Steady State Solution to Atmospheric Circulation Equations with Humidity Effect

Abstract

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.

1. 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)=(𝑥₁,𝑥₂,𝑡)∈Ω×(0,∞) (Ω=(0,2𝜋)×(0,1) is a period of 𝐶^∞ field (−∞,+∞)×(0,1)), 𝜕𝑢𝜕𝑡=𝑃_𝑟Δ𝑢−∇𝑝−𝜎𝑢+𝑃_𝑟𝑅𝑇−𝑅𝑞^→𝜅−(𝑢⋅∇)𝑢,𝜕𝑇𝜕𝑡=Δ𝑇+𝑢₂−(𝑢⋅∇)𝑇+𝑄,𝜕𝑞𝜕𝑡=𝐿_𝑒Δ𝑞+𝑢₂−(𝑢⋅∇)𝑞+𝐺,div𝑢=0, (1.1)(1.2)(1.3)(1.4) where 𝑃_𝑟>0,𝑅>0,𝑅,𝐿_𝑒>0 are constants, 𝑢=(𝑢₁,𝑢₂),𝑇,𝑞,𝑝 denote velocity field, temperature, humidity, and pressure, respectively, 𝑄,𝐺 are known functions, and 𝜎 is a constant matrix 𝜎𝜎=⁰𝜔𝜔𝜎₁. (1.5)

The problems (1.1)–(1.4) are supplemented with the following Dirichlet boundary condition at 𝑥₂=0,1 and periodic condition for 𝑥₁: 𝑢,𝑇,𝑞=0,𝑥₂=0,1,𝑢,𝑇,𝑞0,𝑥₂=𝑢,𝑇,𝑞2𝜋,𝑥₂ (1.6) and initial value conditions: 𝑢,𝑇,𝑞=𝑢₀,𝑇₀,𝑞₀,𝑡=0. (1.7)

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 time-independent 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: Δ𝑢−∇𝑝−𝜎𝑢+𝑅𝑇−𝑅𝑞^→1𝜅−𝑃_𝑟(𝑢⋅∇)𝑢=0,𝑥∈Ω,Δ𝑇+𝑢₂−()(𝑥)𝐿𝑢⋅∇𝑇+𝑄=0,𝑥∈Ω,_𝑒Δ𝑞+𝑢₂−(𝑢⋅∇)𝑞+𝐺(𝑥)=0,𝑥∈Ω,div𝑢=0,𝑥∈Ω,𝑢,𝑇,𝑞=0,𝑥₂=0,1,𝑢,𝑇,𝑞2𝜋,𝑥₂=𝑢,𝑇,𝑞0,𝑥₂. (1.8)(1.9)(1.10)(1.11)(1.12)(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,𝑅⁴)×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).

Let 𝜎=min{𝜎₀,𝜔,𝜎₁}, and ‖⋅‖_𝑋 denote norm of the space 𝑋.

2. Preliminaries

We introduce theory of linear elliptic equation and ADN theory of Stokes equation.

We consider with divergence form of linear elliptic equation: 𝐿𝑢=−𝐷_𝑗𝑎_𝑖𝑗𝐷_𝑖𝑢+𝑏_𝑖𝐷_𝑖𝑢+𝑐𝑢=𝑓, (2.1) where 𝑎_𝑖𝑗,𝑏_𝑖,𝑐∈𝐿^∞(Ω), 𝑓∈𝐿²(Ω), 𝑎_𝑖𝑗=𝑎_𝑗𝑖, (𝑎_𝑖𝑗) is uniformly elliptic, that is, there exist constants 0<𝜆₁≤𝜆₂ such that 𝜆₁𝜉²≤𝑎_𝑖𝑗(𝑥)𝜉_𝑖𝜉_𝑗≤𝜆₂𝜉²,∀𝜉∈𝑅^𝑛,𝑥∈Ω. (2.2)

The problem (1.1) is supplemented with the following Dirichlet boundary condition 𝑢|_𝜕Ω=𝜑. (2.3)

We define three classes of solutions of (2.1) and (2.3). (1)Classical solution: if there is a function 𝑢∈𝐶²(Ω) satisfying (2.1), (2.3), we say 𝑢 is a classical solution to (2.1) and (2.3). (2)Strong solution: if there is a function 𝑢∈𝑊^2,𝑝(Ω) satisfying (2.1), (2.3) almost everywhere for some 𝑝≥1, we say 𝑢 is a strong solution to (2.1) and (2.3). (3)Weak solution: if there is a function 𝑢∈𝑊^1,2(Ω) satisfying _Ω𝑎_𝑖𝑗𝐷_𝑖𝑢𝐷_𝑗𝑣+𝑏_𝑖𝐷_𝑖𝑢𝑣+𝑐𝑢𝑣𝑑𝑥=_Ω𝑓𝑣𝑑𝑥,∀𝑣∈𝑊₀^1,2(Ω), (2.4) and (2.3), we say 𝑢 is a weak solution to (2.1) and (2.3).

Lemma 2.1 2.1 (see [13] (Schauder Theorem)). Let Ω⊂𝑅^𝑛 be a 𝐶^2,𝛼 field, 𝑎_𝑖𝑗,𝑏_𝑖,𝑐,𝑓∈𝐶^0,𝛼(Ω), 𝜑∈𝐶^2,𝛼. If 𝑢∈𝐶^2,𝛼 is a solution to (2.1) and (2.3), then ‖𝑢‖_𝐶^2,𝛼≤𝐶‖𝑢‖_𝐶⁰+𝑓_𝐶0,𝛼+𝜑_𝐶2,𝛼, (2.5) where 𝐶>0 depends on 𝑛,𝛼,𝜆,Ω and 𝐶^0,𝛼-norm of the coefficient functions 𝑎^𝑖𝑗, 𝑏^𝑖, 𝑐.

