The Inviscid Three Dimensional Quasi-Geostrophic System on Bounded Domains

Abstract

We present a formal derivation of the inviscid three dimensional quasi-geostrophic system (QG) from primitive equations on a bounded, cylindrical domain. A key point in the derivation is the treatment of the lateral boundary and the resulting boundary conditions it imposes on solutions. To our knowledge, these boundary conditions are new and differentiate our model from closely related models which have been the object of recent study. These boundary conditions are natural for a variational problem in a particular Hilbert space. We construct solutions and prove an elliptic regularity theorem corresponding to the variational problem, allowing us to show the existence of global weak solutions to QG.

1 Introduction

In this paper we study the inviscid three-dimensional quasi-geostrophic system. The QG model describes stratified flows on a large time scale for which the effect of the rotation of the Earth is significant. The model consists of two coupled transport equations as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \left( \partial _t + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\right) \left( {\mathcal {L}}(\Psi ) + \beta _0 y \right) = a_L &{} \Omega \times [0,h]\times [0,T] \\ \left( \partial _t + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\right) (\partial _{\nu }\Psi ) = a_\nu &{} \Omega \times \{0,h\}\times [0,T].\\ \end{array}\right. } \qquad (QG) \end{aligned}$$

Classically, the model is posed for $\Omega = {\mathbb {R}}^2$ or $\Omega ={\mathbb {T}}^2$. We use the notation

$$\begin{aligned} {\overline{\nabla }}= ( \partial _x, \partial _y, 0 ), \qquad {\overline{\nabla }}^\perp = ( -\partial _y, \partial _x ,0) . \end{aligned}$$

The functions $a_L$ and $a_\nu$ are forcing terms, and $\beta _0$ is a parameter coming from the usual $\beta$-plane approximation. The normal derivative of $\Psi$ on $\Omega \times \{0,h\}$ is denoted by $\partial _{\nu } \Psi$. The operator ${\mathcal {L}}$ is defined by

$$\begin{aligned} {\mathcal {L}}:=\partial _{xx}+\partial _{yy}+ \partial _z \left( \lambda \partial _z \right) , \end{aligned}$$

where $\lambda >0$ is a smooth function depending only on z and is related to the density of the fluid. To ensure ellipticity of ${\mathcal {L}}$ we require that

$$\begin{aligned} \frac{1}{\Lambda } \leqq \lambda (z) \leqq \Lambda \end{aligned}$$

for some $\Lambda \in (0,\infty )$. Throughout the remainder of the paper, the system shall be posed on a fixed cylindrical domain

$$\begin{aligned} \Omega \times [0,h], \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^2$ is a smooth, bounded set, and the height h is fixed and finite. The values of ${\mathcal {L}}(\Psi )$ and $\partial _{\nu } \Psi$ are advected by the fluid velocity field ${\overline{\nabla }}^\perp \Psi$. In order to reconstruct $\Psi$ at each time, it is necessary to supplement the system with a boundary condition on the lateral boundary $\partial \Omega \times [0,h]$.

1.1 Boundary Conditions

The purpose of this paper is to formally derive an appropriate model from the primitive equations while assuming that the lateral boundary is impermeable; that is, we assume only that the fluid velocity ${\overline{\nabla }}^\perp \Psi$ is tangent to $\partial \Omega \times [0,h]$. We then prove that weak solutions exist globally in time for the resulting system. In fact, we show in Section 2.3 that the impermeability produces two constraints on a possible solution. First, we must have that

$$\begin{aligned} \Psi (t,x,y,z)|_{\partial \Omega \times [0,h]} = c(t,z) \end{aligned}$$

(1)

for some unknown function c(t, z). However, this is not enough to define a unique solution to an elliptic problem on $\Omega \times [0,h]$. Crucially, the impermeability condition provides another natural constraint. After defining $\nu _s$ to be the normal derivative to $\partial \Omega \times \{z\}$ and $\,\mathrm{d}\omega$ the Hausdorff measure on $\partial \Omega$, the second constraint is that, for all $z\in [0,h]$,

$$\begin{aligned} \frac{\partial }{\partial t}\int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\Psi \cdot \nu _s \,\mathrm{d}\omega = 0. \end{aligned}$$

(2)

In other words, building a weak solution to (QG) requires choosing a datum $j_0(z):[0,h]\rightarrow {\mathbb {R}}$ such that for all time,

$$\begin{aligned} \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\Psi (t)\cdot \nu _s \,\mathrm{d}\omega = j_0(z). \end{aligned}$$

These two conditions differentiate the model we derive from closely related models which have been studied recently by Constantin and Nguyen [11, 12], Constantin and Ignatova [8, 9], and Constantin, Ignatova, and Nguyen [10]. While we shall explain this distinction in detail in Section 1.3, we first describe a rough sketch of our existence proof, and then state our main results.

In [22] and [18], the authors used the observation that the transport equations for ${\mathcal {L}}(\Psi )$ and $\partial _{\nu } \Psi$ in (QG) formally preserve the norms of the data for an elliptic problem with Neumann boundary condition. Therefore, a sequence of approximate solutions $\Psi _n$ for which ${\mathcal {L}}(\Psi _n)$ and $\partial _{\nu }\Psi _n$ converge weakly in (respectively) $L^\infty _t(L^2(\Omega \times [0,h]))$ and $L^\infty _t(L^2(\Omega \times \{0,h\}))$ will have strong convergence for $\nabla \Psi _n$ in $L_t^\infty (L^2(\Omega \times [0,h]))$. A key property of the (QG) system is a reformulation of the system in terms of $\nabla \Psi$. This reformulation, first utilized extensively by Puel and the second author in [22], can be seen at the level of the primitive equations and draws an analogy to the parallel formulations of the three dimensional Euler equations in terms of the velocity and the vorticity. Unlike Euler, however, the strong convergence then allows one to pass to the limit at the level of $\nabla \Psi _n$ to construct a weak solution. In the setting of the bounded domain $\Omega \times [0,h]$, it is not immediate that imposing (1) and (2) on the lateral boundary will allow for compactness at the level of $\nabla \Psi _n$ in $L_t^\infty (L^2(\Omega \times [0,h]))$. Indeed, it might seem possible that because (2) only controls the average of ${\overline{\nabla }}\Psi \cdot \nu _s$ on the sides, ${\overline{\nabla }}\Psi \cdot \nu _s$ could oscillate quite badly on $\partial \Omega \times [0,h]$. To address this, we must formulate (2) weakly (see Definition 3.1 in Section 3). However, we also prove an elliptic regularity theorem (Theorem 3.2) which implies that in fact ${\overline{\nabla }}\Psi \cdot \nu _s \in L^2(\partial \Omega \times [0,h])$ is well-defined pointwise, and $\nabla \Psi _n$ converges strongly to $\nabla \Psi$ in $L_t^\infty (L^2(\Omega \times [0,h]))$. To the authors’ knowledge, this type of boundary condition and the corresponding elliptic regularity theorem are novel.

1.2 Main Result

Before stating the existence theorem, we must provide several definitions. The first is a natural compatibility condition between the elliptic operator and boundary conditions.

Definition 1.1

Any triple (f, g, j) of functions with $f(x,y,z)\in L^2(\Omega \times [0,h])$, $g(x,y,z)\in L^2(\Omega \times \{0,h\})$, $j(z)\in L^2(0,h)$ is compatible if

$$\begin{aligned} \int _{\Omega \times [0,h]}f(x,y,z) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z = \int _0^h j(z)\,\mathrm{d}z + \int _{\Omega \times \{0,h\}} \lambda (z)g(x,y,z) \,\mathrm{d}x\,\mathrm{d}y. \end{aligned}$$

A pair $(a_L, a_\nu )$ of forcing terms is compatible if $a_L \in L^1\left( [0,T]; L^2(\Omega \times [0,h])\right)$, and $a_\nu \in L^1\left( [0,T]; L^2(\Omega \times \{0,h\})\right)$ for all $T>0$ with

$$\begin{aligned} \int _{\Omega \times [0,h]}a_{L}(x,y,z)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z = \int _{\Omega \times \{0,h\}}\lambda (z) a_\nu (x,y,z) \,\mathrm{d}x\,\mathrm{d}y. \end{aligned}$$

Next, we define the notion of weak solutions to the transport equations in (QG).

Definition 1.2

Let $T>0$ be given and $\Psi (t,x,y,z):[0,T]\times \Omega \times [0,h]\rightarrow {\mathbb {R}}$ be such that $\nabla \Psi , {\mathcal {L}}(\Psi )\in L^\infty ([0,T];L^2(\Omega \times [0,h]))$, $\partial _\nu \Psi \in L^\infty \left( [0,T];L^2(\Omega \times \{0,h\}) \right)$. Then $\Psi$ is a weak solution to the transport equations in (QG) on [0, T] with initial data $f_0$ and $g_0$ and forcing $a_L$, $a_\nu$ if for all ${\tilde{\Omega }}$ compactly contained in $\Omega$ and smooth test functions $\phi (t,x,y,z)$ compactly supported in $[-1,T)\times {\tilde{\Omega }}\times [-1,h+1]$

$$\begin{aligned}&-\int _0^T \int _{{\tilde{\Omega }}\times [0,h]} \left( \left( \partial _t \phi + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\phi \right) \left( {\mathcal {L}}(\Psi ) + \beta _0 y \right) + \phi a_L \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t \\&\quad = \int _{{\tilde{\Omega }}\times [0,h]} \phi |_{t=0}f \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \end{aligned}$$

and

$$\begin{aligned}&\int _0^T \int _{{\tilde{\Omega }}\times \{0,h\}} \left( \left( \partial _t \phi + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\phi \right) \partial _{\nu }\Psi + \phi a_\nu \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}t \\&\quad = -\int _{{\tilde{\Omega }}\times \{0,h\}} \phi |_{t=0}g \,\mathrm{d}x\,\mathrm{d}y. \end{aligned}$$

We can now state our existence result.

Theorem 1.1

Let $(f_0,g_0,j_0)$ and $(a_L,a_\nu )$ satisfy Definition 1.1. Then there exists a global weak solution $\Psi$ to (QG) such that

• (1) ${\mathcal {L}}(\Psi )|_{t=0} = f_0$, $\partial _{\nu } \Psi |_{t=0}=g_0$ and $\Psi$ satisfies Definition 1.2 for any $T>0$;

• (2) There exists c(t, z) such that for almost every time $t>0$, $\Psi (t)|_{\partial \Omega \times [0,h]} = c(t,z)$;

• (3) For all $t>0$, ${\overline{\nabla }}\Psi (t)\cdot \nu _s \in L^2(\partial \Omega \times [0,h])$. If $j_0\in H^\frac{1}{2}(0,h)$, then

  $$\begin{aligned} \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\Psi (t)\cdot \nu _s \,\mathrm{d}\omega = j_0(z), \end{aligned}$$

  with the equality holding pointwise in z;

• (4) For all time t, $\left( {\mathcal {L}}(\Psi )(t), \partial _{\nu } \Psi (t), {\overline{\nabla }}\Psi \cdot \nu _s(t)\right)$ satisfies the compatibility condition in Definition 1.1;

• (5) For all $T>0$ and $t\in [0,T]$, $\Psi$ satisfies the bound

  $$\begin{aligned}&\Vert {\mathcal {L}}(\Psi )(t) \Vert _{L^2(\Omega \times [0,h])} + \Vert \partial _{\nu }\Psi (t) \Vert _{L^2(\Omega \times \{0,h\})} + \Vert \nabla \Psi (t) \Vert _{H^\frac{1}{2}(\Omega \times [0,h])} \\&\quad \leqq C(\Omega ,h,\lambda ) \left( \Vert f \Vert _{L^2}+ \Vert g \Vert _{L^2} + \Vert j \Vert _{L^2} + \Vert a_L \Vert _{L^1\left( [0,T];L^2\right) } \right. \\&\left. \qquad +\, \Vert a_\nu \Vert _{L^1\left( [0,T];L^2\right) } \right) . \end{aligned}$$

1.3 Inviscid Geostrophic Flows

Mathematical inquiry into (QG) is by now quite extensive. Beale and Bourgeois [3] and Desjardins and Grenier [13] provided derivations of the three dimensional system from primitive equations. As we are concerned with the inviscid model, our derivation follows that of Beale and Bourgeois. Puel and the second author [22] proved global existence results for initial data $\Psi _0$ such that ${\mathcal {L}}(\Psi _0), \nabla \Psi _0 \in L^2({\mathbb {R}}^3_+)$, $\partial _{\nu } \Psi _0 \in L^2({\mathbb {R}}^2)$. The first author [18] extended this result to initial data belonging to non-Hilbert Lebesgue spaces and identified the critical regularity at which the system conserves energy. Conversely, the first author showed non-uniqueness of dissipative weak solutions for the model posed on the torus ${\mathbb {T}}^3$ [19], albeit in a regularity regime which leaves open the appropriate Onsager-type conjecture.

Study of the closely related surface quasi-geostrophic equation was initiated by Constantin et al. [6]. To obtain SQG from (QG), one simplifies the model by assuming that $\lambda (z) \equiv 1$, $\beta _0 =0$, $a_L \equiv a_\nu \equiv 0$, and

$$\begin{aligned} \Delta \Psi |_{t=0}=0. \end{aligned}$$

As a result, $\Delta \Psi (t)\equiv 0$ uniformly in time, and the entire dynamic is encoded in the equation for $\theta = -\partial _z\Psi |_{z=0} = (-{\overline{\Delta }})^\frac{1}{2}\Psi$:

$$\begin{aligned} \partial _t \theta + {\mathcal {R}}^\perp \theta \cdot {\overline{\nabla }}\theta = 0. \end{aligned}$$

(3)

Resnick proved global existence of weak solutions for initial data in $L^2({\mathbb {T}}^2)$ [23]. Marchand extended Resnick’s result to initial data belonging to $L^p({\mathbb {R}}^2)$ or $L^p({\mathbb {T}}^2)$ for $p>\frac{4}{3}$ [17]. Both the proofs of Resnick and Marchand are based on a reformulation of the nonlinear term using a Cálderon commutator.

To study (3), it is common to add a dissipative term $(-{\overline{\Delta }})^\alpha \theta$. The case $\alpha = \frac{1}{2}$ is physical and comes from considering viscous effects which produce Ekman layers at the boundary. In the critical case $\alpha = \frac{1}{2}$, global regularity is known by different methods. Proofs are given by Kiselev et al. [16], Caffarelli and the second author [5], Constantin and Vicol [7], and Kiselev and Nazarov [15]. Using the De Giorgi technique from [5] in combination with a bootstrapping argument and an appropriate Beale–Kato–Majda type criterion, the authors established global regularity for the full three dimensional system with critical dissipation in [21]. Buckmaster, Shkoller, and Vicol used the method of convex integration to show that one may prescribe any positive smooth profile for the Hamiltonians of both inviscid and dissipative SQG [4].

The techniques used to produce weak solutions by Resnick and Marchand were adapted to bounded domains in a series of papers. In these works, the Riesz transform on a bounded domain $\Omega$ is defined spectrally using eigenfunctions of the homogenous Dirichlet laplacian. First, Constantin and Ignatova [8, 9] proved nonlinear bounds and commutator estimates for the fractional laplacian and showed the existence of global weak solutions as well as derived interior regularity estimates for (3) with added critical dissipation in bounded domains. Constantin and Nguyen [11, 12] then showed the existence of global weak solutions of (3) in bounded domains as well as local and global strong solutions for supercritical and critical/subcritical versions of (3), respectively. In the paper [10], Constantin, Ignatova, and Nguyen treat the inviscid limit. The weak solutions we construct cannot coincide in general with solutions to (3) constructed using the spectral Riesz transform. The difference lies in the boundary conditions (1) and (2). We emphasize that the discrepancies between models are not related to the relatively low regularity of weak solutions, as we built classical solutions on a short time interval to our model in the recent paper [20]. At each time t, we reconstruct $\Psi$ by solving the elliptic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}(\Psi ) = f &{}\quad \Omega \times [0,h] \\ \partial _{\nu } \Psi = g &{}\quad \Omega \times \{0,h\}\\ \Psi (x,y,z) = c(z) \qquad &{}\quad \partial \Omega \times \{z>0\}\\ \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\Psi \cdot \nu _s =j_0(z) &{}\quad [0,h]. \end{array}\right. } \end{aligned}$$

In particular, we do not require that the stream function $\Psi$ vanishes uniformly on the lateral boundary. While we consider the case of finite height h, the boundary conditions we impose would apply in the case of infinite height as well, which is the most common setting for SQG.

