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The Inviscid 3D Quasi-Geostrophic System on Bounded Domains

We present a formal derivation of the inviscid 3D 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).


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: Classically, the model is posed for Ω = R 2 or Ω = T 2 . We use the notation ∇ = (∂ x , ∂ y , 0), ∇ ⊥ = (−∂ y , ∂ x , 0).
The functions a L and a ν are forcing terms, and β 0 is a parameter coming from the usual β-plane approximation. The normal derivative of Ψ on Ω × {0, h} is denoted by ∂ ν Ψ. The operator L is defined by L := ∂ xx + ∂ yy + ∂ z (λ∂ z ) where λ > 0 is a smooth function depending only on z and is related to the density of the fluid. To ensure ellipticity of L we require 1 Λ ≤ λ(z) ≤ Λ for some Λ ∈ (0, ∞). Throughout the remainder of the paper, the system shall be posed on a fixed cylindrical domain Ω × [0, h] ∂ ∂t ∂Ω×{z} ∇Ψ · ν s dω = 0. (2) In other words, building a weak solution to (QG) requires choosing a datum j 0 (z) : [0, h] → R such that for all time, ∂Ω×{z} ∇Ψ(t) · ν s dω = j 0 (z).
These two conditions differentiate the model we derive from closely related models which have been studied recently by Constantin and Nguyen [10], [11] and Constantin and Ignatova [9], [8]. 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 [19] and [17], the authors used the observation that the transport equations for L(Ψ) and ∂ ν Ψ in (QG) formally preserve the norms of the data for an elliptic problem with Neumann boundary condition. Therefore, a sequence of approximate solutions Ψ n for which L(Ψ n ) and ∂ ν Ψ n converge weakly in (respectively) L ∞ t (L 2 (Ω × [0, h])) and L ∞ t (L 2 (Ω × {0, h})) will have strong convergence for ∇Ψ n in L ∞ t (L 2 (Ω × [0, h])). A key property of the (QG) system is a reformulation of the system in terms of ∇Ψ. This reformulation, first utilized extensively by Puel and the second author in [19], can be seen at the level of the primitive equations and draws an analogy to the parallel formulations of the 3D 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 ∇Ψ n to construct a weak solution. In the setting of the bounded domain Ω × [0, h], it is not immediate that imposing (1) and (2) on the lateral boundary will allow for compactness at the level of ∇Ψ n in L ∞ t (L 2 (Ω × [0, h])). Indeed, it might seem possible that because (2) only controls the average of ∇Ψ · ν s on the sides, ∇Ψ · ν s could oscillate quite badly on ∂Ω × [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 ∇Ψ · ν s ∈ L 2 (∂Ω × [0, h]) is well-defined pointwise, and ∇Ψ n converges strongly to ∇Ψ in L ∞ t (L 2 (Ω × [0, h])). To the authors knowledge, this type of boundary condition and the corresponding elliptic regularity theorem are novel.
As a result, ∆Ψ(t) ≡ 0 uniformly in time, and the entire dynamic is encoded in the equation Resnick proved global existence of weak solutions for initial data in L 2 (T 2 ) [20]. Marchand extended Resnick's result to initial data belonging to L p (R 2 ) or L p (T 2 ) for p > 4 3 [16]. 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 (−∆) α θ. The case α = 1 2 is physical and comes from considering viscous effects which produce Ekman layers at the boundary. In the critical case α = 1 2 , global regularity is known by different methods. Proofs are given by Kiselev, Nazarov, and Volberg [15], Caffarelli and the second author [5], Constantin and Vicol [7], and Kiselev and Nazarov [14]. 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 3D system with critical dissipation in [18]. 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 Ω is defined spectrally using eigenfunctions of the homogenous Dirichlet laplacian. First, Constantin and Ignatova [9], [8] 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 [10], [11] 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.
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 4 (1) and (2). At each time t, we reconstruct Ψ by solving the elliptic problem In particular, we do not require that the stream function Ψ 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 {λ n } for the homogenous Dirichlet laplacian −∆ Ω on Ω, and let θ = n a n (t)e n (x, y) be a solution to (3) posed on the bounded domain Ω. Then the stream function Ψ| z=0 is given by n e n (x, y), and the harmonic extension for z ∈ [0, ∞) is given by With this definition, Ψ vanishes uniformly on ∂Ω × [0, ∞). 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 n Ω e n (x, y) dx dy = 0 for all z > 0. One can see that this is only satisfied if a ′ n (t) Ω e n (x, y) dx dy = 0 for all n and t > 0, which cannot hold for any bounded domain Ω 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 L and prove an elliptic regularity theorem for the solution. Finally, in Section 4, we construct global weak solutions to (QG).

