1.1 Presentation of the problem and related results

In this article, we study 2-D incompressible Navier-Stokes equations in a thin domain where the aspect ratio and the Reynolds number have certain constraints:

(1.1) $$ \begin{align} \left\{ \begin{aligned} &\partial_{t} U+U \cdot \nabla U-\varepsilon^{2} (\partial_x^2+\eta\partial_y^2) U+\nabla P=0,\\ &\mbox{div}~ U=0,\\ &U|_{y=0}=U|_{y=\varepsilon}=0, \end{aligned} \right. \end{align} $$

where $t\geq 0, (x,y)\in \mathcal {S}^{\varepsilon }=\left \{(x, y) \in \mathbb {T}\times \mathbb {R} : 0<y<\varepsilon \right \}.$ Here, $U(t,x,y), P(t,x,y)$ stand for the velocity and pressure function, respectively, and $\eta $ is a positive constant independent of  $\varepsilon $ . The width of domain $\mathcal {S}^{\varepsilon }$ is $\varepsilon ,$ and the boundary condition in (1.1) corresponds to the non-slip condition at the walls $y = 0, \varepsilon .$ In addition, the system is prescribed with the initial data of the form

(1.2) $$ \begin{align} \left.U\right|{}_{t=0}=\left(u_{0}\left(x, \frac{y}{\varepsilon}\right), \varepsilon v_{0}\left(x, \frac{y}{\varepsilon}\right)\right)=U_{0}^{\varepsilon} \quad \text{in}\quad \mathcal{S}^{\varepsilon}. \end{align} $$

This is a classical model with applications to oceanography, meteorology and geophysical flows, where the vertical dimension of the domain is very small compared with the horizontal dimension of the domain.

To study the process $\varepsilon \to 0$ , we first fix the domain independent of $\varepsilon $ . Here, we rescale the system (1.1) as follows:

$$ \begin{align*} U(t, x, y)=\left(u^{\varepsilon}\left(t, x, \frac{y}{\varepsilon}\right), \varepsilon v^{\varepsilon}\left(t, x, \frac{y}{\varepsilon}\right)\right) \quad \text{ and } \quad P(t, x, y)=p^{\varepsilon}\left(t, x, \frac{y}{\varepsilon}\right). \end{align*} $$

We put above relations into (1.1), and then (1.1) is reduced to a scaled anisotropic Navier-Stokes system:

(1.3) $$ \begin{align} \left\{ \begin{aligned} &\partial_t u^{\varepsilon}+u^{\varepsilon}\partial_x u^{\varepsilon}+v^{\varepsilon}\partial_y u^{\varepsilon}-\varepsilon^2\partial_x^2 u^{\varepsilon}-\eta\partial_y^2 u^{\varepsilon}+\partial_x p^{\varepsilon}=0,\\ &\varepsilon^2(\partial_t v^{\varepsilon}+u^{\varepsilon}\partial_x v^{\varepsilon}+v^{\varepsilon}\partial_y v^{\varepsilon}-\varepsilon^2\partial_x^2 v^{\varepsilon}-\eta\partial_y^2 v^{\varepsilon})+\partial_y p^{\varepsilon}=0,\\ &\partial_x u^{\varepsilon}+\partial_y v^{\varepsilon}=0,\\ &(u^{\varepsilon},v^{\varepsilon})|_{y=0,1}=0,\\ &(u^{\varepsilon},v^{\varepsilon})|_{t=0}=(u_0,v_0) , \end{aligned} \right. \end{align} $$

where $(x,y)\in \mathcal {S}=\left \{(x, y) \in \mathbb {T}\times (0,1) \right \}$ .

To simplify the notations, we take $\eta =1$ in this paper and denote $\Delta _{\varepsilon }=\varepsilon ^2\partial _x^2+\partial _y^2$ .