Conversely, let $\{ e_n \}$ be the orthonormal basis of eigenfunctions with corresponding eigenvalues $\{\lambda _n\}$ for the homogenous Dirichlet laplacian $-{\overline{\Delta }}_\Omega$ on $\Omega$, and let

$$\begin{aligned} \theta = \sum _n a_n(t) e_n(x,y) \end{aligned}$$

be a solution to (3) posed on the bounded domain $\Omega$. Then the stream function $\Psi |_{z=0}$ is given by

$$\begin{aligned} \Psi |_{z=0} = \left( -{\overline{\Delta }}_\Omega \right) ^{-\frac{1}{2}} \theta = \sum _n a_n(t) \lambda _n^{-\frac{1}{2}} e_n(x,y), \end{aligned}$$

and the harmonic extension for $z\in [0,\infty )$ is given by

$$\begin{aligned} \Psi (t,x,y,z) = \sum _n a_n(t) e^{-z\sqrt{\lambda _n}} \lambda _n^{-\frac{1}{2}} e_n(x,y). \end{aligned}$$

With this definition, $\Psi$ vanishes uniformly on $\partial \Omega \times [0,\infty )$. In addition, if one were to impose (2) on a solution to (3), then integrating by parts in (x, y) and passing the integral inside the sum gives

$$\begin{aligned} \sum _n a'_n(t) e^{-z\sqrt{\lambda _n}} \lambda _n^\frac{1}{2} \left( \int _{\Omega } e_n(x,y) \,\mathrm{d}x\,\mathrm{d}y \right) =0 \end{aligned}$$

for all $z>0$. One can see that this is only satisfied if

$$\begin{aligned} a'_n(t) \left( \int _{\Omega } e_n(x,y) \,\mathrm{d}x\,\mathrm{d}y \right) =0 \end{aligned}$$

for all n and $t>0$, which cannot hold for any bounded domain $\Omega$ and initial data. The outline of this paper is as follows: in Section 2, we recall the derivation of the system from primitive equations while accounting for the impermeability. In Section 3, we produce a solution to the stationary elliptic problem associated to the operator ${\mathcal {L}}$ and prove an elliptic regularity theorem for the solution. Finally, in Section 4, we construct global weak solutions to (QG).

2 Derivation from Primitive Equations

2.1 Primitive Equations and Re-Scalings

We begin from the so-called primitive equations following the derivation of Bourgeois and Beale [3]. These equations represent the geostrophic balance, which is the balance of the pressure gradient with the Coriolis force. The Boussinesq approximation has been made; that is, changes in density are ignored except when amplified by the effect of gravity. After a re-scaling of the equations, a parameter which varies inversely with the speed of the rotation of the earth called the Rossby number shall appear. Then performing a perturbation expansion in the Rossby number $\varepsilon$ will yield the stratified system and boundary conditions (1) and (2). Given a smooth, bounded set $\Omega \subset {\mathbb {R}}^2$ and a fixed height h, the following equations (after rescaling) will be posed on the cylindrical domain

$$\begin{aligned} \Omega \times [0,h] . \end{aligned}$$

We use the notation $\frac{\mathrm{D}}{\mathrm{D}t}=\partial _t + \vec {u}\cdot \nabla$ for the material derivative, and the Coriolis force ${\mathcal {C}}=2\Theta \sin (\theta )$, where $\Theta$ is the angular velocity of the Earth and $\theta$ is the latitude. Here (u, v, w) is the fluid velocity, p is the pressure and $\rho$ is the variation in density from a known background density profile ${\bar{\varrho }}(z)$. That is, the density $\varrho$ satisfies

$$\begin{aligned} \varrho = {\bar{\varrho }}(z) + \rho (x,y,z,t). \end{aligned}$$

We further assume that the density is decreasing in z and that $-\rho _z$ is bounded above and below away from zero. Throughout, we assume throughout that the fluid velocity is tangent to the boundary.

The primitive equations then are

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\mathrm{D}u}{\mathrm{D}t} - {\mathcal {C}}v = -p_x \\ \frac{\mathrm{D}v}{\mathrm{D}t} + {\mathcal {C}}u = -p_y \\ \frac{\mathrm{D}w}{\mathrm{D}t} +\rho g = -p_z \\ \nabla \cdot u =0 \\ \frac{\mathrm{D}\varrho }{\mathrm{D}t} = 0. \\ \end{array}\right. } \end{aligned}$$

We rescale the equations in such a way so as to remove solutions which vary on a fast time scale. Therefore, we set

$$\begin{aligned} t=\frac{L}{U}t', \quad u=Uu', \quad (x,y,z)=L(x',y',z'). \end{aligned}$$

Letting $\theta _0$ be a central latitude, we estimate ${\mathcal {C}}$ using the linear $\beta$-plane approximation by

$$\begin{aligned}{\mathcal {C}}=2\Theta \sin (\theta _0) + 2\Theta \cos (\theta _0)(\theta -\theta _0):={\mathcal {C}}_0 + 2\Theta \cos (\theta _0)(\theta -\theta _0).\end{aligned}$$

The Rossby number $\varepsilon$ is equal to $\frac{U}{{\mathcal {C}}_0 L}$. Set $\beta _0 = \frac{\cot (\theta _0)}{\varepsilon }\frac{L}{r_0}$. We then have that

$$\begin{aligned} {\mathcal {C}}&=2\Theta \sin (\theta _0) + 2\Theta \cos (\theta _0)(\theta -\theta _0) \\&= {\mathcal {C}}_0(1+\varepsilon \beta _0 y'). \end{aligned}$$

We assume that $\frac{L}{r_0}$ is $O(\varepsilon )$, allowing us to keep the factor of $\varepsilon$ in front of $\beta _0$ even as $\varepsilon \rightarrow 0$. We scale the density variation by

$$\begin{aligned} \rho = \frac{{\mathcal {C}}_0 U}{g} \rho ' = \frac{U^2}{\varepsilon L g} \rho ' \end{aligned}$$

and the reference density by

$$\begin{aligned} {\bar{\varrho }} = \frac{U^2}{\varepsilon ^2 L g} {\bar{\varrho }}' \ . \end{aligned}$$

This allows us to write the density non-dimensionally as

$$\begin{aligned} \varrho = \frac{U^2}{\varepsilon ^2 L g}({\bar{\varrho }}'(z)+ \varepsilon \rho ' ). \end{aligned}$$

Finally, we scale the pressure by $p= {\mathcal {C}}_0 UL p'$. Applying the scalings to the primitive equations, we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\mathrm{D}u'}{\mathrm{D}t'} - \frac{1}{\varepsilon }(1+\epsilon \beta _0 y')v' = -\frac{1}{\varepsilon }p'_{x'} \\ \frac{\mathrm{D}v'}{\mathrm{D}t'} + \frac{1}{\varepsilon }(1+\epsilon \beta _0 y')u' = -\frac{1}{\varepsilon }p'_{y'} \\ \frac{\mathrm{D}w'}{\mathrm{D}t'} +\frac{1}{\varepsilon }\rho ' = -\frac{1}{\varepsilon }p'_{z'} \\ \nabla \cdot u' =0 \\ \frac{\mathrm{D}\rho '}{\mathrm{D}t'} + \frac{1}{\varepsilon } w'{\bar{\varrho }}'_{z'} = 0 .\\ \end{array}\right. } \end{aligned}$$

Let us abuse notation and drop the primes on our scaled equations. Assume that the expansions

$$\begin{aligned} \vec {u} = \vec {u}(\varepsilon ) = \vec {u}^{(0)} + \varepsilon \vec {u}^{(1)} + O(\varepsilon ^2) \end{aligned}$$

and

$$\begin{aligned} \rho = \rho (\varepsilon ) = \rho ^{(0)} + \varepsilon \rho ^{(1)} + O(\varepsilon ^2) \end{aligned}$$

hold. Plugging this ansatz in, we obtain the zero-order equations

$$\begin{aligned} v^{(0)} = p_x^{(0)}, \quad u^{(0)} = -p_y^{(0)}, \quad \rho ^{(0)} = -p_z^{(0)}, \quad w^{(0)}=0. \end{aligned}$$

The last equation follows from the first two equations, the incompressibility (which gives that $w_z^{(0)}=0$), and the assumption that $w^{(0)}\equiv 0$ on the top and bottom of the domain.

We move now to the first order equations. Let us introduce the notation

$$\begin{aligned} d_g = \partial _t - p_y^{(0)} \frac{\partial }{\partial _x} + p_x^{(0)} \frac{\partial }{\partial _y} \end{aligned}$$

for the zero order geostrophic material derivative. The first order equations are then

$$\begin{aligned} {\left\{ \begin{array}{ll} d_g(-p_y^{(0)}) - v^{(1)} - \beta _0 y p_x^{(0)} = -p_{x}^{(1)} \\ d_g(p_x^{(0)}) + u^{(1)} - \beta _0 y p_y^{(0)} = -p_{y}^{(1)} \\ \rho ^{(1)} = -p_{z}^{(1)} \\ \nabla \cdot u^{(1)}=0 \\ d_g(-p_z^{(0)}) + w^{(1)}\varrho _{z} = 0 . \end{array}\right. } \end{aligned}$$

Let us divide the last equation by $-\frac{1}{\varrho _z}$. We introduce the notation

$$\begin{aligned} {\tilde{\nabla }} = \left( \partial _x, \partial _y, -\frac{1}{\varrho _z}\partial _z\right) . \end{aligned}$$

Then we can consolidate the first order equations as

$$\begin{aligned}&d_g({\tilde{\nabla }}p^{(0)}) + \beta _0(p^{(0)},0,0)^t = (-p_y^{(1)}, p_x^{(1)},0)^t - (u^{(1)},v^{(1)}, w^{(1)})^t \nonumber \\&\quad -\, \beta _0y(-p_y^{(0)}, p_x^{(0)}, 0)^t+ \beta _0(p^{(0)},0,0)^t. \end{aligned}$$

(4)

Note that the right-hand side is divergence free and has no vertical component on the top and bottom boundaries of the domain.

2.2 Transporting ${\mathcal {L}}(\Psi )$ and $\partial _{\nu } \Psi$

We now take the divergence of (4) in order to arrive at (QG). As noted, the divergence of the right hand side is zero. The divergence of $\beta _0(p^{(0)},0,0)^t$ is $\beta _0 p_x^{(0)}$. Examining the transport term $d_g({\tilde{\nabla }}p^{(0)})$ and calculating $\partial _z$ of the third component, we obtain

$$\begin{aligned} d_g(\partial _z(e_3 \cdot {\tilde{\nabla }}p^{(0)}))+ \partial _z u^{(0)}\partial _x (e_3 \cdot {\tilde{\nabla }}p^{(0)}) + \partial _z v^{(0)}\partial _y (e_3 \cdot {\tilde{\nabla }}p^{(0)}). \end{aligned}$$

Using the fact that $u^{(0)}=-p_y^{(0)}$ and $v^{(0)}=p_x^{(0)}$, the second two terms cancel each other out. The horizontal divergence $(\partial _x, \partial _y, 0)$ of $d_g({\tilde{\nabla }}p^{(0)})$ is easy to calculate from the stratification and the divergence free nature of the zero-order flow. We arrive at the equation

$$\begin{aligned} \left( \partial _t - p_y^{(0)}\partial _x + p_x^{(0)}\partial _y \right) \left( p_{xx}^{(0)} + p_{yy}^{(0)} + (\lambda p_z^{(0)})_z + \beta _0 y \right) = 0 \end{aligned}$$

after absorbing the $\beta$-plane term into the material derivative and defining $\lambda = -\frac{1}{\varrho _z}$. Note that by the assumptions on the density, there exists $\Lambda$ such that $\frac{1}{\Lambda } \leqq \lambda \leqq \Lambda$. We shall use the notation $\Psi$ for the stream function $p^{(0)}$, allowing us to rewrite the system in the familiar form

$$\begin{aligned} \left( \partial _t + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\right) \left( {\mathcal {L}}(\Psi ) + \beta _0 y \right) =0 . \end{aligned}$$

(5)

Consider now the top and bottom $\Omega \times \{0\}$ and $\Omega \times \{h\}$. Let $\nu$ denote the unit normal vector on the top and bottom. Considering the equation

$$\begin{aligned} d_g(-p_z^{(0)}) + w^{(1)}\varrho _{z} = 0, \end{aligned}$$

using that $w^{(1)}\equiv 0$ on the top and bottom, and substituting the notation $\Psi$ for the stream function, we obtain

$$\begin{aligned} \left( \partial _t + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\right) (\partial _{\nu }\Psi ) =0. \end{aligned}$$

(6)

2.3 The Lateral Boundary

Now consider the sides $\partial \Omega \times [0,h]$ equipped with a horizontal normal vector $\nu _s$. First, the impermeability requires that ${\overline{\nabla }}^\perp p^{(0)}\cdot \nu _s = 0$, implying that $p^{(0)}$ is constant on $\partial \Omega \times \{z\}$. Recalling that the stream function $\Psi =p^{(0)}$, we have that

$$\begin{aligned} \Psi (t,y,x,z)|_{\{\partial \Omega \times [0,h]\}}=c(t,z) \end{aligned}$$

(7)

for some unknown function c(t, z).

Let us next take the dot product of (4) with $\nu _s$. Due to the impermeability of the boundary,

$$\begin{aligned} (u^{(1)}, v^{(1)}, w^{(1)})^t \cdot \nu _s =0. \end{aligned}$$

In addition,

$$\begin{aligned} (-p_y^{(1)}, p_x^{(1)},0)^t \cdot \nu _s = -(p_x^{(1)}, p_y^{(1)},0)^t \cdot \tau , \end{aligned}$$

where $\tau$ is the positively oriented tangent vector perpendicular to $\nu _s$. Then we integrate around the boundary $\partial \Omega \times \{z\} \subset \partial \Omega \times [0,h]$ at a fixed height z. Since $(p_x^{(1)}, p_y^{(1)},0)^t$ is a conservative vector field,

$$\begin{aligned} \int _{\partial \Omega \times \{z\}} (p_x^{(1)}, p_y^{(1)},0)^t \cdot \tau \,\mathrm{d}\omega = 0 . \end{aligned}$$

Notice that

$$\begin{aligned} \beta _0y(-p_y^{(0)}, p_x^{(0)}, 0)^t - \beta _0(p^{(0)},0,0)^t \end{aligned}$$

is also the two-dimensional curl ${\overline{\nabla }}^\perp$ of the scalar field $-\beta _0 yp^{(0)}$. Then we have that

$$\begin{aligned} {\overline{\nabla }}^\perp (-\beta _0 yp^{(0)}) \cdot \nu _s = {\overline{\nabla }}(\beta _0 yp^{(0)}) \cdot \tau . \end{aligned}$$

As this is also a conservative vector field, the integral of this term around the boundary vanishes as well. Thus we are left with

$$\begin{aligned} \int _{\partial \Omega \times \{z\}} (d_g {\tilde{\nabla }}p^{(0)})\cdot \nu _s \,\mathrm{d}\omega = -\int _{\partial \Omega \times \{z\}} (\beta _0 p^{(0)},0,0)\cdot \nu _s \,\mathrm{d}\omega . \end{aligned}$$

(8)

Using (7) shows that

$$\begin{aligned} -\int _{\partial \Omega \times \{z\}} (\beta _0 p^{(0)},0,0)\cdot \nu _s \,\mathrm{d}\omega \end{aligned}$$

is zero. Substituting in the stream function notation and applying the divergence theorem to the nonlinear term on the left hand side of (8), we have that

$$\begin{aligned}&\int _{\partial \Omega \times \{z\}} \left( -p_y^{(0)}\partial _x {\overline{\nabla }}p^{(0)} + p_x^{(0)}\partial _y {\overline{\nabla }}p^{(0)}\right) \cdot \nu _s \,\mathrm{d}\omega \\&\quad = \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\cdot \left( {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}{\overline{\nabla }}\Psi \right) \cdot \nu _s \,\mathrm{d}\omega \\&\quad = \int _{\Omega \times \{z\}} {\overline{\nabla }}{\overline{\nabla }}^\perp \Psi : {\overline{\nabla }}{\overline{\nabla }}\Psi \,\mathrm{d}x\,\mathrm{d}y + \int _{\Omega \times \{z\}} {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}{\overline{\Delta }}\Psi \,\mathrm{d}x\,\mathrm{d}y \\&\quad =\int _{\Omega \times \{z\}} {\overline{\nabla }}\cdot \left( {\overline{\nabla }}^\perp \Psi {\overline{\Delta }}\Psi \right) \,\mathrm{d}x\,\mathrm{d}y \\&\quad = \int _{\partial \Omega \times \{z\}} {\overline{\Delta }}\Psi \left( {\overline{\nabla }}^\perp \Psi \cdot \nu _s\right) \,\mathrm{d}\omega \\&\quad =0. \end{aligned}$$

Utilizing once again the notation $\Psi$ for the stream function, (8) therefore becomes

$$\begin{aligned} \frac{\partial }{\partial t}\int _{\partial \Omega \times \{z\}} ( {\overline{\nabla }}\Psi )\cdot \nu _s \,\mathrm{d}\omega =0. \end{aligned}$$

(9)

Collecting (5), (6), (7), and (9), we have formally derived the following system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \left( \partial _t + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\right) \left( {\mathcal {L}}(\Psi ) + \beta _0 y \right) =0 &{}\quad \Omega \times [0,h] \\ \left( \partial _t + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\right) (\partial _{\nu }\Psi ) =0 &{}\quad \Omega \times \{0,h\}\\ \frac{\partial }{\partial t}\int _{\partial \Omega \times \{z\}} ( {\overline{\nabla }}\Psi )\cdot \nu _s \,\mathrm{d}\omega =0 &{}\quad [0,h]\\ \Psi =c(t,z) &{}\quad \partial \Omega \times [0,h]. \end{array}\right. } \end{aligned}$$

3 The Elliptic Problem

3.1 Building a solution in $L^2$

In order to show global existence of weak solutions to the time-dependent problem, we first solve the stationary elliptic problem which is transported by the fluid velocity ${\overline{\nabla }}^\perp \Psi$. The elliptic operator is given by ${\mathcal {L}}$. The boundary conditions for the elliptic problem will be mixed in nature. We first impose a Neumann condition on the top and bottom of $\Omega \times [0,h]$ coming from the transport equation for $\partial _{\nu } \Psi$. The condition that

$$\begin{aligned} \Psi (t,x,y,z)|_{\partial \Omega \times [0,h]}=c(t,z) \end{aligned}$$

will be structured into the Hilbert space within which we solve the elliptic problem. Finally, the equation

$$\begin{aligned} \frac{\partial }{\partial t}\int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\Psi \cdot \nu _s \,\mathrm{d}\omega =0 \end{aligned}$$

means that

$$\begin{aligned} \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\Psi (t)\cdot \nu _s \,\mathrm{d}\omega =\int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\Psi (0)\cdot \nu _s \,\mathrm{d}\omega =:j(z) \end{aligned}$$

is determined from the initial data, and thus will be incorporated into the data of the elliptic problem. We now provide a weak formulation of this condition for (QG).