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 ǫ will yield the stratified system and boundary conditions (1) and (2). Given a smooth, bounded set Ω ⊂ R 2 and a fixed height h, the following equations (after rescaling) will be posed on the cylindrical domain Ω × [0, h].
We use the notation D Dt = ∂ t + u · ∇ for the material derivative, and the Coriolis force C = 2Θ sin(θ), where Θ is the angular velocity of the Earth and θ is the latitude. Here (u, v, w) is the fluid velocity, p is the pressure and ρ is the variation in density from a known background density profile̺(z). That is, the density ̺ satisfies ̺ =̺(z) + ρ(x, y, z, t).
We further assume that the density is decreasing in z and that −ρ 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 We rescale the equations in such a way so as to remove solutions which vary on a fast time scale. Therefore, we set Letting θ 0 be a central latitude, we estimate C using the linear β-plane approximation by We assume that L r 0 is O(ǫ), allowing us to keep the factor of ǫ in front of β 0 even as ǫ → 0. We scale the density variation by and the reference density by̺ = U 2 ǫ 2 Lg̺ ′ This allows us to write the density non-dimensionally as Finally, we scale the pressure by p = C 0 ULp ′ . Applying the scalings to the primitive equations, we obtain 0. Let us abuse notation and drop the primes on our scaled equations. Assume that the expansions Plugging this ansatz in, we obtain the zero-order equations The last equation follows from the first two equations, the incompressibility (which gives that w (0) z = 0), and the assumption that w (0) ≡ 0 on the top and bottom of the domain. We move now to the first order equations. Let us introduce the notation for the zero order geostrophic material derivative. The first order equations are then Then we can consolidate the first order equations as 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 L(Ψ) and ∂ ν Ψ. We now take the divergence of (4) in order to arrive at (QG). As noted, the divergence of the right hand side is zero.
x . Examining the transport term d g (∇p (0) ) and calculating ∂ z of the third component, we obtain (0) ).
Using the fact that x , the second two terms cancel each other out. The horizontal divergence (∂ x , ∂ y , 0) of d g (∇p (0) ) is easy to calculate from the stratification and the divergence free nature of the zero-order flow. We arrive at the equation z ) z + β 0 y = 0 after absorbing the β-plane term into the material derivative and defining λ = − 1 ̺z . Note that by the assumptions on the density, there exists Λ such that 1 Λ ≤ λ ≤ Λ. We shall use the notation Ψ for the stream function p (0) , allowing us to rewrite the system in the familiar form Consider now the top and bottom Ω × {0} and Ω × {h}. Let ν denote the unit normal vector on the top and bottom. Considering the equation (1) ≡ 0 on the top and bottom, and substituting the notation Ψ for the stream function, we obtain 2.3. The Lateral Boundary. Now consider the sides ∂Ω×[0, h] equipped with a horizontal normal vector ν s . First, the impermeability requires that ∇ ⊥ p (0) · ν s = 0, implying that p (0) is constant on ∂Ω × {z}. Recalling that the stream function Ψ = p (0) , we have that for some unknown function c(t, z).
Let us next take the dot product of (4) with ν s . Due to the impermeability of the boundary, In addition, is also the two-dimensional curl ∇ ⊥ of the scalar field −β 0 yp (0) . Then we have that As this is also a conservative vector field, the integral of this term around the boundary vanishes as well. Thus we are left with Using (7) shows that − ∂Ω×{z} (β 0 p (0) , 0, 0) · ν s dω 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 Utilizing once again the notation Ψ for the stream function, (8) therefore becomes Collecting (5), (6), (7), and (9), we have formally derived the following system: will be structured into the Hilbert space within which we solve the elliptic problem. Finally, the equation

The Elliptic Problem
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).
, and for each time, Ψ has mean value zero. Then we say that Ψ satisfies (2) weakly if there exists j 0 (z) : An integration by parts shows that for smooth functions of time and space, (2) is equivalent to Definition 3.1. Indeed, Thus we consider the elliptic problem for the unknown function u with data f : 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.
Using the notation∇ = (∂ x , ∂ y , λ(z)∂ z ), equip H with the inner product Define the Hilbert space H as the closure of H under the norm induced by this inner product.
