A priori estimates for free boundary problem of 3D incompressible inviscid rotating Boussinesq equations

In this paper, we consider the three-dimensional rotating Boussinesq equations (the “primitive” equations of geophysical fluid flows). Inspired by Christodoulou and Lindblad (Pure Appl Math 53:1536–1602, 2000), we establish a priori estimates of Sobolev norms for free boundary problem of inviscid rotating Boussinesq equations under the Taylor-type sign condition on the initial free boundary. Using the same method, we can also obtain a priori estimates for the incompressible inviscid rotating MHD system with damping.


Introduction
The Boussinesq equations are of relevance to study a number of models coming from atmospheric or oceanographic turbulence where the rotation and stratification play an important role. Referring to [19], we consider the following inviscid rotating Boussinesq equations without heat diffusion in D: where v = (v 1 , v 2 , v 3 ), p and h denote the velocity, the fluid pressure and the deviation of the temperature function from the basic temperature profile, respectively. The Coriolis parameter f = 2A sin φ is assumed to be a nonzero real constant in which A is the angular frequency of rotation and φ is the latitude; e 3 = (0, 0, 1) is the vertical unit vector; the Coriolis force fe 3 × v gives rise to a vertical rigidity in the fluid. The number Υ > 0 is gravity and Γ > 0 is the stratification parameter which represents the Brunt-Väisälä frequency (also buoyancy frequency). The stratification induces the term Γv 3 in the equations, which gives rise to a horizontal rigidity in the fluid. D ⊂ ∪ 0 t T {t} × R 3 is an unknown time-space domain for some constant T > 0. We want to find a set D and (v, h) solving (1.1) and satisfying the initial conditions: Let D t = {x ∈ R n : (t, x) ∈ D}, then the conditions on the free boundary read v N = κ, on ∂D t , p = 0, on ∂D t , (1.3) whereṽ := (−fv 2 , fv 1 , 0) and δ ij is the Kronecker delta symbol such that δ ii = 1 and δ ij = 0 for i = j.
Remark 1.1. Just consider fixed boundary problem, we need to add additional conditions to (1.5). When Γ = Υ and v N = 0 on ∂D 0 , the energy is conserved. In fact, the rotation and stratification do not cause the above energy loss. Remark 1.2. Similar to the fixed boundary problem, in this paper, we do not need to assume the condition of temperature on the boundary. Different from Euler equations, the energy of the system is not conserved, but it can be controlled by the initial data and time T . In the following proof, we can find that the higherorder energy of temperature is actually controlled by velocity and the initial energies, and is not affected by the boundary condition.
In order to define higher-order energies, we introduce the second fundamental form of the free surface and tensor products given in [6]. We want to project the system to the tangent space of the boundary. The orthogonal projection Π to the tangent space of the boundary of a (0,r) tensor α is defined to be the projection of each component along the normal: where θ ij =∂ i N j is the second fundamental form of ∂D t . Then we define the quadratic form Q of the form: Here η is a smooth cut-off function satisfying 0 η(d) 1, η(d) = 1 when d < d 0 /4, and η(d) = 0 when d > d 0 /2. d 0 is a fixed number that is smaller than the injectivity radius ς 0 of the normal exponential map, defined to be the largest number ς 0 such that the map is an injection. Then we define the higher energies for r 1 as where sgn denotes the sign function and In the present paper, we prove the following main result.
There exists a continuous function T > 0 such that if Let us now outline the proof of Theorem 1.1. Firstly, for the rotating Boussinesq Eq. (1.5), we transform the free boundary problem to a fixed boundary problem in the Lagrangian coordinates in Sect. 2. In Sect. 3, we prove the zero-order and the first-order energy estimates. Section 4 is devoted to the higherorder energy estimates by using the identities derived in Sect. 2, then, we justify the a priori assumptions in Sect. 5. Finally, for the rotating MHD equations with damping, we can get a similar conclusion in Sect. 6.

Reformulation in Lagrangian coordinates
We introduce the Lagrangian coordinates to transform the free boundary problem to a fixed boundary problem. Let Ω be a bounded domain in R 3 , and f 0 : Ω → D 0 where f 0 is a diffeomorphism. The connection between the Eulerian coordinates x and the Lagrangian coordinates y is given by The Euclidean metric δ ij in D t , then in Ω for each fixed t, induces a metric and its inverse Furthermore, expressed in the y-coordinates, we have (2.4) Let us introduce the notation for the material derivative If k(t, x) is the (0,r) tensor expressed in the x-coordinates, we have where w a1···ar (t, y) = ∂x i1 ∂y a1 · · · ∂x ir ∂y ar k i1···ir (t, x). Let u(t, y), Θ(t, y), P (t, y) represent the velocity, deviation of the temperature function, pressure in the Lagrangian coordinates, respectively. Then from [17, Lemma 2.1] and (1.5), we have Thus, the system (1.5) can be rewritten in the Lagrangian coordinates as (2.7)