Lemma 2.2 2.2 (see [13] (𝐿^𝑝 Theorem)). Let Ω⊂𝑅^𝑛 be a 𝐶² field, 𝑎_𝑖𝑗∈𝐶⁰(Ω), 𝑏_𝑖,𝑐∈𝐿^∞(Ω), 𝑓∈𝐿^𝑝(Ω), 𝜑∈𝑊^2,𝑝(Ω). If 𝑢∈𝑊^2,𝑝 is a solution of (2.1) and (2.3), then ‖𝑢‖_𝑊^2,𝑝≤𝐶‖𝑢‖_𝐿^𝑝+𝑓_𝐿𝑝+𝜑_𝑊2,𝑝, (2.6) where 𝐶>0 depends on 𝑛,𝑝,𝜆,Ω and 𝐶^0,𝛼-norm or 𝐿^∞-norm of the coefficient functions.

One considers with Stokes equation −𝜇Δ𝑢+∇𝑝=𝑓(𝑥),div𝑢=0,𝑢|_𝜕Ω=𝜑. (2.7)

Lemma 2.3 (see [14, 15] (ADN theory of Stokes equation)). (1) Let 𝑓∈𝐶^𝑘,𝛼(Ω,𝑅^𝑛), 𝜑∈𝐶^𝑘+2,𝛼(Ω,𝑅^𝑛), 𝑘≥0. If (𝑢,𝑝)∈𝐶^2,𝛼(Ω,𝑅^𝑛)×𝐶^1,𝛼(Ω) is a solution of (2.7), then the solution (𝑢,𝑝)∈𝐶^𝑘+2,𝛼(Ω,𝑅^𝑛)×𝐶^𝑘+1,𝛼(Ω), and ‖𝑢‖_𝐶^𝑘+2,𝛼+𝑝_{𝐶𝑘+1,𝛼}≤𝐶𝑓_{𝐶𝑘,𝛼}+𝑢,𝑝_𝐶0+𝜑_{𝐶𝑘+2,𝛼}, (2.8) where 𝐶>0 depends on 𝜇,𝑛,𝑘,𝛼,Ω.
(2) Let 𝑓∈𝑊^𝑘,𝑝(Ω,𝑅^𝑛), 𝜑∈𝑊^𝑘+2,𝑝(Ω,𝑅^𝑛), 𝑘≥0. If (𝑢,𝑝)∈𝑊^2,𝑝(Ω,𝑅^𝑛)×𝑊^1,𝑝(Ω)(1<𝑝<∞) is a solution of (2.7), then the solution (𝑢,𝑝)∈𝑊^𝑘+2,𝑝(Ω,𝑅^𝑛)×𝑊^𝑘+1,𝑝(Ω), and ‖𝑢‖_𝑊^𝑘+2,𝑝+𝑝_{𝑊𝑘+1,𝑝}≤𝐶𝑓_{𝑊𝑘,𝑝}+𝑢,𝑝_𝐿𝑝+𝜑_{𝑊𝑘+2,𝑝}, (2.9) where 𝐶>0 depends on 𝜇,𝑛,𝑘,𝛼,Ω.

Lemma 2.4. The eigenvalue equation: Δ𝑇𝑥₁,𝑥₂=𝜆𝑇𝑥₁,𝑥₂,𝑥₁,𝑥₂∈(0,2𝜋)×(0,1)𝑇=0,𝑥₂𝑇=0,1,0,𝑥₂=𝑇2𝜋,𝑥₂, (2.10) has eigenvalue {𝜆_𝑘}^∞_𝑘=1, and 0<𝜆₁≤𝜆₂≤⋯,𝜆_𝑘→∞,as𝑘→∞. (2.11)

Let 𝑋 be a linear space, 𝑋₁, 𝑋₂ two Banach space, 𝑋₁ separable, and 𝑋₂ reflexive. Let 𝑋⊂𝑋₂. There exists a linear mapping 𝐿∶𝑋→𝑋₁isonetooneanddense. (2.12)

Definition 2.5. A mapping 𝐹∶𝑋₂→𝑋^∗₁ is called weakly continuous provided lim_𝑛→∞𝐹𝑢_𝑛,𝑣=𝐹𝑢₀,𝑣,∀𝑣∈𝑋₁, (2.13) for all {𝑢_𝑛}⊂𝑋₂, 𝑢_𝑛⇀𝑢₀ in 𝑋₂.

Lemma 2.6 (see [2]). If 𝐹∶𝑋₂→𝑋^∗₁ is weakly continuous, 𝑈⊂𝑋₂ is bounded open set, 0∈𝑈, and ⟨𝐹(𝑢),𝐿𝑢⟩≥0,∀𝑢∈𝜕𝑈∩𝑋, (2.14) then the equation 𝐹(𝑢)=0 has a solution in 𝑋₂.
One introduces the Sard-Smale Theorem of infinite dimensional operator. Let 𝑋, 𝑌 be two separable Banach Spaces, 𝐹∶𝑋→𝑌 be a 𝐶¹ mapping. 𝐹 is called a Fredholm operator provided the derivative operator 𝐷𝐹(𝑥)∶𝑋→𝑌 is a Fredholm operator for all 𝑥∈𝑋.