Definition 3.1

Let $T>0$ be given and $\Psi (t,x,y,z):[0,T]\times \Omega \times [0,h]$ be such that $\nabla \Psi , {\mathcal {L}}(\Psi )\in L^\infty ([0,T];L^2(\Omega \times [0,h]))$, and for each time, $\Psi$ has mean value zero. Then we say that $\Psi$ satisfies (2) weakly if there exists $j_0(z):[0,h]\rightarrow {\mathbb {R}}$ such that for each compactly supported smooth function $\phi (t,z): (0,T)\times (0,h)\rightarrow {\mathbb {R}}$,

$$\begin{aligned}&\int _0^T\int _{\Omega \times [0,h]} {\mathcal {L}}(\Psi ) \phi (t,z) - \Psi \partial _z \left( \lambda \partial _z \phi (t,z) \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t\\&\quad = \int _0^T\int _0^h \phi (t,z) j_0(z) \,\mathrm{d}z\,\mathrm{d}t. \end{aligned}$$

An integration by parts shows that for smooth functions of time and space, (2) is equivalent to Definition 3.1. Indeed,

$$\begin{aligned}&\int _0^T\int _{\Omega \times [0,h]} {\mathcal {L}}(\Psi ) \phi (t,z) - \Psi \partial _z \left( \lambda \partial _z \phi (t,z) \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t\\&\quad = \int _0^T\int _{\Omega \times [0,h]} {\mathcal {L}}(\Psi ) \phi (t,z) - \partial _z \left( \lambda \partial _z \Psi \right) \phi (t,z) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t\\&\quad = \int _0^T \int _{\Omega \times [0,h]} \left( \partial _{xx}\Psi + \partial _{yy}\Psi \right) \phi (t,z) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t\\&\quad = \int _0^T \int _0^h \int _{\partial \Omega } \phi (t,z) {\overline{\nabla }}\Psi \cdot \nu _s \,\mathrm{d}\omega \,\mathrm{d}z\,\mathrm{d}t. \end{aligned}$$

Thus we consider the elliptic problem for the unknown function u with data $f:\Omega \times [0,h] \rightarrow {\mathbb {R}}$, $g:\Omega \times \{0,h\}\rightarrow {\mathbb {R}}$, and $j:[0,h]\rightarrow {\mathbb {R}}$.

Let us remark that to formulate (E) variationally, it is not necessary for (f, g, j) to satisfy the compatibility condition Definition 1.1. Indeed our construction of approximate solutions will introduce a small error in the condition of Definition 1.1 which will vanish in the limit. Thus when we say that u is a solution to (E), we generally mean it in the variational sense of (V) (see (12) below). If in addition, (f, g, j) satisifes the compatibility condition so that (V) is equivalent to (E), we shall make note of this. To solve (V) we require a specially constructed Hilbert space.

Definition 3.2

Define H by

$$\begin{aligned} H:= & {} \{ {\nabla }\alpha \in C^\infty \left( {\bar{\Omega }}\times [0,h]\right) : \quad \int _{\Omega \times [0,h]}\alpha \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z=0, \quad \\&\alpha |_{\partial \Omega \times [0,h]}(x,y,z)=\alpha (z) \}. \end{aligned}$$

Using the notation ${\tilde{\nabla }}=(\partial _x,\partial _y, \lambda (z) \partial _z)$, equip H with the inner product

$$\begin{aligned} \langle {\nabla }\alpha , {\nabla }\gamma \rangle _{\mathbb {H}}:= \int _{\Omega \times [0,h]} {\tilde{\nabla }} \alpha \cdot {\nabla }{\gamma } \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z. \end{aligned}$$

Define the Hilbert space ${\mathbb {H}}$ as the closure of H under the norm induced by this inner product.

We remark that we have defined ${\mathbb {H}}$ as the vector space whose elements are gradients of mean zero functions. When there is no risk of confusion, however, we shall freely identify $\nabla u \in {\mathbb {H}}$ with u or ${\tilde{\nabla }} u$ and write $\langle u,v \rangle _{\mathbb {H}}$ or $\Vert u \Vert _{\mathbb {H}}$ instead of $\langle \nabla u,\nabla v \rangle _{\mathbb {H}}$ or $\Vert \nabla u \Vert _{\mathbb {H}}$, respectively. The precision is only needed for an application of the Aubin-Lions lemma in Lemmas 4.2 and 4.3.

By standard trace inequalities and Poincaré’s inequality, we have that for ${\tilde{\nabla }}\gamma \in {\mathbb {H}}$

$$\begin{aligned} \Vert \gamma \Vert _{H^\frac{1}{2}(\partial (\Omega \times [0,h]))}&\leqq C(\Omega ,h) \left( \Vert \gamma \Vert _{L^2(\Omega \times [0,h])} + \Vert \nabla \gamma \Vert _{L^2(\Omega \times [0,h])} \right) \nonumber \\&\leqq C(\Omega ,h,\lambda ) \Vert \gamma \Vert _{\mathbb {H}}. \end{aligned}$$

(10)

We define a bilinear form $B(\alpha ,\gamma ):{\mathbb {H}}\times {\mathbb {H}}\rightarrow {\mathbb {R}}$ and functional $F(\gamma ):{\mathbb {H}}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} B(\alpha ,\gamma ) = \int _{{\Omega }\times [0,h]} {\tilde{\nabla }} \alpha \cdot {\nabla } \gamma \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \end{aligned}$$

and

$$\begin{aligned} F(\gamma ) = -\int _{{\Omega \times [0,h]}}{f \gamma }\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z + \int _{\Omega \times \{0,h\}} \lambda g \gamma \,\mathrm{d}x\,\mathrm{d}y + \int _0^h j(z) \gamma |_{\partial _\Omega \times \{z\}}\,\mathrm{d}z. \end{aligned}$$

The coercivity and continuity of the bilinear form B is immediate from the assumptions on $\lambda (z)$ and the definition of ${\mathbb {H}}$. In addition, we have that

$$\begin{aligned} |F(\gamma )|&\leqq \Vert f\Vert _{L^2(\Omega \times [0,h])} \Vert \gamma \Vert _{L^2(\Omega \times [0,h])} + \Vert \lambda \Vert _{L^\infty (0,h)} \Vert g \Vert _{L^2(\Omega \times \{0,h \})} \Vert \gamma \Vert _{L^2(\Omega \times \{0,h\})} \nonumber \\&\quad +\, \Vert j \Vert _{\left( H^\frac{1}{2}(\partial \Omega \times [0,h])\right) ^*} \Vert \gamma \Vert _{H^\frac{1}{2}(\partial \Omega \times [0,h])}\nonumber \\&\leqq C(\Omega ,h,\lambda ) \left( \Vert f \Vert _{L^2} + \Vert g \Vert _{L^2} + \Vert j \Vert _{\left( H^{\frac{1}{2}}\right) ^*} \right) \Vert \gamma \Vert _{\mathbb {H}}, \end{aligned}$$

(11)

after applying Hölder’s inequality and (10). Applying the Lax-Milgram theorem, we obtain a unique solution $u\in {\mathbb {H}}$ to the variational problem

$$\begin{aligned} B(u, \gamma ) = F(\gamma ) \quad \forall \gamma \in {\mathbb {H}}. \quad (V). \end{aligned}$$

(12)

Let us rigorously state the results of the above argument.

Lemma 3.1

For any data $f\in L^2(\Omega \times [0,h])$, $g \in L^2(\Omega \times \{0,h\})$, and $j \in \left( H^{\frac{1}{2}}([0,h])\right) ^*$ there exists a unique solution $u \in {\mathbb {H}}$ to the variational problem (V) with

$$\begin{aligned} \Vert u\Vert _{{\mathbb {H}}} \leqq C(\Omega ,h,\lambda ) \left( \Vert f \Vert _{L^2} + \Vert g \Vert _{L^2} + \Vert j \Vert _{\left( H^{\frac{1}{2}}\right) ^*} \right) . \end{aligned}$$

If in addition (f, g, j) verifies the compatibility condition in Definition 1.1, then

• (1) (E1) is satisfied in the weak sense;

• (2) (E2) is satisfied in the weak sense;

• (3) (E3) is satisfied pointwise;

• (4) (E4) is satisfied weakly. That is, for $\phi \in C^\infty$ depending only on z,

  $$\begin{aligned}\int _{\Omega \times [0,h]} {\mathcal {L}}(u) \phi (z) - u \partial _z \left( \lambda \partial _z \phi (z) \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z = \langle j, \phi \rangle \end{aligned}$$

  where $\langle \cdot , \cdot \rangle$ denotes duality between $\left( H^\frac{1}{2}\right) ^*$ and $H^\frac{1}{2}$.

Proof

The first claim is simply the above construction of u as the solution to the variational problem (V). For (1)-(4), the compatibility condition implies that constant functions $\gamma$ can be used in the weak formulation, and therefore any $C^\infty$ test function such that $\gamma (x,y,z)|_{\partial \Omega \times [0,h]}=c(z)$ is valid in the weak formulation. Parts (1) and (2) then follow from considering test functions which vanish on the lateral boundary $\partial \Omega \times [0,h]$. Part (3) is a consequence of constructing the solution within ${\mathbb {H}}$. Finally, (4) follows from noticing that when $\phi$ depends only on z and is compactly supported,

$$\begin{aligned} B(u,\phi )&= \int _{\Omega \times (0,h)} \lambda \partial _z \phi \partial _z u \\&= F(\phi )\\&= -\int _{\Omega \times [0,h]} f \phi + \int _{\Omega \times \{0,h\}} \phi \lambda g + \langle j, \phi \rangle \\&= -\int _{\Omega \times [0,h]} {\mathcal {L}} u \phi + \langle j, \phi \rangle . \end{aligned}$$

Integrating by parts to obtain

$$\begin{aligned} \int _{\Omega \times (0,h)} \lambda \partial _z \phi \partial _z u = -\int _{\Omega \times (0,h)} u \partial _z \left( \lambda \partial _{z} \phi \right) \end{aligned}$$

and rearranging finishes the proof. $\quad \square$

3.2 Higher Regularity

In order to build weak solutions, the operator which sends a triple (f, g, j) to the solution of the variational problem (V) must map compactly into ${\mathbb {H}}$. This will be achieved by proving an elliptic regularity theorem which asserts that the solution has strictly more than one derivative in $L^2(\Omega \times [0,h])$. The proof is split up into four preliminary lemmas which correspond to isolating the effects of the compatibility condition, g, f, and j on the regularity of the solution. Specifying a triple of data which does not satisfy Definition 1.1 produces a solution by projecting, in an appropriate sense, the data onto the set of compatible data. Analysis of the effect g is direct because solutions to the extension problem on bounded domains $\Omega$ can be written down explicitly. Once the Neumann derivative has been removed, we analyze the effects of f and j by reflecting the solution over the boundaries $z=0,h$ and utilizing the standard difference quotient technique for elliptic regularity. Each step is proved for the special case $\lambda (z)\equiv 1$, that is when ${\mathcal {L}}=\Delta$. The four lemmas are combined in the proof of the following theorem, where we then provide a description of how to adapt the techniques to general smooth $\lambda$.

Theorem 3.2

Let $f\in L^2(\Omega \times [0,h])$, $g\in L^2(\Omega \times \{0,h\})$, and $j\in L^2([0,h])$. Let $u\in {\mathbb {H}}$ be the unique variational solution to (V) guaranteed by Lemma 3.1. Then

$$\begin{aligned} \Vert \nabla u\Vert _{H^\frac{1}{2}(\Omega \times [0,h])} \leqq C(\Omega ,h,\lambda )\left( \Vert f\Vert _{L^2(\Omega \times [0,h])} + \Vert g\Vert _{L^2(\Omega \times \{0,h\})}+\Vert j\Vert _{L^2([0,h])} \right) . \end{aligned}$$

Before beginning the analysis, we set several notations. Let $\{e_n\}_{n=1}^\infty$ and $\{\lambda _n\}_{n=1}^\infty$ be the sequence of eigenfunctions and corresponding eigenvalues for the operator $-{\overline{\Delta }}$ on $\Omega$ with homogenous Dirichlet boundary conditions; that is,

$$\begin{aligned} {\left\{ \begin{array}{ll} -{\overline{\Delta }}e_n = \lambda _n e_n &{}\quad (x,y)\in \Omega \\ e_n = 0 &{}\quad (x,y)\in \partial \Omega . \\ \end{array}\right. } \end{aligned}$$

For $s\geqq 0$, define

$$\begin{aligned} {\bar{H}}^s(\Omega ) = \left\{ g=\sum _n g_n e_n \in L^2(\Omega ): \sum _n \left( \sqrt{\lambda _n}\right) ^s g_n e_n \in L^2(\Omega ) \right\} . \end{aligned}$$

By duality, we have that

$$\begin{aligned}\left( {\bar{H}}^s(\Omega )\right) ^* \cong \left\{ \{g_n\}_{n=1}^\infty \subset {\mathbb {R}}: \sum _n \frac{1}{\left( \sqrt{\lambda _n}\right) ^{2s}} g_n^2 < \infty \right\} .\end{aligned}$$

Real interpolation of Hilbert spaces $H_1, H_2$ is defined in the classical way (following the book of Bergh and Lofstrom for example [2]). Since the spaces ${\bar{H}}^s(\Omega )$ are isomorphic to $L^2({\mathbb {N}}, \omega )$ where ${\mathbb {N}}$ is equipped with the measure $\omega (n) = \lambda _n^{\frac{s}{2}}$, the Stein-Weiss interpolation theorem (see for example Theorem 5.4.1 from the book of Bergh and Lofstrom [2]) gives that

$$\begin{aligned} {[}{\bar{H}}^{s_1}(\Omega ), {\bar{H}}^{s_2}(\Omega )]_\theta = {\bar{H}}^s \end{aligned}$$

for $s=\theta s_1 + (1-\theta )s_2$ where $s_1,s_2\in {\mathbb {Z}}$. When $s=0$, ${\bar{H}}^s(\Omega )$ coincides with $L^2(\Omega )$. In general, ${\bar{H}}^s(\Omega )\subset H^s(\Omega )$ if $s\geqq 0$, and $H^s(\Omega )$ is defined classically (see for example Constantin and Nguyen [12]).

For $s\in (0,1)$, the fractional Sobolev spaces $H^s(\Omega \times [0,h])$ are defined by

$$\begin{aligned}&H^s(\Omega \times [0,h])\\&\quad := \left\{ h \in L^2(\Omega \times [0,h]): \frac{|h(x_1)-h(x_2)|}{|x_1-x_2|^{\frac{3}{2}+s}} \in L^2\left( (\Omega \times [0,h])\times (\Omega \times [0,h])\right) \right\} . \end{aligned}$$

For $s\in {\mathbb {N}}\cup \{0\}+(0,1)$, $H^s(\Omega \times [0,h])$ is the subset of $L^2(\Omega \times [0,h])$ for which

$$\begin{aligned} \frac{|\nabla ^{\left\lfloor {s}\right\rfloor }( h(x_1)-h(x_2))|}{|x_1-x_2|^{\frac{3}{2}+s-\left\lfloor {s}\right\rfloor }} \in L^2\left( (\Omega \times [0,h])\times (\Omega \times [0,h])\right) . \end{aligned}$$

Classical interpolation results (see for example the work of Triebel [24, 25]) gives that

$$\begin{aligned}{}[{H}^{s_1}(\Omega \times [0,h]), {H}^{s_2}(\Omega \times [0,h])]_\theta = H^{s}(\Omega \times [0,h]) \end{aligned}$$

for $s=\theta s_1 + (1-\theta )s_2$, $s_1, s_2 \geqq 0$.

