}(\widetilde{y},\widetilde{z})=\frac{1}{2|\alpha|}(e^{-|\alpha(\widetilde{y}-\widetilde{z})|}-e^{-|\alpha(\widetilde{y}+\widetilde{z})|})$
be the Green function of the Laplacian
$\partial_{\widetilde{z}}^{2}-\alpha^{2}$ with the Dirichlet boundary
condition. It follows that

$\displaystyle\widetilde{\omega}^{*}_{\alpha}(\widetilde{z})$
$\displaystyle=e^{-|\alpha|\widetilde{z}}\widetilde{\omega}_{\alpha}(0)+\lambda^{2}\int_{0}^{\infty}K_{\alpha}(\widetilde{y},\widetilde{z})(\widetilde{R}_{\Delta}\widetilde{\omega}^{*})_{\alpha}(\widetilde{y})\;d\widetilde{y}$
(2.26)
$\displaystyle\qquad{+\lambda^{-2}\int_{0}^{\infty}K_{\alpha}(\widetilde{y},\widetilde{z})\Big{[}(\Delta\phi^{b}){\omega}^{*}-2\nabla\phi^{b}\cdot\nabla{\omega}^{*}\Big{]}_{\alpha}(\widetilde{y})\;d\widetilde{y}}$

for $\widetilde{z}\geq 0$. The Dirichlet-Neumann operator is thus computed by

$\displaystyle(\widetilde{DN}\widetilde{\omega})_{\alpha}$
$\displaystyle=-\partial_{\widetilde{z}}\widetilde{\omega}^{*}_{\alpha}(0)$
$\displaystyle=|\alpha|\widetilde{\omega}_{\alpha}(0)+\int_{0}^{\infty}e^{-|\alpha|\widetilde{y}}\Big{[}\lambda^{2}(\widetilde{R}_{\Delta}\widetilde{\omega}^{*})_{\alpha}+\lambda^{-2}((\Delta\phi^{b}){\omega}^{*}-2\nabla\phi^{b}\cdot\nabla{\omega}^{*})_{\alpha}\Big{]}(\widetilde{y})\;d\widetilde{y}.$

The decomposition (2.24) thus follows, upon defining $\widetilde{B}$ as the
integral term

$(\widetilde{B}\widetilde{\omega})_{\alpha}:=\int_{0}^{\infty}e^{-|\alpha|\widetilde{y}}\Big{[}\lambda^{2}(\widetilde{R}_{\Delta}\widetilde{\omega}^{*})_{\alpha}+\lambda^{-2}((\Delta\phi^{b}){\omega}^{*}-2\nabla\phi^{b}\cdot\nabla{\omega}^{*})_{\alpha}\Big{]}(\widetilde{y})\;d\widetilde{y}\,,$
(2.27)

for each Fourier variable $\alpha\in{\mathbb{Z}}$. It remains to prove the
boundedness of $\widetilde{B}$. Note that by definition, the last two terms
are defined on the region $\widetilde{y}\geq\delta_{0}+\rho_{0}$ where the
cutoff function $\phi^{b}=1$. Therefore,

$\Big{|}\int_{0}^{\infty}e^{-|\alpha|\widetilde{y}}((\Delta\phi^{b}){\omega}^{*}-2\nabla\phi^{b}\cdot\nabla{\omega}^{*})_{\alpha}(\widetilde{y})\;d\widetilde{y}\Big{|}\lesssim\|\omega^{\star}\|_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}.$

It remains to bound the first integral term in (2.27). In view of (2.17), we
write

$\displaystyle\widetilde{R}_{\Delta}\widetilde{\omega}^{*}$
$\displaystyle=\partial^{2}_{\widetilde{\theta}}[\widetilde{m}\widetilde{\omega}^{*}]-\partial_{\widetilde{\theta}}\Big{[}2\partial_{\widetilde{\theta}}\widetilde{m}\widetilde{\omega}^{*}+\frac{\widetilde{z}\widetilde{\gamma}^{\prime}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{3}}\widetilde{\omega}^{*}\Big{]}+\partial_{\widetilde{z}}\Big{(}\frac{\widetilde{\gamma}}{1+\lambda^{2}\widetilde{z}\widetilde{\gamma}}\widetilde{\omega}^{*}\Big{)}$
$\displaystyle\quad+\Big{[}(\partial^{2}_{\widetilde{\theta}}\widetilde{m})-\partial_{\widetilde{z}}\Big{(}\frac{\widetilde{\gamma}}{1+\lambda^{2}\widetilde{z}\widetilde{\gamma}}\Big{)}+\partial_{\widetilde{\theta}}\Big{(}\frac{\widetilde{z}\widetilde{\gamma}^{\prime}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{3}}\Big{)}\Big{]}\widetilde{\omega}^{*},$

noting the coefficients are analytic near the boundary. We note in particular
that there is no growth in large $\widetilde{z}$: for instance,
$m(\widetilde{z},\widetilde{\theta})\lesssim\lambda^{-2}$ uniformly in large
$\widetilde{z}$. In addition, we note that
$\widetilde{m}=\widetilde{z}\widetilde{m}_{1}$ for some bounded function
$\widetilde{m}_{1}$. Thus, using the fact that
$|\alpha|\widetilde{y}e^{-\frac{1}{2}|\alpha|\widetilde{y}}\lesssim 1$, the
second-order derivative term
$\partial^{2}_{\widetilde{\theta}}[\widetilde{m}\widetilde{\omega}^{*}]$ thus
can be treated as the first order derivative term. Precisely, we can treat the
first integral in (2.27) systematically as follows: for some smooth and
bounded coefficients $b(\widetilde{z},\widetilde{\theta})$,

$\lambda^{2}\int_{0}^{\infty}e^{-\frac{1}{2}|\alpha|\widetilde{y}}|(\alpha,\partial_{\widetilde{y}})(b\widetilde{\omega}^{*})_{\alpha}|(\widetilde{y})\;d\widetilde{y}\lesssim\lambda^{2}|\alpha|^{-1/2}\|(\alpha,\partial_{\widetilde{y}})(b\widetilde{\omega}^{*})_{\alpha}\|_{L^{2}_{\widetilde{y}}}.$

This yields

$\displaystyle|(\widetilde{B}\widetilde{\omega})_{\alpha}|$
$\displaystyle\lesssim\lambda^{2}|\alpha|^{-1/2}\|(\alpha,\partial_{\widetilde{y}})(b\widetilde{\omega}^{*})_{\alpha}\|_{L^{2}_{\widetilde{y}}}+\|\omega^{\star}\|_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}\,.$

Taking $L^{2}_{\alpha}$, we thus obtain

$\displaystyle\sum_{\alpha}|(\widetilde{B}\widetilde{\omega})_{\alpha}|^{2}$
$\displaystyle\lesssim\lambda^{2}\sum_{\alpha}|\alpha|^{-1}\|(\alpha,\partial_{\widetilde{y}})\widetilde{\omega}^{*}_{\alpha}\|_{L^{2}_{\widetilde{y}}}^{2},$
(2.28)

upon noting that the coefficients $b(\widetilde{z},\widetilde{\theta})$, which
in particular have
$\|b_{\alpha}(\widetilde{z})\|_{L^{1}_{\alpha}L^{\infty}_{\widetilde{z}}}<\infty$.
It remains to bound the right-hand side of (2.28). Directly from (2.26), we
compute

$\displaystyle|(\alpha,\partial_{\widetilde{z}})\widetilde{\omega}^{*}_{\alpha}(\widetilde{z})|$
$\displaystyle\lesssim|\alpha|e^{-|\alpha|\widetilde{z}}|\widetilde{\omega}_{\alpha}(0)|+\lambda^{2}\int_{0}^{\infty}e^{-|\alpha(\widetilde{z}-\widetilde{z}^{\prime})|}|(\widetilde{R}_{\Delta}\widetilde{\omega}^{*})_{\alpha}(\widetilde{z}^{\prime})|\;d\widetilde{z}^{\prime}+|\alpha|^{-1/2}\|\omega^{\star}\|_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}.$

Therefore, together with the standard Hausdorff-Young’s inequality, we bound

$\displaystyle\|(\alpha,\partial_{\widetilde{z}})\widetilde{\omega}^{*}_{\alpha}\|_{L^{2}_{\widetilde{z}}}$
$\displaystyle\lesssim|\alpha|^{1/2}|\widetilde{\omega}_{\alpha}(0)|+\lambda^{2}|\alpha|^{-1}\|(\widetilde{R}_{\Delta}\widetilde{\omega}^{*})_{\alpha}\|_{L^{2}_{\widetilde{z}}}+|\alpha|^{-1/2}\|\omega^{\star}\|_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}$

which yields

$\displaystyle\sum_{\alpha}|\alpha|^{-1}\|(\alpha,\partial_{\widetilde{z}})\widetilde{\omega}^{*}_{\alpha}\|_{L^{2}_{\widetilde{z}}}^{2}$
$\displaystyle\lesssim\sum_{\alpha}|\widetilde{\omega}_{\alpha}(0)|^{2}+\lambda^{2}\sum_{\alpha}|\alpha|^{-3}\|(\widetilde{R}_{\Delta}\widetilde{\omega}^{*})_{\alpha}\|_{L^{2}_{\widetilde{z}}}^{2}+\|\omega^{\star}\|_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}^{2}$
$\displaystyle\lesssim\sum_{\alpha}|\widetilde{\omega}_{\alpha}(0)|^{2}+\lambda^{2}\sum_{\alpha}|\alpha|^{-1}\|(\alpha,\partial_{\widetilde{z}})\widetilde{\omega}^{*}_{\alpha}\|_{L^{2}_{\widetilde{z}}}^{2}$
$\displaystyle\quad+\sum_{\alpha}|\alpha|^{-1}\|(\alpha,\partial_{\widetilde{z}})\widetilde{\omega}^{*}_{\alpha}\|^{2}_{L^{2}_{\\{{\widetilde{z}\geq\delta_{0}+\rho_{0}}\\}}}+\|\omega^{\star}\|_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}.$

Taking $\lambda$ sufficiently small so that the second term on the right can
be absorbed into the left. On the other hand, using the standard elliptic
theory, the last term is bounded by

$\sum_{\alpha}|\alpha|^{-1}\|(\alpha,\partial_{\widetilde{z}})\widetilde{\omega}^{\star}_{\alpha}\|^{2}_{L^{2}_{\\{{\widetilde{z}\geq\delta_{0}+\rho_{0}}\\}}}\lesssim\|\omega^{\star}\|^{2}_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0})}\lesssim\|\omega\|_{L^{2}(\partial\Omega)}^{2}.$

Putting these back into (2.28), we obtain the lemma. ∎

##  3 Near boundary analytic spaces

In this section, we introduce the near boundary analytic norm used to control
the vorticity that is analytic near the boundary, but however only has Sobolev
regularity away from the boundary. We then derive sufficient elliptic
estimates, bilinear estimates, as well as the semigroup estimates in these
analytic spaces.

