ON THE ZERO-ROSSBY LIMIT FOR THE PRIMITIVE EQUATIONS OF THE ATMOSPHERE

GUI-QIANG CHEN PING ZHANG

In Memory of Jacques-Louis Lions

Abstract. The zero-Rossby limit for the primitive equations (PE) governing the atmospheric motions is analyzed. The limit is important in geophysics for large scale models (cf. Lions [13]) and is in the level of the zero relaxation limit for nonlinear partial differential equations (cf. [5]). It is proved that, if the initial data appropriately approximate data of geostrophic type, the corresponding solutions of the simplified primitive equations approximate the solutions of the quasigeostrophic equations (QG) with order ϵ accuracy as the Rossby number ϵ goes to zero.

1. Introduction

The primitive equations (PE) governing the atmospheric motions are

(1.1)  \{ \begin{matrix}∂_{t}v + v · ∇v − div V (v) ∂_{η}v − μ_{1}△_{3}v + \frac{1}{ϵ}(v^{⊥} + ∇(Θ(θ) + Φ)) = 0, \\ ∂_{t}θ + v · ∇θ − div V (v) ∂_{η}θ − μ_{2}△_{3}θ − \frac{1}{ϵ}div V (v) = 0, \\ div ∫_{0}^{1} v(t, x, η) dη = 0,  x = (x_{1}, x_{2}) ∈ R^{2}, \end{matrix}

where V (v) = ∫_{0}^{η} v(t, x, s) ds, Θ(θ) = − ∫_{η}^{1} θ(t, x, s)ds; v = (v_{1}, v_{2}) is the horizontal velocity and v^{⊥} = (−v_{2}, v_{1}); θ is the temperature of the atmosphere; Φ(t, x) is a scalar pressure function (depending only on the time variable t and the horizontal variables x); (div, ∇, △) are the divergence, gradient, and Laplace operators in x, respectively; η ∈ [0, 1] is the ratio of pressure difference; Δ_{3} = Δ + ∂_{η}^{2}; ϵ is the Rossby number; μ_{1} is the fluid viscosity, and μ_{2} is the thermal conduction coefficient. The equations for v in (1.1) describe the hydrodynamics of the atmosphere, and the equation for θ describes the thermodynamics of the atmosphere, under the so called “Boussinesq” and hydrostatic assumptions.

For concreteness, in this paper we focus on the primitive equations on Ω = 𝕋^{2} × (0, 1), with the two-dimensional torus 𝕋^{2}, to understand the qualitative behavior of solutions of (1.1). We are mainly interested in the limiting behavior of solutions of (1.1), especially in the possibility of uniform convergence of solutions of (1.1) up to the boundaries, as the Rossby number ϵ → 0. This limit is important in geophysics for large scale models (cf. Lions [13]) and is in the level of the zero relaxation limit which has been addressed for hyperbolic systems of conservation laws in Chen-Levermore-Liu [5]. The zero-Rossby limit addressed here is also interesting from the point of view of nonlinear partial differential equations, since the equations involve the nonlocal terms.

In Lions-Teman-Wang [9], the Leray method [7, 8] was applied to constructing the global L^{2} weak solutions (v^{ϵ}, θ^{ϵ}, Φ^{ϵ}) of an initial-boundary value problem for (1.1); and, in [11], the solutions were asymptotically expanded by

1991 Mathematics Subject Classification. Primary: 35Q30,35Q35,35B25,86A10; Secondary: 35M10, 76E20.

Key words and phrases. Primitive equations, atmosphere, quasigeostrophic model, Rossby number, zero-Rossby limit, energy estimates.

v^{ϵ} = v + ϵv^{1} + ⋯,
θ^{ϵ} = θ + ϵθ^{1} + ⋯,
Φ^{ϵ} = Φ + ϵΦ^{1} + ⋯,

respectively. Then, from the ϵ^{−1} terms,

(1.2)  v^{⊥} + ∇(Θ(θ) + Φ) = 0,  div V (v) = 0,

and, from the ϵ^{0} terms,

(1.3)  \{ \begin{matrix}∂_{t}v + v · ∇v − μ_{1}△_{3}v + (v^{1})^{⊥} + ∇(Θ(θ) + Φ^{1}) = 0, \\ ∂_{t}θ + v · ∇θ − μ_{2}△_{3}θ − div V (v^{1}) = 0. \end{matrix}

A direct computation from (1.2) and (1.3) yields the quasigeostrophic equations (QG):

(1.4)  \{ \begin{matrix}∂_{t}ω + v · ∇ω − μ_{1}△_{3}ω + (μ_{1} − μ_{2})∂_{η}^{2}ω = 0, \\ △_{3}ϕ = ω, v = (−∂_{x_{2}}ϕ, ∂_{x_{1}}ϕ), θ = ∂_{η}ϕ, \end{matrix}

where ω = ∂_{x_{1}}v_{2} − ∂_{x_{2}}v_{1} + ∂_{η}θ is the potential vorticity.

For system (1.4), Bourgeois-Beale [2] established its well-posedness in the torus 𝕋^{3} for the inviscid case, and Chemin [4] showed the well-posedness of the Cauchy problem. In Section 3, we will show the existence and uniqueness of global smooth solutions for the initial-boundary value problem for (1.4) in Ω = 𝕋^{2} × (0, 1).

In [13], Lions conjectured that the solutions of the quasigeostrophic equations (1.4) are the valid approximation of the solutions of the primitive equations (1.1), provided certain restrictions on the initial data are satisfied.

In [4], Chemin verified that (1.4) is the valid approximation of the following system:

(1.5)  \{ \begin{matrix}\frac{Du^{ϵ}}{Dt} − Lu^{ϵ} + \frac{1}{ϵ}Au^{ϵ} = \frac{1}{ϵ}(−∇Φ^{ϵ}, 0),  in ℝ_{+} × ℝ^{3}, \\ div u^{ϵ} = 0, \\ u^{ϵ}|_{t=0} = u_{0}^{ϵ}, \end{matrix}

as ϵ tends to zero, under the assumption that u_{0}^{ϵ} is appropriately close to geostrophic data: there exists some pressure function ϕ_{0} such that u_{0}^{ϵ} approximates (−∂_{2}△^{−1}ϕ_{0}, ∂_{1}△^{−1}ϕ_{0}, 0, ∂_{3}△^{−1}ϕ_{0}) and μ_{2} is sufficiently close to μ_{1}, where A is an anti-symmetric matrix and

\frac{D}{Dt} = ∂_{t} + ∑_{i=1}^{3} v_{i}^{ϵ}∂_{x_{i}},  u^{ϵ} = (v_{1}^{ϵ}, v_{2}^{ϵ}, v_{3}^{ϵ}, θ^{ϵ}),  Lu^{ϵ} = (μ_{1}△v^{ϵ}, μ_{2}△θ^{ϵ}).

In [2], Bourgeois-Beale successfully obtained a similar result in the torus 𝕋^{3} for the inviscid case of (1.5).

In this paper we are concerned with the problem raised by Lions in [13] to analyze whether the quasigeostropic equations (1.4) are the valid approximation of the primitive equations (1.1). This problem for (1.1) has different features from (1.5): In particular, the relaxation terms are nonlocal and do not have the feature as the matrix A in (1.5), which causes additional difficulties. For concreteness, we focus on the following initial-boundary problem of (1.1) in the pseudo-geometrical domain Ω = 𝕋^{2} × [0, 1]:

(1.6)  (v^{ϵ}, θ^{ϵ})|_{t=0} = (v_{0}^{ϵ}, θ_{0}^{ϵ})(x, η),  (v^{ϵ}, ∂_{η}θ^{ϵ})|_{η=l} = 0, for l = 0, 1,

to solve this limit problem. Since η = 0, 1 correspond to the bottom and the top of the atmosphere, respectively, the boundary conditions θ|_{η=l} = 0, l = 0, 1, represent the case that there is no thermal transition at the bottom and the top of the atmosphere, and v|_{η=1} = 0 is the assumption that the horizontal velocity is zero at the top of the atmosphere under consideration. In general, at η = 0, the boundary conditions are much more complicated; they depend upon whether the atmosphere is above solid earth or above solid sea or liquid sea.

The analysis that we present in this paper is another step forward in dealing with the difficulties involving in the zero-Rossby limit problem. One of the main concerns for the zero-Rossby limit in the atmospheric motions is whether the zero-Rossby limit converges strongly even when the Prandtl number (the ratio of the thermal conduction coefficient and the fluid viscosity) is not one. We give a positive answer for (1.1) and (1.6). Another concern is that, in some situations, Ekman-type boundary layers occur in the atmospheric motions. In this paper, we rigorously prove that, for (1.1) and (1.6), the Ekman-type boundary layers do not occur. It would be interesting to study further the zero-Rossby limit problem for various initial-boundary value problems involving the Ekman-type boundary layers.

