Global well-posedness for a rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.


Introduction
In geophysics, thermal convection is often influenced by planetary rotation. Coherence structures in convection under moderate rotation are exclusively cyclonic. However, for rapid rotation, experiments have revealed a transition to equal populations of cyclonic and anticyclonic structures. For instance, the flow visualization experiments of Vorobieff and Ecke [17] identified a striking topological change in the dynamics of the vortices. In the strongly nonlinear and turbulent regimes, plume generation in the thermal boundary layer results in a new population of anticyclonic plumes, in addition to the cyclonic population. Also, the distribution of cyclonic and anticyclonic coherent structures approaches a balance as the Rossby number approaches zero. In order to study such interesting phenomenon numerically, Sprague et al. [14] derived a reduced system of equations for rotationally constrained convection valid in the asymptotic limit of thin columnar structures and rapid rotation. Performing a numerical simulation of Rayleigh-Bénard convection in an infinite layer rotating uniformly about the vertical axis, visualization indicates the existence of cyclonic and anticyclonic vortical population in [14], which is consistent with the experimental results described in [17]. Also see [9,10].
The reduced three-dimensional rapidly rotating convection model of tall columnar structure introduced in [14] is given by the system:
The global regularity for system (1.1)-(1.5) is unknown. The main difficulty of analyzing (1.1)-(1.5) lies in the fact that the physical domain is three-dimensional, whereas the regularizing viscosity acts only on the horizontal variables, and the equations contain troublesome terms ∂φ ∂z and ∂w ∂z involving the derivative in the vertical direction. In our recent paper [4], system (1.1)-(1.5) was regularized by a weak dissipation term and the global well-posedness of strong solutions was established for the regularized system. System (1.1)-(1.5) is a reduced model derived from the three-dimensional Boussinesq equations by using the asymptotic theory. Generally speaking, the Boussinesq approximation for buoyancy-driven flow is applied to problems where the fluid varies in temperature from one place to another, driving a flow of fluid and heat transfer. In particular, the Boussinesq approximation to the Rayleigh-Bénard convection is a system of equations coupling the three-dimensional Navier-Stokes equations to a heat advection-diffusion equation. For small Rossby number (i.e., rapid rotation) and large ratio of the depth of the fluid layer to the horizontal scale (i.e., tall columnar structures), the 3D Boussinesq equations under the influence of a Coriolis force term, can be reduced to system (1.1)-(1.5) asymptotically. The derivation of model (1.1)-(1.5) was motivated by the Taylor-Proudman constraint [13,15] which suggests that rapidly rotating convection takes place in tall columnar structures.
According to the derivation of model (1.1)-(1.5) in [14] from the 3D Boussinesq equations, the state variables, i.e., the velocity, pressure and temperature, are expanded in terms of the small parameter Ro, which stands for the Rossby number. For rapidly rotating flow, i.e., Ro ≪ 1, the leading-order flow is horizontally divergence-free (see [14]). Roughly speaking, if the Rossby number is small in the Boussinesq equations, then the Coriolis force and the pressure gradient force are relatively large, which results in an equation in which the leading order terms are in geostrophic balance: the pressure gradient force is balanced by the Coriolis effect. Then taking the curl of the geostrophic balance equation implies that the horizontal flow is divergence-free, namely, ∇ h · u = u x + v y = 0. In addition, the term ∂ψ ∂z in equation (1.1) also originates from the geostrophic balance.
Since the original Boussinesq equations are considered in a tall column (x, y,z) ∈ [0, 2πL] 2 × [0, 2πℓ], where the aspect ratio ℓ/L ≫ 1, it is natural to introduce the scaled vertical variable z =z/ℓ ∈ [0, 2π] which appears in system (1.1)-(1.5). Set u(x, y, z) =ũ(x, y,z), v(x, y, z) =ṽ(x, y,z) and w(x, y, z) =w(x, y,z), where (ũ,ṽ,w) tr represents the velocity vector field for the original Boussinesq equations. Note, the divergence-free condition of (ũ,ṽ,w) tr reads ∂ũ ∂x + ∂ṽ ∂y + ∂w ∂z = 0, which implies that ∂u ∂x + ∂v ∂y + 1 ℓ ∂w ∂z = 0. Then, for ℓ ≫ 1, one can ignore the term 1 ℓ ∂w ∂z and obtain that ∂u ∂x + ∂v ∂y = 0. In sum, the fast rotation and the tall columnar structure both imply that to leading order terms the horizontal flow is divergence-free. Moreover, the absence of vertical diffusion in system (1.1)-(1.5) is also a consequence of the large aspect ratio of the fluid region. Furthermore, it is remarked in [14] that in the classical small-aspect-ratio (flat) regime, the strong stable stratification permits weak vertical motions only, while in the present large-aspect-ratio (tall) case, the unstable stratification permits substantial vertical motions.