The zero-order and the first-order energy estimates
In this section, we define the zero-order energy as where dμ g = √ detgdy is the Riemannian volume element on Ω in the metric g. In fact, we can easily obtain D t dμ g = 0 and u aũ a = 0 by using divu = 0. Obviously, when Υ = Γ, the energy of the system is conserved. Using the Hölder inequality From the Gronwall inequality, for t ∈ [0, T ] with a constant T > 0, it follows that Due to the initial energy is given, we can get the zero-order energy estimate. Before dealing with the first-order energy estimates, we need the following Identities. From [11, Lemma 2.3], (2.5) and (2.7), we have Now we calculate the first-order energy estimates. From (3.4), [17,Lemma 2.1], and [11, (A.13)], we derive the material derivative of g bd γ ae ∇ a u b ∇ e u d , In fact, γ ae ∇ e u b ∇ aũb = 0 by using symmetry and the definition ofũ. Similarly, by (3.5), we obviously have Next, we shall calculate the material derivative of |curlu| 2 . Indeed, we can get Define the first-order energy as Then we get the following estimates.

Theorem 3.1. For any smooth solution of system (2.7) satisfying
where C depends only on Γ, Υ, |f | and VolΩ.
where dμ γ is the Riemannian volume element on ∂Ω. Since P = 0 on ∂Ω, it follows that γ ae ∇ a P = 0. Thus, the integral on the boundary is zero. Next, from [11, (A.3) and (A.5)], we get By the Hölder inequality and [11, (A.5)], we directly get that . From the Gronwall inequality, it yields the desired estimate.
Remark 3.1. Whether in the lower order or the higher-order energy estimates later in this paper, we can find that the Coriolis forceũ does not affect energy of tangential velocity, but it will affect Θ and the energy of curlu. In fact, the integral involving P is zero, so we do not need to estimate the boundary integral in E 1 . But for the higher-order estimates, we have to introduce boundary integrals for P .

The general r-th order energy estimates
In this section, we establish the higher-order energy estimates. Applying [11, Lemma 2.2] and (1.5), we get and so, we get for r 2, Define the r-th order energy for r 2 as where ϑ = 1/ (−∇ N P ) as before, then we have the following theorem.
where the constants C 1 and where h NN = h ab N a N b and h ab = D t g ab /2. By using [17, Lemma 2.1], (4.1) and (4.2), we can directly get ZAMP A priori estimates for free boundary Page 9 of 21 80 Similarly, and In fact, the difficulty is how to deal with the integration of the higher-order derivatives of P on the boundary.
The higher-order term involving pressure P will vanish by symmetry. For other terms, we can apply the Hölder inequality and the Gauss formula to get that Finally, we only need to estimate the last term in (4.6). By [11, (A.12)], we have Thus, the integrals can be controlled by C(K, Υ, Γ, |f |, M, L, 1/ε)E r (t).
In summary, we obtain which implies the desired result by Gronwall's inequality and the induction argument for r ∈ {2, 3, 4}.

Justification of a priori assumptions
In this section, we justify the a priori assumptions in Sect. 4. At time t, denote .
In fact, our judgment is very similar to those in [6,11], so we only state the results and omit their proofs as follows.

A priori estimates for rotating magnetohydrodynamics with damping
As everyone knows, the rotating MHD has wide application including planetary flows, stellar flows and accretion discs. An incompressible inviscid MHD system with damping under solid body rotation and in the presence of a uniform background magnetic field will be considered. The equations in the rotating frame of reference are: in D (the same as before), where v, b, p denote the velocity, the magnetic field, the total pressure, respectively; α is the rotation rate and η > 0 is a damping coefficient. Obviously, when α = η = 0, (6.1) are the incompressible inviscid MHD equations considered in [11]. The system (6.1) with the conditions p = 0, b · N = 0 and v N = κ (the normal velocity of free surface) on the free boundary ∂D t can be written in the Lagrangian coordinates as whereβ = (−αβ 2 , αβ 1 , 0), Ω = D 0 ; u, β, P denote the velocity, the magnetic field, the total pressure in the new coordinates. Thus, in view of (6.2) and [17, Lemma 2.1], we also have the zero-order energy E 0 (t) = 1 2 Ω |u(t, x)| 2 + |β(t, x)| 2 dμ g + η t 0 Ω |β| 2 dμ g dτ.
Remark 6.2. Because the nonlinear term involves β, then we have to estimate the L 2 norm of ∇ b β a β e ∇ e ∇ a u a when we estimate ∇ s ΔD t P L 2 (Ω) for s 2. Obviously, we have to assume |β| M 1 when r = 2. It is different from the rotating Boussinesq equations.
Similarly, we can obtain the following a priori estimates.  Theorems 6.1, 6.2 and 6.3 can be proved similarly as those of rotating Boussinesq equations and the non-rotating MHD case in [11]. We omit the details of the proof.