On the Effect of Fast Rotation and Vertical Viscosity on the Lifespan of the 3D Primitive Equations

We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega |$$\end{document}|Ω|, we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}L2 in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Omega |$$\end{document}|Ω|, we show that the existence time of the solution can be prolonged, with “well-prepared” initial data. Finally, in the case of two spatial dimensions with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =0$$\end{document}Ω=0, we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.


Introduction
We consider the following 3D viscous primitive equations (PEs) with only vertical viscosity for the largescale oceanic and atmospheric dynamics: (1.1c) in the horizontal channel D := (x, z) = (x, y, z) : x ∈ T 2 , z ∈ (0, 1) , subject to the following initial and boundary conditions: R1 Local well-posedness (see Theorem 3.1): Assume that V 0 is analytic in the horizontal variables x and only L 2 in the vertical variable z. Let Ω ∈ R be arbitrary but fixed. Then there exists a positive time T > 0, independent of Ω, such that there exists a unique Leray-Hopf type weak solution V to system (1.1) (see Definition 3.1, below). Moreover the weak solution V depends continuously on the initial data and in particular it is unique. R2 Instantaneous analyticity in the vertical variable (see Theorem 3.2): With the same assumptions as in R1 above, the unique Leray-Hopf type weak solution V immediately becomes analytic in z for t > 0. Moreover, thanks to the viscous effect the radius of analyticity in z increases in time, at least linearly, for as long as the solution exists. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. R3 Long-time existence (see Theorem 5.1): Let |Ω| ≥ |Ω 0 | with |Ω 0 | large enough, in particular |Ω 0 | > 1.
(a) Under the assumption that the solution V to the 2D Euler equations with initial data V 0 is uniformly-in-time bounded in the analytic space norm, (1.4) can be improved to T = O(log(log(|Ω 0 |))). Let us note that this result is parallel to a similar one in the inviscid case [23]. Roughly speaking, S r,s,τ is the space of functions that are analytic with radius τ in the x-variables, and H s in the z-variable. The space of analytic functions is a special case of Gevrey class. For more details about Gevrey class, we refer readers to [20,21,23,40]. Notice that when τ = 0, one has S r,s,0 = H r x H s z (D).
Remark 1. With abuse of notation, we also write f ∈ S r,0,τ for f = f (x) depending only on the horizontal variables.
The following lemma summarizes the algebraic property of functions with analyticity in the horizontal variables: Lemma 2.1. For τ ≥ 0 and r > 1, we have provided that the right hand side is bounded, where, according to (2.2), The proof of Lemma 2.1 is standard. We refer to [20,23,45] for details.

Projections and Reformulation of the Problem
In this paper, we assume that D V 0 (x, z)dxdz = 0. This assumption is made to simplify the mathematical presentation. In fact, integrating (1.1a) in D leads to, after applying integration by parts, (1. In the remaining of this paper, we will substitute w by its representation (2.10) without explicitly pointing it out.
Since ∇ · V = 0, and V has zero mean over T 2 thanks to (2.7), there exists a stream function ψ(x) such that V = ∇ ⊥ ψ = (−∂ y ψ, ∂ x ψ) . Therefore, the space of solutions to (1.1) is given by for some ψ, T 2 ψ(x) dx = 0 . (2.11) Indeed, S is the analogy of "incompressible function space" for the PEs. Here ϕ and ϕ are the barotropic and baroclinic modes of ϕ, respectively, as in (2.8).

With notations as above, a direct computation shows that
Indeed, owing to (2.11), ϕ = ∇ ⊥ ψ(x) + ϕ ∈ S for some ψ(x). Then Therefore, the kernel of R is given by One can define the projection P 0 : S → ker R by Notice that P 0 can be interpreted as projection to the barotropic mode. The fact that ker R coincides with the space of functions with only the barotropic mode plays an important role in our analysis. Furthermore, let Then it is easy to verify that i.e., P ± are the projection operators to eigenspaces of R with eigenvalues ∓i, respectively. Similarly to [18,23,33], Lemma 2.2-2.3, below, summarize projection properties of P 0 , P ± . For the proof, we refer readers to [23] for details.

Lemma 2.2.
For any ϕ ∈ L 2 (D), we have the following decomposition: (2.16) Moreover, we have the following properties: Here the L 2 inner product is defined as (2.1).
Let I be the identity operator. A direct corollary of Lemma 2.3 is the following: Moreover, after applying P 0 and I − P 0 to equation (1.1a), thanks to (1.3), (2.9), and (2.10), one can derive the evolutionary equations for V and V as follows: Here, we have abused the notation by denoting p − Ωψ with ∇ ⊥ ψ(x, t) = V(x, t) as p, where ψ is the stream function of V (see (2.11)).