By standard trace inequalities and Poincaré's inequality, we have that for γ ∈ H The coercivity and continuity of the bilinear form B is immediate from the assumptions on λ(z) and the definition of H. In addition, we have that after applying Hölder's inequality and (10). Applying the Lax-Milgram theorem, we obtain a unique solution u ∈ H to the variational problem Let us rigorously state the results of the above argument.
there exists a unique solution u ∈ H to the variational problem (V ) with If in addition (f, g, j) verifies the compatibility condition in Definition 1.1, then where ·, · denotes duality between H

11
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 γ can be used in the weak formulation, and therefore any C ∞ test function such that γ(x, y, z)| ∂Ω×[0,h] = c(z) is valid in the weak formulation. Parts (1) and (2) Rearranging the equality B(u, φ) = F (φ) and using (1) finishes the proof.
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 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 (Ω × [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 Ω 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 λ(z) ≡ 1, i.e. when L = ∆. 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 λ. By duality, we have that 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]). For non-integer s ∈ (−∞, ∞), the Stein-Weiss Classical interpolation results (see for example the work of Triebel [21], [22]) give that such that ∆u = f + c and |c| Proof. We define an operator A : maps a triple (f, g, j) to a constant c = A(f, g, j). Since H only contains test functions with mean value zero, given (x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 ) ∈ Ω × [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 ∆u(x 1 , y 1 , z 1 ) − ∆u(x 2 , y 2 , z 2 ) = f (x 1 , y 1 , z 1 ) − f (x 2 , y 2 , z 2 ). Therefore, ∆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, letf ,ḡ,j be given, each with mean value zero. Then A(f ,ḡ,j) satisfies the compatibility condition, implying ∆ū =f in a weak sense, and A(f ,ḡ,j) = 0. Therefore A depends only on Now A is a linear map from R 3 → R, and is therefore bounded. That is, Applying Hölder's inequality finishes the proof. 13 Lemma 3.4 (Effect of g). Consider the equation for g ∈H s (Ω × {0, h}), s ≥ − 1 2 . Then there exists a solution u which satisfies ∇u H s+ 1 2 (Ω×[0,h]) ≤ C(Ω, h) g Hs (Ω×{0,h}) Proof. We begin by assuming that g is smooth so that all calculations with higher derivatives are valid. For arbitrary g ∈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 g(0, x, y) = n b n e n (x, y), g(h, x, y) = n t n e n (x, y).
To sharpen this bound and obtain higher regularity estimates, we split the sum into four pieces corresponding to the four pieces of 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.
for data f ∈ L 2 (Ω × [0, h]). Then Proof. Formally, the assumptions that Regularity of ∆u would then follow from the equality Then we can write that for fixed z, Applying classical elliptic regularity theory z by z shows then that ∂ xy u, ∂ xx u, ∂ yy u ∈ L 2 (Ω × [0, h]). Thus it remains to rigorously show (15). Define Then we can write Rearranging, we have that Examining I, we have that Moving to II, we have that Combining (16) and (17) and repeating the argument but this time with a reflection over z = h, it follows that The uniformity of this inequality in ǫ allows us to pass to a weak limit as ǫ → 0 to conclude that . Regularity of ∂ xx u, ∂ xy u, and ∂ yy u follows as described before, finishing the proof of the lemma.
Lemma 3.6 (Effect of j). Let u ∈ H be a variational solution to The proof for s = 1 2 follows closely that of Lemma 3.5. Define u E and j E on [−h, h] by reflection as before, and define T ǫ and η similarly as well. In addition, let φ ǫ (z) be a one dimensional, smooth, even mollifier supported on a ball of radius ǫ around 0. Note c ′ (z) is yet not well defined as ∇u only belongs to L 2 ([0, h]) for now. However, c should satisfy c ′ (0) = c ′ (h) = 0, and we shall mollify our test function in z to take advantage of this. Thus we choose our test function to be Then we can write Rearranging, we have that Examining I, we have that Before examining II, notice that due to the compact support of η in [−h, h], we can assume without loss of generality that u E * φ ǫ | ∂Ω×[0,h] and j E * φ ǫ are smooth, compactly supported functions on [−h, h] and therefore can be expanded in Fourier series with coefficientŝ u(k)φ ǫ (k) andĵ(k)φ ǫ (k), respectively. Note also that since T ǫ ignores constants, we can assume without loss of generality thatû(0) =ĵ(0) = 0, ensuring that fractional laplacians (as Fourier multipliers) of u E and j E are well-defined on [0, h]. Furthermore, since (c E * φ) ′ (z) vanishes at 0, the reflected function u E * φ(z)| ∂Ω×[0,h] belongs to H 2 ([−h, h]). In addition, |φ ǫ (k)| ≤ 1 for all k and converges to 1 as ǫ → 0. Then we can write The last line follows from applying (10) to T ǫ u E and noticing that Combining (18) and (19) and repeating the argument but this time with a reflection over z = h, it follows that .