###  3.1 Analytic norms

Let $\delta>0$ be small and so that Proposition 2.1 applies for
$\bar{V}_{\delta}=\\{d(x,\partial\Omega)\leq\delta\\}$. In particular,
$\delta$ is small so that the statement of 2.1 still holds for $V_{2\delta}$.
Now for any constant $\lambda\in(0,1)$, we have

$\lambda d(x,\partial\Omega)\leq\lambda\delta$

for all $x\in\bar{V}_{\delta}$. Let $\delta_{0}=\lambda\delta$, which will the
size of the analytic domain for our solution near the boundary. We fix
$\rho_{0}\in(0,1/10)$, and assume that $\rho\in(0,\rho_{0})$. Then

$\Omega_{\rho}=\\{\widetilde{z}\in\mathbb{C}:0\leq\Re\widetilde{z}\leq\delta_{0},|\Im\widetilde{z}|\leq\rho\Re\widetilde{z}\\}\cup\\{\widetilde{z}\in\mathbb{C}:\delta_{0}\leq\Re\widetilde{z}\leq\delta_{0}+\rho,|\Im\widetilde{z}|\leq\delta_{0}+\rho-\Re\widetilde{z}\\}$
(3.1)

denotes the complex domain for functions of the $\widetilde{z}$ variable. We
note that the domain $\Omega_{\rho}$ only contains $\widetilde{z}$ with
$0\leq\Re\widetilde{z}\leq\delta_{0}+\rho$. For a complex valued function $f$
defined on $\Omega_{\rho}$, let

$\|f\|_{L^{1}_{\rho}}=\sup_{0\leq\eta<\rho}\|f\|_{L^{1}(\partial\Omega_{\eta})},\qquad\|f\|_{L^{\infty}_{\rho}}=\sup_{0\leq\eta<\rho}\|f\|_{L^{\infty}(\partial\Omega_{\eta})}$

where the integration is taken over the two directed paths along the boundary
of the domain $\Omega_{\eta}$. Now for an analytic function
$f(\widetilde{\theta},\widetilde{z})$ defined on
$(\widetilde{\theta},\widetilde{z})\in\mathbb{T}\times\Omega_{\rho}$, we
define

$\displaystyle\|f\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle=\sum_{\alpha\in\mathbb{Z}}\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|\alpha|}f_{\alpha}\|_{L^{1}_{\rho}},$
(3.2) $\displaystyle\|f\|_{\mathcal{L}^{\infty}_{\rho}}$
$\displaystyle=\sum_{\alpha\in\mathbb{Z}}\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|\alpha|}f_{\alpha}\|_{L^{\infty}_{\rho}},$

where $f_{\alpha}$ denotes the Fourier transform of $f$ with respect to
variable $\widetilde{\theta}$. The function spaces $\mathcal{L}^{1}_{\rho}$
and $\mathcal{L}_{\rho}^{\infty}$ are to control the scaled vorticity and
velocity, respectively. We stress that the analyticity weight vanishes on
$\Re\widetilde{z}\geq\delta_{0}+\rho$. For convenience, we also introduce the
following analytic norms

$\|f\|_{\mathcal{W}_{\rho}^{k,p}}=\sum_{i+j\leq
k}\|\partial_{\widetilde{\theta}}^{i}(\widetilde{z}\partial_{\widetilde{z}})^{j}f\|_{\mathcal{L}^{p}_{\rho}}$
(3.3)

for $k\geq 0$ and $p=1,\infty$. We observe the following simple algebra.

######  Lemma 3.1.

There hold

$\|fg\|_{\mathcal{L}^{1}_{\rho}}\leq\|f\|_{\mathcal{L}^{\infty}_{\rho}}\|g\|_{\mathcal{L}^{1}_{\rho}}$
(3.4)

and for any $0<\rho^{\prime}<\rho$,

$\|\partial_{\widetilde{\theta}}f\|_{\mathcal{L}^{1}_{\rho^{\prime}}}+\|\widetilde{z}\partial_{\widetilde{z}}f\|_{\mathcal{L}^{1}_{\rho^{\prime}}}\lesssim\frac{1}{\rho-\rho^{\prime}}\|f\|_{\mathcal{L}^{1}_{\rho}}.$
(3.5)

###### Proof.

By definition, we compute

$\displaystyle
e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|\alpha|}|(fg)_{\alpha}(\widetilde{z})|$
$\displaystyle\leq\sum_{\alpha^{\prime}}|f_{\alpha-{\alpha^{\prime}}}(\widetilde{z})g_{\alpha^{\prime}}(\widetilde{z})|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|\alpha|}$
$\displaystyle\leq\sum_{\alpha^{\prime}}|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|\alpha-{\alpha^{\prime}}|}f_{\alpha-{\alpha^{\prime}}}(\widetilde{z})e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|{\alpha^{\prime}}|}g_{\alpha^{\prime}}(\widetilde{z})|$

which gives

$\displaystyle\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|\alpha|}(fg)_{\alpha}(\widetilde{z})\|_{\mathcal{L}_{\rho}^{1}}$
$\displaystyle\leq\sum_{\alpha^{\prime}}\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|\alpha-{\alpha^{\prime}}|}f_{\alpha-{\alpha^{\prime}}}\|_{\mathcal{L}^{\infty}_{\rho}}\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re\widetilde{z})|{\alpha^{\prime}}|}g_{\alpha^{\prime}}\|_{\mathcal{L}^{1}_{\rho}}.$

The estimate (3.4) follows from taking the summation in $\alpha$ over
${\mathbb{Z}}$. The stated bounds on derivatives are classical (e.g., [22,
21]), making use of the fact that
$(\rho-\rho^{\prime})|\alpha|e^{(\rho^{\prime}-\rho)|\alpha|}$ is bounded. ∎

###  3.2 Elliptic estimates in the half-plane

In this section, we derive some basic elliptic estimates in the analytic
spaces $\mathcal{W}^{k,p}_{\rho}$. Precisely, we consider

$\left\\{\begin{aligned} \Delta_{z,\theta}\phi&=f,\qquad\mbox{in
}\quad\mathbb{R}_{+}\times{\mathbb{T}}\\\
\phi_{|_{z=0}}&=0\end{aligned}\right.$ (3.6)

in which we drop titles for sake of presentation. The
$\mathcal{W}^{k,p}_{\rho}$ analytic norm is defined on $\Re
z\leq\delta_{0}+\rho$ as introduced in the previous section. We obtain the
following proposition.

######  Proposition 3.2.

Let $\phi$ be the solution of (3.6). Then, the velocity field
$u=\nabla^{\perp}\phi$ satisfies

$\displaystyle\|u\|_{\mathcal{W}^{k,\infty}_{\rho}}$
$\displaystyle\lesssim\|f\|_{\mathcal{W}^{k,1}_{\rho}}+\|f\|_{H^{k+1}(\\{z\geq\delta_{0}+\rho\\})}$
(3.7)
$\displaystyle\|(\frac{1}{z}\partial_{\theta}\phi)\|_{\mathcal{W}^{k,\infty}_{\rho}}$
$\displaystyle\lesssim\|f\|_{\mathcal{W}^{k,1}_{\rho}}+\|\partial_{\theta}f\|_{\mathcal{W}^{k,1}_{\rho}}+\|f\|_{H^{k+1}(\\{z\geq\delta_{0}+\rho\\})}$
$\displaystyle\|\nabla_{z,\theta}u\|_{\mathcal{W}^{k,\infty}_{\rho}}$
$\displaystyle\lesssim\|f\|_{\mathcal{W}^{k,\infty}_{\rho}}+\|f\|_{H^{k+2}(\\{z\geq\delta_{0}+\rho\\})}$

for $k\geq 0$.

###### Proof.

The elliptic problem (3.6) can be solved explicitly in Fourier space. Indeed,
taking the Fourier transform in $\theta$, we get the elliptic equation

$(\partial_{z}^{2}-\alpha^{2})\phi_{\alpha}=f_{\alpha}$

for the Fourier transform $\phi_{\alpha}$. We focus on the case $\alpha>0$;
the other case is similar. The solution is given by

$\phi_{\alpha}(z)=\int_{0}^{z}K_{-}(y,z)f_{\alpha}(y)dy+\int_{z}^{\infty}K_{+}(y,z)f_{\alpha}(y)dy$

with the Green function defined by

$K_{\pm}(y,z)=-\frac{1}{2\alpha}\Big{(}e^{\pm\alpha(z-y)}-e^{-\alpha(y+z)}\Big{)}.$

This expression may be extended to complex values of $z$. Indeed, for
$z\in\Omega_{\sigma}$, there is a positive $\theta$ so that
$z\in\partial\Omega_{\theta}$. We then write
$\partial\Omega_{\theta}=\gamma_{-}(z)\cup\gamma_{+}(z)$, consisting of
complex numbers $y\in\partial\Omega_{\theta}$ so that $\Re y<\Re z$ and $\Re
y>\Re z$, respectively. Then, the integral is taken over $\gamma_{-}(z)$ and
$\gamma_{+}(z)$, respectively. We note in particular that for
$y\in\gamma_{\pm}(z)$, there hold the same bounds on the Green function

$|K_{\pm}(y,z)|\leq\alpha^{-1}e^{-\alpha|y-z|}.$

This proves that

$\displaystyle|\phi_{\alpha}(z)|$
$\displaystyle\leq\int_{\partial\Omega_{\theta}}\alpha^{-1}e^{-\alpha|y-z|}|f_{\alpha}(y)||dy|.$
(3.8)

By definition of $\mathcal{L}^{1}_{\rho}$ norm, we only need to consider the
case when $0\leq\Re z\leq\delta_{0}+\rho$. Now, for $0\leq\Re
y\leq\delta_{0}+\rho$, we bound

$e^{-\alpha|\Re y-\Re z|}e^{-\varepsilon_{0}(\delta_{0}+\rho-\Re y)\alpha}\leq
e^{-\varepsilon_{0}(\delta_{0}+\rho-\Re
z)\alpha}e^{-(1-\epsilon_{0})\alpha|\Re y-\Re z|}$

noting $\epsilon_{0}\leq 1/2$. On the other hand, for $\Re
y\geq\delta_{0}+\rho$ (recalling $\delta_{0}+\rho\geq\Re z$), we bound

$e^{-\alpha|\Re y-\Re z|}\leq e^{-\epsilon_{0}(\delta_{0}+\rho-\Re
z)\alpha}e^{-(1-\epsilon_{0})\alpha|\Re y-\Re z|}.$

Therefore, we bound

$\displaystyle\int_{\Re
y\leq\delta_{0}+\rho}\alpha^{-1}e^{-\alpha|y-z|}|f_{\alpha}(y)||dy|$
$\displaystyle\lesssim\alpha^{-1}e^{-\varepsilon_{0}(\delta_{0}+\rho-\Re
z)\alpha}\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re
y)\alpha}f_{\alpha}\|_{L^{1}_{\rho}},$ $\displaystyle\int_{\Re
y\geq\delta_{0}+\rho}\alpha^{-1}e^{-\alpha|y-z|}|f_{\alpha}(y)||dy|$
$\displaystyle\lesssim\alpha^{-3/2}e^{-\varepsilon_{0}(\delta_{0}+\rho-\Re
z)\alpha}\|f_{\alpha}\|_{L^{2}(y\geq\delta_{0}+\rho)}.$

