Regularity of global solution to atmospheric circulation equations with humidity effect

Boundary Value Problems20122012:143

DOI: 10.1186/1687-2770-2012-143

Received: 1 June 2012

Accepted: 14 November 2012

Published: 5 December 2012

Abstract

In this article, the regularity of the global solutions to atmospheric circulation equations with humidity effect is considered. Firstly, the formula of the global solutions is obtained by using the theory of linear operator semigroups. Secondly, the regularity of the global solutions to atmospheric circulation equations is presented by using mathematical induction and regularity estimates for the linear semigroups.

MSC:35D35, 35K20, 35Q35.

Keywords

global solution regularity atmospheric circulation equations humidity effect

1 Introduction

This paper is concerned with the regularity of solutions to the following initial-boundary problem of atmospheric circulation equations involving unknown functions ( u , T , q , p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq1_HTML.gif at ( x , t ) = ( x 1 , x 2 , t ) Ω × ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq2_HTML.gif ( Ω = ( 0 , 2 π ) × ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq3_HTML.gif is a period of C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq4_HTML.gif field ( , + ) × ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq5_HTML.gif):
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ1_HTML.gif
(1.1)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ2_HTML.gif
(1.2)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ3_HTML.gif
(1.3)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ4_HTML.gif
(1.4)
where P r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq6_HTML.gif, R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq7_HTML.gif, R ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq8_HTML.gif, L e > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq9_HTML.gif are constants, u = ( u 1 , u 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq10_HTML.gif, 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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equa_HTML.gif
The problems (1.1)-(1.4) are supplemented with the following Dirichlet boundary condition at x 2 = 0 , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq11_HTML.gif and the periodic condition for x 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq12_HTML.gif:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ5_HTML.gif
(1.5)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ6_HTML.gif
(1.6)
and initial value conditions
( u , T , q ) = ( u 0 , T 0 , q 0 ) , t = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ7_HTML.gif
(1.7)

Partial differential equations (1.1)-(1.7) are presented in atmospheric circulation with humidity effect [1]. 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. Moreover, 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 a 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 initial and boundary value problems [25], or an infinite dimensional dynamical system [68]. We deduce atmospheric circulation models which are able to show the 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 [1]. In [9], the steady state solutions to atmospheric circulation equations with humidity effect are studied. A sufficient condition of the existence of the steady state solutions to atmospheric circulation equations is obtained, and the regularity of the steady state solutions is verified. In this article, we investigate the regularity of the solutions to atmospheric circulation equations (1.1)-(1.7).

The paper is organized as follows. In Section 2, we present preliminary results. In Section 3, we present the formula of the solution to the atmospheric circulation equations. In Section 4, we obtain the regularity of the solutions to equations (1.1)-(1.7).

X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq13_HTML.gif denotes the norm of the space X, and C, C i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq14_HTML.gif are variable constants.

2 Preliminaries

We consider the divergence form of the linear elliptic equation
L u = D j ( a i j D i u ) + b i D i u + c u = f , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ8_HTML.gif
(2.1)
where a i j , b i , c L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq15_HTML.gif, f L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq16_HTML.gif, a i j = a j i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq17_HTML.gif, ( a i j ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq18_HTML.gif is uniformly elliptic, i.e., there exist constants 0 < λ 1 λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq19_HTML.gif such that
λ 1 | ξ | 2 a i j ( x ) ξ i ξ j λ 2 | ξ | 2 , ξ R n , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equb_HTML.gif
The problem (2.1) is supplemented with the following Dirichlet boundary condition:
u | Ω = φ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ9_HTML.gif
(2.2)

Lemma 2.1 [10] (Theory of linear elliptic equations)

Let Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq20_HTML.gif be a C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq21_HTML.gif field, a i j C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq22_HTML.gif, b i , c L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq23_HTML.gif, f L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq24_HTML.gif, φ W 2 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq25_HTML.gif. If u W 2 , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq26_HTML.gif is a solution of Eqs. (2.1), (2.2), then
u W 2 , p C ( u L p + f L p + φ W 2 , p ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equc_HTML.gif

where C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq27_HTML.gif depends on n, p, λ, Ω and C 0 , α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq28_HTML.gif-norm or L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq29_HTML.gif-norm of the coefficient functions.

