The Primitive Equations in the scaling invariant space $L^{\infty}(L^1)$

Consider the primitive equations on $\R^2\times (z_0,z_1)$ with initial data $a$ of the form $a=a_1+a_2$, where $a_1 \in BUC_\sigma(\R^2;L^1(z_0,z_1))$ and $a_2 \in L^\infty_\sigma(\R^2;L^1(z_0,z_1))$ and where $BUC_\sigma(L^1)$ and $L^\infty_\sigma(L^1)$ denote the space of all solenoidal, bounded uniformly continuous and all solenoidal, bounded functions on $\R^2$, respectively, which take values in $L^1(z_0,z_1)$. These spaces are scaling invariant and represent the anisotropic character of these equations. It is shown that, if $\|a_2\|_{L^\infty_\sigma(L^1)}$ is sufficiently small, then this set of equations has a unique, local, mild solution. If in addition $a$ is periodic in the horizontal variables, then this solution is a strong one and extends to a unique, global, strong solution. The primitive equations are thus strongly and globally well-posed for these data. The approach depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $L^\infty(L^1)$-setting and can thus be seen as the counterpart of the classical iteration schemes for the Navier-Stokes equations for the situation of the primitive equations.


Introduction
The primitive equations for ocean and atmospheric dynamics serve as a fundamental model for many geophysical flows. This set of equations describing the conservation of momentum and mass of a fluid, assuming hydrostatic balance of the pressure, is given in the isothermal setting by in Ω × (0, T ), div u = 0, in Ω × (0, T ), v(0) = a. (1.1) Here Ω := R 2 × J, where J = (z 0 , z 1 ) is an interval. Denoting the horizontal coordinates by x, y ∈ R 2 and the vertical one by z ∈ (z 1 , z 2 ), we use the notation ∇ H = (∂ x , ∂ y ) T , whereas ∆ denotes the three dimensional Laplacian and ∇ and div the three dimensional gradient and divergence operators. The velocity u of the fluid is described by u = (v, w), where v = (v 1 , v 2 ) denotes the horizontal component and w the vertical one.
In the literature various sets of boundary conditions are considered such as Neumann, Dirichlet and mixed boundary conditions. In this article we choose Neumann boundary conditions for v, i.e. The rigorous analysis of the primitive equations started with the pioneering work of Lions, Temam and Wang [17][18][19], who proved the existence of a global weak solution for this set of equations for initial data a ∈ L 2 . The uniquness problem for weak solutions remains an open problem until today.
The existence of a local, strong solution for this equation with data a ∈ H 1 was proved by Guillén-González, Masmoudi and Rodiguez-Bellido in [7].
In 2007, Cao and Titi [2] proved a breakthrough result for this set of equation which says, roughly speaking, that there exists a unique, global strong solution to the primitive equations for arbitrarily large initial data a ∈ H 1 . Their proof is based on a priori H 1 -bounds for the solution, which in turn are obtained by L ∞ (L 6 ) energy estimates. Kukavica and Ziane [13] proved global strong well-posedness of the primitive equations with respect to arbitrary large H 1 -data for the case of mixed Dirichlet-Neumann boundary conditions. For a different approach see also Kobelkov [11]. We also would like to draw the attention of the reader to the recent survey article by Li and Titi [16] on the primitive equations.
Recently, an new approach to the primitive equations based on the theory of evolution equations has been developed in [9] and [8]. This approach is also valid in the L p -setting for all 1 < p < ∞ and, using this approach, the authors proved global strong well-posedness of the primitive equations for arbitrary large data in H 2/p,p subject to mixed Dirichlet-Neumann boundary conditions. Taking formally the limit p → ∞, it is now tempting to consider initial data a ∈ L ∞ with no differentiability assumption on the initial data. This article aims to find a function space, as large as possible, for the initial data for which the primitive equations are strongly and globally well-posed.
Recent regularity results for weak solutions by Li and Titi [15] and Kukavica, Pei, Rusin and Ziane [12] are also pointing in this direction. More specificially, starting from a weak solution to the primitive equations, these authors investigated regularity criteria for weak solutions for the primitive equations, following hereby in a certain sense the spirit of Serrin's condition in the theory of the Navier-Stokes equations and methods of weak-strong uniqueness. Li and Titi proved in [15] that weak solutions are unique for initial values in C 0 or in {u ∈ L 6 : ∂ z u ∈ L 2 } including a small perturbation belonging to L ∞ . By the weak-strong uniqueness property, it follows that these weak solutions regularize and become strong solutions.