Lemma 3.3

(Effect of the Compatibility Condition) Let a triple (f, g, j) with $f\in L^2(\Omega \times [0,h])$, $g\in L^2(\Omega \times \{0,h\})$, $j\in L^2(0,h)$ be given. Let u be the solution to the variational problem with data (f, g, j). Then there exists a constant c depending only on

$$\begin{aligned} \int _{\Omega \times [0,h]} f,\quad \int _{\Omega \times \{0,h\}} g,\quad \int _0^h j \end{aligned}$$

such that $\Delta u = f + c$ and

$$\begin{aligned} |c| \lesssim \Vert f\Vert _{L^2} + \Vert g\Vert _{L^2} + \Vert j\Vert _{L^2}. \end{aligned}$$

Proof

We define an operator $A: L^2(\Omega \times [0,h])\times L^2(\Omega \times \{0,h\})\times L^2(0,h) \rightarrow {\mathbb {R}}$ which maps a triple (f, g, j) to a constant $c=A(f,g,j)$. Since ${\mathbb {H}}$ only contains test functions with mean value zero, given $(x_1,y_1,z_1), (x_2,y_2,z_2) \in \Omega \times [0,h]$, choose a sequence of test functions which is the difference between two sequences of approximate identities centered at $(x_1,y_1,z_1), (x_2,y_2,z_2)$. Using this sequence of test functions in the variational formulation gives that $\Delta u(x_1,y_1,z_1)-\Delta u (x_2,y_2,z_2) = f(x_1,y_1,z_1)-f(x_2,y_2,z_2)$. Therefore, $\Delta u$ is equal to f up to a constant c, and thus $A(f,g,j)=c$ is well-defined. By the linearity of the variational problem, A is linear. To show that A depends only on the integrals of f, g, and j, let ${\bar{f}}, {\bar{g}}, {\bar{j}}$ be given, each with mean value zero. Then $A({\bar{f}},{\bar{g}},{\bar{j}})$ satisfies the compatibility condition, implying $\Delta {\bar{u}} = {\bar{f}}$ in a weak sense, and $A({\bar{f}},{\bar{g}},{\bar{j}})=0$. Therefore A depends only on

$$\begin{aligned} \int _{\Omega \times [0,h]} f,\quad \int _{\Omega \times \{0,h\}} g,\quad \int _0^h j. \end{aligned}$$

Now A is a linear map from ${\mathbb {R}}^3\rightarrow {\mathbb {R}}$, and is therefore bounded. That is,

$$\begin{aligned} \left| A(f,g,j)\right| ^2 \lesssim \left| \int _{\Omega \times [0,h]} f\right| ^2 + \left| \int _{\Omega \times \{0,h\}} g \right| ^2 + \left| \int _0^h j \right| ^2. \end{aligned}$$

Applying Hölder’s inequality finishes the proof. $\quad \square$

Lemma 3.4

(Effect of g) Consider the equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u = 0 &{}\quad \Omega \times [0,h] \\ \partial _{\nu } u = g &{}\quad \Omega \times \{0,h\}\\ u = 0 \qquad &{}\quad \partial \Omega \times [0,h].\\ \end{array}\right. } \end{aligned}$$

for $g\in {\bar{H}}^s(\Omega \times \{0,h\})$, $s\geqq -\frac{1}{2}$. Then there exists a solution u which satisfies

$$\begin{aligned} \Vert \nabla u \Vert _{H^{s+\frac{1}{2}}(\Omega \times [0,h])} \leqq C(\Omega ,h) \Vert g\Vert _{{\bar{H}}^s(\Omega \times \{0,h\})}. \end{aligned}$$

Proof

We begin by assuming that g is smooth so that all calculations with higher derivatives are valid. For arbitrary $g \in {\bar{H}}^s$, the claim follows from density of smooth functions. By assumption on g, there exist sequences of real numbers $\{ t_n \} , \{ b_n \}$ such that

$$\begin{aligned} g(0,x,y) = \sum _n b_n e_n(x,y), \quad g(h,x,y) = \sum _n t_n e_n(x,y) . \end{aligned}$$

Define

$$\begin{aligned} {u} = \sum _n \left\{ \frac{t_n}{\sqrt{\lambda _n}} \frac{\cosh (z\sqrt{\lambda _n})}{\sinh (h\sqrt{\lambda _n})} + \frac{b_n}{\sqrt{\lambda _n}}\frac{\cosh ((z-h)\sqrt{\lambda _n})}{\sinh (h\sqrt{\lambda _n})} \right\} e_n(x,y). \end{aligned}$$

Using that $\sinh (0) = 0$ and $(\sinh )''=(\cosh )'=\sinh$, we have that

$$\begin{aligned} -\frac{\partial }{\partial z} {u} |_{z=0} = g|_{z=0} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial }{\partial z} {u} |_{z=h} = g|_{z=h}. \end{aligned}$$

In addition, it is immediate that

$$\begin{aligned} \partial _{zz} {u} = -{\overline{\Delta }}u, \end{aligned}$$

and therefore $\Delta {u} \equiv 0$. Since

$$\begin{aligned}\cosh (z\sqrt{\lambda _n}) \approx \sinh (h\sqrt{\lambda _n}) \approx e^{z\sqrt{\lambda _n}}\end{aligned}$$

as $n\rightarrow \infty$, we have that

$$\begin{aligned} (-{\overline{\Delta }})^\frac{1}{2}u \approx \partial _z u \in L^\infty ([0,h];L^2(\Omega )) \subset L^2(\Omega \times [0,h]). \end{aligned}$$

Using the well-known fact that ${\bar{H}}^1(\Omega )=H_0^1(\Omega )$, we have that

$$\begin{aligned} \Vert \nabla u \Vert _{L^2(\Omega \times [0,h])} \leqq C(\Omega ,h) \Vert g \Vert _{L^2(\Omega \times \{0,h\})}, \end{aligned}$$

and thus u is a well-defined function in $\Omega \times [0,h]$ which solves the desired equation.

To sharpen this bound and obtain higher regularity estimates, we split the sum into four pieces corresponding to the four pieces of

$$\begin{aligned} \cosh (z\sqrt{\lambda _n}) = \frac{e^{z\sqrt{\lambda _n}}+e^{-z\sqrt{\lambda _n}}}{2}, \qquad \cosh ((z-h)\sqrt{\lambda _n}) = \frac{e^{(z-h)\sqrt{\lambda _n}}+e^{-(z-h)\sqrt{\lambda _n}}}{2}. \end{aligned}$$

Define

$$\begin{aligned} {\tilde{t}}_n := \frac{t_n}{\sinh (h\sqrt{\lambda _n})} e^{h\sqrt{\lambda _n}}, \qquad {\tilde{g}}:= \sum _n {\tilde{t}}_n e_n \end{aligned}$$

so that

$$\begin{aligned} \Vert {\tilde{g}}\Vert _{{\bar{H}}^s(\Omega )} \leqq C(\Omega ) \Vert g \Vert _{{\bar{H}}^s(\Omega \times \{0,h\})} .\end{aligned}$$

Then

$$\begin{aligned} {\tilde{u}} := \sum _n e^{(z-h)\sqrt{\lambda _n}} \frac{{\tilde{t}}_n}{\sqrt{\lambda _n}} \end{aligned}$$

is the solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta {\tilde{u}} = 0 &{}\quad \Omega \times (-\infty ,h] \\ \partial _{\nu } {\tilde{u}} = {\tilde{g}} &{}\quad \Omega \times \{h\}\\ {\tilde{u}}(x,y,z) = 0 \qquad &{}\quad \partial \Omega \times (-\infty ,h]. \end{array}\right. } \end{aligned}$$

Now we can write that

$$\begin{aligned} \Vert \nabla {\tilde{u}} \Vert _{{\bar{H}}^{s+\frac{1}{2}}(\Omega \times (-\infty ,h))}&= \int _{\Omega \times [0,h]} \nabla \left( (-{\overline{\Delta }})^{\frac{1}{2}(s+\frac{1}{2})} {\tilde{u}} \right) \cdot \nabla \left( (-{\overline{\Delta }})^{\frac{1}{2}(s+\frac{1}{2})} {\tilde{u}} \right) \nonumber \\&= \int _{\Omega \times \{h\}} \partial _\nu (-{\overline{\Delta }})^{\frac{1}{2}(s+\frac{1}{2})} {\tilde{u}} (-{\overline{\Delta }})^{\frac{1}{2}(s+\frac{1}{2})} {\tilde{u}}\nonumber \\&= \Vert \partial _\nu {\tilde{u}}\Vert _{{\bar{H}}^{s}(\Omega )}\nonumber \\&\leqq C(\Omega ,h)\Vert g \Vert _{{\bar{H}}^{s}(\Omega \times \{0,h\})}. \end{aligned}$$

(13)

Arguing in a similar fashion for the other parts of the infinite sum, we conclude that

$$\begin{aligned} \Vert \nabla u \Vert _{{\bar{H}}^{s+\frac{1}{2}}(\Omega \times [0,h])} \leqq C(\Omega ,h) \Vert g \Vert _{{\bar{H}}^{s}(\Omega \times \{0,h\})}. \end{aligned}$$

If $s+\frac{1}{2}\in {\mathbb {N}}$, noticing that $(\partial _z)^{s+\frac{1}{2}} u \approx (-{\overline{\Delta }})^\frac{s+\frac{1}{2}}{2} u$, we have that

$$\begin{aligned} \Vert \nabla u \Vert _{{H}^{s+\frac{1}{2}}(\Omega \times [0,h])} \leqq C(\Omega ,h) \Vert g \Vert _{{\bar{H}}^{s}(\Omega \times \{0,h\})}. \end{aligned}$$

(14)

As noted above, for non-integer $s\in (-\frac{1}{2}, \infty )$, the Stein-Weiss interpolation theorem gives that

$$\begin{aligned} {[}{\bar{H}}^{s_1}(\Omega ), {\bar{H}}^{s_2}(\Omega )]_\theta = {\bar{H}}^s(\Omega ) \end{aligned}$$

for $s=\theta s_1 + (1-\theta )s_2$, and interpolation of Hilbert-Sobolev spaces on Lipschitz domains gives that

$$\begin{aligned} {[}{H}^{s_1+\frac{1}{2}}(\Omega \times [0,h]), {H}^{s_2+\frac{1}{2}}(\Omega \times [0,h])]_\theta = H^{s+\frac{1}{2}}(\Omega \times [0,h]). \end{aligned}$$

Interpolation of (14) then concludes the proof of the lemma. $\quad \square$

In the following two lemmas, we address the effects of f and j. While the solutions we consider are only variational a priori, for the sake of clarity we write each PDE using classical notation rather than the variational form.

Lemma 3.5

(Effect of f) Let $u\in {\mathbb {H}}$ be a variational solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u = f &{}\quad \Omega \times [0,h] \\ \partial _{\nu } u = 0 &{}\quad \Omega \times \{0,h\}\\ u(x,y,z) = c(z) \qquad &{}\quad \partial \Omega \times [0,h]\\ \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}u\cdot \nu _s =0 &{}\quad [0,h] \end{array}\right. } \end{aligned}$$

for data $f\in L^2(\Omega \times [0,h])$. Then

$$\begin{aligned} \Vert u \Vert _{H^2(\Omega \times [0,h])} \leqq C(\Omega ,h) \Vert f\Vert _{L^2(\Omega \times [0,h])}. \end{aligned}$$

Proof

Formally, the assumptions that

$$\begin{aligned} \partial _\nu u \equiv 0, \qquad \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}u \cdot \nu _s \equiv 0 \end{aligned}$$

give

$$\begin{aligned} \int _{\Omega \times [0,h]} \nabla (\partial _z u) \cdot \nabla (\partial _z u)&= -\int _{\Omega \times [0,h]} \partial _z u \Delta (\partial _z u) + \int _{\Omega \times \{0,h\}} \partial _z u \partial _{z}(\partial _\nu u) \\&\quad + \int _0^h \partial _z u \cdot \partial _{z} \left( \int _{\partial \Omega } ({\overline{\nabla }}u \cdot \nu _s) \right) \\&=- \int _{\Omega \times [0,h]} \partial _z u \Delta (\partial _z u)\\&= \int _{\Omega \times [0,h]} \partial _{zz}u f, \end{aligned}$$

implying that

$$\begin{aligned} \Vert \partial _{zz}u \Vert _{L^2(\Omega \times [0,h])} \leqq C\Vert f \Vert _{L^2(\Omega \times [0,h])}. \end{aligned}$$

(15)

Regularity of ${\overline{\Delta }}u$ would then follow from the equality

$$\begin{aligned} {\overline{\Delta }}u = f + A(f,0,0) - \partial _{zz}u. \end{aligned}$$

Then we can write that for fixed z,

$$\begin{aligned} {\left\{ \begin{array}{ll} {\overline{\Delta }}u = f + A(f,0,0) - \partial _{zz}u &{}\quad \Omega \times \{z\} \\ u =c(z) &{}\quad \partial \Omega \times \{z\}.\\ \end{array}\right. } \end{aligned}$$

Applying classical elliptic regularity theory z by z shows then that $\partial _{xy}u, \partial _{xx}u, \partial _{yy}u \in L^2(\Omega \times [0,h])$. Thus it remains to rigorously show (15).

Define

$$\begin{aligned} u_E(x,y,z) = {\left\{ \begin{array}{ll} u(x,y,z) &{}\quad z\in [0,h] \\ u(x,y,-z) &{}\quad z\in [-h,0]\\ \end{array}\right. } \end{aligned}$$

and define $f_E$ similarly. Let $\eta (z)$ be a smooth cutoff function depending only on z such that $\eta \equiv 1$ for all $z\in [-\frac{2h}{3},\frac{2h}{3}]$ and $\eta$ is compactly supported in $[-\frac{3h}{4}, \frac{3h}{4}]$. Define the difference quotient operator

$$\begin{aligned} T_\varepsilon \phi = \frac{\phi (x,y,z+\varepsilon )-\phi (x,y,z)}{\varepsilon }. \end{aligned}$$

Then we can write

$$\begin{aligned} -\int _{\Omega \times [-h,h]} \nabla u_E \cdot \nabla \left( T_{-\varepsilon }(\eta ^2 T_\varepsilon u_E) \right)&= \int _{\Omega \times [-h,h]} \nabla (T_\varepsilon u_E) \cdot \nabla (\eta ^2 T_\varepsilon u_E)\\&= \int _{\Omega \times [-h,h]} \nabla (T_\varepsilon u_E) \cdot \nabla (T_\varepsilon u_E) \eta ^2\\&\quad +\, \int _{\Omega \times [-h,h]} \nabla (T_\varepsilon u_E) \cdot \nabla \eta (2\eta T_\varepsilon u_E). \end{aligned}$$

Rearranging, we have that

$$\begin{aligned}&\int _{\Omega \times [-h,h]} \eta ^2 |\nabla (T_\varepsilon u_E)|^2 \\&\quad \leqq -\int _{\Omega \times [-h,h]} \nabla (T_\varepsilon u_E) \cdot \nabla \eta (2\eta T_\varepsilon u_E) \\&\qquad -\,\int _{\Omega \times [-h,h]} \nabla u_E \cdot \nabla \left( T_{-\varepsilon }(\eta ^2 T_\varepsilon u_E) \right) \\&\quad := I + II. \end{aligned}$$

Examining I, we have that

$$\begin{aligned} I&\leqq \frac{1}{4} \int _{\Omega \times [-h,h]} \eta ^2 | \nabla (T_{\varepsilon } u_E) |^2 + C(\eta ) \Vert f_E \Vert ^2_{L^2(\Omega \times [-h,h])} . \end{aligned}$$

(16)

Moving to II and replacing one of the $T_{-\varepsilon }$ with $\partial _z$, we have that

$$\begin{aligned} II&= \int _{\Omega \times [-h,h]} T_{-\varepsilon }(\eta ^2(T_\varepsilon u_E)) f_E \nonumber \\&\leqq \left\| \partial _z (\eta ^2 (T_{\varepsilon }u_E)) \right\| _{L^2(\Omega \times [-h,h])} \left\| f_E \right\| _{L^2(\Omega \times [-h,h])} \nonumber \\&\leqq \frac{1}{4}\int _{\Omega \times [-h,h]} \eta ^2 |\nabla (T_\varepsilon u_E)|^2 + C(\eta ) \left\| f_E \right\| ^2_{L^2(\Omega \times [-h,h])}. \end{aligned}$$

(17)

Combining (16) and (17), it follows that

$$\begin{aligned} \int _{\Omega \times [-h,h]} \eta ^2 |\nabla (T_\varepsilon u_E) |^2 \leqq C(\eta ) \Vert f_E\Vert _{L^2(\Omega \times [-h,h])}. \end{aligned}$$

The uniformity of this inequality in $\varepsilon$ allows us to pass to a weak limit as $\varepsilon \rightarrow 0$ to conclude that

$$\begin{aligned} \left\| \eta \nabla (\partial _z u_E) \right\| _{L^2(\Omega \times [-h,h])} \leqq C(\Omega ,\eta ) \Vert f_E \Vert _{L^2(\Omega \times [-h,h])}. \end{aligned}$$

Repeating the argument but with a reflection over $z=h$ and then restricting to $z\in [0,h]$, we obtain that

$$\begin{aligned} \left\| \nabla (\partial _z u) \right\| _{L^2(\Omega \times [0,h])} \leqq C(\Omega ,\eta ) \Vert f \Vert _{L^2(\Omega \times [0,h])}. \end{aligned}$$

Regularity of $\partial _{xx} u$, $\partial _{xy}u$, and $\partial _{yy}u$ follows as described before, finishing the proof of the lemma. $\quad \square$

Lemma 3.6

(Effect of j) Let $u\in {\mathbb {H}}$ be a variational solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u = 0 &{}\quad \Omega \times [0,h] \\ \partial _{\nu } u = 0 &{} \quad \Omega \times \{0,h\}\\ u(x,y,z) = c(z) \qquad &{}\quad \partial \Omega \times \{z>0\}\\ \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}u\cdot \nu _s =j(z) &{}\quad [0,h]. \end{array}\right. } \end{aligned}$$

Then the following inequalites hold for $j\in L^2(0,h)$ or $H^\frac{1}{2}(0,h)$:

$$\begin{aligned} \Vert u \Vert _{H^\frac{3}{2}(\Omega \times (0,h))} \leqq C(\Omega ,h) \Vert j \Vert _{L^2(0,h)} \end{aligned}$$

and

$$\begin{aligned} \Vert u \Vert _{H^2(\Omega \times (0,h))} \leqq C(\Omega ,h) \Vert j \Vert _{H^\frac{1}{2}(0,h)}. \end{aligned}$$

Proof

The proof consists of two steps. In the first step, we consider the elliptic problem posed on $\Omega \times [h_1,h_2]$ but for j which has compact support in $(h_1,h_2)$. The second step shows that the general situation on $\Omega \times [0,h]$ can be reduced to the compactly supported setting of the first step using reflections in z over $z=0,h$ and a partition of unity.