For the sequel presentation, we need the following spaces of functions:

V_{1} = {a ∈ C^{∞}(Ω) | a|_{η=l} = 0, ∂_{η}^{2k}a|_{η=0} = ∂_{η}^{2k}a|_{η=1}, div ∫_{0}^{1} a(x, η)dη = 0, l = 0, 1; k = 1, 2, ⋯ },

V_{2} = {b ∈ C^{∞}(Ω) | ∂_{η}^{2k+1}b|_{η=l} = 0, l = 0, 1; k = 0, 1, ⋯ },

V_{3} = {c ∈ C^{∞}(Ω) | ∂_{η}^{2k}c|_{η=l} = 0, l = 0, 1; k = 0, 1, ⋯ }, J^{α} = J_{1}^{α} × J_{2}^{α},

where J_{j}^{α} are the completion of V_{j}, j = 1, 2, in H^{α}(Ω), respectively. We denote J_{3}^{α} the completion of V_{3} in H^{α}(Ω). It has been shown in [9] and [11] that

(1.8)  (J_{1}^{0})^{⊥} = {P = ∇Ψ | Ψ ∈ H^{1}(𝕋^{2})}.

Theorem 1.3 below is our main theorem of this paper. Some related results are formulated in Theorems 1.1 and 1.2. Theorem 1.1. Let (v_{0}^{ϵ}, θ_{0}^{ϵ}) ∈ J^{2}, ∇ ∫_{0}^{1} θ_{0}^{ϵ}(x, η) dη = 0. Then there exists T^{∗} > 0 such that the initial-boundary value problem (1.1) and (1.6) has a strong solution:

u^{ϵ} ≡ (v^{ϵ}, θ^{ϵ}) ∈ L^{∞}([0, T^{∗}); J^{2}) ∩ C([0, T^{∗}); H^{α}(Ω)) ∩ L^{2}([0, T^{∗}); H^{3}(Ω)), for any α < 2,

such that

u_{t}^{ϵ} ∈ L^{2}([0, T^{∗}); L^{2}(Ω)), ∇Φ^{ϵ} ∈ L^{2}([0, T^{∗}); L^{2}(𝕋^{2})), lim_{t→0} ∥u^{ϵ}(t, ·) − u_{0}^{ϵ}(·)∥_{H^{2}(Ω)} = 0.

Theorem 1.2. Let (v_{0}, θ_{0}) = (−∂_{2}ϕ_{0}, ∂_{1}ϕ_{0}, ∂_{η}ϕ_{0}), ϕ_{0} ∈ J_{3}^{5}, and 0 < μ_{2} < 2μ_{1}. Then there exists a unique global solution (ϕ, ω, v, θ)(t, x, η) of the initial-boundary value problem (1.4) with

(1.9)  (v, θ)|_{t=0} = (v_{0}, θ_{0}),  (ϕ, ω, v, ∂_{η}θ)|_{η=l} = 0, for l = 0, 1,

such that

ϕ ∈ L_{loc}^{∞}(ℝ_{+}; J_{3}^{5}) ∩ L_{loc}^{2}(ℝ_{+}; H^{6}(Ω)), ω ∈ L_{loc}^{∞}(ℝ_{+}; J_{3}^{3}) ∩ L_{loc}^{2}(ℝ_{+}; H^{4}(Ω)),
(v, θ) ∈ L_{loc}^{∞}(ℝ_{+}; J^{4}) ∩ L_{loc}^{2}(ℝ_{+}; H^{5}(Ω)), ∂_{t}(v, θ) ∈ L_{loc}^{2}(ℝ_{+}; H^{3}(Ω)).

Theorem 1.3. Let

v_{0}^{ϵ} = v_{0} + ϵv_{0}^{a}, θ_{0}^{ϵ} = θ_{0} + ϵθ_{0}^{a}, (v_{0}, θ_{0}) = (−∂_{x_{2}}ϕ_{0}, ∂_{x_{1}}ϕ_{0}, ∂_{η}ϕ_{0}),

for some pressure function ϕ_{0} ∈ J_{3}^{5}, (v_{0}^{a}, θ_{0}^{a}) ∈ J^{2}, ∇ ∫_{0}^{1} θ_{0}^{a}(x, η) dη = 0, and 0 < μ_{2} < 2μ_{1}. Then, for any T > 0, there exists some ϵ_{0} = ϵ_{0}(T) > 0 such that, for any ϵ ≤ ϵ_{0}, the solutions constructed in Theorem 1.1 can be extended to the time interval [0, T]. Moreover,

v^{ϵ} = v + ϵv^{ϵ,a}, θ^{ϵ} = θ + ϵθ^{ϵ,a},

where (v, θ) is the solution corresponding to (1.4) constructed in Theorem 1.2, and

∥(v^{ϵ,a}, θ^{ϵ,a})(t, ·)∥_{H^{2}(Ω)}^{2} + ∫_{0}^{T} ∥∇_{3}(v^{ϵ,a}, θ^{ϵ,a})(t, ·)∥_{H^{2}(Ω)}^{2} dt ≤ K(T),

where K(T) is independent of ϵ.

Observe that the results in Theorems 1.2 and 1.3 hold even when the Prandtl number is not one, that is, μ_{1} ≠ μ_{2}.

Throughout this paper, ∇_{3} denotes the gradient and derivative with respect to (x, η), (·, ·)_{α} denotes the inner product in H^{α}(Ω) and ∥ · ∥_{α} the norm in H^{α}(Ω), and C > 0 is a generic constant which may be different at each occurrence.

To prove our main theorem (Theorem 1.3), we first show Theorem 1.1 in §2 and Theorem 1.2 in §3.

2. Proof of Theorem 1.1: Solutions of the Primitive Equations

In this section, we use the Galerkin method to establish Theorem 1.1. We first construct the Galerkin approximate solutions for (1.1) and (1.6), and then we make the L^{2} and highorder apriori estimates for the approximate solutions, which yield the convergence of the approximate solutions to the solutions of the initial-boundary value problem (1.1) and (1.6). Although the approach is standard in this section, we will use some ideas and techniques in the proof to show Theorem 1.2; and we hence outline the steps of the proof for selfcontainedness. Since we only use (·, ·)_{0} in this section, we directly denote (·, ·) as (·, ·)_{0}. Before our proof, we recall a lemma due to Lions-Teman-Wang [9, 12].

Consider the Stokes problem with nonlocal constraint:

(2.1)  \{ \begin{matrix}−Δ_{3}v + ∇Φ = f,  Ω = 𝕋^{2} × (0, 1), \\ div ∫_{0}^{1} v(x, η)dη = 0, \\ v|_{η=l} = 0,  for l = 0, 1. \end{matrix}

Lemma 2.1. Let f ∈ H^{k}, k ≥ 0. Then there exists a unique (weak) solution (v, Φ) ∈ J_{1}^{1} × L^{2}(𝕋^{2}) of (2.1) such that

(2.2)  v ∈ H^{k+2}(Ω) ∩ J_{1}, Φ ∈ H^{k+1}(𝕋^{2}), ∥v∥_{H^{k+2}} + ∥Φ∥_{H^{k+1}} ≤ C∥f∥_{H^{k}},

where C is a constant, independent of v, Φ, and f.

Lemma 2.1 indicates that Φ in the solution depends only on the first two independent variables (x_{1}, x_{2}), independent of η ∈ [0, 1]. As a corollary, if f|_{η=0} = f|_{η=1}, then Δ_{3}v|_{η=0} = Δ_{3}v|_{η=1}. Now we first prove Theorem 1.1.

Step 1: Construction of the Galerkin approximate solutions. Let P be the projection operator from L^{2}(Ω) to J_{1}^{0}(Ω). As discussed in [6], we use the Galerkin method with normalized eigenfunctions of the Stokes and Laplace operators as the basis functions in L^{2} to construct the Galerkin approximate solutions. The eigenfunctions {a^{k}} and {b^{k}} are taken orthonormally in J_{1}^{0} and J_{2}^{0}, respectively, with the corresponding eigenvalues {λ_{k}} and {ν_{k}}:

P △_{3}a^{k} = λ_{k}a^{k}, △_{3}b^{k} = ν_{k}b^{k}, ∫_{0}^{1} b^{k}(x, η) dη = 0.

Then Lemma 2.1 and the regularity of the Laplace operator indicate that {a^{k}} and {b^{k}} are in H^{∞}(Ω). Now we take the n^{th} approximate solutions:

(2.3)  v^{ϵ,n}(t, x, η) = ∑_{k=1}^{n} c_{kn}^{ϵ}(t)a^{k}(x, η),  θ^{ϵ,n}(t, x, η) = ∑_{k=1}^{n} d_{kn}^{ϵ}(t)b^{k}(x, η),

where c_{kn}^{ϵ}(t) and d_{kn}^{ϵ}(t), 1 ≤ k ≤ n, are determined by the following initial value problem:

\frac{d}{dt}(v^{ϵ,n}, a^{k}) + ((v^{ϵ,n} · ∇ − div V (v^{ϵ,n})∂_{η})v^{ϵ,n}, a^{k}) − μ_{1}(△_{3}v^{ϵ,n}, a^{k})
(2.4)  + \frac{1}{ϵ}\{((v^{ϵ,n})^{⊥}, a^{k}) + (∇Θ(θ^{ϵ,n}), a^{k})\} = 0,
\frac{d}{dt}(θ^{ϵ,n}, b^{k}) + ((v^{ϵ,n} · ∇ − div V (v^{ϵ,n}) ∂_{η})θ^{ϵ,n}, b^{k})
(2.5)  − μ_{2}(△_{3}θ^{ϵ,n}, b^{k}) − \frac{1}{ϵ}(div V (v^{ϵ,n}), b^{k}) = 0,
(2.6)  c_{kn}^{ϵ}(0) = (v_{0}^{ϵ}, a^{k}), d_{kn}^{ϵ}(0) = (θ_{0}^{ϵ}, b^{k}),  1 ≤ k ≤ n.