Similarly, we also have

$\displaystyle\int_{\Re
y\leq\delta_{0}+\rho}\alpha^{-1}e^{-\alpha|y-z|}|f_{\alpha}(y)||dy|$
$\displaystyle\lesssim\alpha^{-2}e^{-\varepsilon_{0}(\delta_{0}+\rho-\Re
z)\alpha}\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re
y)\alpha}f_{\alpha}\|_{L^{\infty}_{\rho}},$

which gains an extra factor of $\alpha$. This proves

$\displaystyle\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re
z)\alpha}(\alpha,\partial_{z})\phi_{\alpha}\|_{L^{\infty}_{\rho}}$
$\displaystyle\leq\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re
y)\alpha}f_{\alpha}\|_{L^{1}_{\rho}}+\alpha^{-1/2}\|f_{\alpha}\|_{L^{2}(y\geq\delta_{0}+\rho)}$
$\displaystyle\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re
z)\alpha}(\alpha,\partial_{z})^{2}\phi_{\alpha}\|_{L^{\infty}_{\rho}}$
$\displaystyle\leq\|e^{\varepsilon_{0}(\delta_{0}+\rho-\Re
y)\alpha}f_{\alpha}\|_{L^{\infty}_{\rho}}+\alpha^{1/2}\|f_{\alpha}\|_{L^{2}(y\geq\delta_{0}+\rho)}.$

Taking the summation in $\alpha\in{\mathbb{Z}}$ yields the first and last
estimates in (3.7) for $k=0$. For $k\geq 0$, the estimates follow similarly.
For the estimates involving the weight $z^{-1}$, we use the fact that the
Green function vanishes on the boundary $z=0$, and so $|G_{\pm}(y,z)|\leq
ze^{-\alpha|y-z|}.$ ∎

###  3.3 Biot-Savart law in $\Omega$

In this section, we bound the velocity through the Biot-Savart law: namely,
$u=\nabla^{\perp}\phi$, where

$\left\\{\begin{aligned} \Delta\phi&=\omega,\qquad\mbox{in }\quad\Omega\\\
\phi&=0,\qquad\mbox{on}\quad\partial\Omega.\end{aligned}\right.$ (3.9)

Without loss of generality, we will work with the cut-off vorticity
$\omega^{b}$ (see Section 4.1) near the boundary where the rescaled
coordinates introduced in Section 2.3 apply. We obtain the following
proposition.

######  Proposition 3.3.

Let $\phi$ be the solution of (3.9). Then, the velocity field
$u=\nabla^{\perp}\phi$ satisfies

$\displaystyle\|u\|_{\mathcal{W}^{k,\infty}_{\rho}}$
$\displaystyle\lesssim\|\omega\|_{\mathcal{W}^{k,1}_{\rho}}+\|\omega\|_{H^{k+1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})}$ (3.10)
$\displaystyle\|(\frac{1}{\widetilde{z}}\partial_{\widetilde{\theta}}\phi)\|_{\mathcal{W}^{k,\infty}_{\rho}}$
$\displaystyle\lesssim\|\omega\|_{\mathcal{W}^{k,1}_{\rho}}+\|\partial_{\widetilde{\theta}}\omega\|_{\mathcal{W}^{k,1}_{\rho}}+\|\omega\|_{H^{k+1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})}$

for $k\geq 0$.

###### Proof.

Using (2.16) and (3.9), the scaled stream function
$\widetilde{\phi}(\widetilde{t},\widetilde{z},\widetilde{\theta})$ solves

$\Delta_{\widetilde{z},\widetilde{\theta}}\widetilde{\phi}=\lambda^{-2}\widetilde{\omega}-\lambda^{2}\widetilde{R}_{\Delta}\widetilde{\phi},\qquad{\widetilde{\phi}}{}_{|_{\widetilde{z}=0}}=0$

on ${\mathbb{T}}\times\mathbb{R}_{+}$, and so the elliptic theory, Proposition
3.2, developed in the previous section can be applied, yielding

$\displaystyle\|u\|_{\mathcal{W}^{k,\infty}_{\rho}}$
$\displaystyle\lesssim\|\omega\|_{\mathcal{W}^{k,1}_{\rho}}+\|\omega\|_{H^{k+1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho\\})}$ (3.11)
$\displaystyle\quad+\lambda^{2}\|\partial_{\widetilde{\theta}}^{-1}\widetilde{R}_{\Delta}\widetilde{\phi}\|_{\mathcal{W}^{k,\infty}_{\rho}}+\lambda^{2}\|\widetilde{R}_{\Delta}\widetilde{\phi}\|_{H^{k+1}(\\{\widetilde{z}\geq\delta_{0}+\rho\\})}.$

It thus remains to bound $\widetilde{R}_{\Delta}\widetilde{\phi}$. Recall from
(2.17) that

$\widetilde{R}_{\Delta}=\widetilde{m}(\widetilde{z},\widetilde{\theta})\partial^{2}_{\widetilde{\theta}}+\frac{\widetilde{\gamma}}{1+\lambda^{2}\widetilde{z}\widetilde{\gamma}}\partial_{\widetilde{z}}-\frac{\widetilde{z}\widetilde{\gamma}^{\prime}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{3}}\partial_{\widetilde{\theta}},\qquad\widetilde{m}(\widetilde{z},\widetilde{\theta})=-\frac{2\widetilde{z}\widetilde{\gamma}+\lambda^{2}(\widetilde{z}\widetilde{\gamma})^{2}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{2}}.$

Thanks to the analyticity of the boundary, the coefficients are clearly
bounded in $\mathcal{W}^{k,\infty}_{\rho}$. Therefore, using a similar algebra
as in (3.4), we bound

$\lambda^{2}\|\partial_{\widetilde{\theta}}^{-1}\widetilde{R}_{\Delta}\widetilde{\phi}\|_{\mathcal{W}^{k,\infty}_{\rho}}\lesssim\lambda^{2}\|\partial_{\widetilde{\theta}}\widetilde{\phi}\|_{\mathcal{W}^{k,\infty}_{\rho}}+\lambda^{2}\|\partial_{\widetilde{z}}\widetilde{\phi}\|_{\mathcal{W}^{k,\infty}_{\rho}}$
(3.12)

That is, this term can be absorbed into the left hand side of (3.11), upon
taking $\lambda$ sufficiently small. As for the last term in (3.11), we note
that for large $\widetilde{z}$,
$|\widetilde{m}(\widetilde{z},\widetilde{\theta})|\lesssim\lambda^{-2}$, which
in particular proves that there is no growth in $\widetilde{z}$. This gives

$\displaystyle\lambda^{2}\|\widetilde{R}_{\Delta}\widetilde{\phi}\|_{H^{k+1}(\\{\widetilde{z}\geq\delta_{0}+\rho\\})}$
$\displaystyle\lesssim\|\phi\|_{H^{k+3}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho\\})}$ (3.13)
$\displaystyle\lesssim\lambda^{2}\|\widetilde{\phi}\|_{\mathcal{W}^{k,\infty}_{\rho}}+\|\omega\|_{H^{k+1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})},$

in which the last estimate follows from the standard elliptic theory in
Sobolev spaces. The proposition follows. ∎

###  3.4 Bilinear estimates

In this section, we show that the Sobolev-analytic norm is well adapted to
treat the nonlinear $u\cdot\nabla\omega$. We have the following lemma.

######  Lemma 3.4.

For any $\omega$ and $\omega^{\prime}$, denoting by $u$ the velocity related
to $\omega$, we have

$\displaystyle\|u\cdot\nabla\omega^{\prime}\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\leq
C\Big{(}\|\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\omega\|_{H^{1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}\\})}\Big{)}\|\partial_{\widetilde{\theta}}\omega^{\prime}\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\quad+C\Big{(}\|\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\partial_{\widetilde{\theta}}\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\omega\|_{H^{1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}\\})}\Big{)}\|\widetilde{z}\partial_{\widetilde{z}}\omega^{\prime}\|_{\mathcal{L}^{1}_{\rho}}.$

###### Proof.

By definition, the $\mathcal{L}^{1}_{\rho}$ norm is defined near the boundary
$\\{\lambda d(x,\partial\Omega)\leq\delta_{0}+\rho\\}$, on which we can write

$u\cdot\nabla{\omega}^{\prime}=\frac{1}{1+z\gamma(\theta)}\partial_{\theta}\phi\partial_{z}{\omega}^{\prime}-\frac{1}{(1+z\gamma(\theta))^{2}}\partial_{z}\phi\partial_{\theta}{\omega}^{\prime}$

with $\Delta\phi=\omega$. In the rescaled variable
$(\widetilde{z},\widetilde{\theta})$, we get

$u\cdot\nabla{\omega}^{\prime}=\frac{\lambda^{2}}{1+\lambda^{2}\widetilde{z}\widetilde{\gamma}(\widetilde{\theta})}(\partial_{\widetilde{\theta}}\widetilde{\phi})(\partial_{\widetilde{z}}\widetilde{\omega}^{\prime})-\frac{\lambda^{2}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma}(\widetilde{\theta}))^{2}}(\partial_{\widetilde{z}}\widetilde{\phi})(\partial_{\widetilde{\theta}}\widetilde{\omega}^{\prime})$

Note that thanks to the analyticity of $\partial\Omega$, the coefficient
$(1+\lambda^{2}\widetilde{z}\widetilde{\gamma}(\widetilde{\theta}))^{-1}$ is
bounded in $\mathcal{L}^{\infty}_{\rho}$. Using (3.4) and Proposition 3.3, we
bound

$\displaystyle\|(\partial_{\widetilde{z}}\widetilde{\phi})(\partial_{\widetilde{\theta}}\widetilde{\omega}^{\prime})\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\lesssim\|\partial_{\widetilde{z}}\widetilde{\phi}\|_{\mathcal{L}_{\rho}^{\infty}}\|\partial_{\widetilde{\theta}}\widetilde{\omega}^{\prime}\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\lesssim\Big{(}\|\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\omega\|_{H^{1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}\\})}\Big{)}\|\partial_{\widetilde{\theta}}\omega^{\prime}\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\|(\partial_{\widetilde{\theta}}\widetilde{\phi})(\partial_{\widetilde{z}}\widetilde{\omega}^{\prime})\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\lesssim\|\frac{1}{\widetilde{z}}\partial_{\widetilde{\theta}}\widetilde{\phi}\|_{\mathcal{L}_{\rho}^{\infty}}\|\widetilde{z}\partial_{\widetilde{z}}\widetilde{\omega}^{\prime}\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\lesssim\Big{(}\|\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\partial_{\widetilde{\theta}}\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\omega\|_{H^{1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}\\})}\Big{)}\|\widetilde{z}\partial_{\widetilde{z}}\omega^{\prime}\|_{\mathcal{L}^{1}_{\rho}}$

giving the lemma. ∎

###  3.5 Semigroup estimates in the half-plane

In this section, we give bounds on the Stokes semigroup $e^{\nu tS}$ in the
analytic spaces $\mathcal{W}^{k,1}_{\rho}$ on the half-plane
$\mathbb{R}_{+}\times{\mathbb{T}}$. We also denote by $\Gamma(\nu t)=e^{\nu
tS}(\mathcal{H}^{1}_{\\{\widetilde{z}=0\\}\times{\mathbb{T}}})$ the trace of
the semigroup on the boundary, with
$\mathcal{H}^{1}_{\\{\widetilde{z}=0\\}\times{\mathbb{T}}}$ being the one-
dimensional Hausdorff measure restricted on the boundary. The results in this
section are an easy adaptation from those obtained in [21], where the analytic
spaces contained no cutoff in $z$. Precisely, we consider

$\displaystyle(\partial_{t}-\nu\Delta_{z,\theta}){\omega}$ $\displaystyle=0$
(3.14) $\displaystyle\nu(\partial_{z}+|\partial_{\theta}|)\omega_{|_{z=0}}$
$\displaystyle=0$

on $\mathbb{R}_{+}\times{\mathbb{T}}$ (where we drop titles for sake of
presentation). We obtain the following proposition.