Our approach to rough initial data results for the primitive equations is very different: it considers the primitive equation as an evolution equation in an anisotropic function space of the form L ∞ (R 2 ; L 1 (J)). This space is invariant under the scaling to the primitive equations whenever v has this property. For further information on the Navier-Stokes equations in critical spaces see [1,3,14].
Based on L ∞ -type estimates for the underlying hydrostatic Stokes semigroup and as well as on its gradient estimates, we develop an iteration scheme yielding first the existence of a unique, local mild solution for initial data of the form a = a 1 + a 2 with a 1 ∈ BU C σ (R 2 , L 1 (J)) and a 2 being a small perturbation in L ∞ σ (R 2 ; L 1 (J)).
Here BU C(R 2 ) denotes the space of all bounded and uniformly continuous functions on R 2 . The subscript σ means the subspace of solenoidal fields rigorously defined in the next section. Assuming that a 1 , a 2 are periodic with respect to horizontal variables, we are able to prove that the solution regularizes sufficiently and thus, by an appropriate a priori estimate, can be extended to global, strong solution without any restriction on the size of a 1 . Comparing our assumptions on the initial data with the ones given by Li and Titi [15], observe first that our assumptions are slightly less restrictive compared to the case of continuous initial data, while our assumptions are not comparable to their second case.
Our approach may be viewed as the counterpart of the classical iteration schemes for the Navier-Stokes equations due to Giga [4] and Kato [10]. Note that, in contrast to the case of the Navier-Stokes equations, our iteration scheme presented here combined with a suitable a priori estimate yields the existence of a unique, global strong solutions not only for small data as in the case of the Navier-Stokes equations, but for arbitrary large solenoidal data a ∈ BU C σ (R 2 ; L 1 (J)).
As written above, our approach depends crucially on L ∞ (L 1 )-mappig properties of the underlying hydrostatic Stokes semigroup, including gradient estimates. These are collected in Proposition 5.6 and are of independent interest. This article is organized as follows. Section 2 presents the two main results of this article. The following Sections 3, 4 and 5 are devoted to anisotropic estimates for fractional derivatives, the heat semigroup as well as for the hydrostatic semigroup. In Section 6 we present a proof of our main results based on an iteration scheme.

Preliminaries and main results
Let z 0 ∈ R, z 1 = z 0 + h for some h > 0 and J be the interval J = (z 0 , z 1 ). The incompressibility condition div u = 0 in Ω × (0, T ) implies where the boundary condition w = 0 on ∂Ω was taken into account. Also, w = 0 on ∂Ω implies where v denotes the vertical average of v, i.e.
v(x, y) = 1 The linearization of equation (1.1) are the hydrostatic Stokes equations, which are given by (2.1) The name 'hydrostatic Stokes equations' is motivated by the assumption of a hydrostatic balance when deriving the full primitive equations. The equations (2.1) are supplemented by Neumann boundary conditions (1.2) for v.
For a function f : where we use the shorthand notation L q (L p ) for the spaces and · q,p for the norms. The usual modifications hold for the cases p = ∞ or q = ∞. The space L p (R 2 ; L q (J)) consisting of all measurable function f with f p,q < ∞ and equipped with the above norm becomes a Banach space. Following [9], we introduce the hydrostatic Helmholtz projection as follows. For a function f = (f 1 , f 2 ) : R 2 × J → C, we define the hydrostatic Helmholtz projection by The solenoidal subspace L ∞ σ (R 2 ; L p (J)) is then defined for 1 ≤ p ≤ ∞ as the closed subspace of L ∞ (R 2 ; L p (J)) given by where W 1,1 (R 2 ) denotes the homogeneous Sobolev space of the form W 1,1 (R 2 ) = {ϕ ∈ L 1 loc (R 2 ) : ∇ H ϕ ∈ L 1 (R 2 )}, that is, the condition div H v = 0 is understood in the sense of distributions.
If a ∈ L ∞ σ (R 2 ; L p (J)) for some 1 ≤ p ≤ ∞, then the solution of equation (2.1) can be represented as v(t) = S(t)a, where S denotes the hydrostatic Stokes semigroup on L ∞ σ (R 2 ; L p (J)). The latter semigroup may be represented as follows: consider the one-dimensional heat equation Given u 0 ∈ L p (J) for some p ∈ [1, ∞], its solution u is given by u(t) = e t∆N u 0 for (2.2a) and by e t∆DN u 0 for (2.2b), where e t∆N and e t∆DN denote the analytic semigroups on L p (J) generated by the Laplacian subject to Neumann or Dirichlet-Neumann boundary conditions, respectively. For a ∈ L ∞ σ (R 2 ; L p (J)), the solution of (2.1) is thus given by v(t) = S(t)a, where S(t) = e t∆H ⊗ e t∆N , t > 0, and where e t∆H denotes the heat semigroup on L ∞ (R 2 ). The semigroup S is not strongly continuous on L ∞ σ (R 2 ; L p (J)), however, its restriction to BU C σ (L p ) := BU C(R 2 ; L p (J)) ∩ L ∞ σ (L p ) defines for 1 ≤ p < ∞ a bounded analytic C 0 -semigroup on this space.