We consider the Stokes equation
{ μ u + p = f ( x ) , div u = 0 , u | Ω = φ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ10_HTML.gif
(2.3)

Lemma 2.2 [11, 12] (ADN theory of the Stokes equation)

Let f W k , p ( Ω , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq30_HTML.gif, φ W k + 2 , p ( Ω , R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq31_HTML.gif, k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq32_HTML.gif. If ( u , p ) W 2 , p ( Ω , R n ) × W 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq33_HTML.gif ( 1 < p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq34_HTML.gif) is a solution of Eq. (2.3), then the solution ( u , p ) W k + 2 , p ( Ω , R n ) × W k + 1 , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq35_HTML.gif, and
u W k + 2 , p + p W k + 1 , p C ( f W k , p + ( u , p ) L p + φ W k + 2 , p ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equd_HTML.gif

where C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq27_HTML.gif depends on μ, n, k, α, Ω.

Let X be a linear space, X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq36_HTML.gif, X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq37_HTML.gif be two separable reflexive Banach spaces, and H be a Hilbert space. X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq36_HTML.gif, X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq37_HTML.gif, and H are completion spaces of X under the respective norm. X 1 , X 2 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq38_HTML.gif are dense embedding. F : X 2 × ( 0 , ) X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq39_HTML.gif is a continuous mapping. We consider the abstract equation
{ d u d t = F u , 0 < t < , u ( 0 ) = φ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ11_HTML.gif
(2.4)

where φ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq40_HTML.gif, u : [ 0 , + ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq41_HTML.gif is unknown.

Definition 2.3 Let φ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq40_HTML.gif be a given initial value. u L p ( ( 0 , T ) , X 2 ) L ( ( 0 , T ) , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq42_HTML.gif ( 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq43_HTML.gif) is called a global solution of Eq. (2.4) if u satisfies
( u ( t ) , v ) H = 0 t F u , v d t + ( φ , v ) H , v X 1 H . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Eque_HTML.gif
Definition 2.4 Let u n , u 0 L p ( ( 0 , T ) , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq44_HTML.gif. u n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq45_HTML.gif is called uniformly weak convergence in L p ( ( 0 , T ) , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq46_HTML.gif if { u n } L ( ( 0 , T ) , H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq47_HTML.gif is bounded, and
{ u n u 0 , in  L p ( ( 0 , T ) , X 2 ) , lim n 0 T | u n u 0 , v H | 2 d t = 0 , v H . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ12_HTML.gif
(2.5)
Definition 2.5 A mapping F : X 2 × ( 0 , ) X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq48_HTML.gif is called T-weakly continuous if for p = ( p 1 , , p m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq49_HTML.gif, 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq43_HTML.gif and u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq50_HTML.gif uniformly weakly converges to u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq51_HTML.gif, we have
lim n 0 T F u n , v d t = 0 T F u 0 , v d t , v X 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equf_HTML.gif

Lemma 2.6 [3]

Assume F : X 2 × ( 0 , ) X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq52_HTML.gif is T-weakly continuous and satisfies:

(A1) there exists p = ( p 1 , , p m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq49_HTML.gif, p i > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq53_HTML.gif ( 1 i m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq54_HTML.gif), such that
F u , u C 1 u X 2 p + C 2 u H 2 + f ( t ) , u X , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equg_HTML.gif

where C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq55_HTML.gif, C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq56_HTML.gif are constants, f L 1 ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq57_HTML.gif ( 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq43_HTML.gif), X 2 p = i = 1 m | | i p i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq58_HTML.gif, | | i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq59_HTML.gif is a seminorm of X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq37_HTML.gif, X 2 = i = 1 m | | i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq60_HTML.gif,

(A2) there exists 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq61_HTML.gif for any 0 < h < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq62_HTML.gif and u C 1 ( [ 0 , ) , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq63_HTML.gif,
| t t + h F u , v d t | C h α , v X and 0 t < T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equh_HTML.gif

where C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq27_HTML.gif depends only on T, v X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq64_HTML.gif, 0 t u X 2 p d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq65_HTML.gif, and sup 0 t T u H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq66_HTML.gif.