######  Proposition 3.5.

Let $e^{\nu tS}$ be the semigroup of the linear Stokes problem (3.14), and let
$\Gamma(\nu t)g$ be its trace on the boundary. Then, for any $t\geq 0$,
$\rho>0$, and $k\geq 0$, there hold

$\displaystyle\|e^{\nu tS}f\|_{\mathcal{W}^{k,1}_{\rho}}$ $\displaystyle\leq
C_{0}\|f\|_{\mathcal{W}^{k,1}_{\rho}}+\|zf\|_{H^{k+1}(z\geq\delta_{0}+\rho)}$
(3.15) $\displaystyle\|\Gamma(\nu t)g\|_{\mathcal{W}^{k,1}_{\rho}}$
$\displaystyle\leq
C_{0}\sum_{\alpha\in{\mathbb{Z}}}|\alpha^{k}g_{\alpha}|e^{\epsilon_{0}(\delta_{0}+\rho)|\alpha|}$

uniformly in the inviscid limit.

###### Proof.

The proof follows closely from that in [21]. Indeed, taking the Fourier
transform of the semigroup $e^{\nu tS}$ in variable $\theta$, we obtain

$(e^{\nu
tS}f)_{\alpha}(z)=\int_{0}^{\infty}G_{\alpha}(t,y;z)f_{\alpha}(y)\;dy,\qquad(\Gamma(\nu
t)g)_{\alpha}(z)=G_{\alpha}(t,0;z)g_{\alpha},$ (3.16)

for each Fourier variable $\alpha\in{\mathbb{Z}}$, where $G_{\alpha}(t,y;z)$
is the corresponding Green function. We recall the following result of
Proposition 3.3 from [21] that

$G_{\alpha}(t,y;z)=H_{\alpha}(t,y;z)+R_{\alpha}(t,y;z),$ (3.17)

where

$\displaystyle H_{\alpha}(t,y;z)$ $\displaystyle=\frac{1}{\sqrt{\nu
t}}\Big{(}e^{-\frac{|y-z|^{2}}{4\nu t}}+e^{-\frac{|y+z|^{2}}{4\nu
t}}\Big{)}e^{-\alpha^{2}\nu t},$
$\displaystyle|\partial_{z}^{k}R_{\alpha}(t,y;z)|$
$\displaystyle\lesssim\mu_{f}^{k+1}e^{-\theta_{0}\mu_{f}|y+z|}+(\nu
t)^{-\frac{k+1}{2}}e^{-\theta_{0}\frac{|y+z|^{2}}{\nu
t}}e^{-\frac{1}{8}\alpha^{2}\nu t},$

for $y,z\geq 0$, $k\geq 0$, and for some $\theta_{0}>0$ and for
$\mu_{f}=|\alpha|+\frac{1}{\sqrt{\nu}}$. In particular,
$\|G_{\alpha}(t,y;\cdot)\|_{L^{1}_{\rho}}\lesssim 1$, for each fixed $y,t$.

Now, for $z,y\leq\delta_{0}+\rho$, we note that

$\displaystyle e^{-a|y\pm z|}e^{-\epsilon_{0}(\delta_{0}+\rho-y)|\alpha|}$
$\displaystyle=e^{-a|y\pm
z|+\epsilon_{0}|\alpha|(y-z)}e^{-\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}$
(3.18) $\displaystyle\leq e^{-(a-\epsilon_{0}|\alpha|)|y\pm
z|}e^{-\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}$

for any real number $a$ and for $\epsilon_{0}$ sufficiently small. Taking
$a=\frac{1}{2}\theta_{0}\mu_{f}$, we have $a\geq\epsilon_{0}|\alpha|$ and so

$e^{-\theta_{0}\mu_{f}|y+z|}e^{-\epsilon_{0}(\delta_{0}+\rho-y)|\alpha|}\leq
e^{-\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}e^{-\frac{1}{2}\theta_{0}\mu_{f}|y+z|}$

On the other hand, taking $a=\frac{1}{2}\theta_{0}\frac{|y\pm z|}{\nu t}$ in
(3.18), we have either $a\geq\epsilon_{0}|\alpha|$ or
$\frac{1}{2}\theta_{0}\alpha^{2}\nu t\geq\epsilon_{0}|\alpha||y\pm z|$.
Therefore, we have

$e^{-\theta_{0}\frac{|y+z|^{2}}{\nu t}}e^{-\theta_{0}\alpha^{2}\nu
t}e^{-\epsilon_{0}(\delta_{0}+\rho-y)|\alpha|}\leq
e^{-\frac{1}{2}\theta_{0}\frac{|y+z|^{2}}{\nu
t}}e^{-\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}.$

This proves that for $z\leq\delta_{0}+\rho$,

$\displaystyle
e^{\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}\int_{0}^{\delta_{0}+\rho}|G_{\alpha}(t,y;z)f_{\alpha}(y)|\;dy$
$\displaystyle\leq\int_{0}^{\delta_{0}+\rho}\Big{[}(\nu
t)^{-\frac{1}{2}}e^{-\frac{1}{2}\theta_{0}\frac{|y\pm z|^{2}}{\nu
t}}+\mu_{f}e^{-\frac{1}{2}\theta_{0}\mu_{f}|y+z|}\Big{]}|e^{\epsilon_{0}(\delta_{0}+\rho-y)|\alpha|}f_{\alpha}(y)|\;dy.$

Since the term in the bracket is bounded in $L^{1}_{z}$ norm, we have

$\displaystyle\Big{\|}e^{\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}\int_{0}^{\delta_{0}+\rho}G_{\alpha}(t,y;z)f_{\alpha}(y)\;dy\Big{\|}_{\mathcal{L}^{1}_{\rho}}\lesssim\|e^{\epsilon_{0}(\delta_{0}+\rho-y)|\alpha|}f_{\alpha}\|_{\mathcal{L}^{1}_{\rho}}.$

Taking the summation in $\alpha$ yields the stated bounds for this term.

Next, consider the case when $y\geq\delta_{0}+\rho\geq z$. In this case, we
simply use

$e^{-\epsilon_{0}|\alpha||y-z|}\leq
e^{-\epsilon_{0}|\alpha|(\delta_{0}+\rho-z)},$

giving the right analyticity weight in $z$. The control of the weight
$e^{\epsilon_{0}|\alpha||y-z|}$ is done exactly as above, yielding

$\displaystyle
e^{\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}\int_{\delta_{0}+\rho}^{\infty}|G_{\alpha}(t,y;z)f_{\alpha}(y)|\;dy$
$\displaystyle\leq\int_{\delta_{0}+\rho}^{\infty}\Big{[}(\nu
t)^{-\frac{1}{2}}e^{-\frac{1}{2}\theta_{0}\frac{|y\pm z|^{2}}{\nu
t}}+\mu_{f}e^{-\frac{1}{2}\theta_{0}\mu_{f}|y+z|}\Big{]}|f_{\alpha}(y)|\;dy.$

Therefore,

$\displaystyle\sum_{\alpha}\|e^{\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}\int_{\delta_{0}+\rho}^{\infty}|G_{\alpha}(t,y;z)f_{\alpha}(y)|\;dy\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\lesssim\sum_{\alpha}\|f_{\alpha}\|_{L^{1}(z\geq\delta_{0}+\rho)}$
$\displaystyle\lesssim\|zf\|_{H^{1}(z\geq\delta_{0}+\rho)}.$

Similarly, from (3.16), the Fourier transform of the trace operator
$\Gamma(\nu t)g$ is estimated by

$\displaystyle|(\Gamma(\nu t)g)_{\alpha}(z)|$
$\displaystyle\leq|G_{\alpha}(t,0;z)g_{\alpha}|$
$\displaystyle\leq\Big{[}\mu_{f}e^{-\theta_{0}\mu_{f}|z|}+(\nu
t)^{-\frac{1}{2}}e^{-\theta_{0}\frac{|z|^{2}}{\nu
t}}e^{-\frac{1}{8}\alpha^{2}\nu t}\Big{]}|g_{\alpha}|$
$\displaystyle\leq\Big{[}\mu_{f}e^{-\frac{1}{2}\theta_{0}\mu_{f}|z|}+(\nu
t)^{-\frac{1}{2}}e^{-\frac{1}{2}\theta_{0}\frac{|z|^{2}}{\nu
t}}\Big{]}e^{-\epsilon_{0}(\delta_{0}+\rho-z)|\alpha|}|g_{\alpha}|e^{\epsilon_{0}(\delta_{0}+\rho)|\alpha|}$

in which the last inequality is a special case of the previous calculations
for $y=0$ and $z\leq\delta_{0}+\rho$. The bounds $\Gamma(\nu t)g$ are thus
direct. Finally, the bounds on derivatives follow from the similar adaptation
of derivatives bounds provided in [21]. We skip repeating the details. ∎

###  3.6 Semigroup estimates near $\partial\Omega$

In this section, we provide bounds on the Stokes semigroup $e^{\nu tS}$, which
will be used to estimate the vorticity $\omega^{b}$ (see Section 4.1) near the
boundary in the analytic spaces $\mathcal{W}^{k,1}_{\rho}$. Precisely, we
consider

$\left\\{\begin{aligned} &\partial_{t}{\omega}-\nu\Delta{\omega}=0\\\
&\nu(\partial_{n}+DN){\omega}_{|_{\partial\Omega}}=0\end{aligned}\right.$
(3.19)

in $\Omega$. We obtain the following proposition.

######  Proposition 3.6.

Let $e^{\nu tS}$ be the semigroup of the linear Stokes problem (3.19), and let
$\Gamma(\nu t)$ be its trace on the boundary. Fix any finite time $T$. Then,
for sufficiently small $\lambda$, and for any $0\leq t\leq T$, $\rho>0$, and
$k\geq 0$, there hold

$\displaystyle\|e^{\nu tS}f\|_{\mathcal{W}^{k,1}_{\rho}}$ $\displaystyle\leq
C_{0}\|f\|_{\mathcal{W}^{k,1}_{\rho}}+\|f\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}$ (3.20) $\displaystyle\|\Gamma(\nu
t)g\|_{\mathcal{W}^{k,1}_{\rho}}$ $\displaystyle\leq
C_{0}\sum_{\alpha\in{\mathbb{Z}}}|\alpha^{k}g_{\alpha}|e^{\epsilon_{0}(\delta_{0}+\rho)|\alpha|}$

uniformly in the inviscid limit.