Remark 3.
According to (2.13), (2.17) can be viewed as the orthogonal decomposition of (1.1) into ker R and (ker R) ⊥ . As |Ω| → ∞, formal asymptotic analysis of (2.17b) assures that, for well-prepared data (i.e., data ensuring that (2.17b) makes sense), V → 0 in some functional space. Therefore, in the limiting equations, (2.17) converge to the 2D Euler equations at leading order. In particular, in [23], it has been shown that the lifespan of the solutions can be prolonged with well-prepared initial data in the inviscid case. According to (2.15), one has V ⊥ = −iP + V + iP − V. Therefore, after applying P ± to (2.17b), we arrive at Then, for r ≥ 0, τ ≥ 0, s ≥ 0, and s ∈ Z, it is straightforward to check that, (2.20) One can derive from (2.18) that Thanks to Lemma 2.2 and (2.15), we have After applying integration by parts, one has Moreover, thanks to (2.15) and (2.19), V = V + e iΩt + V − e −iΩt . Therefore, the V + part of (2.21) can be written as Similarly, the V − part of (2.21) can be written as (2.23) In addition, (2.17a) can be written as Recalling (2.15) and (2.19), i.e., V ± = e ∓iΩt P ± V = 1 2 e ∓iΩt ( V ± i V ⊥ ), we rewrite the last term of the above equation as which can be combined with ∇p. Therefore, with abuse of notation, one can rewrite (2.17a) as (2.24)

Local Well-posedness
In sections 3.1 and 3.2, below, we will establish the local well-posedness, i.e., the existence, the uniqueness, and the continuous dependency on initial data, of weak solutions to system (1.1), defined as below: Definition 3.1. Let T > 0, r > 2, τ 0 > 0, and suppose that the initial data V 0 ∈ S r,0,τ0 ∩ H. We say V is a Leray-Hopf type weak solution to system (1.1) with initial and boundary conditions ( 2) system (1.1) is satisfied in the distribution sense, 3) and moreover, the following energy inequality holds: The following theorem is the main result in this section. Notice that we do not need to assume (2.7) in Theorem 3.1. Throughout the rest of this section, we assume that (V, p) satisfies (1.1)-(1.3) and is smooth enough such that the following calculation makes sense. The rigid justification can be established through Galerkin approximation arguments (see, e.g., [23,39]). In particular, in section 3.1, we establish the a priori estimates of solutions to system (1.1) with (1.3). In section 3.2, we finish the proof of Theorem 3.1 by establishing the uniqueness and continuous dependency on initial data. In section 3.3, we show that the weak solution immediately becomes analytic in z, and the radius of analyticity in z increases as long as the solution exists.

A Priori Estimates
Direct calculation of (1.1a), V + A r e τA (1.1a), A r e τA V , after applying integration by parts, (1.1c), and (1.3), shows that (3.1) By virtue of Lemma A.1, the Sobolev inequality, and the Hölder inequality, we have Applying Lemma A.2 to I 2 leads to Thus from (3.1), one has (3.2) Choose τ such thatτ Then, one has For T > 0, to be determined, and t ∈ [0, T ], one has, after integrating (3.2) in the t-variable, On the other hand, integrating (3.3) yields Consider, for C r > 0 as in (3.5), that (3.6) which solves Then one has Consequently, (3.4) implies that with T > 0 given as in (3.6) and τ (t) given as in (3.5) (or equivalently (3.3)). Next, in order to obtain the estimate of ∂ t V, testing (1.1a) with ∀φ ∈ V (see (2.5)) leads to where we have substituted, thanks to (1.1b) and (2.5), ∇p, φ = − p, ∇ · φ = 0. Since r > 2, thanks to the Hölder inequality and the Sobolev inequality, we obtain that After applying integration by parts, one has Therefore, one has Since V is dense in V , one has Thanks to (3.7), we have (3.9) Meanwhile, for A r− 1 2 e τA ∂ t V, one has, similarly as in (3.8), With r > 2, thanks to Lemma 2.1, the Hölder inequality, and the Sobolev inequality, we obtain that

After applying integration by parts in the z-variable and the Hölder inequality, one has
Therefore, one has Thanks to (3.7), we have (3.10)