There are two dimensionless numbers in system (1.1)-(1.5). They are the Prandtl number Pr and the Rayleigh number Ra. The Prandtl number Pr represents the ratio of molecular diffusion of momentum to molecular diffusion of heat. More precisely, one defines Here, ν represents momentum diffusivity, i.e., kinematic viscosity. Notice that ν = µ/ρ, where µ is dynamics viscosity and ρ is the constant density. Also, α stands for the thermal diffusivity, which is equal to k/(c p ρ), where k is thermal conductivity and c p represents the specific heat capacity of the fluid. Fluids with small Prandtl numbers are free-flowing liquids with high thermal conductivity and are therefore a good choice for heat transfer liquids. Liquid metals such as mercury have small Prandtl numbers. On the other hand, with increasing viscosity, the Prandtl number also increases, leads to the phenomenon that the momentum transport dominates over the heat transport and acts on a faster time scale. For instance, concerning the engine oil, convection is very effective in transferring energy in comparison to pure conduction, so momentum diffusivity is dominant. Another example is Earth's mantle, which has extremely large Prandtl number. The Rayleigh number Ra is another dimensionless number appeared in model (1.1)-(1.5). It represents the strength of the buoyancy in the fluid driven by the heat gradient.
In this manuscript, we consider rapidly rotating convection in the limit of infinite Prandtl number Pr. Under such scenario, the dynamics of the momentum acts on a much faster time scale than the heat dynamics. For this problem, the appropriate time scale is the horizontal thermal diffusion time. Using the substitution (cf. [14]) 1 Pr Then in the limit of infinite Prandtl number, i.e., letting Pr → ∞ in system (1.7)-(1.11), one formally obtains the following system of equations The system is considered subject to periodic boundary conditions in R 3 with fundamental periodic domain Ω = [0, 2πL] 2 ×[0, 2π]. Here, ω = ∇ h ×u, ψ = ∆ −1 h ω such that its horizontal average ψ = 0. Recall that the horizontal thermal fluctuation θ ′ of the temperature θ is defined as θ ′ = θ − θ, where θ(z, t) = 1 4π 2 L 2 [0,2πL] 2 θ(x, y, z, t)dxdy is the horizontalmean temperature. In system (1.12)-(1.16), the velocity field acting on a very fast time scale, adjusts instantaneously to the dynamics of the thermal fluctuations, demonstrated by the linear equations (1.12)-(1.13). Therefore, the initial condition is imposed on θ ′ only: The purpose of this work is to prove the global well-posedness of weak and strong solutions for system (1.12)-(1.16) defined on a fundamental periodic space domain Ω = [0, 2πL] 2 × [0, 2π]. Also, in order to obtain the uniqueness of the temperature θ, we assume that the average temperature is zero, i.e., Ω θ(x, y, z, t)dxdydz = 0, for all t ≥ 0.
In the literature, there were some analytical studies for the three-dimensional Boussinesq equations in the limit of infinite Prandtl number. Wang [18] rigorously justified the infinite Prandtl number convection model as the limit of the Boussinesq equations when the Prandtl number approaches infinity (see also [19,20]). Also, for infinite Prandtl number convection, there have been several rigorous derivation of upper bounds of the upwards heat flux, as given by the Nusselt number Nu, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime Ra ≫ 1. For example, the work [5] by Constantin and Doering was one of the early papers in the literature for this topic. More recently, by combining the background field method and the maximal regularity in L ∞ , Otto and Seis [12] showed that Nu Ra 1/3 (log log Ra) 1/3 -an estimate that is only a double logarithm away from the supposedly optimal scaling Nu ∼ Ra 1/3 . See also [6,7,8,11,21,22] and references therein.
It is worth mentioning that in [2], Cao, Farhat and Titi established the global regularity for an inviscid three-dimensional slow limiting ocean dynamics model, which was derived as a strong rotation limit of the rotating and stratified Boussinesq equations.
The paper is organized as follows. In section 2, we state main results of the paper, i.e., the global well-posedness of weak solutions and strong solutions for the infinite Prandtl number convection (1.12)-(1.16). In section 3, we provide some auxiliary inequalities and some wellknown identities, which will be used repeatedly in our energy estimates. In section 4, we give a detailed proof for the global well-posedness of weak solutions. Finally, section 5 is devoted to the proof for the global well-posedness of strong solutions.