###### Proof.

In the scaled variables, the Stokes problem for near boundary vorticity
$\omega$ becomes

$\begin{cases}&(\partial_{\widetilde{t}}-\nu\Delta_{\widetilde{z},\widetilde{\theta}})\widetilde{\omega}=-\lambda^{2}\nu\widetilde{R}_{\Delta}\widetilde{\omega}\\\
&\nu(\partial_{\widetilde{z}}+|\partial_{\widetilde{\theta}}|)\widetilde{\omega}|_{\widetilde{s}=0}=-\nu\widetilde{B}\widetilde{\omega}\end{cases}$

where $\widetilde{R}_{\Delta}$ and $\widetilde{B}$ are defined as in (2.17)
and (2.27). Using the Duhamel, the solution with initial data $\omega_{0}$ can
be written as

$\widetilde{\omega}(\widetilde{t})=e^{\nu\widetilde{t}S}\widetilde{\omega}_{0}-\nu\lambda^{2}\int_{0}^{\widetilde{t}}e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S}\widetilde{R}_{\Delta}\widetilde{\omega}(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}-\nu\int_{0}^{\widetilde{t}}\Gamma(\nu(\widetilde{t}-\widetilde{t}^{\prime}))\widetilde{B}\widetilde{\omega}(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}.$
(3.21)

We shall bound the integral terms on the right in term of the initial data.
Recall from (2.17) that

$\widetilde{R}_{\Delta}=\widetilde{m}(\widetilde{z},\widetilde{\theta})\partial^{2}_{\widetilde{\theta}}+\frac{\widetilde{\gamma}}{1+\lambda^{2}\widetilde{z}\widetilde{\gamma}}\partial_{\widetilde{z}}-\frac{\widetilde{z}\widetilde{\gamma}^{\prime}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{3}}\partial_{\widetilde{\theta}},\qquad\widetilde{m}(\widetilde{z},\widetilde{\theta})=-\frac{2\widetilde{z}\widetilde{\gamma}+\lambda^{2}(\widetilde{z}\widetilde{\gamma})^{2}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{2}}.$

We rewrite the operator in the following form

$\displaystyle\widetilde{R}_{\Delta}\widetilde{\omega}$
$\displaystyle=\partial^{2}_{\widetilde{\theta}}[\widetilde{m}\widetilde{\omega}]-\partial_{\widetilde{\theta}}\Big{[}2\partial_{\widetilde{\theta}}\widetilde{m}\widetilde{\omega}+\frac{\widetilde{z}\widetilde{\gamma}^{\prime}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{3}}\widetilde{\omega}\Big{]}+\partial_{\widetilde{z}}\Big{(}\frac{\widetilde{\gamma}}{1+\lambda^{2}\widetilde{z}\widetilde{\gamma}}\widetilde{\omega}\Big{)}$
$\displaystyle\quad+\Big{[}(\partial^{2}_{\widetilde{\theta}}\widetilde{m})-\partial_{\widetilde{z}}\Big{(}\frac{\widetilde{\gamma}}{1+\lambda^{2}\widetilde{z}\widetilde{\gamma}}\Big{)}+\partial_{\widetilde{\theta}}\Big{(}\frac{\widetilde{z}\widetilde{\gamma}^{\prime}}{(1+\lambda^{2}\widetilde{z}\widetilde{\gamma})^{3}}\Big{)}\Big{]}\widetilde{\omega}.$

We now bound each term appearing in the Duhamel formula (3.21). Thanks to the
analyticity of the boundary, the coefficients are bounded in
$\mathcal{W}^{k,\infty}_{\rho}$. Now, recall from (3.17) that the Green
function has two components:

$e^{\nu\widetilde{t}S}=e^{\nu\widetilde{t}S_{H}}+e^{\nu\widetilde{t}S_{R}}$

which corresponds to the Green kernel $H_{\alpha}$ (i.e., the heat kernel) and
the other from the stationary Stokes kernel $R_{\alpha}$.

We first claim that

$\Big{\|}\nu\lambda^{2}\int_{0}^{\widetilde{t}}e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S_{H}}\widetilde{R}_{\Delta}\widetilde{\omega}(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}\Big{\|}_{\mathcal{W}^{k,1}_{\rho}}\lesssim\lambda^{2}\sup_{0\leq\widetilde{t}^{\prime}\leq\widetilde{t}}\|\omega\|_{\mathcal{W}^{k,1}_{\rho}}+\|\omega\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\delta)}.$ (3.22)

For the heat semigroup, we may integrate by parts in $\widetilde{\theta}$ or
$\widetilde{z}$. It follows directly from the representation of the Green
function that derivatives of the semigroup
$\nabla_{\widetilde{\theta},\widetilde{z}}e^{\nu\widetilde{t}S_{H}}$ are of
order $(\nu\widetilde{t})^{-1/2}$ of the semigroup itself. Therefore, the
first-order derivative term in $\widetilde{R}_{\Delta}$ can be treated
systematically as follows:

$\displaystyle\nu\lambda^{2}\Big{\|}\int_{0}^{\widetilde{t}}e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S_{H}}\nabla_{\widetilde{\theta},\widetilde{z}}h(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}\Big{\|}_{\mathcal{W}^{k,1}_{\rho}}$
$\displaystyle\lesssim\nu\lambda^{2}\int_{0}^{\widetilde{t}}(\nu(\widetilde{t}-\widetilde{t}^{\prime}))^{-1/2}\|h(\widetilde{t}^{\prime})\|_{\mathcal{W}^{k,1}_{\rho}}\;d\widetilde{t}^{\prime}$
$\displaystyle\lesssim\sqrt{\nu}\lambda^{2}\sup_{0\leq\widetilde{t}^{\prime}\leq\widetilde{t}}\|h\|_{\mathcal{W}^{k,1}_{\rho}}.$

The zero-order term is treated similarly. The analysis doesn’t apply directly
to the second-order derivative term
$\partial^{2}_{\widetilde{\theta}}[\widetilde{m}\widetilde{\omega}]$ due to
the singularity in time $(\nu t)^{-1}$, if integration by parts was to perform
twice. However, in the Fourier variable $\alpha$, we compute

$\nu\lambda^{2}\int_{0}^{\widetilde{t}}(e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S_{H}}\partial^{2}_{\widetilde{\theta}}[\widetilde{m}\widetilde{\omega}])_{\alpha}(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}=\nu\alpha^{2}\lambda^{2}\int_{0}^{\widetilde{t}}\int_{0}^{\infty}H_{\alpha}(t,\widetilde{y};\widetilde{z})[\widetilde{m}\widetilde{\omega}]_{\alpha}(\widetilde{t}^{\prime})\;d\widetilde{y}d\widetilde{t}^{\prime}.$

Observe that the Green kernel $H_{\alpha}$ has the diffusion term
$e^{-\nu\alpha^{2}\widetilde{t}}$, for which we use

$\nu\alpha^{2}\lambda^{2}\int_{0}^{\widetilde{t}}e^{-\nu\alpha^{2}(\widetilde{t}-\widetilde{t}^{\prime})}d\widetilde{t}^{\prime}\lesssim\lambda^{2}$

yielding the claim (3.22).

Next, we claim that

$\Big{\|}\nu\lambda^{2}\int_{0}^{\widetilde{t}}e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S_{R}}\widetilde{R}_{\Delta}\widetilde{\omega}(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}\Big{\|}_{\mathcal{W}^{k,1}_{\rho}}\lesssim\nu\lambda^{2}\int_{0}^{\widetilde{t}}\|\partial_{\widetilde{\theta}}\omega(\widetilde{t})\|_{\mathcal{W}^{k,1}_{\rho}}\;d\widetilde{t}+\|\omega\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\delta)}.$ (3.23)

It suffices to check for the stationary Green kernel
$\mu_{f}e^{-\theta_{0}\mu_{f}(\widetilde{y}+\widetilde{z})}$ and for the
second-order derivative term
$\partial^{2}_{\widetilde{\theta}}[\widetilde{m}\widetilde{\omega}]$ appearing
in $\widetilde{R}_{\Delta}\widetilde{\omega}(\widetilde{t}^{\prime})$. For
this term, we make use of the fact that $\widetilde{m}$ vanishes at
$\widetilde{z}=0$; namely, we can write
$\widetilde{m}=\widetilde{z}\widetilde{m}_{1}$ and use
$\mu_{f}e^{-\theta_{0}\mu_{f}\widetilde{z}}\widetilde{z}\lesssim 1$, which
controls one spatial derivative, since $\mu_{f}=|\alpha|+\nu^{-1/2}$. This
proves the claim (3.23).

Finally, putting the previous bounds together into the Duhamel representation
(3.21), we have obtained

$\displaystyle\|\omega(\widetilde{t})\|_{\mathcal{W}^{k,1}_{\rho}}$
$\displaystyle\lesssim\|\omega_{0}\|_{\mathcal{W}^{k,1}_{\rho}}+\|\omega_{0}\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\delta)}$ (3.24)
$\displaystyle\quad+\lambda^{2}\sup_{0\leq\widetilde{t}^{\prime}\leq\widetilde{t}}\|\omega(\widetilde{t}^{\prime})\|_{\mathcal{W}^{k,1}_{\rho}}+\nu\lambda^{2}\int_{0}^{\widetilde{t}}\|\partial_{\widetilde{\theta}}\omega(\widetilde{t})\|_{\mathcal{W}^{k,1}_{\rho}}\;d\widetilde{t}$
$\displaystyle\quad+\|\omega\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\delta)}$

for any $k\geq 0$. The standard energy estimates for the heat equation (away
from the boundary) yield

$\|\omega\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\delta)}\lesssim\|\omega_{0}\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}.$ (3.25)

It remains to treat the third and forth terms on the right hand side of
(3.24). We bound these terms by iteration, introducing

$\displaystyle A_{0}(\beta):=$ $\displaystyle\quad\sup_{0\leq k\leq
4}\big{(}\sup_{0<\beta\widetilde{t}<\rho_{0}}\sup_{0<\rho<\rho_{0}-\beta\widetilde{t}}\Bigl{\\{}\|\omega(\widetilde{t})\|_{\mathcal{W}^{k,1}_{\rho}}+\|\partial_{\widetilde{\theta}}\omega(\widetilde{t})\|_{\mathcal{W}^{k,1}_{\rho}}(\rho_{0}-\rho-\beta\widetilde{t})^{\zeta}\Bigr{\\}}\big{)}$

for some $\zeta\in(0,1)$. We bound

$\displaystyle\nu\lambda^{2}\int_{0}^{\widetilde{t}}\|\partial_{\widetilde{\theta}}\omega(\widetilde{t})\|_{\mathcal{W}^{k,1}_{\rho}}\;d\widetilde{t}$
$\displaystyle\leq
C_{0}\nu\lambda^{2}A_{0}(\beta)\int_{0}^{\widetilde{t}}(\rho_{0}-\rho-\beta\widetilde{s})^{-\zeta}\;d\widetilde{s}$
$\displaystyle\leq C_{0}\nu\lambda^{2}\beta^{-1}A_{0}(\beta).$