• Step 1: We consider an elliptic problem identical to that of the statement of Lemma 3.6 but for the spatial domain $\Omega \times [h_1,h_2]$ and data $\alpha$ which is compactly supported in $(h_1,h_2)$. Define $H^{-\frac{1}{2}}_0(h_1,h_2)$ to be the set of all $\alpha \in {\mathcal {D}}'(h_1,h_2)$ such that there exists $\zeta \in H^{-\frac{1}{2}}({\mathbb {R}})$ such that $\zeta |_{(-h_1,h_2)} = \alpha$, with $\Vert \alpha \Vert _{H^{-\frac{1}{2}}(h_1,h_2)}$ defined as the infimum of $\Vert \zeta \Vert _{H^{-\frac{1}{2}}({\mathbb {R}})}$ over such $\zeta$. Define $H_0^{\frac{1}{2}}(h_1,h_2)$ analogously but with the extra assumption that the support of $\zeta$ is contained in $[h_1,h_2]$. Per Theorem 3.5, Definition 3.3, and Definition 2.3 from [25], these two spaces are reflexive and dual to one another. Let $\alpha \in H_0^{-\frac{1}{2}}(h_1,h_2)$ and $\gamma \in {\mathcal {D}}(h_1,h_2)$. Consider the elliptic problem defined by

  $$\begin{aligned} B(u,v)&= \int _{\Omega \times (h_1,h_2)} \nabla u \cdot \nabla v \\&= F(v)\\&= {}_{H_0^{-\frac{1}{2}}(h_1,h_2)}{\langle } \alpha , \gamma v \rangle _{ H_0^\frac{1}{2}(h_1,h_2)}. \end{aligned}$$

  Then F(v) is well-defined and satisfies

  $$\begin{aligned} \left| F(v) \right| \leqq C(\gamma ) \Vert \alpha \Vert _{H_0^{-\frac{1}{2}}(h_1,h_2)} \Vert v \Vert _{H^{\frac{1}{2}}(h_1,h_2)} . \end{aligned}$$

  By Lemma 3.1, there exists a unique variational solution $u\in {\mathbb {H}}$ to the variational problem with $f=g=0$, $j=\gamma \alpha$ satisfying

  $$\begin{aligned} \Vert u \Vert _{\mathbb {H}}\leqq C(\Omega , h, \gamma ) \Vert \alpha \Vert _{H_0^{-\frac{1}{2}}(h_1,h_2)}. \end{aligned}$$

  Now suppose that $\alpha \in {H_0^{\frac{1}{2}}(h_1,h_2)}$, so that

  $$\begin{aligned} F(v) = \int _{h_1}^{h_2} \alpha (z) \gamma (z) v|_{\partial \Omega \times (h_1,h_2)} (z) \,\mathrm{d}z. \end{aligned}$$

  Define $u_E$ and $\alpha _E$ on $(2h_1-h_2,h_2)$ by reflection over $h_1$ as in Lemma 3.5, and define $T_\varepsilon$ similarly as well. Define

  $$\begin{aligned} \frac{1}{2}\left( h_2 + \max _{z'\in {\text {supp}}\gamma }{z'} \right) =: z_0. \end{aligned}$$

  Choose $\eta :(2h_1-h_2,h_2)\rightarrow {\mathbb {R}}$ such that $\eta$ is symmetric about $z=h_1$, and $\eta =1$ for all z such that $h_1 \leqq z \le z_0$, which ensures that $\eta =1$ in the support of $\gamma$. In addition, let $\phi _n(z)$ be a one dimensional, smooth, even mollifier supported on a ball of radius $\frac{1}{n}$ around 0. Thus we choose our test function to be

  $$\begin{aligned} \phi _n *T_{-\varepsilon }\left( \eta ^2 T_\varepsilon (u_E*\phi _n)\right) . \end{aligned}$$

  Then we can write

  $$\begin{aligned}&-\int _{\Omega \times [2h_1-h_2,h_2]} \nabla u_E \cdot \nabla \left( \phi _n *T_{-\varepsilon }(\eta ^2 T_\varepsilon (u_E*\phi _n)) \right) \\&\quad = \int _{\Omega \times [2h_1-h_2,h_2]} \nabla (T_\varepsilon (u_E*\phi _n)) \cdot \nabla (\eta ^2 T_\varepsilon (u_E*\phi _n))\\&\quad = \int _{\Omega \times [2h_1-h_2,h_2]} \nabla (T_\varepsilon (u_E*\phi _n)) \cdot \nabla (T_\varepsilon (u_E*\phi _n)) \eta ^2\\&\qquad +\, \int _{\Omega \times [2h_1-h_2,h_2]} \nabla (T_\varepsilon (u_E*\phi _n)) \cdot \nabla \eta (2\eta T_\varepsilon (u_E*\phi _n)). \end{aligned}$$

  Rearranging, we have that

  $$\begin{aligned}&\int _{\Omega \times [2h_1-h_2,h_2]} \eta ^2 |\nabla (T_\varepsilon (u_E*\phi _n))|^2 \\&\quad \leqq -\int _{\Omega \times [2h_1-h_2,h_2]} \nabla (T_\varepsilon (u_E*\phi _n)) \cdot \nabla \eta (2\eta T_\varepsilon (u_E*\phi _n)) \\&\qquad -\,\int _{\Omega \times [2h_1-h_2,h_2]} \nabla (u_E*\phi _n) \cdot \nabla \left( T_{-\varepsilon }(\eta ^2 T_\varepsilon (u_E*\phi _n)) \right) \\&\quad := I + II. \end{aligned}$$

  Examining I, we have that

  $$\begin{aligned}&I \leqq \frac{1}{4} \int _{\Omega \times [2h_1-h_2,h_2]} \eta ^2 | \nabla (T_\varepsilon u_E *\phi _n) |^2 + C(\eta ) \Vert \alpha _E \Vert ^2_{H_0^\frac{1}{2}(2h_1-h_2,h_2)}. \end{aligned}$$

  (18)

  Before examining II, notice that due to the compact support of $\gamma$, we can assume without loss of generality that in expressions where $c_E*\phi _n$ is integrated against $\gamma$, $c_E *\phi _n$ is a smooth, compactly supported function on $[2h_1-z_0,z_0]$ and therefore can be expanded in Fourier series with coefficients ${\hat{c}}_E(k){\hat{\phi }}_n(k)$. Note also that since $T_\varepsilon$ ignores constants, we can assume without loss of generality that ${\hat{c}}_E(0)=\widehat{\gamma \alpha _E}(0)=0$, ensuring that fractional laplacians (as Fourier multipliers) of $c_E$ and $\gamma \alpha _E$ are well-defined. Finally, we shall freely switch between ${\bar{H}}^s$ norms and $H^s$ or $H^s_0$ norms due to the fact that everything is compactly supported in $(2h_1-z_0,z_0)$. Now we can write that (we recall that $\eta =1$ in the support of $\gamma$)

  $$\begin{aligned}&-\int _{\Omega \times [2h_1-h_2,h_2]} \nabla (u_E *\phi _n) \cdot \nabla \left( T_{-\varepsilon }(\eta ^2 T_\varepsilon (u_E *\phi _n )) \right) \\&\quad = \int _{2h_1-z_0}^{z_0} \left( \phi _n *T_\varepsilon \left( \gamma \alpha _E \right) (z) \right) \left( T_\varepsilon (c_E *\phi _n)(z) \right) \,\mathrm{d}z \\&\quad = \sum _{k\ne 0} \frac{\widehat{\left( \phi _n *T_\varepsilon \left( \gamma \alpha _E \right) \right) }(k)}{|k|^\frac{1}{2}} \cdot \widehat{ T_\varepsilon \left( c_E *\phi _n \right) }(k) |k|^\frac{1}{2} \\&\quad \leqq \left\| \phi _n *T_\varepsilon \left( \gamma \alpha _E \right) \right\| _{{\bar{H}}^{-\frac{1}{2}}(2h_1-z_0,z_0)} \left\| T_\varepsilon \left( c_E *\phi _n \right) \right\| _{{\bar{H}}^\frac{1}{2}(2h_1-z_0,z_0)}\\&\quad \leqq C(\eta ,\gamma ) \left\| \phi _n *\alpha _E \right\| _{{H}_0^\frac{1}{2}(2h_1-z_0,z_0)} \left\| \partial _z (c_E *\phi _n) \right\| _{{H}^{\frac{1}{2}}(2h_1-z_0,z_0)}\\&\quad \leqq C(\eta , \gamma ) \Vert \alpha _E \Vert _{H_0^\frac{1}{2}(2h_1-z_0,z_0)} \\&\qquad \times \,\left( \Vert \partial _z (u_E *\phi _n) \Vert _{H^1(\Omega \times (2h_1-z_0,z_0))} + \Vert u_E *\phi _n \Vert _{H^1(\Omega \times (2h_1-z_0,z_0))} \right) . \end{aligned}$$

  Combining this bound with (18), it follows that

  $$\begin{aligned}&\int _{\Omega \times [2h_1-h_2,h_2]} \eta ^2 |\nabla (T_\varepsilon u_E *\phi _n) |^2 \leqq C(\eta ,\gamma ,\Omega ) \Vert \alpha _E\Vert ^2_{H_0^\frac{1}{2}(2h_1-z_0,z_0)} \\&\quad +\, \frac{1}{4} \Vert \partial _z( u_E *\phi _n ) \Vert ^2_{H^1(\Omega \times (2h_1-z_0,z_0))}. \end{aligned}$$

  The uniformity of this inequality in $\varepsilon$ and n and the fact that $\eta =1$ for $z\in [2h_1-z_0,z_0]$ allows us to pass to a weak limit as $\varepsilon \rightarrow 0$ and $n\rightarrow \infty$ to conclude that

  $$\begin{aligned} \left\| \eta \nabla (\partial _z u_E) \right\| ^2_{L^2\left( \Omega \times [2h_1-h_2,h_2]\right) } \leqq C(\Omega ,\eta ,\gamma ) \Vert \alpha _E \Vert _{H_0^\frac{1}{2}([-h,h])}^2. \end{aligned}$$

  Another reflection over $z=h_2$ and a combination of the estimates shows that

  $$\begin{aligned} \left\| \nabla (\partial _z u) \right\| ^2_{L^2\left( \Omega \times [h_1,h_2]\right) } \leqq C(\Omega ,\eta ,\gamma ) \Vert \alpha \Vert _{H_0^\frac{1}{2}(h_1,h_2 )}^2. \end{aligned}$$

  Regularity of $\partial _{xx} u$, $\partial _{xy}u$, and $\partial _{yy}u$ follows as for Lemma 3.5.

  Thus for $s=-\frac{1}{2},\frac{1}{2}$ we have constructed a bounded linear map $T:H^s_0(0,h) \rightarrow H^{s+\frac{3}{2}}(\Omega \times (0,h))$ with operator norm satisfying

  $$\begin{aligned} \Vert T \Vert _{op} \leqq C(\Omega ,\gamma ) \Vert \alpha \Vert _{H^s_0(0,h)}. \end{aligned}$$

  Applying Theorem 3.5 from [24] to interpolate between $H^s_0(0,h)$ for $s=-\frac{1}{2}$ and $s=\frac{1}{2}$ and using the well-known fact that $H^2$ and $H^1$ can be interpolated to give $H^s$ for $s\in (1,2)$, we obtain that T is bounded from $L^2(0,h)$ to $H^\frac{3}{2}(\Omega \times (0,h))$.

• Step 2: Let $j\in H^s(0,h)$ for $s=0,\frac{1}{2}$. Let $\phi _i:[0,h]\rightarrow {\mathbb {R}}$ for $i=1,2,3$ be bump functions such that $\phi _2$ is compactly supported in (0, h) and $\sum _i \phi _i^2(z) = 1$ for all $z\in [0,h]$. Define $j_i = \phi _i ^2 j$, and let $u_i$ be the solution to the elliptic problem on $\Omega \times [0,h]$ with data $j_i$. By linearity, we have that $u = \sum _i u_i$. We shall prove the desired bounds for $u_i$ and then sum over i to complete the proof of the lemma.

  We begin with $u_2$. It is clear that $\phi _2 j \in H_0^s(0,h)$ with

  $$\begin{aligned} \Vert \phi _2 j \Vert _{H_0^s(0,h)} \leqq C(\phi _2) \Vert j \Vert _{H^s(0,h)} . \end{aligned}$$

  Then we can apply Step 1 with $h_1=0$, $h_2=h$, $\alpha = \phi _2 j$, and $\gamma =\phi _2$ to deduce that

  $$\begin{aligned} \Vert u_2 \Vert _{H^s(\Omega \times (0,h))} \leqq C(\Omega ,h,\phi _2) \Vert j \Vert _{H^s(0,h)}. \end{aligned}$$

  Moving to $u_1$, let us reflect $\phi _1 j$ over $z=0$ to produce $(\phi _1 j)_E \in H^s_0(-h,h)$. We have then that

  $$\begin{aligned} \Vert (\phi _1 j)_E \Vert _{H_0^s(-h,h)} \leqq C(\phi _1) \Vert j \Vert _{H^s(0,h)}. \end{aligned}$$

  We pause briefly to emphasize an important property of the elliptic problem (E) posed on $\Omega \times [h_1,h_2]$. If the data f, g, and j possess a reflection symmetry about $z=\frac{h_1+h_2}{2}$, then the solution u will possess the same symmetry. Then setting $u_{1,E}$ to be the solution to the elliptic problem posed on $[-h,h]$ with datum $(\phi _1^2 j)_E$, we have that $u_{1,E}|_{z\in [0,h]} = u_1$. Now applying Step 1 with $h_1=-h$, $h_2=h$, $\alpha = (\phi _1 j)_E$, and $\gamma = \phi _{1,E}$, we obtain

  $$\begin{aligned} \Vert u_1 \Vert _{H^{s+\frac{3}{2}}(\Omega \times (0,h))} \leqq \Vert u_{1,E} \Vert _{H^{s+\frac{3}{2}}(\Omega \times (-h,h))} \leqq C(\phi _1, \Omega ,h) \Vert j \Vert _{H^s(0,h)}. \end{aligned}$$

  Repeating the argument for $u_3$ and summing the estimates finishes the proof. $\quad \square$

We can now prove Theorem 3.2.