Then for any φ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq40_HTML.gif, Eq. (2.4) has a global weak solution
u L ( ( 0 , T ) , H ) L p ( ( 0 , T ) , X 2 ) , 0 < T < , p in  (A1) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equi_HTML.gif

If F : X 2 × ( 0 , ) X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq67_HTML.gif is Frechét differentiable, then the regular solution can be presented under some condition.

We introduce a space sequence
X H ˜ X 3 X 1 H 1 H , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equj_HTML.gif
where X, X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq36_HTML.gif, H are such as in Lemma 2.6, X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq68_HTML.gif is a Banach space, H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq69_HTML.gif is a Hilbert space, and H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq70_HTML.gif are compact including. There exist a constant C 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq71_HTML.gif and a nonnegative function α L 1 ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq72_HTML.gif ( 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq43_HTML.gif) such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ13_HTML.gif
(2.6)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ14_HTML.gif
(2.7)
Lemma 2.7 In addition to the assumptions about the existence of a global solution in Lemma  2.6, if F : X 2 × ( 0 , ) X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq52_HTML.gif is Frechét differentiable and satisfies (2.6), (2.7), then Eq. (2.4) has a unique global solution
u W 1 , ( ( 0 , T ) , H ) W 1 , 2 ( ( 0 , T ) , H 1 ) , 0 < T < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equk_HTML.gif

for all φ X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq73_HTML.gif.

Lemma 2.8 [13]

Let L be a generator of a strongly continuous semigroup Φ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq74_HTML.gif. If u ( , φ ) L 1 ( ( 0 , T ) , Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq75_HTML.gif is a weak solution to the equation
{ d u d t = L u + F ( u ) , 0 < t < , u ( 0 ) = φ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ15_HTML.gif
(2.8)
and F ( u ( , φ ) ) L 1 ( ( 0 , T ) , X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq76_HTML.gif, then the solution u ( t , φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq77_HTML.gif can be read as
u = Φ ( t ) φ + 0 t Φ ( t τ ) F ( u ) d τ , 0 t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equl_HTML.gif
Note that we used to assume that the linear operator L in (2.8) is a sectorial operator which generates an analytic semigroup Φ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq74_HTML.gif. It is known that there exists a constant λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq78_HTML.gif such that L λ I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq79_HTML.gif generates the fractional power operators L α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq80_HTML.gif and fractional order spaces H α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq81_HTML.gif for α R 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq82_HTML.gif, where L = ( L λ I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq83_HTML.gif. Without loss of generality, we assume that ℒ generates the fractional power operators L α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq80_HTML.gif and fractional order spaces H α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq81_HTML.gif as follows:
L α = ( L ) α : H α H , α R 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equm_HTML.gif

where H α = D ( L α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq84_HTML.gif is the domain of L α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq85_HTML.gif. By the semigroup theory of linear operators (Pazy [13]), we know that H β H α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq86_HTML.gif is a compact inclusion for any β > α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq87_HTML.gif.

Lemma 2.9 [1315] (Imbedding theorem of factional order spaces)

Let Ω R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq20_HTML.gif be a Lipschitz field, L : W m , p L p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq88_HTML.gif be a sectorial operator, m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq89_HTML.gif, and 1 p < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq90_HTML.gif. Then for 0 α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq91_HTML.gif, the fractional order spaces H α = D ( L α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq92_HTML.gif satisfy the following relations:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equn_HTML.gif
and the inequalities
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equo_HTML.gif

For sectorial operators, we also have the following properties which can be found in [13].