Formally, taking $\varepsilon \to 0$ in (1.3), we derive the hydrostatic Navier-Stokes/Prandtl system (see [Reference Lagrée and Lorthois22, Reference Renardy31]):

(1.4) $$ \begin{align} \left\{ \begin{aligned} &\partial_t u_p+u_p\partial_x u_p+v_p\partial_y u_p-\eta\partial_y^2 u_p+\partial_x p_p=0\quad \text{ in } \mathcal{S} \times (0, \infty),\\ &\partial_y p_p=0 \quad \text{ in } \mathcal{S} \times (0, \infty),\\ &\partial_x u_p+\partial_y v_p=0 \quad \text{ in } \mathcal{S} \times (0, \infty),\\ &(u_p,v_p)|_{y=0,1}=0,\\ &u_p|_{t=0}=u_0 \quad \text{ in }\,\, \mathcal{S}. \end{aligned} \right. \end{align} $$

This paper aims to justify the limit from the scaled anisotropic Navier-Stokes system (1.3) to the hydrostatic Navier-Stokes/Prandtl system (1.4) for a class of convex data in the optimal Gevrey class with index $\gamma =\frac 32$ .

Before presenting the precise statement of the main result in this paper, we recall some results on system (1.4). If $\eta =0$ in the system (1.4), we get the hydrostatic Euler system:

(1.5) $$ \begin{align} \left\{ \begin{aligned} &\partial_t u+u\partial_x u+v\partial_y u +\partial_x p=0\quad \text{ in } \mathcal{S} \times (0, \infty) ,\\ &\partial_y p=0\quad \text{ in } \mathcal{S} \times (0, \infty) ,\\ &\partial_x u+\partial_y v=0 \quad \text{ in } \mathcal{S} \times (0, \infty),\\ &v|_{y=0,1}=0,\\ &u|_{t=0}=u_0 \quad \text{ in }\,\, \mathcal{S}. \end{aligned} \right. \end{align} $$

There is a lot of research on the system (1.5), and readers can refer to [Reference Brenier3, Reference Brenier4, Reference Cao, Ibrahim, Nakanishi and Titi5, Reference Grenier16, Reference Kukavica, Temam, Vicol and Ziane20, Reference Kukavica, Masmoudi, Vicol and Wong19, Reference Masmoudi and Wong26, Reference Renardy31, Reference Wong38]. Renardy [Reference Renardy31] proved the linearization of (1.5) has a growth like $e^{|k|t}$ if the initial data is not uniform convexity (or concavity) with respect to the variable y. Local well-posedness in the analytic setting was established in [Reference Kukavica, Temam, Vicol and Ziane20]. Under the convexity condition, Masmoudi and Wong [Reference Masmoudi and Wong26] got the well-posedness of (1.5) in the Sobolev space.

Next, we recall some results on the well-posedness of the hydrostatic Navier-Stokes/Prandtl system (1.4). Similar to the classical Prandtl equation, (1.4) loses one derivative because of the term $v_p\partial _y u_p$ . Paicu, Zhang and Zhang [Reference Paicu, Zhang and Zhang30] obtained the global well-posedness of system (1.4) when the initial data is small in the analytical space. Meanwhile, Renardy [Reference Renardy31] also proved that the linearization of the hydrostatic Navier-Stokes equations at certain parallel shear flows is ill-posed and may have a growth $e^{|k|t}$ which is the same as (1.5) when the initial data is not convex. Thus, to obtain well-posedness results that break through the analytic space, one may need the convexity condition on the velocity. For that, under the convexity condition, Gérard-Varet, Masmoudi and Vicol proved (1.4) is local well-posedness in the Gevrey class with index $9/8$ in [Reference Gérard-Varet, Masmoudi and Vicol15]. In [Reference Gérard-Varet, Masmoudi and Vicol15], they first derive the vorticity equations $\omega =\partial _y u$ :

$$ \begin{align*} \partial_t (\partial_x \omega)+ \partial_x v\partial_y \omega+\cdots=0, \end{align*} $$

where the worst term is $\partial _x v$ leading to one derivative loss. Then, they use the ‘hydrostatic trick’ which means that they take the inner product with $\partial _x \omega /\partial _y \omega $ ( $\partial _y\omega \geq c_0>0$ ) instead of $\partial _x \omega $ to take advantage of the cancellation:

$$ \begin{align*} \int \partial_x v\partial_y \omega\cdot \frac{\partial_x \omega}{\partial_y \omega}=\int \partial_x v\partial_x \omega=-\int \partial_x\partial_y v\partial_x u=0. \end{align*} $$