Main results
In this section, we give definitions of weak solutions as well as strong solutions for system We define the space H 1 h (Ω) of periodic functions on Ω with horizontal average zero by Also, we denote the dual space ofḢ 1 (0, 2π) by H −1 (0, 2π) = (Ḣ 1 (0, 2π)) ′ . Recall θ ′ = θ − θ represents the fluctuation of the temperature θ, about the horizontal average. Also, (u, w) = (u, v, w) is the three-dimensional velocity vector field on the periodic domain Ω.
, and the equations hold in the function spaces specified below: [14], the quantities θ ′ , u and w are "fluctuating" quantities about the horizontal mean, i.e., the original quantities subtracted by their horizontal means. Therefore, in the above definition of weak solutions, all quantities are demanded to have horizontal average zero.
In the next theorem we state the existence and uniqueness of global weak solutions to system (1.12)-(1.16) as well as the continuous dependence on initial data. Theorem 2.2 (Global well-posedness of weak solutions). Assume θ ′ 0 ∈ L 2 (Ω) with θ ′ 0 = 0. Then, system (1.12)-(1.16) has a unique weak solution (θ ′ , θ, u, w) for all t ≥ 0, in the sense of Definition 2.1. Moreover, the solution satisfies the following energy equality: for all t ≥ 0. Also, the following decay estimates are valid: , and (u n , w n ) → (u, w) in L 2 (0, T ; H 1 (Ω)).

Strong solutions.
We define a strong solution of system (1.12)-(1.16). (2.11) and the equations hold in the function spaces specified below: The following theorem states the existence and uniqueness of global strong solutions to system (1.12)-(1.16).

Preliminaries
We state some inequalities which will be useful in our estimates.
The following is an anisotropic Ladyzhenskaya-type inequality which has been proved in [3].
(Ω) and f = 0, the Poincaré inequality is valid: Next, we state some identities which will be employed in the energy estimate. For sufficiently smooth periodic functions u, f and g on Ω, such that ∇ h · u = 0, an integration by parts shows Note that the horizontal velocity u, the vertical component ω of the vorticity, and the horizontal stream function ψ such that ψ = 0 have the following relations: (3.6)

Weak solutions
In this section, we prove the global well-posedness of weak solutions to system (1.12)-(1.16) by using the Galerkin method.
We consider the Galerkin approximation for system (1.12)-(1.16): We assume the horizontal average zero condition: θ ′ m = w m = ω m = ψ m = 0 and u m = 0. In addition, we demand the average value of θ m is zero: Thus, the unknown functions in the Galerkin system can be expressed as finite sums of Fourier modes: (4.7) (4.8) Since equations (4.1)-(4.2) are linear, they can be solved explicitly if θ ′ m is given. Indeed, The above linear system can be written as For the 2 × 2 matrix in (4.9), its determinant k 2 3 + ( Thus we can take the inverse of this matrix. It follows that Consequently, By substituting (4.11)-(4.12) and (4.13) into equation (4.3), we obtain a system of first order nonlinear ordinary differential equations with unknowns { θ ′ m (k, t) : k ∈ Z 3 , k 2 1 + k 2 2 = 0, |k| ≤ m}. By the classical theory of ordinary differential equations, for each m ∈ N, there exists a solution { θ ′ m (k, t) : |k| ≤ m} defined on [0, T max m ) for the system of ODEs. Then, thanks to (4.10)-(4.12), we obtain ψ m (k, t), u m (k, t), v m (k, t) and w m (k, t), for |k| ≤ m. Next, we substitute θ ′ m and w m into (4.13) to get ∂θm ∂z , and along with the assumption 2π 0 θ m (z, t)dz = 0 from (4.5), we obtain θ m . Finally, θ m = θ ′ m + θ m which satisfies Ω θ m dxdydz = 0. Assume the ODE system has finite time of existence, i.e., T max m < ∞. By estimate (4.16) below, we know that θ ′