Lemma 2.7 (see [16, 17] (Sard-Smale Theorem)). Let 𝐹∶𝑋→𝑌 be a 𝐶¹ Fredholm operator with zero index. Then regular value of 𝐹 is dense in 𝑌. If 𝑝∈𝑌 is critical value of 𝐹, then 𝐹⁻¹(𝑝) is discrete set.

3. Existence of Steady State Solution

Theorem 3.1. If 𝜎𝜆₁≥max{(𝑅+1)²,(𝑅−1)²/𝐿_𝑒}, and 𝜆₁ is the first eigenvalue of the elliptic equation (2.10), then for all 𝑄,𝐺∈𝐿^𝑞(Ω), (1.8)–(1.13) have a solution (𝑢,𝑇,𝑞,𝑝)∈𝑊^2,𝑞(Ω,𝑅⁴)×𝑊^1,𝑞(Ω), 𝑞≥2.

Proof. Let 𝑋={𝜙=(𝑢,𝑇,𝑞)∈𝐶^∞(Ω,𝑅⁴)∣𝜙satisfy (1.11)–(1.13)}, and 𝐻₁={𝜙=(𝑢,𝑇,𝑞)∈𝐻¹(Ω,𝑅⁴)∣𝜙satisfy (1.11)–(1.13)}.
Define 𝐹∶𝐻₁→𝐻^∗₁, for all 𝜓=(𝑣,𝑆,𝑧)∈𝐻₁, 𝐹𝜙,𝜓=_Ω∇𝑢∇𝑣+𝜎𝑢⋅𝑣−𝑅𝑇−𝑅𝑞𝑣₂+1𝑃_𝑟(𝑢⋅∇)𝑢⋅𝑣+∇𝑇∇𝑆−𝑢₂𝑆+(𝑢⋅∇)𝑇𝑆−𝑄𝑆+𝐿_𝑒∇𝑞∇𝑧−𝑢₂𝑧+(𝑢⋅∇)𝑞𝑧−𝐺𝑧𝑑𝑥. (3.1) Firstly, we prove the coercivity of 𝐹. 𝐹𝜙,𝜙=_Ω|∇𝑢|²+𝜎𝑢⋅𝑢−𝑅𝑇−𝑅𝑞𝑢₂+1𝑃_𝑟(∇⋅𝑢)𝑢⋅𝑢+|∇𝑇|²−𝑢₂𝑇+(𝑢⋅∇)𝑇𝑇−𝑄𝑇+𝐿_𝑒∇𝑞²−𝑢₂()=𝑞+𝑢⋅∇𝑞𝑞−𝐺𝑞𝑑𝑥_Ω|∇𝑢|²+𝜎𝑢⋅𝑢−(𝑅+1)𝑢₂𝑇+𝑅−1𝑞𝑢₂+|∇𝑇|²−𝑄𝑇+𝐿_𝑒∇𝑞²≥−𝐺𝑞𝑑𝑥_Ω|∇𝑢|²+|∇𝑇|²+𝐿_𝑒∇𝑞²+𝜎|𝑢|²−|𝑅+1|𝑢₂𝑇|−𝑞𝑢|𝑅−1₂−|𝑄𝑇|−|𝐺𝑞≥|||𝑑𝑥_Ω|∇𝑢|²+|∇𝑇|²+𝐿_𝑒∇𝑞²+𝜎|𝑢|²−𝜎𝑢₂²−12𝜎|𝑅+1|²|𝑇|²+𝑅−1²𝑞²−𝜀|𝑇|²−1𝜀|𝑄|²−𝜀𝑞²−1𝜀|𝐺|²≥𝑑𝑥_Ω|∇𝑢|²+12|∇𝑇|²+𝐿_𝑒2∇𝑞²𝑑𝑥+_Ω12|∇𝑇|²−|𝑅+1|²2𝜎|𝑇|²+𝑑𝑥_Ω𝐿_𝑒2∇𝑞²−12𝜎𝑅−1²𝑞²𝑑𝑥−𝜀_Ω|𝑇|²+𝑞²1𝑑𝑥−𝜀_Ω|𝑄|²+|𝐺|²≥𝑑𝑥_Ω|∇𝑢|²+12|∇𝑇|²+𝐿_𝑒2∇𝑞²𝑑𝑥−𝜀_Ω|𝑇|²+𝑞²1𝑑𝑥−𝜀_Ω|𝑄|²+|𝐺|²𝑑𝑥. (3.2)
Let 𝜀>0 be appropriate small. Then 𝐹𝜙,𝜙≥𝐶_1Ω|∇𝑢|²+|∇𝑇|²+∇𝑞²𝑑𝑥−𝐶_2Ω|𝑄|²+|𝐺|²𝑑𝑥. (3.3) From 𝑄,𝑆∈𝐿^𝑞(Ω)(𝑞≥2), it follows that 𝐹𝜙,𝜙≥𝐶₁𝜙_𝐻1−𝐶₃. (3.4)
Then there exists an appropriate large constant 𝑀 such that 𝐹𝜙,𝜙≥0,𝜙∈𝜕𝐵_𝑀∩𝑋. (3.5)
Furthermore, we verify that 𝐹 is weakly continuous.
Let 𝜙_𝑘⇀𝜙 in 𝐻₁, we have from the Sobolev imbedding Theorem 𝜙_𝑘→𝜙in𝐿^𝑝Ω,𝑅⁴,1≤𝑝<∞. (3.6)
By 𝑢₀𝑣∈𝐿², 𝑢_𝑘⇀𝑢₀ in 𝐻¹(Ω,𝑅²), it follows that lim_𝑘→∞Ω𝑢₀⋅∇𝑢_𝑘−𝑢₀⋅∇𝑢₀⋅𝑣𝑑𝑥=0. (3.7)