Uniqueness and Continuous Dependence on the Initial Data
In this section, we show the uniqueness of solutions and the continuous dependence on the initial data. Let V 1 and V 2 be two weak solutions with initial data (V 0 ) 1 and (V 0 ) 2 , respectively. Assume the radius of analyticity of (V 0 ) 1 and (V 0 ) 2 is τ 0 . By virtue of (3.5) and (3.6), for i = 1, 2, let and such that, according to (3.4), (3.7), (3.9), and (3.10), We remind readers that C r,i , i = 1, 2, are independent of Ω and τ 0 . Let Denote by δV := V 1 − V 2 and δp := p 1 − p 2 . Let and T := where C r is a positive constant, to be determined later, satisfying In particular, (3.13) and (3.14) imply thatτ (t) ≤ τ i (t) and T ≤ T i for i ∈ {1, 2} and t ∈ (0, T ]. Therefore, for i = 1, 2, and Notice that from (3.15), one has that A r− 1 2 e τ A δV ∈ L 2 0, T ; V . Thanks to (3.16), similar calculation as in (3.1) leads to (3.17) After applying integration by parts, the Hölder inequality, the Young inequality, and the Sobolev inequality, since r > 2, one has Thanks to Lemmas A.1 and A.2, the Hölder inequality, the Young inequality, and the Sobolev inequality, since r > 2, one has Consequently, combining the calculations between (3.17) and (3.18) yields In addition, from (3.13), and (3.14), and the fact that τ i (t) ≥ τ (t), i = 1, 2, one can derive thaṫ where we have chosen In conclusion, with C r satisfying (3.19), one has Applying the Grönwall inequality to (3.20) results in for t ∈ [0, T ], which establishes the continuous dependence on the initial data as well as the uniqueness of the weak solutions. This, together with section 3.1, finishes the proof of Theorem 3.1.

Instantaneous Analyticity in the z-Variable
In this section, we will show that the weak solution obtained in Theorem 3.1 immediately becomes analytic in the z-variable (and thus analytic in all variables) when t > 0. Moreover, the radius of analyticity in the z-variable increases as long as the solution exists. For simplicity, we consider the even extension for V in the z-variable, which is compatible with (1.3), and work in the unit three-dimensional torus T 3 instead of D. With abuse of notations, we use V to represent both V in D and its even extension with respect to the z-variable in T 3 . We first introduce the following notations that are only used in this subsection. For f ∈ L 2 (T 3 ) even with respect to the z-variable, we consider the following functional space

Denote by
subject to periodic boundary condition, defined by, in terms of the Fourier coefficients, Accordingly, one has With such notations, we establish the following theorem: Theorem 3.2. Assume V 0 ∈ S r,0,τ0,0 with r > 2 and τ 0 > 0. Let Ω ∈ R be arbitrary and fixed. Then there exist T > 0 defined in (3.24), τ (t) > 0 given in (3.23), below, and η(t) = ν 2 t, such that there exists a unique solution and depending continuously on the initial data. In particular, V immediately becomes analytic in all spatial variables for t > 0.

Remark 4.
After restricting V 0 and V in T 2 × (0, 1), the solutions in Theorem 3.2 are the same to the ones in Theorem 3.1, thanks to the uniqueness of solutions. Therefore, the gain of analyticity in the z-variable of Theorem 3.2 can be regarded as a property to solutions in Theorem 3.1.
Remark 5. Theorem 3.2 states the gain of analyticity in the z-variable for solutions to system (1.1). One can then apply the result from [23] to study the effect of rotation on the lifespan of solutions after a initial time layer. However, in order to achieve a longer lifespan, the result from [23] requests smallness of Sobolev norms in the baroclinic mode. In this paper, thanks to the effect of viscosity, we are able to Denote by Observe that H ≤ F . After settingη = ν 2 , one obtains that 1 2 For the nonlinear terms, by applying similar calculations as in Lemma A.1 and Lemma A.2 (we also refer the readers to [23] for detailed calculations in T 3 ), one can obtain that and thanks to the Young inequality, Therefore, combining all the estimates above leads to By takingτ + C r (E Then one has Notice that the radius of analyticity in the z variable satisfies η = ν 2 t. Therefore, (3.22) Based on the estimates above, one is able to show the existence, uniqueness, and continuous dependence on the initial data of the solution V. We omit the details.