Step 2: L^{2} apriori estimates for the approximate solutions. Since ∥P △_{3}a∥_{0} (resp. ∥△_{3}b∥_{0}) is equivalent to ∥a∥_{2} (resp. ∥b∥_{2}) for a ∈ J_{1}^{2} (resp. b ∈ J_{2}^{2}) (cf. Lemma 2.1), as a convention, ∥P △_{3}a∥_{0} (resp. ∥△_{3}b∥_{0}) in this section can be regarded as ∥a∥_{2} (resp. ∥b∥_{2}).

Now, we multiply (2.4) by c_{ln}^{ϵ}(t) and (2.5) by d_{ln}^{ϵ}(t), respectively. Then, summing up and noting that

(2.7)  (∇Θ(θ^{ϵ,n}), v^{ϵ,n}) + (div V (v^{ϵ,n}), θ^{ϵ,n}) = 0,

we have

(2.8)  \frac{1}{2} \frac{d}{dt}∥(v^{ϵ,n}, θ^{ϵ,n})∥_{0}^{2} + μ_{1}∥∇_{3}v^{ϵ,n}∥_{0}^{2} + μ_{2}∥∇_{3}θ^{ϵ,n}∥_{0}^{2} = 0.

On the other hand, since {a^{k}} and {b^{k}} are orthonormal eigenfunctions of P △_{3} and △_{3}, respectively, {P △_{3}a^{k}} × {△_{3}b^{k}} are still orthogonal in J^{0}. Hence, the Bessel inequality implies

(2.9)  ∥(v_{0}^{ϵ,n}, θ_{0}^{ϵ,n})∥_{0} ≤ ∥(v_{0}^{ϵ}, θ_{0}^{ϵ})∥_{0},  ∥(v_{0}^{ϵ,n}, θ_{0}^{ϵ,n})∥_{2} ≤ ∥(v_{0}^{ϵ}, θ_{0}^{ϵ})∥_{2}.

Therefore, integrating (2.8) with respect to t yields

(2.10) ∥(v^{ϵ,n}, θ^{ϵ,n})(t, ·)∥_{0}^{2} + ∫_{0}^{t} (μ_{1}∥∇_{3}v^{ϵ,n}(τ, ·)∥_{0}^{2} + μ_{2}∥∇_{3}θ^{ϵ,n}(τ, ·)∥_{0}^{2}) dτ ≤ ∥(v_{0}^{ϵ}, θ_{0}^{ϵ})∥_{0}^{2}.

Step 3. High-order apriori estimates for the approximate solutions. Multiply (2.4) and (2.5) by λ_{k}^{2}c_{kn}^{ϵ}(t) and ν_{k}^{2}d_{kn}^{ϵ}(t), respectively, and then sum up. Since

(2.11)  ((v^{ϵ,n})^{⊥}, (P △_{3})^{2}v^{ϵ,n}) = 0,

we have

(2.12)

\frac{1}{2} \frac{d}{dt}∥(v^{ϵ,n}, θ^{ϵ,n})∥_{2}^{2} + μ_{1}∥∇_{3}v^{ϵ,n}∥_{2}^{2} + μ_{2}∥∇_{3}θ^{ϵ,n}∥_{2}^{2}

= − (v^{ϵ,n} · ∇v^{ϵ,n} − div V (v^{ϵ,n}) ∂_{η}v^{ϵ,n}, (P △_{3})^{2}v^{ϵ,n}) − \frac{1}{ϵ}(∇Θ(θ^{ϵ,n}), (P △_{3})^{2}v^{ϵ,n})

− (v^{ϵ,n} · ∇θ^{ϵ,n} − div V (v^{ϵ,n})∂_{η}θ^{ϵ,n}, △_{3}^{2}θ^{ϵ,n}) + \frac{1}{ϵ}(div V (v^{ϵ,n}), △_{3}^{2}θ^{ϵ,n}).

To complete the estimates, we will make repeated use of the interpolation inequality:

(2.13)  ∥a∥_{α} ≤ C∥a∥_{α_{1}}^{γ}∥a∥_{α_{2}}^{1−γ},  α = γα_{1} + (1 − γ)α_{2},

and the Sobolev inequality:

(2.14)  ∥a∥_{L^{q}} ≤ C∥a∥_{α},  if α < \frac{3}{2}, \frac{1}{q} = \frac{1}{2} − \frac{α}{3}.

By integration by parts,

(2.15)

((v^{ϵ,n} · ∇ − div V (v^{ϵ,n})∂_{η})v^{ϵ,n}, (P △_{3})^{2}v^{ϵ,n})
≤∥∇_{3}v^{ϵ,n}∥_{L^{4}}^{2}∥∇_{3}v^{ϵ,n}∥_{2} + ∥div v^{ϵ,n}∥_{L^{6}}∥∂_{η}∇_{3}v^{ϵ,n}∥_{L^{3}}∥∇_{3}v^{ϵ,n}∥_{2}
+ ∥v^{ϵ,n}∥_{L^{6}}∥∇_{3}∇v^{ϵ,n}∥_{L^{3}}∥∇_{3}v^{ϵ,n}∥_{2} + ∥∂_{η}v^{ϵ,n}∥_{L^{6}}∥∇_{3}div v^{ϵ,n}∥_{L^{3}}∥∇_{3}v^{ϵ,n}∥_{2}.

Inequalities (2.13) and (2.14) yield that, for any β > 0, there exists C = C_{β} > 0 such that

∥∇_{3}v^{ϵ,n}∥_{L^{4}}^{2}∥∇_{3}v^{ϵ,n}∥_{2} ≤ ∥∇_{3}v^{ϵ,n}∥2_{\frac{3}{4}} ∥∇_{3}v^{ϵ,n}∥_{2}

≤ C∥v^{ϵ,n}∥0^{\frac{1}{4}} ∥v^{ϵ,n}∥2^{\frac{7}{4}} ∥∇_{3}v^{ϵ,n}∥_{2} ≤ C∥v^{ϵ,n}∥2^{\frac{7}{2}} + β∥∇_{3}v^{ϵ,n}∥_{2}^{2}, (2.16)

∥v^{ϵ,n}∥_{L^{6}}∥∇_{3}∇v^{ϵ,n}∥_{L^{3}}∥∇_{3}v^{ϵ,n}∥_{2} ≤ C∥v^{ϵ,n}∥0^{\frac{1}{2}} ∥v^{ϵ,n}∥_{2}∥∇_{3}v^{ϵ,n}∥2^{\frac{3}{2}}

≤ C∥v^{ϵ,n}∥_{2}^{4} + β∥∇_{3}v^{ϵ,n}∥_{2}^{2}, (2.17)

∥∂_{η}v^{ϵ,n}∥_{L^{6}}∥∇_{3}div v^{ϵ,n}∥_{L^{3}}∥∇_{3}v^{ϵ,n}∥_{2} ≤ C∥v^{ϵ,n}∥2^{\frac{3}{2}} ∥∇_{3}v^{ϵ,n}∥2^{\frac{3}{2}}

≤ C∥v^{ϵ,n}∥_{2}^{6} + β∥∇_{3}v^{ϵ,n}∥_{2}^{2}, (2.18) ∥div v^{ϵ,n}∥_{L^{6}}∥∂_{η}∇_{3}v^{ϵ,n}∥_{L^{3}}∥∇_{3}v^{ϵ,n}∥_{2} ≤ C∥v^{ϵ,n}∥_{2}^{6} + β∥∇_{3}v^{ϵ,n}∥_{2}^{2}, (2.19)

where we used the inequality: |ab| ≤ \frac{|a|^{p}}{p} + \frac{|b|^{q}}{q} , \frac{1}{p} + \frac{1}{q} = 1, p, q > 1.

Then, summing up (2.16)-(2.19) yields

(2.20)  \begin{matrix}((v^{ϵ,n} · ∇ − div V (v^{ϵ,n})∂_{η})v^{ϵ,n}, (P △_{3})^{2}v^{ϵ,n}) \\ ≤ & C (∥v^{ϵ,n}∥2^{\frac{7}{2}} + ∥v^{ϵ,n}∥_{2}^{4} + ∥v^{ϵ,n}∥_{2}^{6}) + 4β∥∇_{3}v^{ϵ,n}∥_{2}^{2}. \end{matrix}

By the same argument,

(2.21)  \begin{matrix}((v^{ϵ,n} · ∇ − div V (v^{ϵ,n})∂_{η})θ^{ϵ,n}, △_{3}^{2}θ^{ϵ,n}) \\ ≤ & \frac{3β}{2} ∥∇_{3}θ^{ϵ,n}∥_{2}^{2} + \frac{β}{2} ∥∇_{3}v^{ϵ,n}∥_{2}^{2} \\ + C (∥v^{ϵ,n}∥2^{\frac{7}{4}} ∥θ^{ϵ,n}∥2^{\frac{7}{4}} + ∥v^{ϵ,n}∥_{2}^{2}∥θ^{ϵ,n}∥_{2}^{2} + ∥θ^{ϵ,n}∥_{2}^{4}∥v^{ϵ,n}∥_{2}^{2} + ∥v^{ϵ,n}∥_{2}^{4}∥θ^{ϵ,n}∥_{2}^{2}) . \end{matrix}

By the construction of the approximate solutions, ∇ ∫_{0}^{1} θ^{ϵ,n}(t, x, η) dη = 0. Thus, by integration by parts and (2.10), there is a constant C depending only on ∥(v_{0}^{ϵ}, θ_{0}^{ϵ})∥_{0} such that

(2.22)  |(∇Θ(θ^{ϵ,n}), (P △_{3})^{2}v^{ϵ,n})| ≤ C∥θ∥_{2}^{2} + β(∥∇_{3}v^{ϵ,n}∥_{2}^{2} + C),

and

(2.23)  |(div V (v^{ϵ,n}), △_{3}^{2}θ^{ϵ,n})| ≤ C∥v^{ϵ,n}∥_{2}^{2} + β(∥∇_{3}θ^{ϵ,n}∥_{2}^{2} + C).