Combining the general Hölder inequality and (3.6), we deduce lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑢_𝑘−𝑢₀⋅∇𝑢_𝑘⋅𝑣𝑑𝑥≤lim_𝑘→∞Ω𝑢_𝑘−𝑢₀∇𝑢_𝑘|𝑣|𝑑𝑥≤lim_𝑘→∞Ω𝑢_𝑘−𝑢₀⁴𝑑𝑥^1/4_Ω∇𝑢_𝑘²𝑑𝑥^1/2_Ω|𝑣|⁴𝑑𝑥^1/4=0. (3.8)
Then, lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑢_𝑘−𝑢₀⋅∇𝑢₀⋅𝑣𝑑𝑥=lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑢_𝑘−𝑢₀⋅∇𝑢_𝑘⋅𝑣𝑑𝑥+lim_𝑘→∞Ω𝑢₀⋅∇𝑢_𝑘−𝑢₀⋅∇𝑢₀⋅𝑣𝑑𝑥=0. (3.9) Thus, lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑢_𝑘⋅𝑣𝑑𝑥=_Ω𝑢₀⋅∇𝑢₀⋅𝑣𝑑𝑥. (3.10)
As 𝑢₀𝑆∈𝐿², 𝑇_𝑘⇀𝑇₀ in 𝐻¹, we find lim_𝑘→∞Ω𝑢₀⋅∇𝑇_𝑘−𝑢₀⋅∇𝑇₀𝑆𝑑𝑥=0. (3.11)
Combining the general Hölder inequality and (3.6), we deduce lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑇_𝑘−𝑢₀⋅∇𝑇_𝑘𝑆𝑑𝑥≤lim_𝑘→∞Ω𝑢_𝑘−𝑢₀∇𝑇_𝑘|𝑆|𝑑𝑥≤lim_𝑘→∞Ω𝑢_𝑘−𝑢₀⁴𝑑𝑥^1/4_Ω∇𝑇_𝑘²𝑑𝑥^1/2_Ω|𝑆|⁴𝑑𝑥^1/4=0. (3.12) Then, lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑇_𝑘−𝑢₀⋅∇𝑇₀𝑆𝑑𝑥=lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑇_𝑘−𝑢₀⋅∇𝑇_𝑘𝑆𝑑𝑥+lim_𝑘→∞Ω𝑢₀⋅∇𝑇_𝑘−𝑢₀⋅∇𝑇₀𝑆𝑑𝑥=0.lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑇_𝑘𝑆𝑑𝑥=_Ω𝑢₀⋅∇𝑇₀𝑆𝑑𝑥. (3.13)
By 𝑢₀𝑧∈𝐿², 𝑞_𝑘⇀𝑞₀ in 𝐻¹, we have lim_𝑘→∞Ω𝑢₀⋅∇𝑞_𝑘−𝑢₀⋅∇𝑞₀𝑧𝑑𝑥=0. (3.14)
Combining the general Hölder inequality and (3.6), we deduce lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑞_𝑘−𝑢₀⋅∇𝑞_𝑘𝑧𝑑𝑥≤lim_𝑘→∞Ω𝑢_𝑘−𝑢₀∇𝑞_𝑘|𝑧|𝑑𝑥≤lim_𝑘→∞Ω𝑢_𝑘−𝑢₀⁴𝑑𝑥^1/4_Ω∇𝑞_𝑘²𝑑𝑥^1/2_Ω|𝑧|⁴𝑑𝑥^1/4=0. (3.15) Then, lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑞_𝑘−𝑢₀⋅∇𝑞₀𝑧𝑑𝑥=lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑞_𝑘−𝑢₀⋅∇𝑞_𝑘𝑧𝑑𝑥+lim_𝑘→∞Ω𝑢₀⋅∇𝑞_𝑘−𝑢₀⋅∇𝑞₀𝑧𝑑𝑥=0. (3.16) Thus, lim_𝑘→∞Ω𝑢_𝑘⋅∇𝑞_𝑘𝑧𝑑𝑥=_Ω𝑢₀⋅∇𝑞₀𝑧𝑑𝑥. (3.17)