Such an idea was used previously in [Reference Masmoudi and Wong26]. To close the energy estimates, the ‘hydrostatic trick’ is not enough due to the ‘bad’ boundary condition of $\omega $

$$ \begin{align*} \partial_y\omega|_{y=0}=-\partial_x \int_0^1 u^2dy+\cdots, \end{align*} $$

which loses one derivative too. To overcome that, [Reference Gérard-Varet, Masmoudi and Vicol15] introduce the following decomposition:

$$ \begin{align*} \omega=\omega^{bl}+\omega^{in}, \end{align*} $$

where $\omega ^{bl}$ is the boundary corrector which satisfies that

$$ \begin{align*} \partial_t \omega^{bl}-\partial_{y}^2\omega^{bl}=0,\quad \partial_y\omega^{bl}|_{y=0}=-\partial_x \int_0^1 u^2dy. \end{align*} $$

Following the above decomposition, [Reference Gérard-Varet, Masmoudi and Vicol15] obtain the well-posedness results of (1.4) in the Gevrey class with index $\gamma =\frac 98$ .

To search the best functional space for the system (1.4), based on the Tollmien-Schlichting instabilities for Navier-Stokes [Reference Grenier, Guo and Nguyen17], Gérard-Varet, Masmoudi and Vicol also give the following conjecture: ‘Our conjecture - based on a formal parallel with Tollmien-Schlichting instabilities for Navier-Stokes [18] - is that the best exponent possible should be $\gamma =\frac 32$ , but the such result is for the time being out of reach’.

While studying the anisotropic Navier-Stokes system (1.3) and the hydrostatic Navier-Stokes/Prandtl system (1.4), another important problem is to justify the inviscid limit. Under the analytical setting, Paicu, Zhang and Zhang [Reference Paicu, Zhang and Zhang30] justified the limit from (1.3) to (1.4). Based on the work [Reference Gérard-Varet, Masmoudi and Vicol15], we [Reference Wang, Wang and Zhang35] justified the limit in the Gevrey class with index $\gamma =\frac 98$ .

In this paper, we aim to prove the conjecture of Gérard-Varet, Masmoudi and Vicol. To do that, we use some ideas from the classical inviscid limit theory. Next, we recall the recent development of the classical Prandtl equation and the inviscid limit theory.

There are a lot of papers studying the well-posedness of the Prandtl equation in some special functional space. For monotonic initial data, [Reference Oleinik29, Reference Alexandre, Wang, Xu and Yang1, Reference Masmoudi and Wong27] used a different method to get the local existence and uniqueness of classical solutions to the Prandtl equation in Sobolev space. Without monotonic condition, [Reference Lombardo, Cannone and Sammartino24, Reference Sammartino and Caflisch32] proved that the Prandtl equation is well-posedness in the analytic class; [Reference Gérard-Varet and Masmoudi14, Reference Li and Yang23, Reference Chen, Wang and Zhang6] proved the well-posedness of the Prandtl equations in the Gevrey class for a class of concave initial data. Without any structure assumption, Dietert and Gérard-Varet [Reference Dietert and Gérard-Varet8] proved well-posedness in the Gevrey space with index $\gamma =2$ . According to [Reference Gérard-Varet and Dormy10], $\gamma =2$ may be the optimal index for the well-posedness theory. For more results on the Prandtl equation, see [Reference Grenier and Nguyen18, Reference Wang37, Reference Wang, Xin and Zang36, Reference Xiao and Xin39, Reference Xin and Zhang40].