The uniformity of this inequality in ǫ allows us to pass to a weak limit as ǫ → 0 to conclude that .
Regularity of ∂ xx u, ∂ xy u, and ∂ yy u follows as for Lemma 3.5, finishing the case s = 1 2 . The intermediate cases follow again from interpolation.
We can now prove Theorem 3.2.
Proof of Theorem 3.2. We begin with λ ≡ 1, in which case (V ) is given by First, apply Lemma 3.4 to build a solution u 1 to By the trace estimate (10), (Ω×{0,h}) . Therefore,ũ 1 is the unique variational solution to Now define u 2 := u −ũ 1 ; u 2 is then the unique variational solution to Define u 3 to as the unique variational solution to and u 4 as the unique variational solution to 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 Combining (20) and (21), we conclude that . We now sketch a proof of how to adapt the argument for arbitrary smooth λ satisfying 1 Λ < λ < Λ. Let φ 1 , φ 2 , φ 3 be smooth functions of z such that for δ 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 [13]). Since We focus now on φ 1 u; the argument for φ 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 ∂ z (λ∂ z u) = λ∂ zz u + ∂ z λ∂ z u, notice that we can absorb the first order term ∂ z λ∂ z u into the right hand side, which we renamef . Then consider the ordinary differential equation By the Cauchy-Lipschitz theorem, for δ ′ small enough there exists a unique smooth solution θ which, by the positivity of λ, is a bijection between [0, δ ′ ] and [0, θ(δ ′ )]. Choose δ < θ(δ ′ ) 2 . Then ∂ z ′ z ′ (u(x, y, θ(z ′ ))) = u 33 (x, y, θ(z ′ ))(θ ′ (z)) 2 + u 3 (x, y, θ(z ′ ))θ ′′ (z ′ ) = u 33 (x, y, θ(z ′ ))λ(θ(z ′ )) + u 3 (x, y, θ(z ′ ))θ ′′ (z ′ ).
Absorbing the second term u 3 (x, y, θ(z ′ ))θ ′′ (z ′ ) into the right hand side, (up to the effect of the localization φ 1 ) the elliptic equation becomes and we can repeat the original argument to show that φ 1 u ∈ H

Approximate solutions.
First, we adjust the initial data and forcing terms. Let η ǫ be a standard R 3 mollifier supported in a ball of radius ǫ. Define the extension of f to R 3 by and mollify by setting f ǫ := f E * η ǫ . After similarly extending a L (t) to R 3 and g, a ν (t) to R 2 × {0, h} by zero and mollifying (time by time for the forcing terms), we obtain spatially smooth (for example a L,ǫ ∈ L 1 ([0, T ]; C k (R 3 )) for any k) sequences of functions such that the following convergences hold: Step 1: Let P ∈ C ([0, T ]; H), and let c(z) be the lateral boundary values of P as usual. We extend P (t) to R 3 for each time in a way which allows for a simple construction of a smooth, stratified velocity field from ∇ ⊥ P which is supported in a small neighborhood and Mollify P E by setting P ǫ := P E * η ǫ .
Step 2: Consider the transport equations for F ǫ and G ǫ given by Since the initial data, forcing terms, and velocity fields are all smooth, we can produce global in time solutions F ǫ and G ǫ by the method of characteristics. Notice that F ǫ and G ǫ are defined for (x, y) ∈ R 2 but supported in a neighborhood of order ǫ around Ω.
Step 3: At each time t ≥ 0, apply Lemma 3.1 to define Q ǫ (t) as the solution to Define S ǫ (P ) := Q ǫ . We remark that because F ǫ and G ǫ are defined as solutions to transport equations for (x, y) ∈ R 2 rather than Ω, 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 ǫ } ǫ>0 are compact, continuous operators from C ([0, T ]; H) 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. Define S ǫ (P n ) := Q n,ǫ . Using the notation from the construction of the operators S ǫ , let F n,ǫ and G n,ǫ be the solutions to the transport equations with mollified velocity fields ∇ ⊥ P n,ǫ .