Combining (3.10)–(3.17), we have lim_𝑘→∞Ω𝐹𝜙_𝑘,𝜓𝑑𝑥=lim_𝑘→∞Ω∇𝑢_𝑘∇𝑣+𝜎𝑢_𝑘⋅𝑣−𝑅𝑇_𝑘−𝑅𝑞_𝑘𝑣₂+1𝑃_𝑟𝑢_𝑘⋅∇𝑢_𝑘⋅𝑣+∇𝑇_𝑘∇𝑆−𝑢_𝑘2𝑆+𝑢_𝑘⋅∇𝑇_𝑘𝑆−𝑄𝑆+𝐿_𝑒∇𝑞_𝑘∇𝑧−𝑢_𝑘2𝑧+𝑢_𝑘⋅∇𝑞_𝑘=𝑧−𝐺𝑧𝑑𝑥_Ω∇𝑢₀∇𝑣+𝜎𝑢₀⋅𝑣−𝑅𝑇₀−𝑅𝑞₀𝑣₂+1𝑃_𝑟𝑢₀⋅∇𝑢₀⋅𝑣+∇𝑇₀∇𝑆−𝑢₀₂𝑆+𝑢₀⋅∇𝑇₀𝑆−𝑄𝑆+𝐿_𝑒∇𝑞₀∇𝑧−𝑢₀₂𝑧+𝑢₀⋅∇𝑞₀=𝑧−𝐺𝑧𝑑𝑥_Ω𝐹𝜙₀,𝜓𝑑𝑥,∀𝜓∈𝐻₁, (3.18) which imply that 𝐹∶𝐻₁→𝐻^∗₁ is weakly continuous. According to Lemma 2.6, (1.8)–(1.13) have a solution 𝜙=(𝑢,𝑇,𝑞)∈𝐻¹(Ω,𝑅⁴).
Lastly we prove that (𝜙,𝑝)∈𝑊^2,𝑞(Ω,𝑅⁴)×𝑊^1,𝑞(Ω), 2≤𝑞<∞.
From the Hölder inequality, we see |(𝑢⋅∇)_Ω𝑢|^3/2𝑑𝑥≤_Ω|𝑢|^3/2|𝐷𝑢|^3/2≤𝑑𝑥_Ω|𝐷𝑢|²𝑑𝑥^3/4_Ω|𝑢|⁶𝑑𝑥^1/4. (3.19) Then, (𝑢⋅∇)𝑢∈𝐿^3/2(Ω,𝑅²).
For the Stokes equation: −Δ𝑢+∇𝑝=𝑔,div𝑢=0,𝑢=0,𝑥₂𝑢=0,1,0,𝑥₂=𝑢2𝜋,𝑥₂, (3.20) since 𝑔=−𝜎𝑢+(𝑅𝑇−𝑅𝑞)^→𝜅−(1/𝑃_𝑟)(𝑢⋅∇)𝑢∈𝐿^3/2(Ω), according to ADN theory, (3.20) has a solution: 𝑢,𝑝∈𝑊^2,3/2Ω,𝑅²×𝑊^1,3/2(Ω). (3.21)
By the Hölder inequality, we have |(𝑢⋅∇)_Ω𝑇|^3/2𝑑𝑥≤_Ω|𝑢||𝐷𝑇|^3/2≤𝑑𝑥_Ω|𝐷𝑇|²𝑑𝑥^3/4_Ω|𝑢|⁶𝑑𝑥^1/4≤𝐶_Ω|𝐷𝑇|²𝑑𝑥, (3.22) thus, (𝑢⋅∇)𝑇∈𝐿^3/2(Ω).
For the elliptic equation: −Δ𝑇=𝑓₁,𝑇=0,𝑥₂𝑇=0,1,0,𝑥₂=𝑇2𝜋,𝑥₂, (3.23) as 𝑓₁=𝑢₂−(𝑢⋅∇)𝑇+𝑄∈𝐿^3/2(Ω), according to theory of linear elliptic equation, (3.23) has a solution 𝑇∈𝑊^2,3/2(Ω). (3.24)
From the Hölder inequality, we see (𝑢⋅∇)_Ω𝑞^3/2𝑑𝑥≤_Ω|𝑢|^3/2𝐷𝑞^3/2𝑑𝑥≤_Ω∇𝑞²𝑑𝑥^3/4_Ω|𝑢|⁶𝑑𝑥^1/4, (3.25) thus, (𝑢⋅∇)𝑞∈𝐿^3/2(Ω).
For the elliptic equation: −Δ𝑞=𝑓₂,𝑞=0,𝑥₂𝑞=0,1,0,𝑥₂=𝑞2𝜋,𝑥₂, (3.26) as 𝑓₂=𝑢₂−(𝑢⋅∇)𝑞+𝐺∈𝐿^3/2(Ω), 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 𝑊^2,(3/2)(Ω)↪𝑊^1,6(Ω)↪𝐿⁶(Ω). (3.28) Then 𝑇,𝑞∈𝐿⁶, and |(𝑢⋅∇)_Ω𝑢|³𝑑𝑥≤_Ω|𝐷𝑢|⁶𝑑𝑥^1/2_Ω|𝑢|⁶𝑑𝑥^1/2≤𝐶‖𝑢‖⁶_𝑊2,(3/2). (3.29) Thus, (𝑢⋅∇)𝑢∈𝐿³(Ω,𝑅²). Consequently 𝑔∈𝐿³(Ω). According to ADN theory, (3.20) has a solution 𝑢,𝑝∈𝑊^2,3Ω,𝑅²×𝑊^1,3(Ω). (3.30)
Similarly, we deduce 𝑇,𝑞∈𝑊^2,3Ω,𝑅². (3.31) By doing the same procedures as above, (1.8)–(1.13) have a solution (𝑢,𝑇,𝑞,𝑝)∈𝑊^2,𝑞(Ω,𝑅⁴)×𝑊^1,𝑞, 𝑞≥2.

