Steady State Solution to Atmospheric Circulation Equations with Humidity Effect

The steady state solution to atmospheric circulation equations with humidity effect is studied. A sufficient condition of existence of steady state solution to atmospheric circulation equations is obtained, and regularity of steady state solution is verified.


Introduction
This paper is concerned with steady state solution of the following initial-boundary problem of atmospheric circulation equations involving unknown functions u, T, q, p at x, t x 1 , x 2 , t ∈ Ω × 0, ∞ Ω 0, 2π × 0, 1 is a period of C ∞ field −∞, ∞ × 0, 1 , where P r > 0, R > 0, R, L e > 0 are constants, u u 1 , u 2 , T, q, p denote velocity field, temperature, humidity, and pressure, respectively, Q, G are known functions, and σ is a constant matrix σ σ 0 ω ω σ 1 . 1.5 The partial differential equations 1.1 -1.7 were presented in atmospheric circulation with humidity effect.Atmospheric circulation is one of the main factors affecting the global climate, so it is very necessary to understand and master its mysteries and laws.Atmospheric circulation is an important mechanism to complete the transports and balance of atmospheric heat and moisture and the conversion between various energies.On the contrary, it is also the important result of these physical transports, balance, and conversion.Thus it is of necessity to study the characteristics, formation, preservation, change, and effects of the atmospheric circulation and master its evolution law, which is not only the essential part of human's understanding of nature, but also the helpful method of changing and improving the accuracy of weather forecasts, exploring global climate change, and making effective use of climate resources.
The atmosphere and ocean around the earth are rotating geophysical fluids, which are also two important components of the climate system.The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity, and a lot of them cannot be produced by laboratory experiments.The atmosphere or the ocean or the couple atmosphere and ocean can be viewed as an initial-and boundary-value problem 1-4 or an infinite dimensional dynamical system 5-7 .We deduce atmospheric circulation models which are able to show features of atmospheric circulation and are easy to be studied from the very complex atmospheric circulation model based on the actual background and meteorological data, and we present global solutions of atmospheric circulation equations with the use of the T weakly continuous operator 8 .
We investigate steady state solution of the atmospheric circulation equations in this paper.The steady state solution is a special state of evolution equations and the timeindependent solution, which plays a very important role on understanding the dynamical behavior of the evolution equations and is the main directions and important content in studying evolution equations.Steady state solutions of some systems are studied 9-12 .The purpose to consider with the steady state solution of atmospheric circulation equations is to seek the conditions under which atmospheric circulation is stable and to understand structure of the circulation cell.
We discuss the existence and regularity of steady state solution to atmospheric circulation equations 1.1 -1.4 with the boundary condition 1.6 .In other words, we discuss the following equations: u, T, q 0, x 2 0, 1, 1.12 u, T, q 2π, x 2 u, T, q 0, x 2 .

1.13
The paper is organized as follows.In Section 2 we present preliminary results.In Section 3, we prove that the systems 1.8 -1.13 possess steady state solutions in W 2, q Ω, R 4 × W 1, q Ω , q ≥ 2 by using space sequence method.In Section 4, by using Sard-Smale Theorem and energy method, we obtain regularity of the solutions to the models 1.8 -1.13 .

Preliminaries
We introduce theory of linear elliptic equation and ADN theory of Stokes equation.We consider with divergence form of linear elliptic equation:

2.2
The problem 1.1 is supplemented with the following Dirichlet boundary condition We define three classes of solutions of 2.1 and 2.3 . where where C > 0 depends on n, p, λ, Ω and C 0, α -norm or L ∞ -norm of the coefficient functions.

2.10
has eigenvalue {λ k } ∞ k 1 , and Let X be a linear space, X 1 , X 2 two Banach space, X 1 separable, and X 2 reflexive.Let X ⊂ X 2 .There exists a linear mapping L : X −→ X 1 is one to one and dense.
2.12 Definition 2.5.A mapping F : then the equation F u 0 has a solution in X 2 .One introduces the Sard-Smale Theorem of infinite dimensional operator.Let X, Y be two separable Banach Spaces, F : X → Y be a C 1 mapping.F is called a Fredholm operator provided the derivative operator DF x : X → Y is a Fredholm operator for all x ∈ X.