Set Y (t) = ∥(v^{ϵ,n}, θ^{ϵ,n})(t, ·)∥_{2}^{2}. Then, using (2.12), (2.20)-(2.23), and taking β = \frac{μ}{12}, we have

(2.24) Y′(t) + μ_{1}∥∇_{3}v^{ϵ,n}(t, ·)∥_{2}^{2} + μ_{2}∥∇_{3}θ^{ϵ,n}(t, ·)∥_{2}^{2} ≤ C(1 + Y (t) + Y ^{\frac{7}{2}} (t) + Y^{4}(t) + Y^{6}(t)),

where the constant C may depend on ϵ.

The comparison principle of ordinary differential equations yields

(2.25)  Y (t) ≤ F (t, y(0)), ∫_{0}^{t} ∥∇_{3}(v^{ϵ,n}, θ^{ϵ,n})(τ, ·)∥_{2}^{2} dτ < ∞, for 0 ≤ t < T^{∗},

where F (t, y(0)) and T^{∗} are respectively the solution and lifespan of the following Cauchy problem:

\frac{dy}{dt} = C(1 + y + y^{\frac{7}{2}} + y^{4} + y^{6}), y|_{t=0} = y(0) ≡ ∥(v_{0}^{ϵ}, θ_{0}^{ϵ})∥_{2}^{2}.

Step 4. Further apriori estimates for the approximate solutions. Similar to the arguments in establishing (2.8) and (2.12), we multiply (2.4) and (2.5) by \frac{d}{dt}c_{kn}^{ϵ}(t) and \frac{d}{dt}d_{kn}^{ϵ}(t), respectively, and sum up to obtain

∥∂_{t}(v^{ϵ,n}, θ^{ϵ,n})∥_{0}^{2} =μ_{1}(P △_{3}v^{ϵ,n}, ∂_{t}v^{ϵ,n}) + μ_{2}(△_{3}θ^{ϵ,n}, ∂_{t}θ^{ϵ,n})
− ((v^{ϵ,n} · ∇ − div V (v^{ϵ,n})∂_{η})v^{ϵ,n}, ∂_{t}v^{ϵ,n})
− ((v^{ϵ,n} · ∇ − div V (v^{ϵ,n})∂_{η})θ^{ϵ,n}, ∂_{t}θ^{ϵ,n}),

which implies

∥∂_{t}(v^{ϵ,n}, θ^{ϵ,n})∥_{0} ≤μ_{1}∥v^{ϵ,n}∥_{2} + μ_{2}∥θ^{ϵ,n}∥_{2} + ∥(v^{ϵ,n} · ∇ − div V (v^{ϵ,n})∂_{η})(v^{ϵ,n}, θ^{ϵ,n})∥_{0}
≤μ_{1}∥v^{ϵ,n}∥_{2} + μ_{2}∥θ^{ϵ,n}∥_{2} + C∥v^{ϵ,n}∥0^{\frac{1}{2}} ∥v^{ϵ,n}∥2^{\frac{1}{2}} (∥v^{ϵ,n}∥0^{\frac{1}{4}} ∥v^{ϵ,n}∥2^{\frac{3}{4}} + ∥θ^{ϵ,n}∥0^{\frac{1}{4}} ∥θ^{ϵ,n}∥2^{\frac{3}{4}} )

+ C∥v^{ϵ,n}∥0^{\frac{1}{8}} ∥v^{ϵ,n}∥2^{\frac{7}{8}} (∥v^{ϵ,n}∥0^{\frac{1}{8}} ∥v^{ϵ,n}∥2^{\frac{7}{8}} + ∥θ^{ϵ,n}∥0^{\frac{1}{8}} ∥θ^{ϵ,n}∥2^{\frac{7}{8}} ),

where we have used (2.13), (2.14), and the Hölder inequality. Then, by (2.25), we have

(2.26)  ∥∂_{t}(v^{ϵ,n}, θ^{ϵ,n})(t, ·)∥_{0} ≤ G(t),  0 ≤ t < T^{∗},

for some continuous function G(t) defined in [0, T^{∗}).

Step 5. Construction of solutions. Estimates (2.25) and (2.26) imply that (v^{ϵ,n}, θ^{ϵ,n}) is compact in L_{loc}^{p}([0, T^{∗}); H^{α}(Ω)), for any 1 < p < ∞, α < 3. Let (v^{ϵ}, θ^{ϵ}) be a limit of (v^{ϵ,n}, θ^{ϵ,n}). Then (v^{ϵ}, θ^{ϵ}) must be a weak solution of (1.1). By (2.25) and (2.26), (v^{ϵ}, θ^{ϵ}) ∈ L^{∞}([0, T^{∗}); J^{2})∩L^{2}([0, T^{∗}); H^{3}(Ω)) and (v_{t}^{ϵ}, θ_{t}^{ϵ}) ∈ L^{∞}([0, T^{∗}); L^{2}(Ω)). Moreover, by Lemma 2.1, there exists some Φ^{ϵ} ∈ L_{loc}^{2}([0, T^{∗}); H^{1}(𝕋)) such that (v^{ϵ}, θ^{ϵ}) and Φ^{ϵ} satisfy (1.1) in the weak sense.

For arbitrary 0 ≤ t_{1} < t_{2} < T^{∗},

(2.27)  ∥v^{ϵ}(t_{1}, ·) − v^{ϵ}(t_{2}, ·)∥_{0} = ∥ ∫_{t_{1}}^{t_{2}} \frac{∂v^{ϵ}}{∂t} (t, ·) dt∥_{0} ≤ ∥\frac{∂v^{ϵ}}{∂t} ∥_{L^{2}([t_{2},t_{1}]×Ω)}|t_{1} − t_{2}|^{\frac{1}{2}} .

By the Rellich Theorem, H^{2}(Ω) ↪ H^{α}(Ω) is compact, for arbitrary α < 2. Thus, by the Lions-Aubin Lemma, for any δ > 0, there exists C_{δ} > 0 such that

(2.28)  ∥v^{ϵ}(t_{1}, ·) − v^{ϵ}(t_{2}, ·)∥_{α} ≤ δ∥v^{ϵ}∥_{L^{∞}([t_{1},t_{2}];H^{2}(Ω))} + C_{δ}∥v^{ϵ}(t_{1}, ·) − v^{ϵ}(t_{2}, ·)∥_{0}.

Then we conclude from (2.27) and (2.28), similarly for θ^{ϵ}, that

(2.29)  (v^{ϵ}, θ^{ϵ}) ∈ C([0, T^{∗}); H^{α}(Ω)), for any α < 2.

Step 6. Claim: lim_{t→0} ∥(v^{ϵ}, θ^{ϵ})(t, ·) − (v_{0}^{ϵ}, θ_{0}^{ϵ})(·)∥_{2} = 0.

Taking t = 0 in (2.25), we have

lim_{t→0} ∥(v^{ϵ}, θ^{ϵ})(t, ·)∥_{2} ≤ lim_{t→0} \overline{lim}_{n→∞}∥(v^{ϵ,n}, θ^{ϵ,n})(t, ·)∥_{2} ≤ ∥(v_{0}^{ϵ}, θ_{0}^{ϵ})∥_{2}.

Then, taking t_{1} = 0 in (2.27), we conclude

lim_{t→0} v^{ϵ}(t, ·) = v_{0}^{ϵ}(·) weakly in H^{2}(Ω),

which implies

lim_{t→0} ∥v^{ϵ}(t, ·) − v_{0}^{ϵ}(·)∥_{2} = 0.

Similarly, we have

lim_{t→0} ∥θ^{ϵ}(t, ·) − θ_{0}^{ϵ}(·)∥_{2} = 0.

This completes the proof of Theorem 1.1.

Remark 2.1. Following the proof of Step 6, we can show

lim_{t→t′+0} ∥(v^{ϵ}, θ^{ϵ})(t, ·) − (v^{ϵ}, θ^{ϵ})(t′, ·)∥_{2} = 0.

One can also establish the corresponding case of Theorem 1.1 for initial data (v_{0}^{ϵ}, θ_{0}^{ϵ})(x, η) ∈ J^{α}, with α > 2 an integer. Remark 2.2. By the construction of the approximate solutions for (1.1), we have

(2.30)  △_{3}v^{ϵ,n}(t, x, 0) = △_{3}v^{ϵ,n}(t, x, 1).