Next, we check the bound on
$\|\partial_{\widetilde{\theta}}\omega(\widetilde{t})\|_{\mathcal{W}^{k,1}_{\rho}}$.
We focus only the worst term as in (3.23). Note that
$\rho<\rho_{0}-\beta\widetilde{t}\leq\rho_{0}-\beta\widetilde{s}$. Thus, we
take $\rho^{\prime}=\frac{\rho+\rho_{0}-\beta s}{2}$ and bound

$\displaystyle\Big{\|}$
$\displaystyle\nu\lambda^{2}\partial_{\widetilde{\theta}}\int_{0}^{\widetilde{t}}e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S_{R}}\widetilde{R}_{\Delta}\widetilde{\omega}(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}\Big{\|}_{\mathcal{W}^{k,1}_{\rho}}$
$\displaystyle\lesssim\nu\lambda^{2}\int_{0}^{\widetilde{t}}\frac{1}{\rho^{\prime}-\rho}\|\partial_{\widetilde{\theta}}\omega(\widetilde{t})\|_{\mathcal{W}^{k,1}_{\rho^{\prime}}}\;d\widetilde{t}+\|\omega\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\delta)}$ $\displaystyle\leq
C_{0}\nu\lambda^{2}\int_{0}^{\widetilde{t}}(\rho_{0}-\rho-\beta
s)^{-1-\zeta}\;ds+\|\omega_{0}\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}$ $\displaystyle\leq
C_{0}\nu\lambda^{2}\beta^{-1}A_{0}(\beta)(\rho_{0}-\rho-\beta\widetilde{t})^{-\zeta}+\|\omega_{0}\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}.$

This proves that

$A_{0}(\beta)\lesssim\|\omega_{0}\|_{\mathcal{W}^{k,1}_{\rho}}+\|\omega_{0}\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}+\Big{(}\lambda^{2}+\nu\lambda^{2}\beta^{-1}\Big{)}A_{0}(\beta).$

Taking $\lambda$ and $\nu$ small, the last term can be absorbed into the left
hand side, completing the bounds on $A_{0}(\beta)$ or the
$\mathcal{W}^{k,1}_{\rho}$ norm for the vorticity. Note that we do not require
$\beta$ to be sufficiently large (compared with the nonlinear iteration
provided in the next section). As a consequence, the proposition holds for any
given finite time. ∎

##  4 Nonlinear analysis

As already mentioned in the introduction, we construct the solutions to the
Navier-Stokes equation via the vorticity formulation

$\displaystyle\partial_{t}\omega+u\cdot\nabla\omega=\nu\Delta\omega$ (4.1)

together with the nonlocal boundary condition (1.8) and with initial data
$\omega_{|_{t=0}}=\omega_{0}$ satisfying

$\|\omega_{0}\|_{\mathcal{W}^{2,1}_{\rho}}+\|\omega_{0}\|_{H^{4}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})}<\infty.$ (4.2)

Introduce the smooth cutoff function $\phi^{b}$ as in (2.18), and write

$\omega=\omega^{b}+\omega^{i},\qquad\omega^{b}=\phi^{b}\omega,\qquad\omega^{i}=(1-\phi^{b})\omega.$
(4.3)

We also define the corresponding velocity field through the Biot-Savart law

$u=u^{b}+u^{i},\qquad u^{b}=\nabla^{\perp}\Delta^{-1}\omega^{b},\qquad
u^{i}=\nabla^{\perp}\Delta^{-1}\omega^{i}.$ (4.4)

This yields

$\left\\{\begin{aligned}
\partial_{t}{\omega}^{b}+u\cdot\nabla{\omega}^{b}&=\nu\Delta{\omega}^{b}\\\
\nu(\partial_{n}+DN)\omega^{b}_{|_{\partial\Omega}}&=[\partial_{n}\Delta^{-1}(u\cdot\nabla\omega)]_{|_{\partial\Omega}}\end{aligned}\right.$
(4.5)

for the vorticity near the boundary, and

$\left\\{\begin{aligned}
\partial_{t}{\omega}^{i}+u\cdot\nabla\omega^{i}&=\nu\Delta{\omega}^{i}\\\
{\omega}^{i}_{|{\partial\Omega}}&=0\end{aligned}\right.$ (4.6)

for the vorticity away from the boundary. Here, we note that the boundary
condition on $\omega^{i}$ follows directly from the definition (4.3), while
the boundary condition on $\omega^{b}$ was due to the fact that
$DN\omega^{i}=0$ by Lemma 2.2. We also note that the velocity field $u$ that
appears in both the systems is the full velocity, which is the summation of
$u^{b}$ and $u^{i}$ generated by $\omega^{b}$ and $\omega^{i}$, respectively.

We shall construct the near boundary vorticity solving (4.5) through the
semigroup of the Stokes problem. Indeed, we have the following standard
Duhamel’s integral representation, written in the scaled variables,

$\widetilde{\omega}(\widetilde{t})=e^{\nu\widetilde{t}S}\widetilde{\omega}_{0}+\int_{0}^{\widetilde{t}}e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S}f(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}+\int_{0}^{\widetilde{t}}\Gamma(\nu(\widetilde{t}-\widetilde{t}^{\prime}))g(\widetilde{t}^{\prime})\;d\widetilde{t}^{\prime}$
(4.7)

where

$\displaystyle f(\widetilde{t})=-\lambda^{-2}u\cdot\nabla\omega^{b},\qquad
g(\widetilde{t})$
$\displaystyle=\lambda^{-1}[\partial_{n}\Delta^{-1}(u\cdot\nabla\omega)]_{|_{\partial\Omega}}.$
(4.8)

Here, $e^{\nu\widetilde{t}S}$ denotes the semigroup of the corresponding
Stokes problem and $\Gamma(\nu\widetilde{t})$ being its trace on the boundary;
see Section 3.6.

###  4.1 Global Sobolev-analytic norm

We now introduce Sobolev-analytic norms to control global vorticity. Let us
fix positive numbers $\rho_{0},\delta_{0},$ and $\zeta\in(0,1)$. Introduce the
following family of nonlinear iterative norms for vorticity:

$\displaystyle A(\beta):=$ $\displaystyle\quad\sup_{0<\lambda^{2}\beta
t<\rho_{0}}\Big{[}\sup_{0<\rho<\rho_{0}-\beta\lambda^{2}t}\Bigl{\\{}\|\omega(t)\|_{\mathcal{W}^{1,1}_{\rho}}+\|\omega(t)\|_{\mathcal{W}^{2,1}_{\rho}}(\rho_{0}-\rho-\lambda^{2}\beta
t)^{\zeta}\Bigr{\\}}$ (4.9)
$\displaystyle\quad+\|\omega(t)\|_{H^{4}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})}\Big{]}$

for a parameter $\beta>0$, with recalling

$\|\omega(t)\|_{\mathcal{W}^{k,1}_{\rho}}=\sum_{j+\ell\leq
k}\|\partial_{\widetilde{\theta}}^{j}(\widetilde{z}\partial_{\widetilde{z}})^{\ell}\omega(t)\|_{\mathcal{L}^{1}_{\rho}}.$

Note that by definition the norm $\|\cdot\|_{\mathcal{W}^{k,1}_{\rho}}$
controls the analyticity of the vorticity near the boundary, precisely in the
region $\lambda d(x,\partial\Omega)\leq\delta_{0}+\rho,$ while the $H^{4}$
norm is to control the Sobolev regularity away from the boundary. We shall
show that the vorticity norm remains finite for sufficiently large $\beta$.
The weight $(\rho_{0}-\rho-\lambda^{2}\beta t)^{\zeta}$, with a small
$\zeta>0$, is standard in the literature to avoid time singularity when
recovering the loss of derivatives ([2, 6]). See also [14] for an alternative
framework to construct analytic solutions through generator functions.

Our goal is to prove the following key proposition.

######  Proposition 4.1.

For $\beta>0$, there holds

$A(\beta)\leq
C_{0}\|\omega_{0}\|_{\mathcal{W}^{2,1}_{\rho}}+C_{0}\|\omega_{0}\|_{H^{4}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})}+C_{0}\beta^{-1}A(\beta)^{2}.$

In Section 4.4, we will show that our main theorem, Theorem 1.1, follows
straightforwardly from Proposition 4.1.

###  4.2 Analytic bounds near the boundary

In this section, we bound the vorticity near the boundary $\lambda
d(x,\partial\Omega)\leq\delta_{0}+\rho_{0}$, on which by definition
$\omega=\omega^{b}$ and therefore the Duhamel representation (4.7) holds. Let
$\rho<\rho_{0}-\lambda^{2}\beta t$. Recalling the notation
$\widetilde{t}=\lambda^{2}t$ and using (4.7), we bound

$\|\widetilde{\omega}(\widetilde{t})\|_{\mathcal{W}_{\rho}^{k,1}}\leq\|e^{\nu\widetilde{t}S}\widetilde{\omega}_{0}\|_{\mathcal{W}_{\rho}^{k,1}}+\int_{0}^{\widetilde{t}}\|e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S}f(\widetilde{t}^{\prime})\|_{\mathcal{W}_{\rho}^{k,1}}\;d\widetilde{t}^{\prime}+\int_{0}^{\widetilde{t}}\|\Gamma(\nu(\widetilde{t}-\widetilde{t}^{\prime}))g(\widetilde{t}^{\prime})\|_{\mathcal{W}_{\rho}^{k,1}}\;d\widetilde{t}^{\prime}$
(4.10)

for $0<k\leq 4$ and for $f,g$ defined as in (4.8). Let us bound each term on
the right. Using the semigroup estimates, Proposition 3.5, we have

$\displaystyle\|e^{\nu\widetilde{t}S}\widetilde{\omega}_{0}\|_{\mathcal{W}^{k,1}_{\rho}}$
$\displaystyle\leq
C_{0}\|\widetilde{\omega}_{0}\|_{\mathcal{W}^{k,1}_{\rho}}+\|\widetilde{z}\widetilde{\omega}_{0}\|_{H^{k+1}(\widetilde{z}\geq\delta_{0}+\rho)}$
$\displaystyle\leq
C_{0}\|\widetilde{\omega}_{0}\|_{\mathcal{W}^{k,1}_{\rho}}+\|\omega_{0}\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho)}.$

While for the second integral term in (4.10), we have

$\int_{0}^{\widetilde{t}}\|e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S}f(\widetilde{t}^{\prime})\|_{\mathcal{W}_{\rho}^{k,1}}\;d\widetilde{t}^{\prime}\lesssim\int_{0}^{\widetilde{t}}\Big{[}\|f(\widetilde{t}^{\prime})\|_{\mathcal{W}_{\rho}^{k,1}}+\|\widetilde{z}f(\widetilde{t}^{\prime})\|_{H^{k+1}(\widetilde{z}\geq\delta_{0}+\rho)}\Big{]}\;d\widetilde{t}^{\prime}.$

Then, we use (4.8), in the above formula with $f(\widetilde{t})$ replaced by
$-\lambda^{-2}u\cdot\nabla\omega^{b}.$ First, using the standard elliptic
theory for $k=0,1,2$, we bound