On the inviscid limit problem, we refer to [Reference Sammartino and Caflisch33, Reference Wang, Wang and Zhang34, Reference Kukavica, Vicol and Wang21, Reference Nguyen and Nguyen28, Reference Maekawa25, Reference Fei, Tao and Zhang9] for the analytical class. Note that going from analytic to Gevrey data is a challenging problem. The first result in the Gevrey class is given by [Reference Gérard-Varet, Maekawa and Masmoudi12]. Gérard-Varet, Masmoudi and Maekawa [Reference Gérard-Varet, Maekawa and Masmoudi12] proved the stability of the Prandtl expansion for the perturbations in the Gevrey class when $U^{BL}(t, Y)$ is a monotone and concave function where the boundary layer is the shear type like

$$ \begin{align*} u_s^{\nu}=(U^e(t, y), 0)+(U^{BL}(t, \frac{y}{\sqrt\nu}),0), \end{align*} $$

where $\nu $ is the viscosity coefficient. Later, Chen, Wu and Zhang [Reference Chen, Wu and Zhang7] improved the results in [Reference Gérard-Varet, Maekawa and Masmoudi12] to get the $L^2\cap L^{\infty }$ stability. Very recently, Gérard-Varet, Masmoudi and Maekawa [Reference Gérard-Varet, Maekawa and Masmoudi13] used a very clever decomposition to get the optimal Prandtl expansion around the concave boundary layer. Their results generalized the one obtained in [Reference Gérard-Varet, Maekawa and Masmoudi12, Reference Chen, Wu and Zhang7], which restricted to expansions of shear flow type. In their paper, they decompose the stream function $\phi $ as follows:

$$ \begin{align*} \phi=\phi_{slip}+\phi_{bc}, \end{align*} $$

where $\phi _{slip}$ enjoys a “good” boundary condition and $\phi _{bc}$ is a corrector which recover the boundary condition. This kind of decomposition is also used in [Reference Chen, Wu and Zhang7]. To estimate $\phi _{bc}$ , they also need the following decomposition

$$ \begin{align*} \phi_{bc}=\phi_{bc, S}+\phi_{bc, T}+\phi_{bc, R}, \end{align*} $$

where $\phi _{bc, S}$ satisfies the Stokes equation, $\phi _{bc, T}$ is to correct the stretching term with ‘good’ boundary condition and $\phi _{bc, R}$ solves formally the same system as $\phi _{slip}$ . In this paper, we apply the decomposition in [Reference Gérard-Varet, Maekawa and Masmoudi13] to justify the limit from (1.3) to (1.4).

1.2 Statement of the main results

Before starting the main results, we give some assumptions on the initial data. Assume that initial data belong to the following Gevrey class:

(1.6) $$ \begin{align} \|e^{\langle D_x\rangle^{\frac23}}\partial_y u_0\|_{H^{14,0}}+\|e^{\langle D_x\rangle^{\frac23}}\partial_y^3 u_0\|_{H^{10,0}}:=M<+\infty, \end{align} $$

where $H^{r,s}$ is the anisotropic Sobolev space defined by

$$ \begin{align*} \|f\|_{H^{r,s}}=\|\|f\|_{H^r_x(\mathbb{T})}\|_{H^s_y(0,1)}. \end{align*} $$

More precisely, we consider the initial data of the form

$$ \begin{align*} u^{\varepsilon}(0,x,y)=u_0(x,y),\quad v^{\varepsilon}(0,x,y)=v_0(x,y), \end{align*} $$

which satisfies the compatibility conditions

(1.7) $$ \begin{align} &\partial_xu_0+\partial_y v_0(t,x,y)=0, \quad u_0(t,x,0)=u_0(t,x,1)=v_0(t,x,0)=v_0(t,x,1)=0, \end{align} $$

(1.8) $$ \begin{align} &\int_0^1\partial_x u_0dy=0,\quad \partial_y^2u_0|_{y=0,1}=\int_0^1(-\partial_xu_0^2+\partial_y^2 u_0)-\int_{\mathcal{S}}\partial_y^2 u_0 dxdy.\qquad\quad\ \end{align} $$

Moreover, we assume the initial velocity satisfies the convex condition

(1.9) $$ \begin{align} \inf_{\mathcal{S}}\partial_y^2 u_0\geq 2 c_0>0. \end{align} $$

Now, we are in the position to state the main results of our paper.