Thus, using the Trace Theorem and the arguments at the beginning of Step 5, we obtain for all 0 < t < T^{∗} that

∫_{0}^{t} ∥△_{3}v^{ϵ}(τ, ·, 1) − △_{3}v^{ϵ}(τ, ·, 0)∥_{0} dτ

= lim_{n→∞} ∫_{0}^{t}{∥△_{3}(v^{ϵ} − v^{ϵ,n})(τ, ·, 1)∥_{0} + ∥△_{3}(v^{ϵ} − v^{ϵ,n})(τ, ·, 0)∥_{0}} dτ

≤ C lim_{n→∞} ∫_{0}^{t} ∥(v^{ϵ} − v^{ϵ,n})(τ, ·, ·)∥_{\frac{5}{2}} dτ = 0,

which implies

(2.31)  ∫_{0}^{t} ∥P △_{3}v^{ϵ}(τ, ·, η)|_{η=l}∥_{0}^{q} dτ = 0, for l = 0, 1, 0 ≤ t < T^{∗}, q < ∞.

3. Proof of Theorem 1.2: Solutions of the Quasigeostrophic Equations

The proof of Theorem 1.2 is similar to that of Theorem 1.1; and we only outline it here. A special care is taken since the Prandtl number is not one.

We first take the n^{th} approximate solutions:

(3.1)  \begin{matrix}ϕ^{n}(t, x, η) = ∑_{k=1}^{n} g_{kn}(t)c^{k}(x, η), \\ v_{1}^{n}(t, x, η) = − ∑_{k=1}^{n} g_{kn}(t)∂_{2}c^{k}(x, η), v_{2}^{n}(t, x, η) = ∑_{k=1}^{n} g_{kn}(t)∂_{1}c^{k}(x, η), \\ θ^{n}(t, x, η) = ∑_{k=1}^{n} g_{kn}(t)∂_{η}c^{k}(x, η), ω^{n}(t, x, η) = ∑_{k=1}^{n} g_{kn}(t)△_{3}c^{k}(x, η), \end{matrix}

satisfying the boundary conditions: (v^{n}, ∂_{η}θ^{n})|_{η=l} = 0, l = 0, 1, where c^{k}(x, η) are the normalized eigenfunctions of the Laplace operator, Δ_{3}c^{k} = γ_{k}c^{k}, which are orthonormal in J_{3}^{0}, and g_{kn}(t), 1 ≤ k ≤ n, are determined by the following system:

(3.2) \{ \begin{matrix}\frac{d}{dt}(ω^{n}, c^{k})_{0} + (v^{n} · ∇ω^{n}, c^{k})_{0} − μ_{1}(△_{3}ω^{n}, c^{k})_{0} + (μ_{1} − μ_{2})(∂_{η}^{2}ω^{n}, c^{k})_{0} = 0, \\ g_{kn}(0) = (ϕ_{0}, c^{k})_{0}. \end{matrix}

The regularity of the Laplace operator ensures that {c^{k}} are in H^{∞}(Ω). Then we will obtain some a-priori estimates for the approximations (ϕ^{n}, ω^{n}, v^{n}, θ^{n}).

Now we multiply (3.2) by γ_{k}g_{kn}(t) and γ_{k}^{3}g_{kn}(t), respectively, and sum up to obtain

\frac{1}{2} \frac{d}{dt}∥ω^{n}∥_{0}^{2} + μ_{1}∥∇_{3}ω^{n}∥_{0}^{2} = (μ_{2} − μ_{1})(∂_{η}^{2}ω^{n}, ω^{n})_{0}, (3.3)

\frac{1}{2} \frac{d}{dt}∥△_{3}ω^{n}∥_{0}^{2} + μ_{1}∥∇_{3}△_{3}ω^{n}∥_{0}^{2} = (v^{n} · ∇ω^{n}, △_{3}^{2}ω^{n})_{0} + (μ_{2} − μ_{1})(∂_{η}^{2}ω^{n}, △_{3}^{2}ω^{n})_{0}. (3.4)

On the other hand, similar to the proof of (2.10), the Bessel inequality implies

(3.5)  ∥ω_{0}^{n}∥_{0} ≤ ∥ϕ_{0}∥_{2},  ∥△_{3}ω_{0}^{n}∥_{0} ≤ ∥ϕ_{0}∥_{4}.

Since (3.6) (v^{n}·∇ω^{n}, △_{3}^{2}ω^{n})_{0} ≤ C∥(v^{n}, ∇v^{n})∥_{L^{6}}∥ω^{n}∥_{2}∥∇_{3}△_{3}ω^{n}∥_{0} ≤ C∥(v^{n}, ∇v^{n})∥_{1}∥ω^{n}∥_{2}∥∇_{3}△_{3}ω^{n}∥_{0},

we follow (3.1) to have

(3.7) ∥v^{n}(t, ·)∥_{α} ≤ ∥ω^{n}(t, ·)∥_{α−1} ≤ ∥ϕ^{n}(t, ·)∥_{α+1},  ∥∇_{3}v^{n}(t, ·)∥_{α} ≤ ∥∇_{3}ω(t, ·)∥_{α−1}.

Hence

(3.8)  (v^{n} · ∇ω^{n}, △_{3}^{2}ω^{n})_{0} ≤ C∥(ω^{n}, ∇_{3}ω^{n})∥_{0}^{2}∥ω^{n}∥_{2}^{2} + \frac{μ}{2}∥∇_{3}ω^{n}∥_{2}^{2},

with μ = μ_{1} − |μ_{1} − μ_{2}| > 0. Furthermore,

(3.9)  \begin{matrix}|(∂_{η}^{2}ω^{n}, ω^{n})_{0}| = ∥∂_{η}ω^{n}∥_{0}^{2} ≤ ∥∇_{3}ω^{n}∥_{0}^{2}, \\ |(∂_{η}^{2}△_{3}ω^{n}, △_{3}^{2}ω^{n})_{0}| ≤ ∥∇_{3}△_{3}ω^{n}∥_{0}^{2}. \end{matrix}

Thus, by (3.3), (3.4), and (3.8),

\frac{d}{dt}∥ω^{n}(t, ·)∥_{0}^{2} + μ∥∇_{3}ω^{n}(t, ·)∥_{0}^{2} ≤ 0, (3.10)

\frac{d}{dt}∥△_{3}ω^{n}(t, ·)∥_{0}^{2} + μ∥∇_{3}△_{3}ω^{n}(t, ·)∥_{0}^{2} ≤ C∥(ω^{n}, ∇_{3}ω^{n})(t, ·)∥_{0}^{2}∥△_{3}ω^{n}(t, ·)∥_{0}^{2}. (3.11)

Using the Gronwall inequality and integrating (3.10) yield

(3.12)  ∥△_{3}ω(t, ·)∥_{0}^{2} ≤ C exp (C ∫_{0}^{t} ∥(ω^{n}, ∇_{3}ω^{n})(τ, ·)∥_{0}^{2} dτ) ≤ C exp(Ct).

Now, multiplying (3.2) by γ_{k}g′_{kn}(t) and summing up yield

(3.13) ∥ω_{t}^{n}(t, ·)∥_{0} ≤ μ_{1}∥△_{3}ω^{n}(t, ·)∥_{0}+μ_{2}∥△_{3}∂_{η}θ^{n}(t, ·)∥_{0}+∥v^{n}(t, ·)∥_{1}∥ω^{n}(t, ·)∥0^{\frac{1}{4}} ∥ω^{n}(t, ·)∥2^{\frac{3}{4}} ,

with a similar argument as in the proof of (2.26). Similarly, we can establish (3.14)

\frac{d}{dt}∥∇_{3}△_{3}ω^{n}(t, ·)∥_{0}^{2} + μ_{1}∥△_{3}^{2}ω^{n}(t, ·)∥_{0}^{2} = (v^{n} · ∇ω^{n}, △_{3}^{3}ω^{n})_{0} + (μ_{1} − μ_{2})(△_{3}∂_{η}θ^{n}, △_{3}^{3}ω^{n})_{0}.

Trivially, by (3.1), we have

(3.15)  |(△_{3}∂_{η}θ^{n}, △_{3}^{3}ω^{n})_{0}| ≤ ∥△_{3}^{2}ω^{n}∥_{0}^{2},

and

(3.16) \begin{matrix}(v^{n} · ∇ω^{n}, △_{3}^{3}ω^{n})_{0} = −([∇_{3}△_{3}, v^{n}]∇ω^{n}, ∇_{3}△_{3}ω^{n})_{0} \\ ≤ & C(∥∇v^{n}(t, ·)∥_{L^{∞}}∥D^{3}ω^{n}(t, ·)∥_{0} + ∥∇ω^{n}(t, ·)∥_{L^{∞}}∥D^{3}v^{n}(t, ·)∥_{0})∥∇_{3}△_{3}ω^{n}(t, ·)∥_{0}, \end{matrix}

where we have used the inequality of Moser type (see [14]).

Thus, by the Sobolev and Poincaré inequalities, we have

(3.17)  (v^{n} · ∇ω^{n}, △_{3}^{3}ω^{n})_{0} ≤ (∥∇v^{n}(t, ·)∥_{L^{∞}} + ∥D^{3}v^{n}(t, ·)∥_{0})∥∇_{3}△_{3}ω^{n}(t, ·)∥_{0}^{2}.

Using (3.7), (3.12), and ∥∇v^{n}(t, ·)∥_{L^{∞}} + ∥D^{3}v^{n}(t, ·)∥_{0} ≤ Ce^{Ct}, we obtain

(3.18)  ∥∇_{3}△_{3}ω^{n}(t, ·)∥_{0}^{2} + μ ∫_{0}^{t} ∥△_{3}ω(τ, ·)∥_{0}^{2}dτ ≤ Ce^{e^{Ct}}.