Proof of Theorem 3.2

We begin with $\lambda \equiv 1$, in which case (V) is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u = f &{}\quad \Omega \times [0,h] \\ \partial _{\nu } u = g &{}\quad \Omega \times \{0,h\}\\ u(x,y,z) = c(z) &{}\quad \partial \Omega \times [0,h]\\ \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}u\cdot \nu _s =j(z) &{}\quad [0,h]. \end{array}\right. } \qquad (V) \end{aligned}$$

First, apply Lemma 3.4 to build a solution $u_1$ to

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u_1 = 0 &{}\quad \Omega \times [0,h]\\ \partial _{\nu } u_1 = g &{}\quad \Omega \times \{0,h\}\\ u_1(x,y,z) = 0 &{}\quad \partial \Omega \times [0,h],\\ \end{array}\right. } \end{aligned}$$

which satisfies

$$\begin{aligned} \Vert \nabla u_1 \Vert _{H^\frac{1}{2}(\Omega \times [0,h])} \leqq C(\Omega ,h) \Vert g \Vert _{L^2(\Omega \times \{0,h\})}. \end{aligned}$$

Now choose $c_1$ such that ${\tilde{u}}_1 = u_1 + c_1$ has mean value zero on $\Omega \times [0,h]$; then

$$\begin{aligned} \Vert \nabla {\tilde{u}}_1\Vert _{H^\frac{1}{2}(\Omega \times [0,h])}=\Vert \nabla {u}_1\Vert _{H^\frac{1}{2}(\Omega \times [0,h])} \leqq C(\Omega ,h) \Vert g \Vert _{L^2(\Omega \times \{0,h\})}. \end{aligned}$$

(19)

By the trace estimate (10),

$$\begin{aligned} {j}_1(z) := \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}{\tilde{u}}_1 \cdot \nu _s \end{aligned}$$

is well-defined in $L^2([0,h])$ and satisfies

$$\begin{aligned} \Vert j_1\Vert _{L^2([0,h])} \leqq C(\Omega ,h) \Vert \nabla {\tilde{u}}_1\Vert _{H^\frac{1}{2}(\Omega \times [0,h])} \leqq C(\Omega ,h) \Vert g \Vert _{L^2(\Omega \times \{0,h\})}. \end{aligned}$$

Therefore, ${\tilde{u}}_1$ is the unique variational solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta {\tilde{u}}_1 = 0 &{}\quad \Omega \times [0,h] \\ \partial _{\nu } {\tilde{u}}_1 = g &{}\quad \Omega \times \{0,h\}\\ {\tilde{u}}_1(x,y,z) = c_1 &{}\quad \partial \Omega \times [0,h]\\ \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}{\tilde{u}}_1\cdot \nu _s =j_1(z) &{}\quad [0,h]. \end{array}\right. } \end{aligned}$$

Now define $u_2:= u - {\tilde{u}}_1$; $u_2$ is then the unique variational solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta {u}_2 = f &{}\quad \Omega \times [0,h] \\ \partial _{\nu } {u}_2 = 0 &{}\quad \Omega \times \{0,h\}\\ {u}_2(x,y,z) = c_2(z) &{}\quad \partial \Omega \times [0,h]\\ \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}{u}_2\cdot \nu _s =j(z)-j_1(z) &{}\quad [0,h]. \end{array}\right. } \end{aligned}$$

Define $u_3$ to as the unique variational solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta {u}_3 = f &{}\quad \Omega \times [0,h] \\ \partial _{\nu } {u}_3 = 0 &{}\quad \Omega \times \{0,h\}\\ {u}_3(x,y,z) = c_3(z) &{}\quad \partial \Omega \times [0,h]\\ \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}{u}_3\cdot \nu _s =0 &{}\quad [0,h] \end{array}\right. } \end{aligned}$$

and $u_4$ as the unique variational solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta {u}_4 = 0 &{} \quad \Omega \times [0,h] \\ \partial _{\nu } {u}_4 = 0 &{}\quad \Omega \times \{0,h\}\\ {u}_4(x,y,z) = c_4(z) &{}\quad \partial \Omega \times [0,h]\\ \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}{u}_4\cdot \nu _s =j(z)-j_1(z) &{}\quad [0,h] \end{array}\right. } \end{aligned}$$

so that $u_2 = u_3+u_4$. Applying Lemma 3.5 to $u_3$ and Lemma 3.6 to $u_4$, we conclude that

$$\begin{aligned} \Vert \nabla u_2 \Vert _{H^\frac{1}{2}(\Omega \times [0,h])} \leqq C(\Omega ,h) \left( \Vert f\Vert _{L^2(\Omega \times [0,h])} + \Vert j-j_1\Vert _{L^2([0,h])} \right) . \end{aligned}$$

(20)

Combining (19) and (20), we conclude that

$$\begin{aligned} \Vert \nabla u \Vert _{H^\frac{1}{2}(\Omega \times [0,h])}= & {} \Vert \nabla ({\tilde{u}}_1 + u_2) \Vert _{H^\frac{1}{2}(\Omega \times [0,h])} \\\leqq & {} C(\Omega ,h) \left( \Vert f\Vert _{L^2} + \Vert g\Vert _{L^2} + \Vert j\Vert _{L^2} \right) . \end{aligned}$$

We now sketch a proof of how to adapt the argument for arbitrary smooth $\lambda$ satisfying $\frac{1}{\Lambda }<\lambda <\Lambda$. Let $\phi _1,\phi _2,\phi _3$ be smooth functions of z such that

$$\begin{aligned} \phi _1+\phi _2+\phi _3\equiv 1 \quad \forall z\in [0,h] \end{aligned}$$

and

$$\begin{aligned} \phi _1\in C_c^\infty (-\delta ,2\delta ),\quad \phi _2\in C_c^\infty (\delta ,h-\delta ),\quad \phi _3\in C_c^\infty (h-2\delta ,h+\delta ) \end{aligned}$$

for $\delta$ to be chosen later. Because the proofs of Lemma 3.5 and Lemma 3.6 rely only on the variational structure, the difference quotient technique applies as well to general elliptic operators in divergence form (see for example sections 6.3 or 8.3 of Evans [14]). Since $\partial _\nu (\phi _2 u) \equiv 0$, it follows that $\phi _2 u \in H^\frac{3}{2}(\Omega \times [\delta ,h-\delta ])$.

We focus now on $\phi _1 u$; the argument for $\phi _3 u$ is similar. The goal is to perform a change of variables in z such that the elliptic operator after changing variables is given by the standard Laplacian plus lower order terms depending on the change of variables. By writing

$$\begin{aligned} \partial _z ( \lambda \partial _z u) = \lambda \partial _{zz}u + \partial _z \lambda \partial _z u, \end{aligned}$$

notice that we can absorb the first order term $\partial _z \lambda \partial _z u$ into the right hand side, which we rename ${\tilde{f}}$. Then consider the ordinary differential equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta '(z') = \root \of {\lambda (\theta (z'))} &{}\quad z\in [0,\delta '] \\ \theta (0)=0. \end{array}\right. } \end{aligned}$$

By the Cauchy-Lipschitz theorem, for $\delta '$ small enough there exists a unique smooth solution $\theta$ which, by the positivity of $\lambda$, is a bijection between $[0,\delta ']$ and $[0,\theta (\delta ')]$. Choose $\delta <\frac{\theta (\delta ')}{2}$. Then

$$\begin{aligned} \partial _{z'z'}(u(x,y,\theta (z')))&= u_{33}(x,y,\theta (z'))(\theta '(z))^2 + u_3(x,y,\theta (z'))\theta ''(z') \\&= u_{33}(x,y,\theta (z'))\lambda (\theta (z')) + u_3(x,y,\theta (z'))\theta ''(z'). \end{aligned}$$

Absorbing the second term $u_3(x,y,\theta (z'))\theta ''(z')$ into the right hand side, (up to the effect of the localization $\phi _1$) the elliptic equation becomes

$$\begin{aligned} {\overline{\Delta }}(u\circ \theta ) + \partial _{z'z'}(u\circ \theta ) = {\tilde{f}}\circ \theta - (u_3\circ \theta )\theta '', \end{aligned}$$

and we can repeat the original argument to show that $\phi _1 u \in H^\frac{3}{2}(\Omega \times [0,2\delta ])$. Repeating the argument for $\phi _3 u$ and summing finishes the proof. $\quad \square$

4 Proof of Theorem 1.1

4.1 Approximate solutions

First, we adjust the initial data and forcing terms. Let $\eta _\varepsilon$ be a standard ${\mathbb {R}}^3$ mollifier supported in a ball of radius $\varepsilon$. Define the extension of f to ${\mathbb {R}}^3$ by

$$\begin{aligned} f_E(x,y,z) = {\left\{ \begin{array}{ll} f_0(x,y,z) &{}\quad (x,y,z)\in \Omega \times [0,h] \\ 0 &{}\quad \text {otherwise},\\ \end{array}\right. } \end{aligned}$$

and mollify by setting $f_\varepsilon := f_E *\eta _\varepsilon$. After similarly extending $a_L(t)$ to ${\mathbb {R}}^3$ and $g, a_\nu (t)$ to ${\mathbb {R}}^2\times \{0,h\}$ by zero and mollifying (time by time for the forcing terms), we obtain spatially smooth (for example $a_{L,\varepsilon }\in L^1([0,T];C^k({\mathbb {R}}^3))$ for any k) sequences of functions such that the following convergences hold:

$$\begin{aligned} \begin{array}{ll} f_\varepsilon \rightarrow f_0 &{}\quad \text {in} \quad L^2(\Omega \times [0,h]) \\ g_\varepsilon \rightarrow g_0&{} \quad \text {in} \quad L^2(\Omega \times \{0,h\}) \\ a_{L,\varepsilon } \rightarrow a_L &{}\quad \text {in} \quad L^1 \left( [0,T];L^2(\Omega \times [0,h])\right) \\ a_{\nu ,\varepsilon } \rightarrow a_\nu &{}\quad \text {in} \quad L^1 \left( [0,T];L^2(\Omega \times \{0,h\})\right) . \end{array} \end{aligned}$$

We define the approximate (QG) solution operators $S_\varepsilon : C\left( [0,T];{\mathbb {H}}\right) \rightarrow C\left( [0,T];{\mathbb {H}}\right)$ for $\varepsilon >0$ in several steps. These operators shall provide solutions to linear transport equations with mollified velocity fields.

• Step 1: Let $P\in C\left( [0,T];{\mathbb {H}}\right)$, and let c(z) be the lateral boundary values of P as usual. We extend P(t) to ${\mathbb {R}}^3$ for each time in a way which allows for a simple construction of a smooth, stratified velocity field from ${\overline{\nabla }}^\perp P$ which is supported in a small neighborhood of $\Omega \times [0,h]$.

  $$\begin{aligned} P_e(x,y,z) = {\left\{ \begin{array}{ll} P(x,y,z) &{} (x,y,z)\in \Omega \times [0,h] \\ c(z) &{} (x,y,z)\in \Omega ^{{\mathsf {C}}}\times [0,h]\\ \end{array}\right. } \end{aligned}$$

  and

  $$\begin{aligned} P_E(x,y,z) = {\left\{ \begin{array}{ll} P_e(x,y,z) &{} (x,y,z)\in {\mathbb {R}}^2\times [0,h] \\ P_e(x,y,0) &{} (x,y,z)\in {\mathbb {R}}^2\times [-\infty ,0] \\ P_e(x,y,h) &{} (x,y,z)\in {\mathbb {R}}^2\times [h,\infty ]. \\ \end{array}\right. } \end{aligned}$$

  Mollify $P_E$ by setting $P_\varepsilon := P_E *\eta _\varepsilon$.

• Step 2: Consider the transport equations for $F_\varepsilon$ and $G_\varepsilon$ given by

  $$\begin{aligned} {\left\{ \begin{array}{ll} \left( \partial _t + {\overline{\nabla }}^\perp P_{\varepsilon } \cdot {\overline{\nabla }}\right) \left( F_\varepsilon + \beta _0 y\right) = a_{L,\varepsilon } &{}\quad {\mathbb {R}}^2\times [0,h]\times [0,\infty ) \\ \left( \partial _t + {\overline{\nabla }}^\perp P_{\varepsilon }\cdot {\overline{\nabla }}\right) G_\varepsilon =a_{\nu ,\varepsilon } &{}\quad {\mathbb {R}}^2\times \{0,h\}\times [0,\infty )\\ F_\varepsilon = f_\epsilon &{}\quad t=0 \\ G_\varepsilon = g_\epsilon &{}\quad t=0. \\ \end{array}\right. } \end{aligned}$$

  Since the initial data, forcing terms, and velocity fields are all smooth, we can produce global in time solutions $F_\varepsilon$ and $G_\varepsilon$ by the method of characteristics. Notice that $F_\varepsilon$ and $G_\varepsilon$ are defined for $(x,y)\in {\mathbb {R}}^2$ but supported in a neighborhood of order $\varepsilon$ around $\Omega$.

• Step 3: At each time $t\geqq 0$, apply Lemma 3.1 to define $Q_\varepsilon (t)$ as the solution to

  $$\begin{aligned} B(Q_\varepsilon (t), v)&:= \int _{\Omega \times [0,h]} {\tilde{\nabla }} Q_\varepsilon (t) \cdot \nabla v \\&= -\int _{\Omega \times [0,h]} -F_\varepsilon (t) v + \int _{\Omega \times \{0,h\}} \lambda G_\varepsilon (t) v + \int _0^h v|_{\partial \Omega \times [0,h]} j_0.\\&=: F(v) \end{aligned}$$

  Define $S_\varepsilon (P) := Q_\varepsilon$. We remark that because $F_\varepsilon$ and $G_\varepsilon$ are defined as solutions to transport equations for $(x,y)\in {\mathbb {R}}^2$ rather than $\Omega$, the compatibility condition is lost. However, Lemma 3.1 still produces a solution to the abstract variational problem, and we will recover the compatibility condition in the limit.

In search of fixed points, we will show that the operators $\{ S_\varepsilon \}_{\epsilon >0}$ are compact, continuous operators from $C\left( [0,T];{\mathbb {H}}\right)$ to itself with bounded range. Continuity of the operators results from examining the characteristics of the mollified transport equations, while the proof of compactness will require Theorem 3.2 and the Aubin-Lions lemma. We split the argument into three lemmas.

Lemma 4.1

(Continuity) The operator $S_\varepsilon$ is continuous from $C\left( [0,T];{\mathbb {H}}\right)$ to itself, with modulus of continuity dependent on $\varepsilon$.

Proof

Let

$$\begin{aligned} P_n \rightarrow P \quad \text {in} \quad C\left( [0,T];{\mathbb {H}}\right) . \end{aligned}$$

Define $S_\varepsilon (P_n):= Q_{n,\varepsilon }$. Using the notation from the construction of the operators $S_\varepsilon$, let $F_{n,\varepsilon }$ and $G_{n,\varepsilon }$ be the solutions to the transport equations with mollified velocity fields ${\overline{\nabla }}^\perp P_{n,\varepsilon }$. Applying Lemma 3.1, for fixed $t\in [0,T]$,

$$\begin{aligned}&\Vert \left( Q_{n_1,\varepsilon }-Q_{n_2,\varepsilon } \right) (t) \Vert _{{\mathbb {H}}}\\&\quad \lesssim \bigg ( \Vert (F_{n_1,\varepsilon }-F_{n_2,\varepsilon })(t)\Vert _{L^2(\Omega \times [0,h])} + \Vert (G_{n_1,\varepsilon }-G_{n_2,\varepsilon })(t)\Vert _{L^2(\Omega \times \{0,h\})} \bigg ). \end{aligned}$$

Therefore, it suffices to show that

$$\begin{aligned} \sup _{t\in [0,T]} \left\{ \Vert (F_{n_1,\varepsilon }-F_{n_2,\varepsilon })(t)\Vert _{L^2(\Omega \times [0,h])} + \Vert (G_{n_1,\varepsilon }-G_{n_2,\varepsilon })(t)\Vert _{L^2(\Omega \times \{0,h\})} \right\} \rightarrow 0 \end{aligned}$$

(21)

as $n_1,n_2 \rightarrow \infty$.