4. Regularity of Steady State Solution

Theorem 4.1. If 𝜎𝜆₁≥max{(𝑅+1)²,(𝑅−1)²/𝐿_𝑒}, and 𝜆₁ is the first eigenvalue of elliptic equation (2.10), then there exists a dense open set ℱ⊂𝐿^𝑞(Ω,𝑅²)(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): ‖𝑢‖_𝑊^2,𝑞+‖𝑇‖_𝑊^2,𝑞+𝑞_𝑊2,𝑞+𝑝_𝑊1,𝑞≤𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1³,𝑞≥2. (4.1)
As 𝜙=(𝑢,𝑇,𝑞) is a solution to (1.8)–(1.13), we have ⟨𝐹𝜙,𝜙⟩=0. Then |∇𝑢|_Ω²+𝜎𝑢⋅𝑢−𝑅𝑇−𝑅𝑞𝑢₂+|∇𝑇|²−𝑢₂𝑇−𝑄𝑇+𝐿_𝑒∇𝑞²−𝑢₂𝑞−𝐺𝑞𝑑𝑥=0. (4.2) Thus, |∇𝑢|_Ω²+𝜎𝑢⋅𝑢−(𝑅+1)𝑇𝑢₂+𝑅−1𝑞𝑢₂+|∇𝑇|²−𝑄𝑇+𝐿_𝑒∇𝑞²−𝐺𝑞𝑑𝑥=0. (4.3) Consequently, |∇𝑢|_Ω²+|∇𝑇|²+𝐿_𝑒∇𝑞²+𝜎𝑢²≤𝑑𝑥_Ω|𝑅+1|𝑢₂|𝑇|+𝑅−1𝑞𝑢₂+|𝑄𝑇|+|𝐺𝑞≤|||𝑑𝑥_Ω𝜎𝑢₂²+|𝑅+1|²2𝜎|𝑇|²+𝑅−1²2𝜎𝑞²+𝜀|𝑇|²+1𝜀|𝑄|²+𝜀𝑞²+1𝜀|𝐺|²𝑑𝑥. (4.4) Choosing an appropriate constant 𝜀, we see |∇𝑢|_Ω²+|∇𝑇|²+∇𝑞²𝑑𝑥≤𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1². (4.5) According to the Sobolev imbedding Theorem and (4.5), we deduce ‖𝑢‖_𝐿^𝑞≤𝐶‖𝑢‖_𝐿^2𝑞≤𝐶‖𝐷𝑢‖_𝐿²≤𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1. (4.6) Using the Gagliardo-Nironberg inequality and Young inequality, we have ‖𝐷𝑢‖_𝐿^2𝑞≤𝐶‖𝑢‖_𝑊^1/2_2,𝑞‖𝑢‖_𝑊^1/2_1,2≤𝜀𝐷²𝑢_𝐿𝑞+𝐶𝜀⁻¹‖𝐷𝑢‖_𝐿²,‖𝐷𝑇‖_𝐿^2𝑞≤𝐶‖𝑇‖_𝑊^1/2_2,𝑞‖𝑇‖_𝑊^1/2_1,2≤𝜀𝐷²𝑇_𝐿𝑞+𝐶𝜀⁻¹‖𝐷𝑇‖_𝐿²,𝐷𝑞_𝐿2𝑞≤𝐶𝑞_𝑊^1/2_2,𝑞𝑞_𝑊^1/2_1,2≤𝜀𝐷²𝑞_𝐿𝑞+𝐶𝜀⁻¹𝐷𝑞_𝐿2. (4.7)(4.8)(4.9)
Combining the Hölder inequality and (4.6)–(4.9), we see ‖𝑢⋅∇𝑢‖_𝐿^𝑞+‖𝑢⋅∇𝑇‖_𝐿^𝑞+𝑢⋅∇𝑞_𝐿𝑞≤‖𝑢‖_𝐿^2𝑞‖𝐷𝑢‖_𝐿^2𝑞+‖𝐷𝑇‖_𝐿^2𝑞+𝐷𝑞_𝐿2𝑞≤𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞×𝜀𝐷+1₂𝑢_𝐿𝑞+𝐷²𝑇_𝐿𝑞+𝐷²𝑞_𝐿𝑞+𝜀⁻¹‖𝐷𝑢‖_𝐿²+‖𝐷𝑇‖_𝐿²+𝐷𝑞_𝐿2. (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 ‖𝑢‖_𝑊^2,𝑞+‖𝑇‖_𝑊^2,𝑞+𝑞_𝑊2,𝑞+𝑝_𝑊1,𝑞≤𝐶𝑔_𝐿𝑞+𝑓_1𝐿𝑞+𝑓_2𝐿𝑞≤𝐶‖𝑢‖_𝐿^𝑞+‖𝑇‖_𝐿^𝑞+𝑞_𝐿𝑞+‖𝑢⋅∇𝑢‖_𝐿^𝑞+𝑢_2𝐿𝑞+‖𝑢⋅∇𝑇‖_𝐿^𝑞+𝑢⋅∇𝑞_𝐿𝑞+‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞≤𝐶‖𝑢‖_𝐿^𝑞+‖𝑇‖_𝐿^𝑞+𝑞_𝐿𝑞+𝐶‖𝑢⋅∇𝑢‖_𝐿^𝑞+|𝑢⋅∇𝑇‖_𝐿^𝑞+𝑢⋅∇𝑞_𝐿𝑞+𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞. (4.11) From (4.6) and (4.10), it follows that ‖𝑢‖_𝑊^2,𝑞+‖𝑇‖_𝑊^2,𝑞+𝑞_𝑊2,𝑞+𝑝_𝑊1,𝑞≤𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1+𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1𝜀𝐷²𝑢_𝐿𝑞+𝐷²𝑇_𝐿𝑞+𝐷²𝑞_𝐿𝑞+𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1𝜀⁻¹‖𝐷𝑢‖_𝐿²+‖𝐷𝑇‖_𝐿²+𝐷𝑞_𝐿2+𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞. (4.12) Let 𝐶𝜀(‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1)=1/2. Then ‖𝑢‖_𝑊^2,𝑞+‖𝑇‖_𝑊^2,𝑞+𝑞_𝑊2,𝑞+𝑝_𝑊1,𝑞≤𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1+𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1³≤𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1³, (4.13) which imply (4.1).
We introduce the mappings: 𝐺=𝐿+𝐻∶𝑊^2,𝑞Ω,𝑅⁴×𝑊^1,𝑞(Ω)→𝐿^𝑞Ω,𝑅⁴,𝐿𝑢,𝑝=−Δ𝑢+∇𝑝−Δ𝑇−𝐿_𝑒,𝐻=Δ𝑞𝑢,𝑝𝜎𝑢−𝑅𝑇−𝑅𝑞^→1𝜅+𝑃_𝑟(𝑢⋅∇)𝑢−𝑢₂+(𝑢⋅∇)𝑇−𝑢₂+()𝑞.𝑢⋅∇ (4.14) Let 𝑓(𝑥)=0𝑄(𝑥)𝐺(𝑥)∈𝐿^𝑞Ω,𝑅⁴. (4.15)
Then, (1.8)–(1.13) can be rewrite as the following mapping 𝐹𝑢,𝑇,𝑞,𝑝=𝑓(𝑥). (4.16)
Clearly, 𝐹∶𝑊^2,𝑞(Ω,𝑅⁴)×𝑊^1,𝑞(Ω)→𝐿^𝑞(Ω,𝑅⁴) 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,𝑅⁴), and 𝐹⁻¹(𝑓) is discrete in 𝑊^2,𝑞(Ω,𝑅⁴)×𝑊^1,𝑞(Ω) for all 𝑓∈ℱ. 𝐹⁻¹(𝑓) in 𝑊^2,𝑞(Ω,𝑅⁴)×𝑊^1,𝑞(Ω) is finite for 𝑞≥2 from (4.1). Consequently, 𝑓∈ℱ is interior point and ℱ is an open set.

Theorem 4.2. If 𝜎𝜆₁≥max{(𝑅+1)²,(𝑅−1)²/𝐿_𝑒}, and 𝜆₁ is the first eigenvalue of elliptic equation (2.10), then (1)the Equations (1.8)–(1.13) have a classical solution (𝑢,𝑇,𝑞,𝑝)∈𝐶^2,𝛼(Ω,𝑅⁴)×𝐶^1,𝛼(Ω) for all 𝑄,𝐺∈𝐶^𝛼(Ω), (2)there exists a tense open set ℱ⊂𝐶^𝛼(Ω,𝑅²), such that the solution to (1.8)–(1.13) is finite for all (𝑄,𝐺)∈ℱ, (3)the solution (𝑢,𝑇,𝑞,𝑝) to (1.8)–(1.13) is in 𝐶^∞(Ω,𝑅⁵) if 𝑄,𝐺∈𝐶^∞(Ω).

Proof. We prove the assertion (1). As 𝐶^𝛼(Ω)↪𝐿^𝑞(Ω), for all 𝑞≤∞, 𝑄,𝐺∈𝐿^𝑞(Ω) for all 𝑄,𝐺∈𝐶^𝛼(Ω). From Theorem 4.1, (1.8)–(1.13) have a strong solution (𝑢,𝑇,𝑞,𝑝)∈𝑊^2,𝑞(Ω,𝑅⁴)×𝑊^1,𝑞, 𝑞≥2.
When 𝑞≥2, 𝑊^2,𝑞(Ω)↪𝐶^1,𝛼(Ω), 𝛼=1−(2/𝑞) from the Sobolev imbedding theorem. Then 𝑢,𝑇,𝑞∈𝐶^1,𝛼Ω,𝑅⁴. (4.17)
Thus, (𝑢⋅∇)𝑢,(𝑢⋅∇)𝑇,(𝑢⋅∇)𝑞∈𝐶^𝛼(Ω). (4.18)
For Stokes equation: −Δ𝑢+∇𝑝=𝑔,div𝑢=0,𝑢=0,𝑥₂𝑢=0,1,0,𝑥₂=𝑢2𝜋,𝑥₂, (4.19) as 𝑔=−𝜎𝑢+(𝑅𝑇−𝑅𝑞)^→𝜅−(1/𝑃_𝑟)(𝑢⋅∇)𝑢∈𝐶^𝛼(Ω), according to ADN theory, (4.19) has a solution: 𝑢,𝑝∈𝐶^2,𝛼Ω,𝑅²×𝐶^1,𝛼(Ω). (4.20)
For the elliptic equation: −Δ𝑇=𝑓₁,𝑇=0,𝑥₂𝑇=0,1,0,𝑥₂=𝑇2𝜋,𝑥₂, (4.21) as 𝑓₁=𝑢₂−(𝑢⋅∇)𝑇+𝑄∈𝐶^𝛼(Ω), according to theory of linear elliptic equation, (4.21) has a solution: 𝑇∈𝐶^2,𝛼(Ω). (4.22)
For the elliptic equation: −Δ𝑞=𝑓₂,𝑞=0,𝑥₂𝑞=0,1,0,𝑥₂=𝑞2𝜋,𝑥₂, (4.23) as 𝑓₂=𝑢₂−(𝑢⋅∇)𝑞+𝐺∈𝐶^𝛼(Ω), according to theory of linear elliptic equation, (4.23) has a solution: 𝑞∈𝐶^2,𝛼(Ω). (4.24) Thus, (1.8)–(1.13) have a solution (𝑢,𝑇,𝑞,𝑝)∈𝐶^2,𝛼(Ω,𝑅⁴)×𝐶^1,𝛼(Ω).