Theorem 1.1. Let initial data $u_0$ satisfy (1.6)-(1.9). Then there exist $T>0$ and $C> 0$ independent of $\varepsilon $ such that there exists a unique solution of the scaled anisotropic Navier–Stokes equations (1.3) in $[0,T]$ , which satisfies that for any $t\in [0,T],$ it holds that

$$ \begin{align*} \|(u^{\varepsilon}-u_p, \varepsilon v^{\varepsilon}-\varepsilon v_p)\|_{L^2_{x,y}\cap L^{\infty}_{x,y}}\leq C\varepsilon^2, \end{align*} $$

where $(u_p, v_p)$ is the solution to (1.4).

1.3 Sketch of the proof

In this subsection, we sketch the main ingredients in our proof.

1. (1) Introduce the error equations. In Section 3, we deduce the error equations. We introduce the error

   $$ \begin{align*} u^R=u^{\varepsilon}-u^p,\quad v^R=v^{\varepsilon}-v^p, \quad p^R=p^{\varepsilon}-p^p, \end{align*} $$

   which satisfies

   (1.10) $$ \begin{align} \left\{ \begin{aligned} &\partial_t u^R-{\Delta_{\varepsilon} u^R}+v^R\partial_y u^p+\partial_x p^R=\cdots,\\ &\varepsilon^2(\partial_t v^R-\Delta_{\varepsilon} v^R)+\partial_y p^R=\cdots. \end{aligned} \right. \end{align} $$

   Here, $(u^p, v^p, p^p)$ is an approximate solution given in (3.1). The key point in this paper is to obtain the uniform estimate (in $\varepsilon $ ) of $(u^R, \varepsilon v^R)$ in the Gevrey class with index $\gamma =\frac 32.$ In view of (1.10), since $v^R$ is controlled via the relation $v^R=-\int _0^y \partial _x u^R dy'$ , the main difficulty comes from the term $v^R\partial _y u^p$ , which loses one tangential derivative. In [Reference Wang, Wang and Zhang35], we justify the limit in Gevrey class $\frac 98.$ For the data in the Gevrey class with optimal index $\gamma =\frac 32$ , we need to introduce new ideas.

2. (2) Introduce the vorticity formulation. In order to eliminate $p^R,$ we introduce vorticity $\omega ^R=-\varepsilon ^2\partial _x v^R+\partial _y u^R$ and rewrite the equation of $\omega ^R$ by stream function $\phi $ which satisfies

   $$ \begin{align*} v^R=-\partial_x\phi, \quad u^R=\partial_y \phi+C(t), \quad C(t)=\frac1{2\pi}\int_{{\mathcal S}} u^Rdxdy. \end{align*} $$

   Thus, we get

   (1.11) $$ \begin{align} \left\{ \begin{aligned} &(\partial_t-\Delta_{\varepsilon})\Delta_{\varepsilon} \phi-\partial_x\phi\partial_y \omega^p=\cdots,\qquad (x,y)\in\mathcal{S},\\ &\phi|_{y=0,1}=0, \quad \partial_y\phi|_{y=0,1}=C(t),\qquad x\in\mathbb{T}. \end{aligned} \right. \end{align} $$

   We notice the term $\partial _x\phi \partial _y \omega ^p$ also loses one tangential derivative. But under the convexity condition $\partial _y\omega ^p\geq c_0>0,$ one can use the ‘hydrostatic trick’ to deal with this term. Testing $\frac {\omega ^R}{\partial _y\omega ^p}$ to the (1.11) instead of $\omega ^R,$ we have the following cancellation:

   $$ \begin{align*} -\int_{{\mathcal S}}\partial_x\phi \partial_y\omega^p \cdot\frac{\omega^R}{\partial_y\omega^p}dxdy=-\int_{{\mathcal S}}\partial_x\phi \triangle_{\varepsilon} \phi dxdy=\int_{{\mathcal S}}\partial_x|\nabla_{\varepsilon}\phi|^2 dxdy=0, \end{align*} $$

   where we use $\phi |_{y=0,1}=0.$ However, the boundary condition of $\phi $ is $\partial _y\phi |_{y=0,1}= C(t)$ , which brings an essential difficulty.