First, notice that due to the mollification, given $k\in {\mathbb {N}}$, there exist constants $C(\varepsilon ,k)$ depending on $\varepsilon , k$ such that

$$\begin{aligned} \Vert {\overline{\nabla }}^\perp ( P_{n_1,\varepsilon }-P_{n_2,\varepsilon } )\Vert _{L^\infty \left( [0,T];C^k(\Omega \times [0,h])\right) } \leqq C(\varepsilon ,k) \Vert P_{n_1}-P_{n_2}\Vert _{C\left( [0,T];{\mathbb {H}}\right) }. \end{aligned}$$

(22)

Fix $(t,x,y,z)\in [0,T]\times {\mathbb {R}}^2\times [0,h]$, and let $\Gamma _{n_i}$ for $i=1,2$ solve

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{\Gamma }}_{n_i}(s) = {\overline{\nabla }}^\perp P_{n_i,\varepsilon }\left( s,\Gamma _{n_i}(s)\right) &{}\quad s\in [0,t] \\ \Gamma _{n_i}(t)=(x,y,z). \\ \end{array}\right. } \end{aligned}$$

Then

$$\begin{aligned} F_{n_i,\varepsilon }(t,x,y,z) = f_\varepsilon (\Gamma _{n_i}(t)) + \int _0^t a_{L,\varepsilon }(\Gamma _{n_i}(s)) \,\mathrm{d}s, \end{aligned}$$

and

$$\begin{aligned} F_{n_1}(t,x,y,z) - F_{n_2}(t,x,y,z)= & {} f_\varepsilon (\Gamma _{n_1}(t)) - f_\varepsilon (\Gamma _{n_2}(t))\\&+ \int _0^t a_{L,\varepsilon }(\Gamma _{n_1}(s)) - a_{L,\varepsilon }(\Gamma _{n_2}(s)) \,\mathrm{d}s. \end{aligned}$$

Applying (22) and using the smoothness of $f_\varepsilon$ and $a_{L,\varepsilon }$ shows that as $n_1,n_2 \rightarrow \infty$, $F_{n_1}(t,x,y,z) - F_{n_2}(t,x,y,z)$ converges to 0 uniformly for $(t,x,y,z) \in [0,T]\times {\mathbb {R}}^2\times [0,h]$. Arguing similarly for $G_{n_1,\varepsilon },G_{n_2,\varepsilon }$ then shows (21). $\quad \square$

Before continuing, we remind the reader that the precise definition of ${\mathbb {H}}$ is as a vector space of gradients of mean-zero functions, and so for the sake of clarity we will write $\nabla u$ for members of ${\mathbb {H}}$ or ${\mathbb {H}}^*$ throughout the next two lemmas.

Lemma 4.2

(Time Derivative Bounds) Let $\nabla P\in C\left( [0,T];{\mathbb {H}}\right)$ with mollified velocity field ${\overline{\nabla }}^\perp P_\varepsilon$, and put $S_\varepsilon (\nabla P):={\nabla }Q_\varepsilon$. Considering ${\nabla }Q_\varepsilon (t)$ as an element of ${\mathbb {H}}^*$ acting by the rule

$$\begin{aligned} \nabla v \rightarrow \langle \nabla Q(t), \nabla v \rangle _{\mathbb {H}}\quad \forall \nabla v \in {\mathbb {H}}, \end{aligned}$$

we have that $\partial _t {\nabla } Q_\varepsilon$ is a bounded linear functional in $L^\infty ([0,T];({\mathbb {H}}\cap H^2(\Omega \times [0,h]))^*)$ and

$$\begin{aligned}&\Vert \partial _t {\nabla }Q_\varepsilon \Vert _{L^\infty \left( [0,T];({\mathbb {H}}\cap H^2(\Omega \times [0,h]))^*\right) }\\&\quad \leqq C\left( f_0, g_0, a_L, a_\nu , \beta _0,h,\Vert {\overline{\nabla }}^\perp P_{\varepsilon }\Vert _{L^\infty ([0,T]\times [0,h];L^2(\Omega ))} \right) . \end{aligned}$$

Proof

The distributional time derivative of ${\nabla }Q_\varepsilon (t)$ is defined by the equality

$$\begin{aligned} \langle \partial _t {\nabla }Q_\varepsilon , \phi \rangle := -\int _0^T \phi '(t) {\nabla }Q_\varepsilon (t) \,\mathrm{d}t \end{aligned}$$

for all $\phi \in C_c^\infty (0,T)$. To show that $\partial _t {\nabla }Q_\varepsilon (t) \in L^\infty ([0,T];({\mathbb {H}}\cap H^2(\Omega \times [0,h]))^*))$, we test against functions $\nabla v\in {\mathbb {H}}\cap H^2(\Omega \times [0,h])$. First, recall the definitions of $F_{\varepsilon }$ and $G_{\varepsilon }$ as the solutions to the linear transport equations with mollified velocity ${\overline{\nabla }}^\perp P_{\varepsilon }$ as in Step 2. Then we have

$$\begin{aligned}&-\int _0^T \phi '(t) \langle \nabla Q_{\varepsilon }(t), \nabla v \rangle _{\mathbb {H}}\,\mathrm{d}t \nonumber \\&\quad = -\int _0^T \int _{\Omega \times [0,h]} {\tilde{\nabla }} Q_{\varepsilon }(t,x,y,z) \cdot \nabla v(x,y,z)\phi '(t) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t \nonumber \\&\quad = -\int _0^T B(\nabla Q_{\varepsilon }(t), \phi '(t)\nabla v) \,\mathrm{d}t \nonumber \\&\quad = -\int _0^T F(\phi '(t)\nabla v) \,\mathrm{d}t \nonumber \\&\quad = -\int _0^T \bigg ( -\int _{\Omega \times [0,h]} F_{\varepsilon }(t,x,y,z) \phi '(t)v(x) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \nonumber \\&\qquad +\,\int _{\Omega \times \{0,h\}} \lambda G_{\varepsilon }(t,x,y,z) \phi '(t)v(x,y,z) \,\mathrm{d}x\,\mathrm{d}y \nonumber \\&\qquad + \int _0^h j_0(z)v|_{\partial \Omega }(z)\phi '(t)\,\mathrm{d}z\bigg )\,\mathrm{d}t. \end{aligned}$$

(23)

Since $F_{\varepsilon }$ and $G_{\varepsilon }$ are classical solutions to transport equations, we have that

$$\begin{aligned} \int _0^T \int _{{\Omega }\times [0,h]} \left( \left( v \phi ' + {\overline{\nabla }}^\perp P_{\varepsilon } \cdot {\overline{\nabla }}v \phi \right) ( F_{\varepsilon } + \beta _0 y ) + \phi v a_{L,\varepsilon } \right) = 0 \end{aligned}$$

(24)

and

$$\begin{aligned} \int _0^T \int _{{\Omega }\times \{0,h\}} \left( \left( v \phi ' + {\overline{\nabla }}^\perp P_{\varepsilon } \cdot {\overline{\nabla }}v \phi \right) G_{\varepsilon } + \phi v a_{\nu ,\varepsilon } \right) =0 . \end{aligned}$$

(25)

Plugging (24) and (25) into (23) and noticing that

$$\begin{aligned} \int _0^T \int _0^h j_0 v \phi ' = - \int _0^T \int _0^h (j_0 v)' \phi = 0 \end{aligned}$$

gives

$$\begin{aligned}&-\int _0^T \phi '(t) \langle \nabla Q_{\varepsilon }(t), \nabla v \rangle _{\mathbb {H}}\,\mathrm{d}t \\&\quad = -\int _0^T \int _{{\Omega }\times [0,h]} \left( \left( {\overline{\nabla }}^\perp P_{\varepsilon } \cdot {\overline{\nabla }}v \phi \right) ( F_{\varepsilon } + \beta _0 y ) + \phi v a_{L,\varepsilon } \right) \\&\qquad +\, \int _0^T \int _{{\Omega }\times \{0,h\}} \lambda \left( \left( {\overline{\nabla }}^\perp P_{\varepsilon } \cdot {\overline{\nabla }}v \phi \right) G_{\varepsilon } + \phi v a_{\nu ,\varepsilon } \right) \\&\quad \leqq \Vert {\overline{\nabla }}^\perp P_{\varepsilon }\Vert _{L^\infty ([0,T]\times [0,h];L^2(\Omega ))} \Vert {\overline{\nabla }}v \Vert _{L^\infty (\Omega \times [0,h])} \Vert \phi \Vert _{L^\infty (0,T)}\\&\qquad \qquad \left( \Vert F_{\varepsilon }\Vert _{L^\infty \left( [0,T];L^2(\Omega \times [0,h])\right) }+ \beta _0 h \right) \\&\qquad +\,\Vert \phi \Vert _{L^\infty (0,T)}\Vert v\Vert _{L^\infty (\Omega \times [0,h])} \Vert a_{L,\varepsilon }\Vert _{L^1([0,T];L^2(\Omega \times [0,h]))} \\&\qquad +\, \Lambda \Vert {\overline{\nabla }}^\perp P_{\varepsilon }\Vert _{L^\infty ([0,T]\times [0,h];L^2(\Omega ))} \Vert {\overline{\nabla }}v \Vert _{L^\infty (\Omega \times \{0,h\})} \\&\qquad \times \Vert \phi \Vert _{L^\infty (0,T)} \Vert G_{\varepsilon }\Vert _{L^\infty ([0,T];L^2(\Omega \times \{0,h\}))} \\&\qquad +\,\Vert \phi \Vert _{L^\infty (0,T)}\Vert v\Vert _{L^\infty (\Omega \times \{0,h\})} \Vert a_{\nu ,\varepsilon }\Vert _{L^1([0,T];L^2(\Omega \times [0,h]))} \\&\quad \leqq \Vert \nabla v \Vert _{H^2(\Omega \times [0,h])}\Vert \phi \Vert _{L^\infty (0,T)} \left( 1+\Vert {\overline{\nabla }}^\perp P_{\varepsilon }\Vert _{L^\infty ([0,T]\times [0,h];L^2(\Omega ))}\right) \\&\qquad \times \, \left( 1+\Vert a_{L}\Vert _{L^1([0,T];L^2(\Omega \times [0,h]))} +\Vert a_{\nu }\Vert _{L^1([0,T];L^2(\Omega \times [0,h]))}\right) \\&\qquad \times \,\left( 1+ \Vert f_0\Vert _{L^2(\Omega \times [0,h])}+ \beta _0 h + \Lambda \Vert g_0\Vert _{L^2(\Omega \times \{0,h\})} \right) . \end{aligned}$$

$\square$

Lemma 4.3

(Compactness) Let $\{\varepsilon _n\}_{n=1}^\infty$ be a sequence of positive numbers, $\nabla P_n$ be a sequence of functions in $C([0,T];{\mathbb {H}})$, and $S_{\varepsilon _n}(\nabla P_n):= \nabla Q_n$. If there exists M such that the mollified velocity fields ${\overline{\nabla }}^\perp P_{n,\varepsilon _n}$ satisfy

$$\begin{aligned} \sup _n \Vert {\overline{\nabla }}^\perp P_{n,\varepsilon _n}\Vert _{L^\infty ([0,T]\times [0,h];L^2(\Omega ))} < M, \end{aligned}$$

then, up to a subsequence, there exists $\nabla Q\in C([0,T];{\mathbb {H}})$ such that $\nabla Q_n$ converges strongly in $C([0,T];{\mathbb {H}})$ to $\nabla Q$.

Proof

To set notation, $\nabla Q_{n}$ is the solution to the variational problem

$$\begin{aligned} B_n(\nabla Q_{n}, \nabla v) = F_n(\nabla v), \quad \nabla v \in {\mathbb {H}}\end{aligned}$$

described in Step 3. Define the Banach spaces

$$\begin{aligned} {\mathcal {B}}_1 = {\mathbb {H}}^*,\quad {\mathcal {B}}_0 = {\mathbb {H}}\cap H^\frac{1}{2}(\Omega \times [0,h]),\quad {\mathcal {B}}_2 = \left( {\mathbb {H}}\cap H^2(\Omega \times [0,h]) \right) ^*. \end{aligned}$$

We set $\nabla u^*\in {\mathbb {H}}^*$ as the linear functional on ${\mathbb {H}}$ defined by $\nabla v \rightarrow \langle \nabla u,\nabla v \rangle _{\mathbb {H}}$. This identification provides an isometric linear bijection between ${\mathbb {H}}$ and ${\mathbb {H}}^*$. Then by the Rellich-Kondrachov theorem and the observed isomorphism, the embedding of ${\mathcal {B}}_0$ into ${\mathcal {B}}_1$ is compact. The inclusion map from ${\mathcal {B}}_1$ to ${\mathcal {B}}_2$ is continuous as well. Applying Lemma 3.1, invoking the isomorphism between ${\mathbb {H}}$ and ${\mathbb {H}}^*$, and using the divergence free property of the mollified transport equations, we have that $\nabla Q_{n}^* \in C\left( [0,T]; {\mathbb {H}}^* \right)$, and for $t\in [0,T]$,

$$\begin{aligned} \Vert \nabla Q_{n}^*(t) \Vert _{{\mathbb {H}}^*}&\leqq C(\Omega ,h,\lambda ,\beta _0)\left( 1 + \Vert f_0 \Vert _{L^2} + \Vert g_0 \Vert _{L^2} + \Vert j_0 \Vert _{L^2} \nonumber \right. \\&\left. \quad +\Vert a_L \Vert _{L^1\left( [0,T];L^2\right) } + \Vert a_\nu \Vert _{L^1\left( [0,T];L^2\right) } \right) . \end{aligned}$$

(26)

In addition, Theorem 3.2 provides the bound

$$\begin{aligned} \Vert {\nabla }Q_{n}(t) \Vert _{H^\frac{1}{2}(\Omega \times [0,h])} \leqq&C(\Omega ,h,\lambda ,M) \left( \Vert f_0 \Vert _{L^2} + \Vert g_0 \Vert _{L^2} + \Vert j_0 \Vert _{L^2} \nonumber \right. \\&\left. +\Vert a_L \Vert _{L^1\left( [0,T];L^2\right) } + \Vert a_\nu \Vert _{L^1\left( [0,T];L^2\right) } \right) , \end{aligned}$$

(27)

showing that $\nabla Q_{n} \in L^\infty ([0,T];{\mathcal {B}}_0)$. Note that the constant in (26) does not depend on M. By Lemma 4.2 and the existence of the constant M, $\partial _t (\nabla Q_{n}^*)$ is a sequence of operators bounded in $L^\infty ([0,T];{\mathcal {B}}_2)$, and the assumptions of the Aubins-Lions lemma are satisfied. We have then that $\nabla Q_n^*$ is precompact in $C \left( [0,T];{\mathbb {H}}^*\right)$, and thus $\nabla Q_{n}$ is precompact in $C \left( [0,T];{\mathbb {H}}\right)$. $\quad \square$

Corollary 4.4

(Fixed Points) Each operator $S_\varepsilon$ has a fixed point $\Psi _\varepsilon$.

Proof

Lemma 4.1 shows that $S_\varepsilon$ is continuous. By the mollification of the velocity fields, there exists $C(\varepsilon )$ such that for all $P\in {C([0,T];{\mathbb {H}})}$,

$$\begin{aligned} \Vert {\overline{\nabla }}^\perp P_{\varepsilon }\Vert _{L^\infty ([0,T]\times [0,h];L^2(\Omega ))} \leqq C(\varepsilon ) \Vert P \Vert _{C([0,T];{\mathbb {H}})}. \end{aligned}$$

Then by Lemma 4.3 with $\varepsilon _n=\epsilon$ for all n, $S_\varepsilon$ is a compact operator. By (26), the range of $S_\varepsilon$ is bounded. Therefore, we can apply the Leray-Schauder fixed point theorem (see Evans [14]) to obtain a fixed point $\Psi _\varepsilon$. $\quad \square$

4.2 Passing to the Limit

Consider the sequence of fixed points $\Psi _\varepsilon$ to the operators $S_\varepsilon$. By definition, $S_\varepsilon (\Psi _\epsilon ) = \Psi _\varepsilon$, and therefore $\Psi _\varepsilon$ solves the variational problem

$$\begin{aligned} B_{\Psi _\varepsilon }(\Psi _\epsilon (t), v)&= \int _{\Omega \times [0,h]} {\tilde{\nabla }} \Psi _\varepsilon (t) \cdot \nabla v \\&= -\int _{\Omega \times [0,h]} -F_\varepsilon (t) v + \int _{\Omega \times \{0,h\}} \lambda G_\varepsilon (t) v + \int _0^h v|_{\partial \Omega \times [0,h]} j_0.\\&= F_{\Psi _\varepsilon }(v) \end{aligned}$$

Let us extract a subsequence which we index by $n \in {\mathbb {N}}$ such that $F_{\varepsilon _n}$ converges weakly to F in $L^\infty \left( [0,T];L^2(\Omega \times [0,h])\right)$, and $G_{\varepsilon _n}$ converges weakly to G in $L^\infty ([0,T];L^2(\Omega \times \{0,h\}) )$.

Recall that in Step 1, $\Psi _{\varepsilon _n}:\Omega \times [0,h]\rightarrow {\mathbb {R}}$ was extended to $\Psi _{\varepsilon _n, E}:{\mathbb {R}}^3\rightarrow {\mathbb {R}}$ and then mollified at length scale ${\varepsilon _n}$ to produce a smooth velocity field $\Psi _{\varepsilon _n,E}*\eta _{\varepsilon _n}$. The following technical lemma regarding both the convergence of $\Psi _{\varepsilon _n}$ and the mollified velocity fields $\Psi _{\varepsilon _n,E}*\eta _{\varepsilon _n}$ shall be useful:

Lemma 4.5

• (1) Up to a subsequence, $\Psi _{\varepsilon _n}$ converges strongly to $\Psi$ in $C([0,T];{\mathbb {H}})$

• (2) For any compact subdomain ${\tilde{\Omega }} \subset \Omega$, ${\overline{\nabla }}^\perp \Psi _{\varepsilon _n,E}*\eta _{\varepsilon _n}$ converges strongly up to a subsequence to ${\overline{\nabla }}^\perp \Psi$ in $C([0,T];L^2({\tilde{\Omega }}\times [0,h]))$.