$\displaystyle\|\widetilde{z}(u\cdot\nabla\omega^{b})(\widetilde{t}^{\prime})\|_{H^{k+1}(\widetilde{z}\geq\delta_{0}+\rho)}$
$\displaystyle\lesssim\|\omega\|_{H^{4}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})}^{2}\lesssim A(\beta)^{2}.$

Next, for the analytic norm, with the bilinear estimates from Lemma 3.4, we
have:

$\displaystyle\|u\cdot\nabla\omega^{b}\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\leq
C\Big{(}\|\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\omega\|_{H^{1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}\\})}\Big{)}\|\partial_{\widetilde{\theta}}\omega^{b}\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\quad+C\Big{(}\|\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\partial_{\widetilde{\theta}}\omega\|_{\mathcal{L}^{1}_{\rho}}+\|\omega\|_{H^{1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}\\})}\Big{)}\|\widetilde{z}\partial_{\widetilde{z}}\omega^{b}\|_{\mathcal{L}^{1}_{\rho}}$
$\displaystyle\lesssim\|\omega\|_{\mathcal{W}^{1,1}_{\rho}}^{2}+\|\omega\|_{H^{1}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}\\})}^{2}$ $\displaystyle\lesssim
A(\beta)^{2}$
$\displaystyle\|u\cdot\nabla\omega^{b}\|_{\mathcal{W}^{1,1}_{\rho}}$
$\displaystyle\lesssim\|\omega\|_{\mathcal{W}^{1,1}_{\rho}}\|\omega\|_{\mathcal{W}^{1,2}_{\rho}}+\|\omega\|_{H^{2}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}\\})}^{2}$ $\displaystyle\lesssim
A(\beta)^{2}(\rho_{0}-\rho-\beta\widetilde{t})^{-\zeta}.$

Therefore,

$\displaystyle\int_{0}^{\widetilde{t}}\|u\cdot\nabla\omega^{b}\|_{\mathcal{W}^{1,1}_{\rho}}\;d\widetilde{s}$
$\displaystyle\leq
C_{0}A(\beta)^{2}\int_{0}^{\widetilde{t}}(\rho_{0}-\rho-\beta\widetilde{s})^{-\zeta}\;d\widetilde{s}$
$\displaystyle\leq C_{0}\beta^{-1}A(\beta)^{2}.$

Similarly, we consider the case when $k=2$. Noting $\rho<\rho_{0}-\beta
t\leq\rho_{0}-\beta s$, we take $\rho^{\prime}=\frac{\rho+\rho_{0}-\beta
s}{2}$ and compute

$\displaystyle\int_{0}^{t}\|u\cdot\nabla\omega^{b}\|_{\mathcal{W}^{2,1}_{\rho}}\;ds$
$\displaystyle\leq
C_{0}\int_{0}^{t}\frac{1}{\rho^{\prime}-\rho}\|u\cdot\nabla\omega^{b}\|_{\mathcal{W}^{1,1}_{\rho^{\prime}}}\;ds$
$\displaystyle\leq C_{0}A(\beta)^{2}\int_{0}^{t}(\rho_{0}-\rho-\beta
s)^{-1-\zeta}\;ds$ $\displaystyle\leq
C_{0}\beta^{-1}A(\beta)^{2}(\rho_{0}-\rho-\beta t)^{-\zeta}.$

Finally, we treat the last integral term in (4.10). Precisely, we will show
that, for $k\leq 2$:

$\displaystyle\|\Gamma(\nu(\widetilde{t}-\widetilde{t}^{\prime}))g(\widetilde{t}^{\prime})\|_{\mathcal{W}_{\rho}^{k,1}}\leq$
$\displaystyle\quad
C_{0}\|u\cdot\nabla{\omega}^{b}(\widetilde{t}^{\prime})\|_{\mathcal{W}^{k,1}_{\rho}}+C_{0}\|{\omega}(\widetilde{t}^{\prime})\|^{2}_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}$ (4.11)
$\displaystyle\quad+C_{0}\|{\omega}(\widetilde{t}^{\prime})\|_{\mathcal{W}^{k,1}_{\rho}}\|{\omega}(\widetilde{t}^{\prime})\|_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}$

which would then imply

$\int_{0}^{\widetilde{t}}\|\Gamma(\nu(\widetilde{t}-\widetilde{t}^{\prime}))g(\widetilde{t}^{\prime})\|_{\mathcal{W}_{\rho}^{2,1}}d\widetilde{t}^{\prime}\leq
C_{0}\left(A(\beta)^{2}+\beta^{-1}A(\beta)^{2}(\rho_{0}-\rho-\beta
t)^{-\zeta}\right).$

Here the constant $C_{0}$ may change from line to line. It remains to give the
proof for the inequality (4.11). First, by Proposition 3.5, we have

$\|\Gamma(\nu(\widetilde{t}-\widetilde{t}^{\prime}))g(\widetilde{t}^{\prime})\|_{\mathcal{W}_{\rho}^{k,1}}\leq
C_{0}\sum_{\alpha}|\alpha|^{k}|g_{\alpha}|e^{\varepsilon_{0}(\delta_{0}+\rho)|\alpha|},$

where $g_{\alpha}$ is given by

$g_{\alpha}=\lambda^{-1}\partial_{n}\Delta^{-1}(u\cdot\nabla{\omega})_{\alpha}|_{\partial\Omega}.$

Let $\Phi=\Delta^{-1}(u\cdot\nabla{\omega})$. By definition, $\Phi$ solves

$\begin{cases}&\Delta\Phi=u\cdot\nabla{\omega},\qquad x\in\Omega\\\
&\Phi|_{\partial\Omega}=0.\end{cases}$

In the rescaled geodesic coordinates, we have
$g_{\alpha}=\partial_{\widetilde{z}}\Phi_{\alpha}(0)$. Let
$\Phi^{b}=\Phi(x)\phi^{b}(x)$, we have

$\begin{cases}&\Delta\Phi^{b}=2\nabla_{x}\phi^{b}\cdot\nabla_{x}\Phi^{b}+\Delta\phi^{b}\Phi+\phi^{b}u\cdot\nabla{\omega}\\\
&\Phi^{b}|_{z=0}=0.\end{cases}$

By a direct calculation, we have

$\displaystyle
e^{\varepsilon_{0}(\delta_{0}+\rho)|\alpha|}g_{\alpha}(\widetilde{t}^{\prime})=\partial_{z}\Phi_{\alpha}^{b}|_{\widetilde{z}=0}$
$\displaystyle=\int_{0}^{\infty}e^{|\alpha|(\varepsilon_{0}(\delta_{0}+\rho)-\widetilde{z})}\left\\{\lambda^{2}\left(\widetilde{R}_{\Delta}\widetilde{\Phi}_{b}\right)_{\alpha}(\widetilde{z})-\lambda^{-2}\left(2\nabla_{x}\phi^{b}\cdot\nabla_{x}\Phi^{b}-\Phi\Delta\phi^{b}-\phi^{b}u\cdot\nabla{\omega}\right)_{\alpha}\right\\}d\widetilde{z}$
$\displaystyle=I_{1,\alpha}+I_{2,\alpha}+I_{3,\alpha}+I_{4,\alpha}.$

Treating $I_{1,\alpha}$. As in the proof of Proposition 3.20 for
$\widetilde{R}_{\Delta}$, we have

$\displaystyle|I_{1,\alpha}|\leq$ $\displaystyle
C_{0}|\alpha|^{2}\lambda^{2}\int_{0}^{\infty}e^{|\alpha|(\varepsilon_{0}(\delta_{0}+\rho)-\widetilde{z})}|\widetilde{z}\Phi^{b}_{\alpha}(\widetilde{z})|d\widetilde{z}$
$\displaystyle+C_{0}\lambda^{2}\int_{0}^{\infty}e^{|\alpha|(\varepsilon_{0}(\delta_{0}+\rho)-\widetilde{z})}\left(|\alpha||\Phi^{b}_{\alpha}(\widetilde{z})|+|\partial_{\widetilde{z}}\Phi_{\alpha}^{b}|\right)d\widetilde{z}.$

First, we use the inequality
$\widetilde{z}|\alpha|e^{-|\alpha|\widetilde{z}}\leq
e^{-\frac{1}{2}|\alpha|\widetilde{z}}$ to get

$\displaystyle|I_{1,\alpha}|$ $\displaystyle\leq
C_{0}\lambda^{2}\int_{0}^{\infty}e^{|\alpha|\left(\varepsilon_{0}(\delta_{0}+\rho)-\frac{1}{2}\widetilde{z}\right)}\left(|\alpha||\Phi^{b}_{\alpha}(\widetilde{z})|+|\partial_{\widetilde{z}}\Phi_{\alpha}^{b}|\right)d\widetilde{z}$
$\displaystyle\leq
C_{0}\lambda^{2}\int_{0}^{\delta_{0}+\rho}e^{|\alpha|\varepsilon_{0}(\delta_{0}+\rho-\widetilde{z})}\left(|\alpha||\Phi^{b}_{\alpha}(\widetilde{z})|+|\partial_{\widetilde{z}}\Phi_{\alpha}^{b}|\right)d\widetilde{z}$
$\displaystyle\quad+C_{0}\lambda^{2}\int_{\delta_{0}+\rho}^{\infty}\left(|\alpha||\Phi^{b}_{\alpha}(\widetilde{z})|+|\partial_{\widetilde{z}}\Phi_{\alpha}^{b}|\right)d\widetilde{z}.$

For the first term, we use the $L^{1}_{\mu}$ elliptic estimate for the
velocity (since the kernel $K_{\alpha}\in L^{1}$), to get

$\displaystyle\sum_{\alpha}|\alpha|^{k}\int_{0}^{\delta_{0}+\rho}e^{|\alpha|\varepsilon_{0}(\delta_{0}+\rho-\widetilde{z})}\left(|\alpha||\Phi^{b}_{\alpha}(\widetilde{z})|+|\partial_{\widetilde{z}}\Phi_{\alpha}^{b}|\right)d\widetilde{z}$
(4.12) $\displaystyle\leq
C\|\phi^{b}u\cdot\nabla{\omega}\|_{\mathcal{W}_{\rho}^{k,1}}+C\|\Phi\|_{H^{k+2}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}.$

Now we have

$\displaystyle\|\phi^{b}u\cdot\nabla{\omega}\|_{\mathcal{W}^{k,1}_{\rho}}$
$\displaystyle=\|u\cdot\nabla{\omega}^{b}-(u\cdot\nabla\phi^{b}){\omega}\|_{\mathcal{W}^{k,1}_{\rho}}$
(4.13) $\displaystyle\leq
C\left(\|u\cdot\nabla{\omega}^{b}\|_{\mathcal{W}^{k,1}_{\rho}}+\|u{\omega}\|_{H^{k}(\lambda
d(x,\partial\Omega)\geq\delta_{0})}\right)$ $\displaystyle\leq
C\|u\cdot\nabla{\omega}^{b}\|_{\mathcal{W}_{\rho}^{k,1}}+C\|{\omega}\|_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}\left(\|{\omega}\|_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}+\|{\omega}\|_{\mathcal{W}^{k,1}_{\rho}}\right).$

By standard elliptic estimate, we have

$\displaystyle\|\Phi\|_{H^{k+2}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}\leq$ $\displaystyle
C\|\phi^{b}u\cdot\nabla{\omega}\|_{\mathcal{W}_{\rho}^{k,1}}+C\|u\cdot\nabla{\omega}\|_{H^{k}(\lambda
d(x,\partial\Omega)\geq\delta_{0})}$ (4.14) $\displaystyle\leq$ $\displaystyle
C\|u\cdot\nabla{\omega}^{b}\|_{\mathcal{W}_{\rho}^{k,1}}+\|{\omega}\|^{2}_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}+\|{\omega}\|_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}\|{\omega}\|_{\mathcal{W}^{k,1}_{\rho}}.$

Combining (4.12),(4.14) and (4.13), we have

$\sum_{\alpha}|\alpha|^{k}|I_{1,\alpha}|\leq
C_{0}\left(\|u\cdot\nabla{\omega}^{b}(\widetilde{t}^{\prime})\|_{\mathcal{W}^{k,1}_{\rho}}+\|{\omega}\|^{2}_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}+\|{\omega}\|_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}\|{\omega}\|_{\mathcal{W}^{k,1}_{\rho}}\right)$

as claimed in (4.11). The proof for $I_{1,\alpha}$ is complete.