   By the energy estimates, taking the inner product in $X^{r}$ (the definition is given in section 2) with $-\partial _t\phi ,$ we get

   (1.12) $$ \begin{align} &\sup_{s\in[0,t]}(\lambda\|\nabla_{\varepsilon}\phi(s)\|_{X^{\frac73}}^2+\| \Delta_{\varepsilon}\phi(s)\|_{X^2}^2) \end{align} $$
   $$\begin{align*}\leq &\ C\int_0^t(\varepsilon^{-2}\|\varphi\Delta_{\varepsilon}\phi\|_{X^2}^2+\varepsilon^{-2}\| \nabla_{\varepsilon}\phi\|_{X^2}^2+\cdots)ds,\end{align*}$$

   where $\varphi (y)=y(1-y).$ All we need to do is to control $\varepsilon ^{-1}\|\varphi \Delta _{\varepsilon }\phi \|_{X^2}$ and $\varepsilon ^{-1}\|\nabla _{\varepsilon }\phi \|_{X^2}$ by the left-hand side of (1.12).

   Motivated by [Reference Gérard-Varet, Maekawa and Masmoudi13], we expect to achieve that by a decomposition of stream function, $\phi =\phi _{slip}+\phi _{bc}$ in Gevrey $\frac 32$ regularity. Here, $\phi _{slip}$ enjoys a ‘good’ boundary condition and $\phi _{bc}$ is a corrector which recovers the boundary condition. In the following, we present the decomposition precisely.

3. (3) Gevrey estimate under artificial boundary conditions. $\phi _{slip}$ enjoying a good boundary condition is defined by

   (1.13) $$ \begin{align} \left\{ \begin{aligned} &(\partial_t-\Delta_{\varepsilon})\omega_{slip}-\partial_x\phi_{slip}\partial_y \omega^p=\cdots,\qquad (x,y)\in\mathcal{S}\\ &\phi_{slip}|_{y=0,1}=0, \quad \omega_{slip}|_{y=0,1}=0,\qquad x\in\mathbb{T}, \end{aligned} \right. \end{align} $$

   where $\omega _{slip}=\Delta _{\varepsilon }\phi _{slip}.$ By ‘hydrostatic trick’ and Navier-slip boundary conditions, we obtain

   (1.14) $$ \begin{align} \lambda\int_0^t (\|\omega_{slip}\|_{X^{\frac73}}^2+\|\nabla_{\varepsilon}\phi_{slip}\|_{X^{\frac73}}^2+|\nabla_{\varepsilon}\phi_{slip}|_{y=0,1}|_{X^{\frac73}}^2)ds\leq \frac{C}{\lambda}\int_0^t\| \varepsilon\Delta_{\varepsilon}\phi \|_{X^{2}}^2 ds+\cdots. \end{align} $$

   The full study of the Orr-Sommerfeld formulation (1.13) with Navier-slip boundary conditions is given in Section 7.

4. (4) Recovery the non-slip boundary condition. In Step (3), we use the slip boundary condition, not the real boundary condition $\partial _y\phi |_{y=0, 1}=C(t).$ To recover the boundary condition, we introduce the following system:

   (1.15) $$ \begin{align} \left\{ \begin{aligned} &(\partial_t-\Delta_{\varepsilon})\Delta_{\varepsilon}\phi_{bc}-\partial_x\phi_{bc}\partial_y \omega^p=0, \qquad (x,y)\in\mathcal{S}\\ &\phi_{bc}|_{y=0,1}=0, \quad \partial_y\phi_{bc}|_{y=i}=h^i,\qquad x\in\mathbb{T}, \end{aligned} \right. \end{align} $$

   where $\omega _{bc}=\Delta _{\varepsilon } \phi _{bc}$ and $i=0,1.$ And we need to choose a suitable $h^i$ such that

   $$ \begin{align*} \partial_y \phi_{bc}|_{y=0,1}=-\partial_y\phi_{slip}|_{y=0,1}+C(t). \end{align*} $$