Proof

To show (1), we consider Lemma 4.3 with $P_n=\Psi _{\varepsilon _n}=Q_n$. By (27)

$$\begin{aligned}&\sup _n \Vert {\tilde{\nabla }}\Psi _{\varepsilon _n} \Vert _{L^\infty ([0,T];H^\frac{1}{2}(\Omega \times [0,h]))} \leqq C(\Omega ,h,\lambda ) \\&\qquad \times \left( \Vert f_0 \Vert _{L^2} + \Vert g_0 \Vert _{L^2} + \Vert j_0 \Vert _{L^2} +\Vert a_L \Vert _{L^1\left( [0,T];L^2\right) } + \Vert a_\nu \Vert _{L^1\left( [0,T];L^2\right) } \right) . \end{aligned}$$

Taking the trace (of $\Psi _{\varepsilon _n}$ as a function in $H^\frac{3}{2}(\Omega \times (0,h))$ to $\Psi |_{z} \in H^1(\{z\}\times \Omega )$ for $z\in [0,h]$) then shows that

$$\begin{aligned}&\sup _n \Vert {\overline{\nabla }}^\perp \Psi _{\varepsilon _n}\Vert _{L^\infty ([0,T]\times [0,h];L^2(\Omega ))} \leqq C(\Omega ,h,\lambda )\times \nonumber \\&\qquad \left( \Vert f_0 \Vert _{L^2} + \Vert g_0 \Vert _{L^2} + \Vert j_0 \Vert _{L^2} +\Vert a_L \Vert _{L^1\left( [0,T];L^2\right) } + \Vert a_\nu \Vert _{L^1\left( [0,T];L^2\right) } \right) . \end{aligned}$$

(28)

By construction of the extension $\Psi _{\varepsilon _n,E}$,

$$\begin{aligned} {\overline{\nabla }}^\perp \Psi _{\varepsilon _n,E}(x,y,z) = {\left\{ \begin{array}{ll} {\overline{\nabla }}^\perp \Psi _{\varepsilon _n}(z) &{} (x,y,z)\in {\tilde{\Omega }}\times [0,h] \\ {\overline{\nabla }}^\perp \Psi _{\varepsilon _n}(0) &{} (x,y,z)\in {\tilde{\Omega }}\times (-\infty ,0]\\ {\overline{\nabla }}^\perp \Psi _{\varepsilon _n}(h) &{} (x,y,z)\in {\tilde{\Omega }}\times [h,\infty ), \end{array}\right. } \end{aligned}$$

showing that ${\overline{\nabla }}^\perp \Psi _{\varepsilon _n, E}$ is uniformly bounded in n in $L^\infty ([0,T]\times [-\varepsilon _n,h+\epsilon _n];L^2(\Omega ))$. Therefore,

$$\begin{aligned} \sup _{n} \Vert {\overline{\nabla }}^\perp \Psi _{\varepsilon _n,E}(t)*\eta _{\varepsilon _n}\Vert _{L^\infty ([0,T]\times [0,h];L^2(\Omega ))}< \infty . \end{aligned}$$

(29)

Thus the assumptions of Lemma 4.3 are satisfied, and up to a subsequence, $\Psi _{\varepsilon _n}$ converges to $\Psi$ strongly in $C([0,T];{\mathbb {H}})$.

Moving to (2), let ${\tilde{\Omega }}$ be a fixed compact subdomain of $\Omega$. We have that for $t\in [0,T]$,

$$\begin{aligned}&\limsup _{n\rightarrow \infty }\Vert {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n,E}*\eta _{\varepsilon _n}(t)-\Psi (t))\Vert _{L^2({\tilde{\Omega }}\times [0,h])}\\&\quad \leqq \limsup _{n\rightarrow \infty }\Vert {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n,E}*\eta _{\varepsilon _n}(t)-\Psi _{\varepsilon _n}(t))\Vert _{L^2({\tilde{\Omega }}\times [0,h])}\\&\qquad +\, \limsup _{n\rightarrow \infty }\Vert {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n}(t)-\Psi (t))\Vert _{L^2({\tilde{\Omega }}\times [0,h])} \\&\quad \leqq \sup _{n} \Vert {\overline{\nabla }}^\perp \Psi _{\varepsilon _n,E}*{\eta _{\varepsilon _n}}(t)\Vert _{L^2({\tilde{\Omega }}\times ([0,\delta )\cup (h-\delta ,h]))}\\&\qquad +\sup _{n} \Vert {\overline{\nabla }}^\perp \Psi _{\varepsilon _n}(t)\Vert _{L^2({\tilde{\Omega }}\times ([0,\delta )\cup (h-\delta ,h]))} \\&\qquad +\,\limsup _{n\rightarrow \infty }\Vert {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n,E}*\eta _{\varepsilon _n}(t)-\Psi _{\varepsilon _n}(t)) \Vert _{L^2({\tilde{\Omega }}\times [\delta ,h-\delta ])}. \end{aligned}$$

By (28) and (29), the first two terms go to zero as $\delta \rightarrow 0$. Thus it suffices to show that for fixed $\delta$,

$$\begin{aligned} \limsup _{n\rightarrow \infty }\Vert {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n,E}*\eta _{\varepsilon _n}(t)-\Psi _{\varepsilon _n}(t))\Vert _{L^2({\tilde{\Omega }}\times [\delta ,h-\delta ])} =0. \end{aligned}$$

For n large enough,

$$\begin{aligned} \Psi _{\varepsilon _n,E}*\eta _{\varepsilon _n} = \Psi _{\varepsilon _n}*\eta _{\varepsilon _n} \qquad \forall (x,y,z)\in ({\tilde{\Omega }}\times [\delta ,h-\delta ]). \end{aligned}$$

By extending $\Psi _{\varepsilon _n}$ from ${\tilde{\Omega }}\times [\delta ,h-\delta ]$ to ${\mathbb {R}}^3$ using a standard Sobolev extension operator, it suffices to prove the claim for functions defined on all of ${\mathbb {R}}^3$. Using the Fourier characterization of $H^\frac{3}{2}({\mathbb {R}}^3)$, we can write

$$\begin{aligned}&\Vert {\overline{\nabla }}^\perp \Psi _{\varepsilon _n}*\eta _{\varepsilon _n}(t) - {\overline{\nabla }}^\perp \Psi _{\varepsilon _n}(t)\Vert _{L^2({\mathbb {R}}^3)}^2 \\&\quad \leqq \int _{{\mathbb {R}}^3} |\xi |^2|{\hat{\Psi }}_{\varepsilon _n}(t,\xi )|^2 |{\hat{\eta }}_{\varepsilon _n}(\xi )-1|^2 \,d\xi \\&\quad \leqq \int _{{\mathbb {R}}^3} |\xi |^2 (1+|\xi |^2)^\frac{1}{2}|{\hat{\Psi }}_{\varepsilon _n}(t,\xi )|^2 \frac{|{\hat{\eta }}(\varepsilon _n\xi )-1|^2}{(1+|\xi |^2)^\frac{1}{2}} \,d\xi \\&\quad \leqq \sup _{n} \Vert \Psi _{\varepsilon _n}(t)\Vert _{H^\frac{3}{2}({\mathbb {R}}^3)}^2 \cdot \sup _{\xi } \frac{|{\hat{\eta }}(\varepsilon _n\xi )-1|^2}{(1+|\xi |^2)^\frac{1}{2}}, \end{aligned}$$

which goes to zero uniformly in t as $n\rightarrow \infty$ since ${\hat{\eta }}$ is smooth and ${\hat{\eta }}(0)=1$, concluding the proof. $\quad \square$

We now pass to the limit to show that $\Psi$ is the solution we seek. As first utilized in [22] and then again in [18], the strong convergence at the level of $\nabla \Psi _{\varepsilon _n}$ and the reformulation of the system in terms of $\nabla \Psi _{\varepsilon _n}$ give compactness in the nonlinear term of the reformulation. Fix a test function $\phi$ as in Definition 1.2. Then

$$\begin{aligned}&-\int _0^T \int _{{\tilde{\Omega }}\times [0,h]} \left( \left( \partial _t \phi + {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n}*\eta _{\varepsilon _n}) \cdot {\overline{\nabla }}\phi \right) (F_{\varepsilon _n}+\beta _0 y) + \phi a_{L,\varepsilon _n} \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t \nonumber \\&\quad = \int _{{\tilde{\Omega }}\times [0,h]} \phi |_{t=0}f_{\varepsilon _n} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \end{aligned}$$

(30)

and

$$\begin{aligned}&\int _0^T \int _{{\tilde{\Omega }}\times \{0,h\}} \left( \left( \partial _t \phi + {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n}*\eta _{\varepsilon _n}) \cdot {\overline{\nabla }}\phi \right) G_{\varepsilon _n} + \phi a_{\nu ,\varepsilon _n} \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}t \nonumber \\&\quad = -\int _{{\tilde{\Omega }}\times \{0,h\}} \phi |_{t=0}g_{\varepsilon _n} \,\mathrm{d}x\,\mathrm{d}y. \end{aligned}$$

(31)

For each time $t>0$, let $A_{\varepsilon _n}(t)\in {\mathbb {H}}$ be the solution to

$$\begin{aligned} B_{A,n}(A_{\varepsilon _n}(t), v)&:= \int _{\Omega \times [0,h]} {\tilde{\nabla }} A_{\varepsilon _n}(t) \cdot \nabla v \\&= -\int _{\Omega \times [0,h]} -a_{L,\varepsilon _n}(t) v + \int _{\Omega \times \{0,h\}} a_{\nu ,\varepsilon _n}(t)v \\&= F_{A,n}(v). \end{aligned}$$

Using $\partial _t \phi + {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n}*\eta _{\varepsilon _n})\cdot {\overline{\nabla }}\phi$ and $\phi$ as test functions in the variational formulations for $\Psi _{\varepsilon _n}$ and $A_{\varepsilon _n}$, respectively, turns (30) and (31) into

$$\begin{aligned}&-\int _0^T \int _{{\tilde{\Omega }}\times [0,h]} \left( \left( \partial _t \nabla \phi + {\overline{\nabla }}^\perp (\Psi _{\varepsilon _n}*\eta _{\varepsilon _n}):{\overline{\nabla }}\nabla \phi \right) \cdot {\tilde{\nabla }} \Psi _{\varepsilon _n} + \nabla \phi \cdot {\tilde{\nabla }} A_{\varepsilon _n} \right) \,\nonumber \\&\qquad \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t \nonumber \\&\quad = \int _{{\tilde{\Omega }}\times [0,h]} \nabla \phi |_{t=0}\cdot {\tilde{\nabla }}\Psi _{\varepsilon _n}|_{t=0} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z. \end{aligned}$$

(32)

Applying Lemma 4.5, we pass to the limit to obtain

$$\begin{aligned}&-\int _0^T \int _{{\tilde{\Omega }}\times [0,h]} \left( \left( \partial _t \nabla \phi + {\overline{\nabla }}^\perp \Psi :{\overline{\nabla }}\nabla \phi \right) \cdot {\tilde{\nabla }} \Psi + \nabla \phi \cdot {\tilde{\nabla }} A \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t \\&\quad = \int _{{\tilde{\Omega }}\times [0,h]} \nabla \phi |_{t=0}\cdot {\tilde{\nabla }}\Psi |_{t=0} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z. \end{aligned}$$

Rearranging the variational formulation now for $\Psi$ gives

$$\begin{aligned}&-\int _0^T \int _{{\tilde{\Omega }}\times [0,h]} \left( \left( \partial _t \phi + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\phi \right) F + \phi a_L \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t \\&\quad = \int _{{\tilde{\Omega }}\times [0,h]} \phi |_{t=0}f \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \end{aligned}$$

and

$$\begin{aligned} \int _0^T \int _{{\tilde{\Omega }}\times \{0,h\}} \left( \left( \partial _t \phi + {\overline{\nabla }}^\perp \Psi \cdot {\overline{\nabla }}\phi \right) G + \phi a_\nu \right) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}t = -\int _{{\tilde{\Omega }}\times \{0,h\}} \phi |_{t=0}g\,\mathrm{d}x\,\mathrm{d}y. \end{aligned}$$

The final part of the proof consists of showing that $\Psi (t)$ solves a variational problem for all $t\in [0,T]$ which verifies the compatibility condition Definition 1.1. By construction of the approximate solution operators, $\Psi _{\varepsilon _n}(t)$ solves the variational problem with data

$$\begin{aligned} \left( F_{\varepsilon _n}(t)|_{\Omega \times [0,h]} , G_{\varepsilon _n}(t)|_{\Omega \times \{0,h\}}, j_0 \right) . \end{aligned}$$

In addition, $F_{\varepsilon _n}(t)$ and $G_{\varepsilon _n}(t)$ are supported in a neighborhood of order ${\varepsilon _n}$ around $\Omega$ for all time $t\in [0,T]$ and have uniformly bounded $L^2$ norms. Therefore, for all $\phi (t)$ smooth and compactly supported on (0, t),

$$\begin{aligned}&\int _0^t \phi (\tau ) \cdot \int _{\Omega \times [0,h]} F(\tau ) \\&\quad = \lim _{n\rightarrow \infty } \int _0^t \phi (\tau ) \cdot \int _{\Omega \times [0,h]} F_{\varepsilon _n}(\tau ) \\&\quad = \lim _{n\rightarrow \infty } \int _0^t \phi (\tau ) \cdot \left( \int _{\Omega _{\varepsilon _n}\times [0,h]} F_{\varepsilon _n}(\tau ) - \int _{(\Omega _{\varepsilon _n} \cap \Omega ^c) \times [0,h]} F_{\varepsilon _n}(\tau ) \right) \\&\quad = \lim _{n\rightarrow \infty } \int _0^t \phi (\tau ) \cdot \left( \int _{\Omega _{\varepsilon _n}\times [0,h]} f_{\varepsilon _n} + o(\varepsilon ^\frac{1}{2}) \right) \\&\quad = \int _0^t \phi (\tau )) \cdot \int _{\Omega \times [0,h]} f_0. \end{aligned}$$

A similar argument holds for G, and since $\phi$ was arbitrary, we deduce that for all t,

$$\begin{aligned} \int _{\Omega \times [0,h]} F(t) - \int _0^h j - \int _{\Omega \times \{0,h\}} \lambda G(t) = \int _{\Omega \times [0,h]} f_0 - \int _0^h j_0 - \int _{\Omega \times \{0,h\}} \lambda g_0. \end{aligned}$$

Using the assumption that $(f_0,g_0,j_0)$ and $(a_L, a_\nu )$ satisfy Definition 1.1 shows that $\Psi (t)$ solves an elliptic problem with compatible data. Then by Lemma 3.1, ${\mathcal {L}}(\Psi )=F$ and $\partial _{\nu } \Psi =G$ in the traditional weak sense.

We have thus shown that $\Psi$ satisfies part (4) of Theorem 1.1, and therefore Definition 1.2 and part (1) of Theorem 1.1. For part (2), the choice of $\Psi$ as a weak limit of functions belonging to $L^\infty \left( [0,T];{\mathbb {H}}\right)$ implies that $\Psi (t)\in {\mathbb {H}}$ for almost every t. Therefore, $\Psi$ must depend only on z on the lateral boundary, and there exists c(t, z) such that $\Psi |_{\partial \Omega \times [0,h]}=c(t,z)$ for almost every time. To show part (3), first note that in light of the $H^\frac{1}{2}\left( \Omega \times [0,h]\right)$ bound on $\nabla \Psi$, ${\overline{\nabla }}\Psi \cdot \nu _s(t)$ is well-defined in $L^2(\partial \Omega \times [0,h])$ for almost every time (using the trace of $\Psi \in H^\frac{3}{2}(\Omega \times [0,h])$ again). Assuming now that $j_0\in H^\frac{1}{2}(0,h)$, let $\alpha _n(z)$ be a compactly supported smooth function in $(\frac{1}{n},h-\frac{1}{n})$ such that $\alpha _n(z)=1$ for all $z\in (\frac{2}{n},h-\frac{2}{n})$. Applying Lemmas 3.4, 3.5, and 3.6 to $\alpha _n \Psi (t)$ (using the fact that $\alpha _n \Psi (t)$ solves an elliptic problem) shows that $\alpha _n \Psi (t) \in H^2(\Omega \times [0,h])$, and therefore ${\overline{\Delta }}\Psi (t,z)\in L^2(\Omega )$ for $z\in (\frac{2}{n},h-\frac{2}{n})$. Then

$$\begin{aligned} \int _{\Omega \times \{z\}} {\overline{\Delta }}\Psi = \int _{\partial \Omega \times \{z\}} {\overline{\nabla }}\Psi \cdot \nu _s, \end{aligned}$$

and therefore the calculation following Definition 3.1 can be justified, showing that asserted pointwise equality holds. Finally, the bounds in part (5) follow from the divergence free property of the flow and Theorem 3.2, completing the proof of the theorem.