Our first main result concerns the existence of a unique, mild solution to the primitive equations with initial value a, i.e. a function v satisfying for some constant C > 0 independent of a 1 and a 2 .
The mild solution constructed in Theorem 2.1 exists at least for some nontrivial time interval [0, T ) where T depends on a. We are further able to estimate the existence time T > 0 from below in terms of the following |||·|||-norm defined for a ∈ L ∞ σ (R 2 ; L 1 (J)) by |||a||| :

Proposition 2.3 (Local existence for p > 1).
Let T > 0 be as in Theorem 2.1. If in addition to the assumptions of Theorem 2.1 the initial data a satisfies The following theorem on the global strong well-posedness of the primitive equation with rough and arbitrary data is the second main result of this article. Theorem 2.4 (Global existence). Suppose in addition to the assumptions of Theorem 2.1 that a is periodic with respect to the horizontal variables. Then, for any T * > 0 the unique, mild solution v obtained in Theorem 2.1 can be extended to a unique, strong solution of (1.1) on the interval (0, T * ).

Interpolation Inequalities for Fractional Derivatives
In this section we consider the Caputo fractional derivative of a function f ∈ L ∞ (J; C), where J = (z 0 , z 1 ) for z 0 ∈ R and z 1 = z 0 + h for some h > 0. To this end, we denote for α > 0 by I α z0 the Riemann-Liouville operator of the form where Γ denotes the usual Gamma function, i.e. Γ(α) = ∞ 0 e −ζ ζ α−1 dζ. Considering the zero extension of f to R still denoted by f and the zero extension of z from (0, h) by zero onto R denoted by z + , the Riemann-Liouville operator is defined as convolution Then I α z0 f is the α-times integral of f from z 0 whenever α > 0 and I α1+α2 z f as an integrable function or for f ∈ W 1,∞ (J) and thus in particular for Lipschitz continuous f . Indeed, by the Hausdorff-Young inequality for convolutions we have Here we identified ∂ z f with ∂ z f χ (z0,z0+µ) and z + with zχ (0,µ) denoting by χ (z0,z0+µ) and χ (0,µ) the corresponding characteristic functions.
We next state an interpolation inequality for ∂ α Proof. We may assume that ∂ z f 1 = 0 and f 1 = 0. Given µ ∈ (0, h) and z ∈ (z 0 + µ, z 1 ], we subdivide the integral into two parts and integrate by parts to obtain Since f (z 0 ) = 0, applying the Hausdorff-Young inequality yields Combining (3.1) and (3.3), we get and since by Poincaré's inequality Remark 3.2. The estimate (3.2) remains valid if the L 1 norm is replaced by the L p norm for some p ∈ [1, ∞], and we obtain in this case with essentially the same proof We next derive an interpolation inequality for the horizontal Laplace operator in the space L ∞ (L 1 ).
. Then there exists a constant C > 0, depending only on α, such that Proof. We may assume that ∇ H f ∞,1 = 0 and f ∞,1 = 0. Denoting by G t the 2-dimensional Gauss kernel, i.e., G t (x) = (4πt) −1 exp −|x| 2 /4t for x ∈ R 2 and t > 0 and setting e t∆H f = G t * H f , where * H denotes convolution in the horizontal variables, only, and the negative fractional powers of −∆ H are defined as We now employ the estimates By a direct calculation, and an application of Fubini's theorem yields where the constants C j depends only on α. Choosing µ = ( f ∞,1 / ∇f ∞,1 ) 2 , we obtained the desired estimate.

Anisotropic Estimates for the Heat Semigroup
In this section we derive various estimates on the semigroups e t∆N and e t∆DN introduced in Section 2. We denote by e t∆ * one of these semigroups and start with a regularizing decay estimate for e t∆ * . Lemma 4.1. Given α ∈ (0, 1), there exists a constant C > 0 such that f, e t∆ * ϕ = − I α z0 f, ∂ z e t∆ * ϕ , where in the last identity we used the fact that (I α z0 f )(z 1 ) = 0 and (I α z0 f )(z 0 ) = 0; the latter is trivial by definition. Since ∂ z e t∆ * ϕ .