   Next, we give the main idea for proving the existence of $h^i$ :

   1. 1. We define $\phi _{bc,S}=\phi _{bc,S}^0+\phi _{bc,S}^1$ , where $\phi _{bc,S}^i$ solve

      (1.16) $$ \begin{align} \left\{ \begin{aligned} &(\partial_t -\Delta_{\varepsilon})\Delta_{\varepsilon}\phi^i_{bc, S}=0,\\ &\phi^i_{bc, S}|_{y=i}=0,\quad \partial_y\phi^i_{bc, S}|_{y=i}=h^i,\\ &\phi^i_{bc, S}|_{t=0}=0, \end{aligned} \right. \end{align} $$

      where $x\in \mathbb {T}$ , $y\in (0,+\infty )$ for $i=0$ and $y\in (-\infty ,1)$ for $i=1.$ Taking Fourier transformation on t and x, we can write the precise expression of the solution to obtain the Gevrey estimate for $\phi _{bc,S}^i$ :

      (1.17) $$ \begin{align} &\int_0^t\|\nabla_{\varepsilon}\phi^i_{bc,S}\|_{X^{\frac{5}{2}}_i}^2+\|\varphi^i\Delta_{\varepsilon}\phi^i_{bc,S}\|_{X^{\frac52}_i}^2 + \|\partial_x\phi^i_{bc,S}\|_{X^{\frac53}}^2ds\leq \frac{C}{\lambda^{\frac12}}\int_0^t|h^i|_{X^{\frac73}}^2ds, \end{align} $$

      where $\varphi ^0(y)=y,~\varphi ^1(y)=1-y.$ Compared with the decomposition in [Reference Wang, Wang and Zhang35], we get more regularity of $\partial _x\phi ^i_{bc,S}$ , which is a key point to get the optimal Gevrey regularity. The details for this step are given in Section 8.1.

   2. 2. We correct the nonlocal term constructed in the above step by considering the following equations:

      (1.18) $$ \begin{align} \left\{ \begin{aligned} &(\partial_t -\Delta_{\varepsilon})\Delta_{\varepsilon}\phi^i_{bc, R}-\partial_x\phi_{bc,R}^i\partial_y\omega^p=\partial_x\phi_{bc,S}^i\partial_y\omega^p,\quad (x,y)\in {\mathcal S}\\ &\phi^i_{bc, R}|_{t=0}=0, \quad (x,y)\in {\mathcal S} \end{aligned} \right. \end{align} $$

      with Navier-slip conditions. By the same process as Step (3) and combining with the sharp estimate (1.17) to get an estimate for $\phi ^i_{bc,R},$

      (1.19) $$ \begin{align} \lambda\int_0^t \|\omega_{bc,R}^i\|_{X^{\frac73}}^2ds&+\int_0^t (\|\nabla_{\varepsilon}\phi_{bc,R}^i\|_{X^{\frac73}}^2+|\partial_y\phi_{bc,R}^i|_{y=0,1}|_{X^{\frac73}}^2)ds \end{align} $$
      $$\begin{align*}\leq&\frac{C}{\lambda^{\frac12}} \int_0^t|h^i|_{X^{\frac73}}^2ds+\cdots,\nonumber \qquad t\in[0, T]. \end{align*}$$

      More details are given in Section 8.3.