Thus, by (3.12), (3.13), and (3.18), {ω^{n}} is compact in L_{loc}^{p}([0, ∞); H^{α}(Ω)), and {(v^{n}, θ^{n})} is compact in L_{loc}^{p}([0, ∞); H^{α+1}(Ω)), for any 1 < p < ∞, α < 4. Without loss of generality, we assume that (ϕ, ω, v, θ)(t, x, η) is a limit of (ϕ^{n}, ω^{n}, v^{n}, θ^{n})(t, x, η). Then ω = △_{3}ϕ, and (ϕ, ω, v, θ)(t, x, η) is a smooth solution of (1.4) and (3.19)

∥(v, θ)(t, ·)∥_{4} + μ ∫_{0}^{t} ∥∇_{3}(v, θ)(τ, ·)∥_{4}^{2} dτ < ∞, μ = μ_{1} − |μ_{1} − μ_{2}|, for any t < ∞.

Hence, (ϕ, ω, v, θ)(t, x, η) satisfies all the properties in Theorem 1.2. The uniqueness is straightforward, and hence we omit it here. Remark 3.1. As Theorem 1.1, we can obtain further regularity results for (ϕ, ω, v, θ)(t, x, η) with respect to t. Remark 3.2. By the same proof of Theorem 1.1, we can prove the local existence of strong solution (v, θ)(t, x, η) ∈ L^{∞}((0, T^{∗}); J^{4}) ∩ L^{2}([0, T^{∗}); H^{5}(Ω)), for some positive constant T^{∗} ≤ ∞, to the following equations:

(3.20)  \{ \begin{matrix}∂_{t}v + v · ∇v − μ_{1}△_{3}v + ∇Φ(t, x) = 0, \\ div v = 0, \\ ∂_{t}θ + v · ∇θ − μ_{2}△_{3}θ = 0, \\ (v, θ)|_{t=0} = (v_{0}, θ_{0}) = (−∂_{x_{2}}ϕ_{0}, ∂_{x_{1}}ϕ_{0}, ∂_{η}ϕ_{0}), \\ (v, θ_{η})|_{η=l} = 0,  for l = 0, 1, \end{matrix}

with ϕ_{0} ∈ J_{3}^{5}. Moreover, it is trivial to calculate that, if we set ω = ∂_{x_{1}}v_{2} − ∂_{x_{2}}v_{1} + ∂_{η}θ, and △_{3}ϕ = ω, then v = ∇^{⊥}ϕ, θ = ∂_{η}ϕ, and, by the first equation of (3.20),

(3.21)  \{ \begin{matrix}∂_{t}ω + v · ∇ω − μ_{1}△_{3}ω + (μ_{1} − μ_{2})△_{3}∂_{η}θ = 0, \\ ω|_{η=l} = 0, for l = 0, 1. \end{matrix}

This shows that the solution (v, θ)(t, x, η) of (1.4) and (1.9) must be a solution of (3.20). On the other hand, by (1.4) and (1.9), we have △_{3}v|_{η=l} = ∇^{⊥}ω|_{η=l} = 0 for l = 0, 1. Thus, by the first equation of (3.20), we have ∇Φ(t, x) = 0, which implies

(3.22)  ∂_{t}v + v · ∇v − μ_{1}△_{3}v = 0.

Furthermore, by a direct calculation, we have

(3.23)  v^{⊥} + ∇Θ(θ) = 0.

4. Proof of Theorem 1.3: Zero-Rossby Limit

We rewrite the solutions (ϕ^{ϵ}, v^{ϵ}, θ^{ϵ})(t, x, η) of (1.1) as follows:

(ϕ^{ϵ}, v^{ϵ}, θ^{ϵ})(t, x, η) = (ϕ, v, θ)(t, x, η) + ϵ(\overline{ϕ}^{ϵ,a}, v^{ϵ,a}, θ^{ϵ,a})(t, x, η),

where (ϕ, v, θ)(t, x, η) is the same as the correspondence in Theorem 1.2, and

ϕ^{ϵ}(t, x, η) = ∫_{1}^{η} θ^{ϵ}(t, x, s) ds + Φ^{ϵ}(t, x).

Using Theorem 1.2, it suffices for Theorem 1.3 to prove that, for every T > 0, there exist ϵ_{0} > 0 and some continuous function K(t) defined on [0, T] depending only on T and ϵ_{0} such that, when ϵ ≤ ϵ_{0},

(4.1)  ∥v^{ϵ,a}(t, ·)∥_{2} + ∥θ^{ϵ,a}(t, ·)∥_{2} ≤ K(t),  for 0 ≤ t < T.

To achieve this, we first use Theorems 1.1 and 1.2 and Remarks 2.2 and 3.2 to derive the equations and boundary conditions of (v^{ϵ,a}, θ^{ϵ,a}). Then we perform the energy estimates by carefully employing the boundary conditions for (v^{ϵ,a}, θ^{ϵ,a}) to derive an integral inequality for the H^{2}-norm of (v^{ϵ,a}, θ^{ϵ,a}). Finally, using the comparison principle of ordinary differential equations yields (4.1).

Proof of (4.1). In the sequel, we will omit ϵ in (v^{ϵ,a}, θ^{ϵ,a}), and we may regard ∥△_{3}a∥_{0} as ∥a∥_{2} and (·, ·) as (·, ·)_{2}.

By Theorems 1.1 and 1.2 and Remark 3.2, we have

(4.2)  \{ \begin{matrix}θ^{a} = ∂_{η}ϕ^{a}, \\ \frac{Dv}{Dt} + ϵ\frac{Dv^{a}}{Dt} − μ_{1}△_{3}(v + ϵv^{a}) + ((v^{a})^{⊥} + ∇\overline{ϕ}^{a}) = 0, \\ \frac{Dθ}{Dt} + ϵ\frac{Dθ^{a}}{Dt} − μ_{2}△_{3}(θ + ϵθ^{a}) − div V (v^{a}) = 0, \\ (v^{a}, ∂_{η}θ^{a})|_{η=l} = 0,  l = 1, 2, \end{matrix}

where the material derivative \frac{D}{Dt} = ∂_{t} + v^{ϵ} · ∇ − div V (v^{ϵ})∂_{η}. On the other hand, notice that, by (1.8), there is some Ψ^{a}(t, x) ∈ L^{2}((0, T_{ϵ}); H^{2}(𝕋^{2})) such that

(4.3)  △_{3}v^{a} = P △_{3}v^{a} + ∇Ψ^{a}.

Set ϕ^{a}(t, x, η) = \overline{ϕ^{a}}(t, x, η) − ϵμΨ^{a}(t, x). Then we can write the second equation of (4.2) in the following form:

(4.4)  \frac{Dv}{Dt} + ϵ\frac{Dv^{a}}{Dt} − μ_{1}(△_{3}v + ϵP △_{3}v^{a}) + ((v^{a})^{⊥} + ∇ϕ^{a}) = 0.

Thus, by Remarks 2.2 and 3.2, we have

(4.5)  ∇ϕ^{a}(t, x, η)|_{η=l} = 0,  for l = 0, 1, (t, x) ∈ (0, T^{∗}) × 𝕋^{2}.

Taking H^{2}(Ω) inner product of (4.4) with v^{a} and the third equation in (4.2) with θ^{a}, respectively, yields

(4.6)  \begin{matrix}ϵ\{ & (\frac{Dv^{a}}{Dt} − μ_{1}P △_{3}v^{a}, v^{a}) + (\frac{Dθ^{a}}{Dt} − μ_{2}△_{3}θ^{a}, θ^{a})\} + (\frac{Dv}{Dt} − μ_{1}△_{3}v, v^{a}) \\ + (\frac{Dθ}{Dt} − μ_{2}△_{3}θ, θ^{a}) + (∇ϕ^{a}, v^{a}) − (div V (v^{a}), θ^{a}) = 0. \end{matrix}

Now we perform the following integration by parts:

(4.7)

(div V (v^{a}), θ^{a}) = ∫_{0}^{1} ∫_{𝕋^{2}} div△_{3} ∫_{0}^{η} v^{a}(t, x, s) ds△_{3}θ^{a} dx dη
= ∫_{0}^{1} ∫_{𝕋^{2}} ∂_{η} (div△_{3} ∫_{0}^{η} v^{a}(t, x, s) ds△_{3}ϕ^{a}) dx dη − (div v^{a}, ϕ^{a}) = (v^{a}, ∇ϕ^{a}),

since ϕ^{a}(t, x, η) = ∂_{η}θ^{a}(t, x, η) and △_{3}ϕ^{a}|_{η=l} = △ϕ^{a}|_{η=l} + ∂_{η}θ^{a}|_{η=l} = 0 by (4.5). Then (4.8)

(\frac{Dv}{Dt} − μ_{1}△_{3}v, v^{a}) + (\frac{Dθ}{Dt} − μ_{2}△_{3}θ, θ^{a})

=(\frac{Dv_{1}}{Dt} − μ_{1}△_{3}v_{1}, −∂_{x_{2}}ϕ^{a} + μ_{1}△_{3}v_{2} − \frac{Dv_{2}}{Dt} − ϵ(\frac{Dv_{2}^{a}}{Dt} − μ_{1}△_{3}v_{2}^{a}))

