THE STEADY STATE SOLUTIONS TO THERMOHALINE CIRCULATION EQUATIONS

. In the article, we study the existence and the regularity of the steady state solutions to thermohaline circulation equations. Firstly, we obtain a suﬃcient condition of the existence of weak solutions to the equations by acute angle theory of weakly continuous operator. Secondly, we prove the existence of strong solutions to the equations by ADN theory and iteration procedure. Furthermore, we study the generic property of the solutions by Sard-Smale theorem and the existence of classical solutions by ADN theorem.

We consider the following boundary condition for the problem (1)-(4) : Here u n and u 3 are normal and tangential component of the velocity field on the vertical side of cylinder Ω respectively.
The thermohaline circulation equations(1)-(4) describe the motion and the states of the thermohaline circulation. The thermohaline circulation is one important source of internal climate variability and the greatest oceanic current on the earth. The thermohaline circulation is also called the ocean conveyor belt, the great ocean conveyor, or the global conveyor belt. It works in a fashion similar to a conveyor belt transporting enormous volume of cold, salty water from the North Atlantic to the North Pacific, and bringing warmer fresher water in return. Physically speaking, the buoyancy fluxes at the ocean surface give rise to gradients in temperature and salinity, which produce, in turn, density gradients. These gradients are, overall, sharper in the vertical than in the horizontal and are associated therefore with an overturning or thermohaline circulation.
The thermohaline circulation varies on time-scales of decades or longer, so there have been extensive observational, physical and numerical studies [1,2,3,7,10,18,19]. Henk A. Dijkstra and Michael Ghil [1] reviewed recent theoretical and numerical results that helped explain the physical processes governing the large-scale ocean circulation and its intrinsic variability by applying systematically the methods of dynamical systems theory. Ma and Wang studied dynamic stability and transitions by using the above equations (1)-(4) [10]. The equations governing the thermohaline circulation is the Navier-Stokes equations with the Coriolis force generated by the earth's rotation, coupled with the first law of thermodynamics. Ma and Wang have received the equations (1)-(4) by nondimensional process and have obtained the condition for the transitions of the thermohaline circulation in [10], where they considered the equations on the square region Ω = (0, L 1 ) × (0, L 2 ) × (0, 1) because of eigenvalue problem. They gave the trivial steady state solution because of the neglect of the heat source. However, the heat source is a important factor for the thermohaline circulation, so we consider the heat source. Furthermore, we discuss the equations in the region Ω = D × (0, 1) because the thermohaline circulation is periodic, where D is a section and [0, 1] is a length of a period. So the tube region Ω is C ∞ .
In this paper, we study the steady state solutions to the thermohaline circulation equations. The steady state solution is a special state of evolution equations and time-independent solution. The steady state solutions to the thermohaline circulation equations are significant to understand the dynamical behavior of the thermohaline circulation and are the main directions and important content in studying thermohaline circulation equations. The steady state solutions of other system have been studied in [5,6,9,15]. In paper [6], the author discussed the existence of the equilibrium solutions to the semilinear reaction diffusion system by fixed point theorem. In papers [5] and [15], the authors discussed the existence of stationary solutions to the Navier-Stokes equations by Galerkin method and the generic properties by Sard-Smale theorem.
We discuss the existence, regularity and generic property of the steady state solutions to thermohaline circulation equations(1)-(4) with the boundary condition(5)- (6). That is to say that we discuss the following equations: The paper is organized as follows. In section 2, we present preliminary results. In section 3, using an iteration procedure, we prove that these equations(7)-(12) possess steady state solutions in W 2,q (Ω, R 5 ) × W 1,q (Ω), q ≥ 2 by acute angle theory of weakly continuous operator and ADN theorem. In section 4, we study the regularity and generic property of the solutions by Sard-Smale theorem and ADN theorem.

2.
Preliminaries. Firstly, we present the eigenvalue results of the elliptic operator.
Furthermore, we introduce the theory of weakly continuous operator. Let X be a linear space, X 1 , X 2 two Banach spaces, X 1 separable, and X 2 reflexive. Let X ⊂ X 2 . There exists a linear mapping which is one to one and dense.
Lemma 2.4. [13](Acute angle theory) Let F : X 2 → X * 1 be weakly continuous, and then the equation F (u) = 0 has a solution in X 2 .
Definition 2.5. Let X, Y be two separable Banach spaces, and F : X → Y be C 1 . F is called a Fredholm operator provided the derivative operator DF : X → Y is a Fredholm operator for all x ∈ X.
Lemma 2.6. [11,12] 3. Existence of steady state solution. In this paper, we will introduce the spaces as follows We write L = I : X → H 1 , which is a containing mapping.
First, we prove that the definition F is reasonable. In other words, we will show that ||F Φ|| H * 1 < ∞ , ∀Ψ = (v, W, Z) ∈ H 1 . By using the Hölder inequality, we can get and and According to the Hölder inequality and the Sobolev imbedding theorem [17], we have (17) Similarly, by the Hölder inequality and the Sobolev imbedding theorem, we have and (19) and (20) Hence, combining (14)-(20), we have As a result, we get which implies that the definition F is reasonable. Second, we show F Φ, Φ ≥ 0. According to the Young inequality, we have Let > 0 and be appropriate small. Then we have Then there exists an appropriate large constant M such that At last, we prove that F is weakly continuous. Assume Φ k Φ in H 1 , we have from the Sobolev compactness imbedding theorem [8,14,17] Φ k → Φ in L p (Ω, R 5 ), 1 ≤ p < 6.
4. Regularity of steady state solution.
Therefore, we have which implies that Third, we introduce the mappings Let Then, the equations(7)- (12) can be rewritten as It is obvious that the mappings F : W 2,q (Ω, R 5 ) × W 1,q (Ω) → L q (Ω, R 5 ) is completely continuous field.
(ii) If Q = 0, it is obvious that the equations (7)-(12) have a trivial classical solution (u, T, S, p) = 0 as Ma and Wang pointed out in [10].