First, notice that due to the mollification, given k ∈ N, there exist constants C(ǫ, k) depending on ǫ, k such that Applying (23) and using the smoothness of f ǫ and a L,ǫ shows that as n 1 , n 2 → ∞, F n 1 (t, x, y, z)− F n 2 (t, x, y, z) converges to 0 uniformly for (t, x, y, z) Arguing similarly for G n 1 ,ǫ , G n 2 ,ǫ then shows (22). Proof. The distributional time derivative of∇Q ǫ (t) is defined by the equality . First, recall the definitions of F ǫ and G ǫ as the solutions to the linear transport equations with mollified velocity ∇ ⊥ P ǫ as in Step 2. Then we have Since F ǫ and G ǫ are classical solutions to transport equations, we have that Plugging (25) and (26) into (24) and noticing that  Proof. To set notation, Q n is the solution to the variational problem Step 3. Define the Banach spaces We set u * ∈ H * as the linear functional on H defined by v → u, v H . This identification provides an isometric linear bijection between H and H * . Then by the Rellich-Kondrachov theorem and the observed isomorphism, the embedding of B 0 into B 1 is compact. The inclusion map from B 1 to B 2 is continuous as well. Applying Lemma 3.1, invoking the isomorphism between H and H * , and using the divergence free property of the mollified transport equations, we have that Q * n ∈ C ([0, T ]; H * ), and for t ∈ [0, T ], In addition, Theorem 3.2 provides the bound ∇ Q n (t)  Then by Lemma 4.3 with ǫ n = ǫ for all n, S ǫ is a compact operator. By (27), the range of S ǫ is bounded. Therefore, we can apply the Leray-Schauder fixed point theorem (see Evans [13]) to obtain a fixed point Ψ ǫ .

4.2.
Passing to the Limit. Consider the sequence of fixed points Ψ ǫ to the operators S ǫ . By definition, S ǫ (Ψ ǫ ) = Ψ ǫ , and therefore Ψ ǫ solves the variational problem Let us extract a subsequence which we index by n ∈ N such that F ǫn converges weakly to Recall that in Step 1, Ψ ǫn : Ω × [0, h] → R was extended to Ψ ǫn,E : R 3 → R and then mollified at length scale ǫ n to produce a smooth velocity field Ψ ǫn,E * η ǫn . The following technical lemma regarding both the convergence of Ψ ǫn and the mollified velocity fields Ψ ǫn,E * η ǫn shall be useful. Lemma 4.5.
By extending Ψ ǫn fromΩ×[δ, h−δ] to R 3 using a standard Sobolev extension operator, it suffices to prove the claim for functions defined on all of R 3 . Using the Fourier characterization of H 3 2 (R 3 ), we can write .
which goes to zero uniformly in t as n → ∞ sinceη is smooth andη(0) = 1, concluding the proof.
We now pass to the limit to show that Ψ is the solution we seek. As first utilized in [19] and then again in [17], the strong convergence at the level of ∇Ψ ǫn and the reformulation of the system in terms of ∇Ψ ǫn give compactness in the nonlinear term of the reformulation. Using ∂ t φ + ∇ ⊥ (Ψ ǫn * η ǫn ) · ∇φ and φ as test functions in the variational formulations for Ψ ǫn and A ǫn , respectively, turns (31)  ∂ t φ + ∇ ⊥ Ψ · ∇φ G + φa ν dx dy dt = − Ω ×{0,h} φ| t=0 g dx dy.
The final part of the proof consists of showing that Ψ(t) solves a variational problem for all t ∈ [0, T ] which verifies the compatibility condition Definition 1.1. By construction of the approximate solution operators, Ψ ǫn (t) solves the variational problem with data F ǫn (t)| Ω×[0,h] , G ǫn (t)| Ω×{0,h} , j 0 .
In addition, F ǫn (t) and G ǫn (t) are supported in a neighborhood of order ǫ n around Ω for all time t ∈ [0, T ]. Then using the weak convergence of F n | Ω×[0,h] to F and G n | Ω×{0,h} to G, we have  Using the assumption that (f 0 , g 0 , j 0 ) and (a L , a ν ) satisfy Definition 1.1 shows that Ψ(t) solves an elliptic problem with compatible data. Then by Lemma 3.1, L(Ψ) = F and ∂ ν Ψ = G in the traditional weak sense.