+ (\frac{Dv_{2}}{Dt} − μ_{1}△_{3}v_{2}, ∂_{x_{1}}ϕ^{a} − μ_{1}△_{3}v_{1} + \frac{Dv_{1}}{Dt} + ϵ(\frac{Dv_{1}^{a}}{Dt} − μ_{1}△_{3}v_{1}^{a})) + (\frac{Dθ}{Dt} − μ_{2}△_{3}θ, ∂_{η}ϕ^{a})

=(\frac{Dv_{1}}{Dt} , −∂_{x_{2}}ϕ^{a}) + (\frac{Dv_{2}}{Dt} , ∂_{x_{1}}ϕ^{a}) + (\frac{Dθ}{Dt} , ∂_{η}ϕ^{a}) + μ_{1}(△_{3}v_{1}, ∂_{x_{2}}ϕ^{a})

− μ_{1}(△_{3}v_{2}, ∂_{x_{1}}ϕ^{a}) − μ_{2}(△_{3}θ, ∂_{η}ϕ^{a}) + μ_{1}(\frac{Dv_{1}}{Dt} , △_{3}v_{2}) − μ_{1}(\frac{Dv_{2}}{Dt} , △_{3}v_{1})

− μ_{1}^{2}(△_{3}v_{1}, △_{3}v_{2}) + μ_{1}^{2}(△_{3}v_{2}, △_{3}v_{1}) − (\frac{Dv_{1}}{Dt} , \frac{Dv_{2}}{Dt}) + (\frac{Dv_{2}}{Dt} , \frac{Dv_{1}}{Dt})

+ μ_{1}(△_{3}v_{1}, \frac{Dv_{2}}{Dt}) − μ_{1}(△_{3}v_{2}, \frac{Dv_{1}}{Dt}) + ϵ(\frac{Dv_{1}}{Dt} − μ_{1}△_{3}v_{1}, \frac{Dv_{2}^{a}}{Dt} − μ_{1}△_{3}v_{2}^{a})

− ϵ(\frac{Dv_{2}}{Dt} − μ_{1}△_{3}v_{2}, \frac{Dv_{1}^{a}}{Dt} − μ_{1}△_{3}v_{1}^{a}),

where we have substituted (v^{a}, θ^{a}) in (4.2) and (4.4) to (\frac{Dv}{Dt} − μ_{1}△_{3}v, v^{a}) and (\frac{Dθ}{Dt} − μ_{2}△_{3}θ, θ^{a}). Note that

(4.9)  \begin{aligned}(\frac{Dv_{1}}{Dt} , & − ∂_{x_{2}}ϕ^{a}) + (\frac{Dv_{2}}{Dt} , ∂_{x_{1}}ϕ^{a}) + (\frac{Dθ}{Dt} , ∂_{η}ϕ^{a}) \\ = & − (d_{g}v_{1} + ϵd_{a}v_{1}, ∂_{x_{2}}ϕ^{a}) + (d_{g}v_{2} + ϵd_{a}v_{2}, ∂_{x_{1}}ϕ^{a}) + (d_{g}θ + ϵd_{a}θ, ∂_{η}ϕ^{a}) \\ = & − (d_{g}v_{1}, ∂_{x_{2}}ϕ^{a}) + (d_{g}v_{2}, ∂_{x_{1}}ϕ^{a}) + (d_{g}θ, ∂_{η}ϕ^{a}) \\ & − ϵ\{−(d_{a}v_{1}, ∂_{x_{2}}ϕ^{a}) + (d_{a}v_{2}, ∂_{x_{1}}ϕ^{a}) + (d_{a}θ, ∂_{η}ϕ^{a})\}, \end{aligned}

where we have adopted the notations d_{g} = ∂_{t}+v·∇ and d_{a} = v^{a}·∇−div V (v^{a})∂_{η}. Moreover, it is obvious to see

(4.10)  (d_{g}v_{1}, ∂_{x_{2}}ϕ^{a}) = −(∂_{x_{2}}d_{g}v_{1}, ϕ^{a}),  (d_{g}v_{2}, ∂_{x_{1}}ϕ^{a}) = −(∂_{x_{1}}d_{g}v_{1}, ϕ^{a}),

and

(4.11)  \begin{matrix}(d_{g}θ, & ∂_{η}ϕ^{a}) = ∫_{0}^{1} ∫_{𝕋^{2}} △_{3}(d_{g}θ)△_{3}(∂_{η}ϕ^{a}) dx dη \\ = ∫_{0}^{1} ∫_{𝕋^{2}} ∂_{η}(△_{3}(d_{g}θ)△_{3}ϕ^{a}) dx dη − (∂_{η}d_{g}θ, ϕ^{a}) = −(∂_{η}d_{g}θ, ϕ^{a}), \end{matrix}

where we have used again the observation following (4.7). Summing up (4.10) and (4.11) and using (3.18) lead to

(4.12) \begin{matrix}− (d_{g}v_{1}, & ∂_{x_{2}}ϕ^{a}) + (d_{g}v_{2}, ∂_{x_{1}}ϕ^{a}) + (d_{g}θ, ∂_{η}ϕ^{a}) \\ = & (d_{g}(∂_{x_{2}}v_{1} − ∂_{x_{1}}v_{2} − ∂_{η}θ), ϕ^{a}) = (μ_{1}△_{3}(∂_{x_{2}}v_{1} − ∂_{x_{1}}v_{2}) − μ_{2}△_{3}∂_{η}θ, ϕ^{a}) \\ = & − μ_{1}(△_{3}v_{1}, ∂_{x_{2}}ϕ^{a}) + μ_{1}(△_{3}v_{2}, ∂_{x_{1}}ϕ^{a}) + μ_{2}(△_{3}θ, ∂_{η}ϕ^{a}). \end{matrix}

Using (4.7)-(4.12), we finally rewrite (4.6) as (4.13)

\begin{aligned}( & \frac{Dv^{a}}{Dt} − μ_{1}P △_{3}v^{a}, v^{a}) + (\frac{Dθ^{a}}{Dt} − μ_{2}△_{3}θ^{a}, θ^{a}) − (d_{a}v_{1}, ∂_{x_{2}}ϕ^{a}) + (d_{a}v_{2}, ∂_{x_{1}}ϕ^{a}) + (d_{a}θ, ∂_{η}ϕ^{a}) \\ & + (\frac{Dv_{1}}{Dt} − μ_{1}△_{3}v_{1}, \frac{Dv_{2}^{a}}{Dt} − μ_{1}△_{3}v_{2}^{a}) + (\frac{Dv_{2}}{Dt} − μ_{1}△_{3}v_{2}, \frac{Dv_{1}^{a}}{Dt} − μ_{1}△_{3}v_{1}^{a}) = 0, \end{aligned}

while

(4.14) \begin{aligned}(\frac{Dv_{1}}{Dt} & − μ_{1}△_{3}v_{1}, \frac{Dv_{2}^{a}}{Dt} − μ_{1}△_{3}v_{2}^{a}) + (\frac{Dv_{2}}{Dt} − μ_{1}△_{3}v_{2}, \frac{Dv_{1}^{a}}{Dt} − μ_{1}△_{3}v_{1}^{a}) \\ = & − ϵ(d_{a}v_{1}, \frac{Dv_{2}^{a}}{Dt} − μ_{1}△_{3}v_{2}^{a}) + ϵ(d_{a}v_{2}, \frac{Dv_{1}^{a}}{Dt} − μ_{1}△_{3}v_{1}^{a}) \\ & − (d_{g}v_{1} − μ_{1}△_{3}v_{1}, \frac{Dv_{2}^{a}}{Dt} − μ_{1}△_{3}v_{2}^{a}) − (d_{g}v_{2} − μ_{1}△_{3}v_{2}, \frac{Dv_{1}^{a}}{Dt} − μ_{1}△_{3}v_{1}^{a}). \end{aligned}

Using (4.4) again yields

(4.15)  \begin{matrix}−(d_{a}v_{1}, & ∂_{x_{2}}ϕ^{a}) + (d_{a}v_{2}, ∂_{x_{1}}ϕ^{a}) \\ = & ϵ(d_{a}v_{1}, \frac{Dv_{2}^{a}}{Dt} − μ_{1}△_{3}v_{2}^{a}) + (d_{a}v_{1}, \frac{Dv_{2}}{Dt} − μ_{1}△_{3}v_{2} + v_{1}^{a}) \\ − ϵ(d_{a}v_{2}, \frac{Dv_{1}^{a}}{Dt} − μ_{1}△_{3}v_{1}^{a}) − (d_{a}v_{2}, \frac{Dv_{1}}{Dt} − μ_{1}△_{3}v_{1} + v_{2}^{a}). \end{matrix}

Thus, (4.13) turns to the following equality:

(4.16)  \begin{matrix}(\frac{Dv^{a}}{Dt} − μ_{1}P △_{3}v^{a}, v^{a}) + (\frac{Dθ^{a}}{Dt} − μ_{2}△_{3}θ^{a}, θ^{a}) \\ = & − (d_{a}v_{1}, v_{1}^{a}) − (d_{a}v_{2}, v_{2}^{a}) − (d_{a}θ, θ^{a}) + (d_{g}v_{1} − μ_{1}△_{3}v_{1}, \frac{Dv_{2}^{a}}{Dt} − μ_{1}△_{3}v_{2}^{a}) \\ − (d_{g}v_{2} − μ_{1}△_{3}v_{2}, \frac{Dv_{1}^{a}}{Dt} − μ_{1}△_{3}v_{1}^{a}) \\ − (d_{a}v_{1}, \frac{Dv_{2}}{Dt} − μ_{1}△_{3}v_{2}) + (d_{a}v_{2}, \frac{Dv_{1}}{Dt} − μ_{1}△_{3}v_{1}). \end{matrix}

On the other hand, noticing the fact from Remark 3.2 that d_{g}v − μ_{1}△_{3}v = 0 and integrating (4.16) with respect to t, we obtain

(4.17)  \begin{matrix}∥(v^{a}, & θ^{a})(t, ·)∥_{2}^{2} + 2 ∫_{0}^{t}(μ_{1}∥∇_{3}P △_{3}v^{a}(τ, ·)∥_{0}^{2} + μ_{2}∥∇_{3}θ^{a}(τ, ·)∥_{2}^{2}) dτ \\ ≤ & ∥(v_{0}^{a}, θ_{0}^{a})∥_{2}^{2} + 2| ∫_{0}^{t}\{−(d_{a}v_{1}, v_{1}^{a}) + (d_{a}v_{2}, v_{2}^{a}) − (d_{a}θ, θ^{a}) \\ + ((div V (v^{ϵ})∂_{η} − v^{ϵ} · ∇)v^{a}, v^{a}) + ((div V (v^{ϵ})∂_{η} − v^{ϵ} · ∇)θ^{a}, θ^{a}) \\ − (d_{a}v_{1}, \frac{Dv_{2}}{Dt} − μ_{1}△_{3}v_{2}) + (d_{a}v_{2}, \frac{Dv_{1}}{Dt} − μ_{1}△_{3}v_{1})\} dτ|. \end{matrix}

Next we estimate term by term in (4.17). First, from the discussion at the beginning of this section and the proof of (2.20), we find

(4.18)

((div V (v^{ϵ})∂_{η} − v^{ϵ} · ∇)v^{a}, v^{a})
≤∥∂_{η}v^{a}∥_{L^{6}}∥∇_{3}div V (v^{ϵ})∥_{L^{3}}∥∇_{3}v^{a}∥_{2} + ∥div V (v^{ϵ})∥_{L^{6}}∥∇_{3}∂_{η}v^{a}∥_{L^{3}}∥∇_{3}v^{a}∥_{2}

+ ∥∇_{3}v^{ϵ}∥_{L^{4}}∥∇v^{a}∥_{L^{4}}∥∇_{3}v^{a}∥_{2} + ∥v^{ϵ}∥_{L^{6}}∥∇_{3}∇v^{a}∥_{L^{3}}∥∇_{3}v^{a}∥_{2}

≤C(3ϵ∥v^{a}∥2^{\frac{3}{2}} ∥∇_{3}v^{a}∥2^{\frac{3}{2}} + ϵ∥v^{a}∥2_{\frac{7}{4}} ∥∇_{3}v^{a}∥_{2} + ∥∇v^{ϵ}∥_{L^{4}}∥v^{a}∥_{\frac{7}{4}} ∥∇v^{a}∥_{2}

+ ∥v∥_{L^{6}}∥v^{a}∥_{2}∥∇_{3}v^{a}∥2^{\frac{3}{2}} )

≤C(∥v^{a}∥_{2}^{2} + ϵ^{2}∥v^{a}∥_{2}^{4} + ϵ^{4}∥v^{a}∥_{2}^{6}) + α∥∇_{3}v^{a}∥_{2}^{2}.

Similarly,

(4.19) \begin{matrix}((div V & (v^{ϵ})∂_{η} − v^{ϵ} · ∇)θ^{a}, θ^{a}) \\ ≤ & C (∥θ^{a}∥_{2}^{2} + ϵ^{2}∥v^{a}∥_{2}^{2}∥θ^{a}∥_{2}^{2} + ϵ^{4}(∥θ^{a}∥_{2}^{4}∥v^{a}∥_{2}^{2} + ∥v^{a}∥_{2}^{4}∥θ^{a}∥_{2}^{2})) + α∥∇_{3}θ^{a}∥_{2}^{2}, \end{matrix}

and

(4.20)

(d_{a}v_{1},v_{1}^{a}) = (△_{3}(v^{a} · ∇ − div V (v^{a})∂_{η})v_{1}, △_{3}v_{1}^{a})_{0}
≤∥∇_{3}v^{a}∥_{L^{6}}∥∇v_{1}∥_{L^{3}}∥∇v_{1}^{a}∥_{2} + ∥v^{a}∥_{L^{∞}}∥∇_{3}∇v_{1}∥_{0}∥∇_{3}v_{1}^{a}∥_{2}
+ ∥∂_{η}v_{1}∥_{L^{6}}∥div V (v^{a})∥_{L^{3}}∥∇v_{1}^{a}∥_{2} + ∥div V (v^{a})∥_{L^{∞}}∥∇_{3}∂_{η}v_{1}∥_{0}∥∇_{3}v_{1}^{a}∥_{2}
≤C∥v_{1}∥_{2}^{2}∥v^{a}∥_{2}^{2} + α∥∇_{3}v_{1}^{a}∥_{2}^{2}.

Similar estimates for (d_{a}v_{2}, v_{2}^{a}) and (d_{a}θ, θ^{a}) can be obtained.

By the definition of \frac{D}{Dt}, d_{g}, and d_{a}, we have (4.21)

\begin{matrix}− (d_{a}v_{1}, \frac{Dv_{2}}{Dt} − μ_{1}△_{3}v_{2}) + (d_{a}v_{2}, \frac{Dv_{1}}{Dt} − μ_{1}△_{3}v_{1}) \\ = −(d_{a}v_{1}, d_{g}v_{2} − μ_{1}△_{3}v_{2}) − ϵ(d_{a}v_{1}, d_{a}v_{2}) + (d_{a}v_{2}, d_{g}v_{1} − μ_{1}△_{3}v_{1}) + ϵ(d_{a}v_{2}, d_{a}v_{1}) = 0. \end{matrix}

Summing up (4.17)-(4.21), together with Theorem 1.2, Lemma 2.1, and taking α = \frac{min(μ_{1},μ_{2})}{6}, we finally have

(4.22)  \begin{matrix}∥(v^{a}, & θ^{a})(t, ·)∥_{2}^{2} + min(μ_{1}, μ_{2}) ∫_{0}^{t} ∥∇_{3}(v^{a}, θ^{a})(τ, ·)∥_{2}^{2} dτ \\ ≤ & C + C ∫_{0}^{t}(∥v^{a}∥_{2}^{2} + 2ϵ^{2}∥v^{a}∥_{2}^{4} + ϵ^{4}∥v^{a}∥_{2}^{6}) dτ,  for 0 ≤ t < T^{∗}. \end{matrix}

Set y(t) = ∥(v^{a}, θ^{a})(t, ·)∥_{2}^{2}. Then

(4.23)  y(t) ≤ C (1 + ∫_{0}^{t} (y(τ) + 2ϵ^{2}y(τ)^{2} + ϵ^{4}y(τ)^{3}) dτ) , for 0 ≤ t < T^{∗}.

Now we compare (4.23) with the following initial value problem:

(4.24)  \frac{df(t)}{dt} = C(f(t) + 2ϵ^{2}f(t)^{2} + ϵ^{4}f(t)^{3}),  f|_{t=0} = C.

For this equation, we can choose ϵ small enough to suppress the nonlinear growth terms so that its solution exists up to the pre-specified time T. In fact, if f(t) is the solution of (4.24), then

t = \frac{1}{C} ln(\frac{f(t)}{1 + ϵ^{2}f(t)})(\frac{1 + ϵ^{2}C}{C}) + \frac{1}{1 + ϵ^{2}f(t)} − \frac{1}{1 + Cϵ^{2}} .

Hence, if we can take ϵ_{0} ≪ 1 such that T ≤ \frac{1}{C} ln\frac{1}{2ϵ_{0}^{2}C} − 1, then, for any ϵ ∈ (0, ϵ_{0}), f(t) exists at least up to T. Thus, by the comparison principle of ordinary differential equation, (4.23), and (4.24),

\begin{matrix}∥(v^{a}, θ^{a})(t, ·)∥_{2}^{2} + min(μ_{1}, μ_{2}) ∫_{0}^{t} ∥∇_{3}(v^{a}, θ^{a})(τ, ·)∥_{2}^{2} dτ \\ ≤ & C (1 + ∫_{0}^{t}(f(τ) + 2ϵ^{2}f(τ)^{2} + ϵ^{4}f(τ)^{3}) dτ) ≡ K(t), \end{matrix}

which implies (4.1). Thus, we complete the proof of Theorem 1.3. Remark 4.1. To justify (4.17) from (4.2) rigorously, we need a mollifying process; and, for simplicity of the presentation, we omit it here.

Acknowledgments

The authors would like to thank J. Y. Chemin for drawing their attention to [4, 13] and S. Wang for helpful conversations. Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-9971793, INT-9987378, and INT-9726215. Ping Zhang’s research was supported in part by the Chinese Youth Foundation and the innovation grants from the Chinese Academy of Sciences.

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Gui-Qiang Chen Department of Mathematics Northwestern University Lunt Hall, 2033 Sheridan Road Evanston, IL 60208-2730, USA gqchen@math.northwestern.edu

Ping Zhang Academy of Mathematics and System Sciences The Chinese Academy of Sciences Beijing 100080, PRC zp@math03.math.ac.cn