3.1
Firstly, we prove the coercivity of F.

3.5
Furthermore, we verify that F is weakly continuous.Let φ k φ in H 1 , we have from the Sobolev imbedding Theorem Combining the general H ölder inequality and 3.6 , we deduce Combining the general H ölder inequality and 3.6 , we deduce

3.13
By u 0 z ∈ L 2 , q k q 0 in H 1 , we have Combining the general H ölder inequality and 3.6 , we deduce

3.18
which imply that F : From the H ölder inequality, we see For the Stokes equation: Ω , according to ADN theory, 3.20 has a solution:

3.21
By the H ölder inequality, we have For the elliptic equation:

3.24
From the H ölder inequality, we see For the elliptic equation: , according to theory of linear elliptic equation, 3.26 has a solution q ∈ W 2, 3/2 Ω .

3.27
By the Sobolev imbedding Theorem, we see Then T, q ∈ L 6 , and

Regularity of Steady State Solution
Theorem 4.1.
/L e }, and λ 1 is the first eigenvalue of elliptic equation 2.10 , then there exists a dense open set F ⊂ L q Ω, R 2 q ≥ 2 , the solution to 1.8 -1.13 is finite for all Q, G ∈ F.
Proof.There are the following estimates for 1.8 -1.13 : As φ u, T, q is a solution to 1.8 -1.13 , we have Fφ, φ 0. Then

4.4
Choosing an appropriate constant ε, we see According to the Sobolev imbedding Theorem and 4.5 , we deduce Using the Gagliardo-Nironberg inequality and Young inequality, we have Combining the H ölder inequality and 4.6 -4.9 , we see

4.10
Since u, T, q are solutions to 3.20 , 3.23 , and 3.26 , according to ADN theory and theory of linear elliptic equation, we have 4.11 From 4.6 and 4.10 , it follows that which imply 4.1 .

Journal of Applied Mathematics
We introduce the mappings:

4.15
Then, 1.8 -1.13 can be rewrite as the following mapping F u, T, q, p f x .

4.16
Clearly, F : W 2, q Ω, R 4 × W 1, q Ω → L q Ω, R 4 is a completely continuous field.Thus F is a Fredholm operator with zero index.According to the Sard-Smale Theorem, the regular value of F is dense in F ⊂ L q Ω, R 4 , and 2 /L e }, and λ 1 is the first eigenvalue of elliptic equation 2.10 , then Proof.We prove the assertion 1 .As 4.17 Thus, For Stokes equation: For the elliptic equation: T ∈ C 2, α Ω .

Journal of Applied Mathematics
Secondly, we prove the assertion 2 .Combining ADN theory, theory of linear elliptic equation, and 4.19 -4.23 , we have

4.25
We introduce the mappings:

4.26
Let 4.27 then, 1.8 -1.13 can be rewritten as F u, T, q, p f x .

4.28
Clearly, F : C 2, α Ω, R 4 × C 1, α Ω → C α Ω, R 4 is complete continuous field.Then F is a Fredholm operator with zero index.The regular value F ⊂ C α Ω, R 4 is dense from Sard-Smale theorem, and F −1 f is discrete in C 2, α Ω, R 4 × C 1, α Ω for all f ∈ F. From 4.25 , we find that F −1 f is finite in C 2, α Ω, R 4 × C 1, α Ω .Thus, f ∈ F is an interior point and F is an open set.
Finally we prove the assertion 3 .Since Q, G ∈ C ∞ Ω , it is true that Q, G ∈ W k, q Ω k is arbitrary integer .According to Theorem 4.1, we conclude that u, T, q, p ∈ W k 2, q Ω, R 4 × W k 1, q Ω, R k is arbitrary integer .From the Sobolev imbedding theorem, u, T, q ∈ C k 1 Ω, R 4 × C k Ω, R k is arbitrary integer .Then u, T, q, p ∈ C ∞ Ω, R 5 .

Remark
/L e } is a sufficient condition, not a necessary condition.In fact, if the condition does not hold, 1.8 -1.13 may have not solution for some Q, G.
Returning to the problem of atmospheric circulation, as the temperature source and the moisture source are changed, the state of the atmospheric circulation changes, but there is still a corresponding steady state.
according to theory of linear elliptic equation, 4.21 has a solution: according to theory of linear elliptic equation, 4.23 has a solution:

F L H : C 2 ,
α Ω, R 4 × C 1, α Ω −→ C α Ω, R 4 , and λ 1 is the first eigenvalue of the elliptic equation 2.10 , then for all according to theory of linear elliptic equation, 3.23 has a solution according to ADN theory, 4.19 has a solution: