Global Wellposedness of the Primitive Equations with Nonlinear Equation of State in Critical Spaces

This article investigates the primitive equations with nonlinear equations of state. A global, strong well-posedness result for this set of equations is established for initial data lying in critical spaces provided that the density, depending on temperature, salinity and pressure, satisfies certain regularity assumptions. These assumptions are in particular satisfied for the TEOS-80 formulation of the equation of state.


Introduction
The primitive equations for the ocean and the atmosphere with linear equation of state are derived from the Navier-Stokes equations by assuming a hydrostatic balance for the pressure term in the vertical direction. The analysis of these equations was pioneered by Lions, Temam and Wang in their articles [12][13][14], where they proved the existence of a global, weak solution to these equations. Their uniqueness remains an open problem until today. A breakthrough result concerning global strong well-posedness of the primitive equations for initial data in H 1 was shown by Cao and Titi [2] using energy methods. Different approaches based on the theory of evolution equations and maximal regularity are due to Hieber and Kashiwabara [8], and Giga, Gries, Hieber, Hussein and Kashiwabara [5]. The vertical momentum equation of the fluid is represented due to the hydrostatic balance as where π denotes the pressure and ρ the density of the fluid, respectively, and g > 0 is the acceleration of gravity. The special case of the linear equation of state was investigated in [2,7,11]. Here τ and σ denote temperature and salinity, and ρ 0 , τ and σ given constants. In this article we consider the primitive equations with a general nonlinear equation of state that relates temperature τ , salinity σ and pressure π to the density ρ. Following Section 1.8 of [18], the physical properties of seawater in thermodynamical equilibrium are described by an equation of state for ρ of the form ρ = R(π, τ, σ).
In this article we consider general nonlinear equations of state and consider densities ρ belonging to the class We then prove global strong well-posedness for the set of equations (1.2) subject to (1.3) for initial data lying in critical spaces, which in the given situation are the Besov spaces B μ qp for p, q ∈ (1, ∞) with The above spaces are critical spaces and they correspond in the situation of the Navier-Stokes equations to the function spaces B n/p−1 qp introduced by Cannone [1]. Choosing in particular p = q = 2 and μ = 1 and noting that B 1 22 = H 1 , we obtain the result by Korn for the above mentioned special equations of state. This paper is organized as follows. Section 2 presents preliminaries on the approach to the primitive equations by evolution equations; in Sect. 3 we discuss typical examples of densities arising in ocean dynamics. Our main result of strong, global well-posedness of the primitive equations subject to equations of state of the form (1.3) is presented in Sect. 4. In Sect. 5 we give estimates needed for the proof of the local existence, whereas in Sect. 6 we prove a priori estimates in the situation of equations of state satisfying (1.3). Local and global well-posedness for the system (1.2) are proved in Sect. 7. Finally, we summarize results on semilinear evolution equations needed for our proof in the Appendix A.

Preliminaries
Let Ω = G × (−h, 0) a cylindrical domain with G = (0, 1) 2 and h > 0. The velocity u of the fluid is given by where v and w denote the horizontal and vertical components of u, respectively. Furthermore, τ , σ and π denote the temperature, the salinity and the pressure of the fluid, respectively. The primitive equations are given by Here horizontal periodicity is modelled by the closure of the functions spaces C ∞ per (Ω) and C ∞ per (G) as defined in [8,Section 2], where periodicity is only with respect to x, y coordinates and not necessary in the z coordinate. To simplify our notation we use ζ := (τ, σ), The hydrostatically solenoidal vector fields introduced in [8] are considered as a subspace of L q (Ω) 2 for q ∈ (1, ∞) and are defined by For q ∈ (1, ∞) and s ∈ [0, ∞) we define the Bessel potential spaces and W 0,q per := L q , where W s,q (Ω) denotes the Bessel potential spaces, see e.g. The hydrostatic Helmholtz projection P q is a continuous projection from L q (Ω) 2 onto L q σ (Ω), see [8] or [5]. Note that it annihilates the pressure term ∇ H π s . Applying it to the first line of (2.4) we obtain The Eq. (2.5) can be thus rewritten as In [3] it is shown that the Laplacian subject to the boundary condition described above has the maximal L p -regularity on X τ 0 = X σ 0 := L q (Ω). We set X ζ 0 := X τ 0 × X σ 0 . Further, it was shown in in [7] that A v q has maximal L p -regularity on X v 0 := L q σ (Ω). This means that is an isomorphism, where γ 0 v := v(0) denotes the trace operator. Here and the space of maximal regularity is given by . for η ∈ {v, τ, σ, ζ}, where X η 1 denotes the domain of the operator involved. Both spaces are Banach spaces equipped with its natural norms. Further, the trace space is given by X η θ,p := (X η 1 , X η 0 ) θ,p , where (·, ·) θ,p denotes the real interpolation functor for θ ∈ (0, 1) and q ∈ (1, ∞) and η ∈ {v, τ, σ, ζ}. Further, we denote the product space X θ,p := X v θ,p × X ζ θ,p . For more information on maximal regularity and the Navier-Stokes equations, we refer to [3,16,19]. Following [16,Theorem 3.5.4] the maximal regularity is equivalent to the maximal regularity in time-weighted spaces, which are defined for a time-weight μ ∈ ( 1 p , 1] and for k ∈ N recursively by τ, σ, ζ} and consider initial data within the real interpolation space The trace spaces can be computed explicitly in terms of Besov spaces.

Typical Ocean Densities
The set of equations (2.1) couples the velocity of the fluid to two advection-diffusion equations for the temperature and the salinity. It is closed by the equation of state expressing the density as a function of temperature, salinity and the pressure. In contrast to atmospheric dynamics where the equation of state can be expressed in the simplest case by the ideal gas law, the oceanic equation of state lacks a rigorous derivation. The latter needs to represent measurements of τ, σ, π and should also be thermodynamically consistent. In the TEOS formulation of the equation of state (see [20]), thermodynamical consistency is implemented by the introduction of a certain Gibbs-potential. We use thus the TEOS formulation of the equation of state and variants hereof as our main examples for the equations of state satisfying Assumption (A).

Linear Density
The density ρ considered in [7,11] is linear and given by ) and Assumption (A) is satisfied by Remark 2.1.

Remark 3.1.
We point out that the density ρ above can be replaced by any smooth function.

Equation of State by TEOS-10
The equation of state TEOS-10 for the description of seawater by the Intergovernmental Oceanographic Commission is today the most accurate equation of state with respect to measurements and it rests on the Gibbs formalism of thermodynamics, see [20]. In this formulation, the density ρ is given by , see also [10, (20)]. Note that h :

Equation of State by McDongall-Jacket-Wright-Feistel
Another example of an equation of state of interest is the so-called McDongall-Jacket-Wright-Feistel equation, see [15]. In this case the density ρ is given by where Q 1 , Q 2 are polynomials such that there are constants c, C > 0 with , see also [10, (20)]. Denote again by h : C \ {0} → C the inversion. As in the last subsection we conclude that Q 2 does not vanish on (−h, 0) × (K 1 , K 2 ) × (L 1 , L 2 ). Since Q 1 and Q 2 are polynomials and hence smooth it follows that , R)) and Assumption (A) is satisfied.

Equation of State by UNESCO-80
The UNESCO-80 equation of state is a classical equation of state, see [22]. The density ρ is given in this case by where Q and K are polynomials and such that there are constants c, C > 0 satisfying The precise form of the polynomials are formulated e.g. in [10, (23)] and in [10, (25)]. Denote again by h : C \ {0} → C the inversion. As in the last subsection we conclude that 1 − cz K (z,a,b) does not vanish on (−h, 0) × (K 1 , K 2 ) × (L 1 , L 2 ). Since Q and K are polynomials and hence smooth and the inversion h is smooth it follows that ρ is smooth. In particular, L 2 ), R)) and Assumption (A) is satisfied.

Main Result
The following theorem is the main result of this article. We set and T ∈ (0, ∞). Assume Assumption (A) and that ) for some δ > 0 sufficiently small. Then there exists a unique, strong global solution ) to the primitive equations (2.1) subject to (2.2). [8, (6.2)] the surface pressure π s can be recovered from the velocity v by

Remark 4.2. (a) As in
It follows π s ∈ L p μ ((0, T ), W 1,q per,0 (G)), where W 1,q per,0 (G) = π ∈ W 1,q per (G): G π = 0 . (b) The above solution satisfies belongs to the regularity classes (e) We remark that Δ v can be replaced by L v given by where Re 1 , Re 2 > 0 denotes the horizontal and vertical Reynolds number, f the Coriolis parameter and k ∈ R 3 . Further, Δ τ and Δ σ can be replaced by L τ and L σ given by where Rt 1 , Rt 2 > 0 and Rs 1 , Rs 2 > 0 denotes the horizontal and vertical mixing coefficients for the temperature and salinity.

Estimates for the Local Existence
We start this section by showing that the operator −A q + λ admits a bounded H ∞ -calculus on L q σ (Ω) × L q (Ω) 2 .

Lemma 5.1. Let q ∈ (1, ∞). Then the operator −A q + λ has a bounded H ∞ -calculus for
Proof. By [5, Theorem 3.1] the hydrostatic Stokes operator −A v q + λ has a bounded H ∞ -calculus for λ > 0 with φ ∞ A = 0 on L q σ (Ω). Furthermore, by [5, Lemma 4.1] the Laplacians −Δ τ q + λ and −Δ σ q + λ for λ > 0 has bounded H ∞ -calculus with φ ∞ A = 0 on L q (Ω). We conclude that We continue by providing properties of the nonlinearities F and Π given by The following lemma is crucial for our approach. It yields L q -estimates and Lipschitz type estimates for the right hand side.
(i) Using Jensen's inequality we obtain (ii) By the same arguments as in part (i) we obtain . In order to bound the second term, let r = (3q) 2 and p = 3q and note that 1 p Hence, Hölder's inequality and the mean value theorem imply . .
Combining these estimates we obtain ).
We now set X β := [L q σ (Ω), D(A q )] β for 0 < β < 1 and note that The results given in [ a continuous bilinear map, i.e. there exists a constant C > 0, depending only on Ω and q, such that

A Priori Estimates
In this section we prove a priori estimates in the space of maximal regularity. Throughout this section we assume that p = q = 2. We start with energy estimates.

Lemma 6.3. Assume Assumption (A) and let
Proof. The proof is divided into two steps.
We conclude the following a priori bounds in maximal regularity spaces.