Energy estimate.
Taking the inner product of (4.3) with θ ′ m and using (3.5), one has 1 2 Recall that the horizontal mean of a function f is defined as for all t ≥ 0. Therefore, Integrating over [0, t] yields Next, we estimate u m and w m . By (4.12), we calculate In addition, by using Young's inequality, one has By (4.11), we see that (4.20) Also, using (4.11), one has Moreover, applying Young's inequality, we obtain Therefore, on a subsequence, we have the following weak convergences as m → ∞: ∂ z θ m → θ z weakly in L 2 ((0, 2π) × (0, T )). (4.32) Using these weak convergences and inequality (4.16), we obtain the energy inequality: for all t ∈ [0, T ].
Also, using these weak convergences, we can pass to the limit for the linear equations (4.1)-(4.2) in the Galerkin approximation system to obtain Then we can use the same calculations as (4.17)-(4.22), to derive that for all t ∈ [0, T ].
By (4.61), we have where we have used Lemma 3.3 and (4.37) as well as the decay estimate (4.59). Since 2π 0 θdz = 0, then we can use the Poincaré inequality to conclude

Uniqueness of weak solutions and continuous dependence on initial data.
This section is devoted to proving the uniqueness of weak solutions. Assume there are two weak solutions (θ ′ 1 , θ 1 , u 1 , w 1 ) and (θ ′ 2 , θ 2 , u 2 , w 2 ) on [0, T ], in the sense of Definition 2.1.
Next we estimate each term on the right-hand side of (4.68). By employing identity (3.4) and Lemma 3.1 as well as estimate (4.66), we deduce where we use Hölder's inequality.

Strong solutions
In this section, we prove the global well-posedness of strong solutions for system (1.12)-(1.16). The uniqueness of strong solutions follows from the uniqueness of weak solutions. It remains to show the existence of global strong solutions, when the initial value θ ′ 0 ∈ H 1 (Ω).

5.1.
Existence of strong solutions. We use the method of Galerkin approximation. Let us consider the Galerkin system (4.1)-(4.4) and perform energy estimate as follows.
Thanks to Lemma 3.1, we have where we have used estimates (5.1)-(5.2). Next, using the Cauchy-Schwarz inequality, we estimate where we have used Lemma 3.3 and estimates (5.1) and (5.3). By applying estimates (5.8) and (5.9) into (5.7), we infer Applying Grönwall's inequality to (5.10) and using estimate (5.6), we obtain Taking the L 2 inner product of (5.13) with ∂ z θ ′ m , we obtain where we have used Ω (u m · ∇ h ∂ z θ ′ m ) ∂ z θ ′ m dxdydz = 0 since ∇ h · u m = 0. We evaluate every integral on the right-hand side of (5.14). First, by using Lemma 3.1 as well as estimate (5.4)-(5.5), we have Applying the Cauchy-Schwarz inequality yields where we have used Lemma 3.3 and estimate (5.5).
Now we estimate the third integral on the right-hand side of (5.14). Since ∂ z (θ ′ m w m ) = ∂ zz θ m due to (4.4), we have where the last inequality is due to the Cauchy-Schwarz inequality and the Young's inequality. By using the Cauchy-Schwarz inequality, and Lemma 3.3 as well as estimate (5.2), then On account of (5.6), (5.11) and (5.21), we conclude that where C( θ ′ 0 H 1 (Ω) ) is a constant depending on θ ′ 0 H 1 (Ω) but independent of t. where C( θ ′ 0 H 1 (Ω) ) is a constant depending on θ ′ 0 H 1 (Ω) but independent of t.

Estimate for
5.1.6. Passage to the limit. Let T > 0. In order to pass to the limit for nonlinear terms in the Galerkin system as m → ∞, we shall show that ∂ t θ ′ m is uniformly bounded in L 2 (Ω × (0, T )).