Treating $I_{2,\alpha}$. For $I_{2,\alpha}$, we note that the domain of
integration is $\widetilde{z}\geq\delta_{0}+\rho_{0}>\delta_{0}+\rho$, we have

$|\alpha|^{k}e^{|\alpha|(\varepsilon_{0}(\delta_{0}+\rho)-\widetilde{z})}\leq
C.$

Thus we have

$\displaystyle\sum_{\alpha}|\alpha|^{k}|I_{2,\alpha}|$ $\displaystyle\leq
C\sum_{\alpha}\|\nabla_{x}\Phi^{b}_{\alpha}\|_{L^{1}(\widetilde{z}\geq\delta_{0}+\rho_{0})}\leq
C\|d(x,\partial\Omega)\nabla_{x}\Phi\|_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}$ $\displaystyle\leq
C\|d(x,\partial\Omega)\Phi\|_{H^{2}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}$ $\displaystyle\leq
C\|\phi^{b}u\cdot\nabla{\omega}\|_{\mathcal{W}_{\rho}^{k,1}}+C\|u\cdot\nabla{\omega}\|_{L^{2}(\lambda
d(x,\partial\Omega)\geq\delta_{0})},$

which is bounded by the right hand side of (4.11). The proof for
$I_{2,\alpha}$ is complete.

Treating $I_{3,\alpha}$. Similarly, for $I_{3,\alpha}$, we get

$\displaystyle\sum_{\alpha}|\alpha|^{k}|I_{3,\alpha}|$ $\displaystyle\leq
C\|d(x,\partial\Omega)\Phi\|_{H^{1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}$ $\displaystyle\leq
C\|\phi^{b}u\cdot\nabla{\omega}\|_{\mathcal{W}^{k,1}_{\rho}}+C\|u\cdot\nabla{\omega}\|_{L^{2}(\lambda
d(x,\partial\Omega)\geq\delta_{0})}.$

This is also bounded by the right hand side of (4.11). The proof for
$I_{3,\alpha}$ is complete.

Treating $I_{4,\alpha}$. For $I_{4,\alpha}$ we have

$\sum_{\alpha}|\alpha|^{k}|I_{4,\alpha}|\leq\|\phi^{b}u\cdot\nabla{\omega}\|_{\mathcal{W}^{k,1}_{\rho}}.$

We rewrite
$\phi^{b}u\cdot\nabla{\omega}=u\cdot\nabla(\phi^{b}{\omega})-u\cdot\nabla\phi^{b}{\omega}=u\cdot\nabla{\omega}^{b}-(u\cdot\nabla\phi^{b}){\omega}$.
Hence we obtain

$\displaystyle\sum_{\alpha}|\alpha|^{k}|I_{4,\alpha}|$ $\displaystyle\leq
C\left(\|u\cdot\nabla{\omega}^{b}\|_{\mathcal{W}^{k,1}_{\rho}}+\|u{\omega}\|_{H^{k+1}(\lambda
d(x,\partial\Omega)\geq\delta_{0}+\rho_{0})}\right)$ $\displaystyle\leq
C\|u\cdot\nabla{\omega}^{b}\|_{\mathcal{W}^{k,1}_{\rho}}+C\|{\omega}\|^{2}_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}+C\|{\omega}\|_{\mathcal{W}^{k,1}_{\rho}}\|{\omega}\|_{H^{4}(\lambda
d(x,\partial\Omega)\geq\delta_{0}/2)}.$

This completes the bound for $I_{4,\alpha}$.

Combining all of the above, we obtain bounds on $A(\beta)$ in the analytic
norm.

###  4.3 Sobolev bounds away from the boundary

Finally, we bound the vorticity away from the boundary. Recall that

$\left\\{\begin{aligned}
\partial_{t}{\omega}^{i}+u\cdot\nabla\omega^{i}&=\nu\Delta{\omega}^{i}\\\
{\omega}^{i}_{|{\partial\Omega}}&=0\end{aligned}\right.$ (4.15)

Note that by definition, $\omega^{i}$ vanishes in the region when $\lambda
d(x,\partial\Omega)\leq\delta_{0}$. We perform the standard energy estimates,
for $k\geq 3$ so that the standard Sobolev embedding applies, yielding

$\displaystyle\frac{d}{dt}\|\omega^{i}\|_{H^{k}}^{2}+\nu\|\nabla\omega^{i}\|_{H^{k}}^{2}$
$\displaystyle\lesssim\|u\|_{H^{k}}\|\omega^{i}\|_{H^{k}}^{2}$
$\displaystyle\lesssim\|\omega^{i}\|^{3}_{H^{k}}+\|u^{b}\|^{3}_{H^{k}(\lambda
d(x,\partial\Omega)\geq\delta_{0})}.$

Using the elliptic theory for the Biot-Savart law
$u^{b}=\nabla^{\perp}\Delta^{-1}\omega^{b}$, we have

$\|u^{b}\|_{H^{k}(\lambda
d(x,\partial\Omega)\geq\delta_{0})}\lesssim\|\omega^{b}\|_{\mathcal{W}^{k,1}_{\rho}}+\|\omega^{b}\|_{H^{k}(\lambda
d(x,\partial\Omega)\geq\delta_{0})}.$

This proves that

$\displaystyle\frac{d}{dt}\|\omega^{i}\|_{H^{k}}^{2}$
$\displaystyle\lesssim\|\omega^{b}\|^{3}_{\mathcal{W}^{k,1}_{\rho}}+\|\omega^{b}\|^{3}_{H^{k}(\lambda
d(x,\partial\Omega)\geq\delta_{0})}.$

Integrating in time and recalling the iterative norm $A(\beta)$, we arrive at

$\|\omega^{i}\|_{H^{4}}^{2}\lesssim\|\omega_{0}\|_{H^{4}}^{2}+TA(\beta)^{2}.$

This bounds the Sobolev norm in $A(\beta)$, completing the proof of
Proposition 4.1.

###  4.4 Proof of Theorem 1.1

Finally, we show that our main theorem, Theorem 1.1, follows from Proposition
4.1. Indeed, taking $\beta$ sufficiently large in Proposition 4.1, we obtain
uniform bounds on the iterative norm (4.9) in term of initial data, which
gives the local solution in $\mathcal{W}^{1,1}_{\rho}+H^{4}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})$ for $t\in[0,T]$, with
$T=\beta^{-1}\lambda^{-2}\rho_{0}$. In particular, by definition of the
iterative norm $A(\beta)$, we have

$\|\omega(t)\|_{\mathcal{W}^{1,1}_{\rho}}+\|\omega(t)\|_{H^{4}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})}\leq C_{0}$

for $t\in[0,T]$. To prove the stated bound (1.9) on vorticity, we note that

$\displaystyle\|\omega\|_{L^{\infty}(\partial\Omega)}$
$\displaystyle\lesssim\|\partial_{\widetilde{z}}\omega\|_{\mathcal{L}_{\rho}^{1}}+\|\omega(t)\|_{H^{2}(\\{\lambda
d(x,\partial\Omega)\geq\delta_{0}/2\\})}.$

It thus suffices to prove that
$\|\partial_{\widetilde{z}}\omega\|_{\mathcal{L}_{\rho}^{1}}\lesssim\nu^{-1/2}$.
Indeed, similar to (4.10), we bound

$\|\partial_{\widetilde{z}}\omega(\widetilde{t})\|_{\mathcal{L}_{\rho}^{1}}\leq\|\partial_{\widetilde{z}}e^{\nu\widetilde{t}S}\omega_{0}\|_{\mathcal{L}_{\rho}^{1}}+\int_{0}^{\widetilde{t}}\|\partial_{\widetilde{z}}e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S}f(\widetilde{t}^{\prime})\|_{\mathcal{L}_{\rho}^{1}}\;d\widetilde{t}^{\prime}+\int_{0}^{\widetilde{t}}\|\partial_{\widetilde{z}}\Gamma(\nu(\widetilde{t}-\widetilde{t}^{\prime}))g(\widetilde{t}^{\prime})\|_{\mathcal{L}_{\rho}^{1}}\;d\widetilde{t}^{\prime}$

for the same $f,g$ defined as in (4.8). It follows directly from the
construction, see Section 3.6, that the $\widetilde{z}$-derivative of the
semigroup $\partial_{\widetilde{z}}e^{\nu\widetilde{t}S}$ satisfies the same
bounds as does $e^{\nu\widetilde{t}S}$, up to an extra factor of
$(\nu\widetilde{t})^{-1/2}$ or $|\partial_{\widetilde{\theta}}|+\nu^{-1/2}$.
Therefore, using the previous bounds on $f(\widetilde{t})$, we have

$\displaystyle\int_{0}^{\widetilde{t}}\|\partial_{\widetilde{z}}e^{\nu(\widetilde{t}-\widetilde{t}^{\prime})S}f(\widetilde{t}^{\prime})\|_{\mathcal{L}_{\rho}^{1}}\;d\widetilde{t}^{\prime}$
$\displaystyle\lesssim\int_{0}^{\widetilde{t}}(\nu(\widetilde{t}-\widetilde{t}^{\prime}))^{-1/2}\Big{[}\|f(\widetilde{t}^{\prime})\|_{\mathcal{W}_{\rho}^{1,1}}+\|\widetilde{z}f(\widetilde{t}^{\prime})\|_{H^{1}(\widetilde{z}\geq\delta_{0}+\rho)}\Big{]}\;d\widetilde{t}^{\prime}$
$\displaystyle\lesssim\int_{0}^{\widetilde{t}}(\nu(\widetilde{t}-\widetilde{t}^{\prime}))^{-1/2}\;d\widetilde{t}^{\prime}$
$\displaystyle\lesssim\nu^{-1/2}.$

Other terms are estimated similarly, giving
$\|\partial_{\widetilde{z}}\omega\|_{\mathcal{L}_{\rho}^{1}}\lesssim\nu^{-1/2}$
as claimed.

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