The Limit Resonant System
In this section, we derive the formal limit resonant system, i.e., the limit system of system (1.1) (or, equivalently, system (2.17)) as |Ω| → ∞, and discuss some properties of the limit resonant system. Recall that from (2.22), we have (4.1) We can further rewrite (4.1) as Denote by the formal limits of V + , V − , and V to be V + , V − , and V , respectively. By taking limit Ω → ∞, we obtain the limit resonant equation for V + is Similarly, one has and Notice that (4.4) is nothing but the 2D Euler equations. Accordingly, we consider the initial conditions for equations (4.2)-(4.4). Since V 0 , V 0 , and V 0 are real valued, one has that (V Therefore, provided solutions exist and are well-posed, one has Notice that, according to (2.19), V is the formal limit of It is easy to verify that and or, thanks to ∇ · V = 0, equivalently, In summary, to solve the limit equations (4.2)-(4.4) with (4.5) is equivalent to solve the following equations: Notice that, thanks to our choice of V 0 and V 0 , one has P 0 V = V and P 0 V = 0. In addition, (4.11a)-(4.11b) is the 2D Euler system, and (4.11c) is a linear transport equation with a stretching term and vertical dissipation. In the rest of this section, we summarize the well-posedness theory of (4.11). The global well-posedness of solutions to the 2D Euler system (4.11a)-(4.11b) in Sobolev spaces H r (T 2 ) = S r,0,0 with r > 3 is a classical result (see, e.g., [7]). Moreover, from equation (3.84) in [7], for r > 3, we Let V 0 r,0,0 ≤ M for some M ≥ 0. Denote by W (t) := V (t) r,0,0 + e. Thanks to ln + x + 1 ≤ 2 ln(x + e), from (4.12), we have Therefore, one can obtain that The authors in [40] proved the global existence of solutions to system (4.11a)-(4.11b) for initial data in the space of analytic functions. For completion, we state it here, with slight modifications to meet our settings. See also [23]. to system (4.11a)-(4.11b). Moreover, there exist constants C M > 1 and C r > 1 such that The solution is continuously depending on the initial data.
Proof. (Sketch of proof) We will consider the case when s = 1 and only show the a priori estimates. The construction of solutions, uniqueness, and continuous dependency of solutions on initial data, as well as the case when s = 0, are left to readers as exercises. The global well-posedness of the 2D Euler equations in Sobolev spaces and corresponding growth estimate have been reviewed in the previous subsection. From (4.13), we obtain that with some constants C M,1 , C r,1 > 1.
Remark 8. Proposition 4.2 is for the general initial data. However, by considering special solutions to the 2D Euler equations, one has the following: • Supposed that V is uniformly-in-time bounded in S r+1,0,τ , i.e., sup 0≤t<∞ V (t) r+1,0,τ ≤ C M,r for some positive constant C M,r , then one can control the growth of V (t) r,1,τ by one exponential in time. • Supposed that sup 0≤t<∞ V (t) r+1,0,τ ≤ ν 4Cr,α is small enough, by applying the Poincaré inequality and with τ chosen suitably, from (4.20) one can derive that After applying the Grönwall inequality to the above, we obtain In particular, the estimate above holds when V ≡ 0, i.e., zero solutions to the 2D Euler equations.

Effect of Fast Rotation
In this section, we investigate the effect of rotation on the lifespan T of solutions to system (1.1). We show that the existing time of the solution in S r,0,τ (t) can be prolonged for large |Ω| provided that the Sobolev norm V 0 5 2 +δ,1,0 is small, while the analytic-Sobolev norm V 0 r,0,τ0 can be large. Such initial data is referred to as "well-prepared" initial data.

Remark 9.
Recall that the result in [23] requires the initial baroclinic mode V 0 to be small in the H 3+δ space instead of (5.2) in Theorem 5.1. This relaxation on the requirement of V 0 is due to the vertical viscosity.
In Theorem 5.1, we consider general initial data V 0 for the barotropic mode, where the vertical viscosity helps relax the requirement on the initial baroclinic mode, but does not help prolong the lifespan. By virtue of Remark 8, when the solution V to the 2D Euler equations with initial condition V 0 satisfies certain conditions, the smallness condition (5.2) can be relaxed and the result (5.3) can be improved. The following theorem is the summary of these results: Remark 10. Compared to [23], the main improvement in Theorem 5.1 is that the initial data is analytic in the horizontal variables but only L 2 in the vertical variable. The main improvements in Theorem 5.2 are points (ii) and (iii), where the smallness assumption does not depend on Ω 0 , and the lifespan is growing faster with respect to Ω 0 . For more details, we refer readers to R3 and R4 in the introduction (pages 2 and 3).
In this section, we focus on equations (2.22)-(2.24), which are equivalent to system (1.1). To prove Theorem 5.1, in section 5.1, we rewrite (2.22)-(2.24) as the perturbation of (4.2)-(4.4). In section 5.2, we establish a series of a priori estimates on the solutions to the perturbation system. This together with Proposition 4.2 will finish the proof of Theorem 5.1. In section 5.3, the proof of Theorem 5.2 is provided.
Remark 11. In this section, we only focus on the long-time existence of the weak solution. By virtue of Theorem 3.2, the weak solution is analytic in all spatial variables. where Recalling that (V, V ± ) and (V , V ± ) are complemented with the same initial data. Hence, we have φ| t=0 = 0 and φ ± | t=0 = 0. (5.8)

Estimates of Type 1 -Type 4 Terms.
We start with Type 1 terms. Applying Lemmas A.1-A.3 yields where we have used the embedding L ∞ z → H 1 z in the z-variable and the Hölder inequality. Notice that, for φ = φ, the estimate is similar with obvious modification. Therefore, hereafter, unless pointed out explicitly, we omit the estimates in the case of φ = φ and, similarly, V = V .
Remark 12. For the interested readers, we refer to [23] for an alternative estimate of Tp4 1 , where some cancellations are taking care of. However, in this paper, such cancellations are not necessary and thus omitted. Notably, the terms V ± 3 2 +δ,1,0 in the estimate of Tp4 1 is the reason for the requirement (5.2).

Estimates of Type 5 Terms.
In this case, j = 0 and e jΩit = 1 jΩi d dt e jΩit . Therefore, Tp5 can be written as, with abuse of notations, It is straightforward to check that Meanwhile, one has It follows that, thanks to Lemma 2.1 and similar arguments as in section 5.2.1, After applying the Leray projection (2.12) to (4.4), together with (4.2) and (4.3), for V = V ± or V , one has Here we use B to represent a generic bilinear term with respect to both of its arguments. With such notations, after applying integration by parts, one can derive Therefore, R 1,3 can be estimated as The estimate of R 2 is the same as R 1 (see (5.15), (5.17), and (5.19)). To estimate R 3 , one has, after applying integration by parts, As before, The estimate of R 3,2 is the same as that of R 1,2 in (5.17). Meanwhile, substituting representation (5.16) in R 3,3 leads to After substituting (5.18), R 3,4 can be estimated as We emphasize that, in the estimates above, we do not distinguish V ± and V , φ ± and φ, i.e., we treat all V and φ as if they are three-dimensional. The estimates in the case when they are two-dimensional are similar with obvious modifications, and thus omitted. Consequently, combining (5.15)- (5.22) leads to the estimate of R.

Proof of Theorem 5.2
In this section, we prove Theorem 5.2. We only sketch the proof for the first two parts, and will provide detailed proof for the third part.
For the first part of the theorem, thanks to Remark 8, we know that when sup 0≤t<∞ V (t) r+3,0,τ (t) ≤ C M,r the growth of V (t) r+2,1,τ (t) will only be exponentially in time. Thus, the function K(t) appears in the proof of Theorem 5.1 (e.g., (5.29) and (5.35)) becomes only exponentially in time. This reduces two logarithms in the estimate of existence time and gives This can be seen as in (5.
in which we will ask for provided that C r,ν,M and C r,α are large enough. From this, one can conclude that the smallness assumption can be relaxed and replaced by V 0 3 2 +δ,0,0 ≤ τ0 Cr,ν,M . Next we give the detailed proof to the third part of Theorem 5.2. Consider the initial data satisfying V 0 r+3,0,τ0 ≤ M |Ω|0 . We set V = 0 and replace the initial condition (5.8) of the perturbed system to With more careful estimates, (5.23) becomes   After integrating the above equation in time and recalling that F (t = 0) ≤ M |Ω0| , since |Ω| > |Ω 0 | > 1, one obtains The estimate of B 3 is similar to that of B 1 , and one can obtain that B 3 ≤ C r A r+ 1 2 e τA f A r e τA ∂ z g A r e τA h .
Combine the estimates of B 1 , B 2 , and B 3 , we obtain the desired result.
The proof of Lemma A.3 is almost the same as Lemma A.1, so we omit it. We will show lemmas which are essential in the study of effect of rotation. Lemma A.4 to Lemma A.6 are concerning the commutator estimates.
We start with the proof of Theorem A.4. The proof of Theorem A.5 will be similarly.