   3. 3. We define $\phi _{bc}=\phi _{bc,S}+\phi _{bc,R}$ , where $\phi _{bc,S}=\sum _{i=0, 1}\phi ^i_{bc,S} $ and $\phi _{bc,R}=\sum _{i=0, 1}\phi ^i_{bc,R}$ , which solves system (1.15). To match the boundary condition on the derivative of $\partial _y\phi |_{y=0,1}=C(t)$ , we need

      $$ \begin{align*} \partial_y\phi_{bc,S}|_{y=0,1}+\partial_y\phi_{bc,R}|_{y=0,1}=\partial_y\phi_{bc}|_{y=0,1}=- \partial_y\phi_{slip}|_{y=0,1}+C(t). \end{align*} $$

      On one hand, $\phi _{bc,S}$ and $\phi _{bc,R}$ are defined by $h^i$ . We define a $0$ -order operator $R_{bc}$ given in (8.71) such that

      $$ \begin{align*} (1+R_{bc}) h^i= - \partial_y\phi_{slip}|_{y=0,1}+C(t). \end{align*} $$

      Moreover, by the estimate in Step 1 and Step 2, we can get

      $$ \begin{align*} \int_0^t\Big|R_{bc}[h^0, h^1]\Big|_{X^{\frac73}}^2 ds\leq&\frac{C}{\lambda^{\frac12}}\int_0^t|(h^0,h^1)|_{X^{\frac 73}}^2ds, \end{align*} $$

      which means that $(1+R_{bc})$ is an invertible operator when $\lambda $ is large. That means that $\phi _{bc,S}$ and $\phi _{bc,R}$ are well-defined and (1.15) is well-posedness. Details are given in Section 8.4.

      Due to the transport terms, we need to introduce a new auxiliary function $\phi _{bc, T}$ between Step 1 and Step 2. For more details, see Section 8.2.

5. (5) Close the energy estimates (1.12). Summing estimates (1.17) and (1.19) in Step (4), we get an estimate for $\phi _{bc}:$

   $$ \begin{align*} \int_0^t\|\nabla_{\varepsilon}\phi_{bc}\|_{X^{\frac{7}{3}}}^2+\|\varphi\Delta_{\varepsilon}\phi_{bc}\|_{X^{\frac 73}}^2ds\leq &\ \frac{C}{\lambda^{\frac12}}\int_0^t|(h^0, h^1)|_{X^{\frac73}}^2ds\\ \leq &\ C\int_0^t |\nabla_{\varepsilon}\phi_{slip}|_{y=0,1}|^2_{X^{\frac73}}ds+\cdots, \end{align*} $$
   which along with (1.14), we have
   $$ \begin{align*} \int_0^t (\|\varphi\omega\|_{X^{\frac73}}^2+\|\nabla_{\varepsilon}\phi\|_{X^{\frac73}}^2)ds\leq \frac{C}{\lambda}\int_0^t\| \varepsilon\Delta_{\varepsilon}\phi \|_{X^{2}}^2 ds+\cdots, \end{align*} $$
   and then we put the above estimate into (1.12) to close the estimate for system (1.11).

1.4 Notations

• – $\mathcal {S}^{\varepsilon }=\left \{(x, y) \in \mathbb {T}\times \mathbb {R} : 0<y<\varepsilon \right \}$ and $\mathcal {S}=\left \{(x, y) \in \mathbb {T}\times \mathbb {R} : 0<y<1\right \}.$

• – $\nabla _{\varepsilon }=(\varepsilon \partial _x, \partial _y)$ and $\Delta _{\varepsilon }=\varepsilon ^2 \partial _x^2+\partial _y^2$ .

• – Vorticity of Prandtl part $\omega ^p$ is defined by $ \omega ^p=\partial _y u^p$ .

• – Vorticity of reminder part $\omega ^R=\Delta _{\varepsilon } \phi $ is defined by $\omega ^R=\varepsilon ^2\partial _x v^R-\partial _y u^R$ . In this paper, we also define $\omega ^i_{bc, j}=\Delta _{\varepsilon } \phi _{bc, j}^i$ , where $i=0, 1$ and $j\in \{R, T\}$

• – Cut-off functions $\varphi (y)=y(1-y)$ and $\varphi ^i(y)=i+(-1)^iy$ .

• – $C(t)=\frac 1{2\pi }\int _{{\mathcal S}} u^Rdxdy$ .

• – The Fourier transform of $f_{\Phi }$ is defined by $e^{(1-\lambda t)\langle k\rangle ^{\frac 23}} \widehat {f}(k)$ .