Secondly, we prove the assertion (2). Combining ADN theory, theory of linear elliptic equation, and (4.19)–(4.23), we have ‖𝑢‖_𝐶^2,𝛼+‖𝑇‖_𝐶^2,𝛼+𝑞_𝐶2,𝛼+𝑝_𝐶1,𝛼≤𝐶𝑔_𝐶𝛼+𝑓_1𝐶𝛼+𝑓_2𝐶𝛼≤𝐶‖𝑢‖_𝐶^𝛼+‖𝑇‖_𝐶^𝛼+𝑞_𝐶𝛼+‖𝑢⋅∇𝑢‖_𝐶^𝛼+𝑢_2𝐶𝛼+‖𝑢⋅∇𝑇‖_𝐶^𝛼+𝑢⋅∇𝑞_𝐶𝛼+‖𝑄‖_𝐶^𝛼+‖𝐺‖_𝐶^𝛼≤𝐶‖𝑢‖_𝐶^𝛼+‖𝑇‖_𝐶^𝛼+𝑞_𝐶𝛼+𝐶‖𝑢⋅∇𝑢‖_𝐶^𝛼+|𝑢⋅∇𝑇‖_𝐶^𝛼+𝑢⋅∇𝑞_𝐶𝛼+𝐶‖𝑄‖_𝐶^𝛼+‖𝐺‖_𝐶^𝛼≤𝐶‖𝑢‖_𝑊^1,𝑞+‖𝑇‖_𝑊^1,𝑞+𝑞_𝑊1,𝑞+𝐶‖𝑢⋅∇𝑢‖_𝑊^1,𝑞+|𝑢⋅∇𝑇‖_𝑊^1,𝑞+𝑢⋅∇𝑞_𝑊1,𝑞+𝐶‖𝑄‖_𝐶^𝛼+‖𝐺‖_𝐶^𝛼≤𝐶‖𝑢‖_𝑊^2,𝑞+‖𝑇‖_𝑊^2,𝑞+𝑞_𝑊2,𝑞+𝐶‖𝑢‖_𝑊^2,𝑞‖𝑢‖_𝑊^2,𝑞+‖𝑇‖_𝑊^2,𝑞+𝑞_𝑊2,𝑞+𝐶‖𝑄‖_𝐶^𝛼+‖𝐺‖_𝐶^𝛼≤𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1+𝐶‖𝑄‖_𝐿^𝑞+‖𝐺‖_𝐿^𝑞+1⁶+𝐶‖𝑄‖_𝐶^𝛼+‖𝐺‖_𝐶^𝛼≤𝐶‖𝑄‖_𝐶^𝛼+‖𝐺‖_𝐶^𝛼+1+𝐶‖𝑄‖_𝐶^𝛼+‖𝐺‖_𝐶^𝛼+1⁶. (4.25)
We introduce the mappings: 𝐹=𝐿+𝐻∶𝐶^2,𝛼Ω,𝑅⁴×𝐶^1,𝛼(Ω)→𝐶^𝛼Ω,𝑅⁴,𝐿𝑢,𝑝=−Δ𝑢+∇𝑝−Δ𝑇−𝐿_𝑒,𝐻=Δ𝑞𝑢,𝑝𝜎𝑢−𝑅𝑇−𝑅𝑞^→1𝜅+𝑃_𝑟(𝑢⋅∇)𝑢−𝑢₂+(𝑢⋅∇)𝑇−𝑢₂+()𝑞.𝑢⋅∇ (4.26) Let 𝑓(𝑥)=0𝑄(𝑥)𝐺(𝑥)∈𝐶^𝛼Ω,𝑅⁴, (4.27)
then, (1.8)–(1.13) can be rewritten as 𝐹𝑢,𝑇,𝑞,𝑝=𝑓(𝑥). (4.28)
Clearly, 𝐹∶𝐶^2,𝛼(Ω,𝑅⁴)×𝐶^1,𝛼(Ω)→𝐶^𝛼(Ω,𝑅⁴) is complete continuous field. Then 𝐹 is a Fredholm operator with zero index. The regular value ℱ⊂𝐶^𝛼(Ω,𝑅⁴) is dense from Sard-Smale theorem, and 𝐹⁻¹(𝑓) is discrete in 𝐶^2,𝛼(Ω,𝑅⁴)×𝐶^1,𝛼(Ω) for all 𝑓∈ℱ. From (4.25), we find that 𝐹⁻¹(𝑓) is finite in 𝐶^2,𝛼(Ω,𝑅⁴)×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,𝑅⁴)×𝑊^𝑘+1,𝑞(Ω,𝑅) (𝑘 is arbitrary integer). From the Sobolev imbedding theorem, (𝑢,𝑇,𝑞)∈𝐶^𝑘+1(Ω,𝑅⁴)×𝐶^𝑘(Ω,𝑅) (𝑘 is arbitrary integer). Then (𝑢,𝑇,𝑞,𝑝)∈𝐶^∞(Ω,𝑅⁵).

5. Remark

𝜎𝜆₁≥max{(𝑅+1)²,(𝑅−1)²/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.

Funding

The National Natural Science Foundation of China (no. 11271271) and the NSF of Sichuan Education Department of China (no. 11ZA102).