Since I α z1 ∂ z resembles the Caputo derivative and ∂ z e t∆ ϕ(z 1 , t) = 0 by (2.2a) or (2.2b), we are able to adapt Lemma 3.1 to obtain Notice that e t∆ * ϕ ∞ ≤ ϕ ∞ for all t > 0 and ∂ z e t∆ * ϕ ∞ ≤ C 0 t −1/2 ϕ ∞ for all t > 0 for some constant C 0 > 0. This can be seen by extending the problem to the whole space problem by extending ϕ periodically in a suitable way to R to obtain e t∆ * ϕ = G t * φ, whereφ is the extension of ϕ.
with C 1 depending on α, only. We thus conclude that which gives the desired estimate.
In the following, we derive further regularity estimates for the heat semigroup e t∆ on L ∞ (R d ) for d ≥ 1. We denote by where H t ∈ L 1 (R d ) such that H t 1 ≤ C α with a constant C α independent of t > 0. In particular, withH t having the same properties as H t . In particular, withH t having the same property as H t . In particular, Proof. i) Using the Bochner representation formula for fractional powers of the Laplacian (see e.g. [20] p. 260) where we used the identities A calculation similar to the one given in the proof of Lemma 3.3 yields |∂ i ∂ j G t | ≤ Ct −1 G 2t for all t > 0 and proceeding as above we obtain iii) As above we have e t∆ R i R j ∂ k f = ∂ i ∂ j ∂ k e t∆ (−∆) −1 f and using the estimate |∂ i ∂ j ∂ k G t | ≤ Ct −3/2 G 2t for t > 0 yields the desired result by the same arguments.
for all f ∈ L p (J) satisfying I α z0 f (z 1 ) = 0 with z 1 = z 0 + h. The proof parallels then the one given above provided we have the estimates at hand. These estimates are well known and we give an elementary and self-contained proof of them by a periodization method, which is explained in the next section, see Lemma 5.5.

Estimates for the hydrostatic Stokes semigoup
Consider the hydrostatic Stokes semigroup S on L ∞ σ (L p (J)) for p ∈ [1, ∞] as introduced in Section 2 and given by S(t) = e t∆H ⊗ e t∆N , t > 0.
This semigroup admits an extension to the space L ∞ (L p ) for all p ∈ [1, ∞], which we denote by S ∞ . The following estimates with respect to the · ∞,1 -norm for the semigroup S ∞ and hence in particular for the hydrostatic Stokes semigroup S will be of crucial importance in our iteration argument in the subsequent Section 6.
Lemma 5.1. Let α ∈ (0, 1). Then there is a constant C such that (i) for all f ∈ L ∞ (L 1 ) and all t > 0 Moreover, for all f ∈ L ∞ (L 1 ) and all t > 0 (iv) for all f ∈ L ∞ (L 1 ) and all t > 0 Remark 5.2. We note that assertion ii) remains true also for the cases α = 0 and α = 1 by Remark 4.3. The later implies that S ∞ is a bounded semigroup in L ∞ (L 1 ).

Proof. i) This assertion follows from the estimates
∂ z e t∆ * f 1 ≤ Ct −1/2 f 1 , t > 0 and e t∆ * f 1 ≤ f 1 , t > 0, and from the pointwise estimates compare (3.7), as well as e t∆H ∂ xi f = ∂ xi e t∆H f for i = 1, 2 as in the proof of Proposition 5.6. ii) Since , t > 0, which allows us to conclude that The proof is also valid for the case α = 0 yielding S ∞ (t)∂ z f ∞,1 ≤ Ct −1/2 f ∞,1 for all t > 0.
(iii) We verify by Lemma 4.2 i) and ii) that for almost all x ′ ∈ R 2 since f is independent of z. By Fubini's theorem which allows us to conclude that (iv) As above we have The first term was already estimated and the second one is treated in the same way as in iii).
In order to extend the above estimates to L ∞ (L q ) space, it is convenient to investigate the periodic heat semigroup. Lemma 5.3. Let T = R/ω 0 Z for some ω 0 > 0 and f ∈ L 1 (T). Then If for all p ∈ [1, ∞]. Proof. The above representation for G t * f follows by noting that where f (y + kω 0 ) = f (y) for all k ∈ Z by periodicity. The estimate claimed follows by Young's inequality since Lemma 5.4 (Derivative estimate for the periodic heat semigroup). Under the assumption of Lemma 5.3, there exists a constant C > 0, independent of ω 0 , such that In particular, Proof. By (3.7) |∂ z G t (z)| ≤ Ct −1/2 G 2t (z), t > 0, z ∈ T, which implies the first assertion. The second one follows by Young's inequality.
Lemma 5.5 (Periodization). Given 1 ≤ p ≤ ∞, there exists a constant C > 0 such that Proof. Consider the case, where ∆ * = ∆ N . We first extend f ∈ L p (J) to (z 0 − h, z 0 ) by even extension, i.e., by setting f (z) = f (−z) for z ∈ (z 0 − h, z 0 ). and extend then f to a periodic function f per with period ω 0 = 2h by f per (z) = f (z − kω 0 ) for z ∈ (kω 0 , (k + 1)ω 0 ) and k ∈ Z, see Figure 1 . It then follows that e t∆N f = e t∆ f per J , These estimates are important in order to extend Lemma 5.1 to the situation of L ∞ (L q ) spaces.
Proposition 5.6. Let S ∞ be the semigroup on L ∞ (R 2 ; L q (J)) given by S ∞ (t) = e t∆H ⊗ e t∆N . Furthermore let α ∈ [0, 1) and q ∈ [1, ∞]. Then there exists a constant C > 0 such that (i) for all f ∈ L ∞ (L q ) Remark 5.7. Note that also the following L ∞ (L 1 )-L ∞ (L q ) smoothing holds for q ∈ [1, ∞] We also remark that, if the case α = 0 is considered in assertion ii), the term I 0 z0 is interpreted as the identity operator and there is no restriction for f other than f ∈ L ∞ (L q ).
for almost all x ′ ∈ R 2 . By Minkowski's inequality and due to the positivity of e t∆H As in Lemma 5.4, we observe that and applying Minkowski's inequality yields We thus conclude that The second part of i) follows from S(t)∂ xi f = ∂ xi S(t)f for i = 1, 2. The remaining assertions ii) and iii) follow from the L q versions of Lemma 5.1 and Remark 4.4.
In order to estimate II ∞,1 we observe that Lemma 5.1 (ii) and Lemma 3.1 yield Here we invoked the fact that we are able to estimate II ∞,1 in the same way as I. This completes the proof.
Our next step consists in proving a similar estimate for the above integral term, however, without assuming that the vertical average of the functions involved is vanishing. To this end, we set v 1,∞,1 := v ∞,1 + ∇v ∞,1 . Proposition 6.2. There exists a constant C > 0 such that for all α ∈ (0, 1) for all v ∈ L ∞ σ (L 1 ) satisfying ∇v ∈ L ∞ (L 1 ) and allũ = (ṽ,w) withṽ ∈ L ∞ σ (L 1 ) and ∇ṽ ∈ L ∞ (L 1 ) and andw = z1 z div Hṽ dx 3 . The statement stil holds true for α = 0. Proof. We argue similarly as in the proof of Lemma 6.1. In order to estimate v ∞,∞ we write Thus and the desired estimate follows as in the proof of Lemma 6.1.
We now give a proof of our main results.
The proof of the uniqueness follows a similar line of arguments. Let v,ṽ be two solutions, then and one obtains as above using Lemma 5.1 (i) and Proposition 6.2 with α = 1/2, and setting Hence one obtains By assumption, if t is small we have Thus supposing that t and a 2 ∞,1 · a ∞,1 are small enough, one has for all m ≥ 0, where C 1 > 0 and C 2 > 0 are constants independent of m. Then there exists ε 0 = ε 0 (C 1 , C 2 , A) > 0 such that {K m } and {H m } are bounded sequences provided that ε ≤ ε 0 .
The solution v constructed in Theorem 2.1 exists at least for some nontrivial time interval [0, T ), where T > 0 depends on a. Given a ∈ BU C σ (L p ) for some p > 1 we are unfortunately unable to estimate T from below by terms involving the norm of a, only. However, in Proposition 2.3 we claim that v ∈ C ([0, T ), BU C σ (L p )) for all p ∈ (1, ∞) in the same time interval.
Since e t∆H as well as P and the nonlinearity leave horizontal periodicity invariant we obtain the following.
Lemma 6.4. If a is in addition to the assumption of Theorem 2.1 periodic with respect to the horizontal variables, so is the solution v(t) for t > 0.
Using this for p ≥ 2 as new initial value, it follows using e.g. [6] that v extends to a global strong solution.