Lemma 2.10 Let L : H 1 H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq93_HTML.gif be a sectorial operator which generates an analytic semigroup T ( t ) = e t L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq94_HTML.gif. If all eigenvalues λ of L satisfy Re λ < λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq95_HTML.gif for some real number λ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq96_HTML.gif, then for L α ( L = L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq97_HTML.gif, we have
  1. (1)

    T ( t ) : H H α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq98_HTML.gif is bounded for all α R 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq82_HTML.gif and t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq99_HTML.gif,

     
  2. (2)

    T ( t ) L α x = L α T ( t ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq100_HTML.gif, x H α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq101_HTML.gif,

     
  3. (3)
    for each t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq99_HTML.gif, L α T ( t ) : H H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq102_HTML.gif is bounded and
    L α T ( t ) C α t α e δ t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equp_HTML.gif
     
where some δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq103_HTML.gif, C α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq104_HTML.gif is a constant only depending on α,
  1. (4)
    the H α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq81_HTML.gif-norm can be defined by
    x H α = L α x H , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ16_HTML.gif
    (2.9)
     
  2. (5)
    ifis symmetric, for any α , β R 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq105_HTML.gif, we have
    L α u , v H = L α β u , L β v H . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equq_HTML.gif
     

3 Formula of global solutions

We introduce the spaces
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equr_HTML.gif
Let
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equs_HTML.gif
Then Eqs. (1.1)-(1.7) can be rewritten as an abstract equation
d ϕ d t = L ϕ + F ( ϕ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equt_HTML.gif
Theorem 3.1 If ϕ 0 = ( u 0 , T 0 , q 0 ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq106_HTML.gif, Q , G L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq107_HTML.gif, then the global solution ϕ of Eqs. (1.1)-(1.7) can be read as
ϕ ( x , t ) = Φ ( t ) ϕ 0 + 0 t Φ ( t τ ) P ( F ( ϕ ) ) d τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ17_HTML.gif
(3.1)

where Φ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq74_HTML.gif is an analytic semigroup generated by L, and P : L 2 ( Ω , R 4 ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq108_HTML.gif is a Leray projection.

Proof As ϕ = ( u , T , q ) L ( ( 0 , T ) , H ) L 2 ( ( 0 , T ) , H 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq109_HTML.gif is a weak solution to Eqs. (1.1)-(1.7) [1], from the Hölder inequality and the Sobolev imbedding theorem, it follows that
0 T [ Ω | ( u ) u | 3 2 d x ] 2 3 d t 0 T [ Ω | u | 3 2 | u | 3 2 d x ] 2 3 d t 0 T [ ( Ω | u | 6 d x ) 1 6 ( Ω | D u | 2 d x ) 1 2 ] d t C 0 T Ω | D u | 2 d x d t C u L 2 ( 0 , T ; H 1 ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equu_HTML.gif
Then ( u ) u L 1 ( ( 0 , T ) , L 3 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq110_HTML.gif. Hence,
P r σ u + P r ( R T R ˜ q ) κ ( u ) u L 1 ( ( 0 , T ) , L 3 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ18_HTML.gif
(3.2)
From the Hölder inequality and the Sobolev imbedding theorem, we see
0 T [ Ω | ( u ) T | 3 2 d x ] 2 3 d t 0 T [ Ω | u | 3 2 | D T | 3 2 d x ] 2 3 d t 0 T [ ( Ω | u | 6 d x ) 1 6 ( Ω | D T | 2 d x ) 1 2 ] d t C 0 T [ ( Ω | D u | 2 d x ) 1 2 ( Ω | D T | 2 d x ) 1 2 ] d t C ( 0 T Ω | D u | 2 d x d t ) 1 2 ( 0 T Ω | D T | 2 d x d t ) 1 2 C T L 2 ( 0 , T ; H 1 ) u L 2 ( 0 , T ; H 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equv_HTML.gif
Then ( u ) T L 1 ( ( 0 , T ) , L 3 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq111_HTML.gif. Thus,
u 2 ( u ) T + Q L 1 ( ( 0 , T ) , L 3 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ19_HTML.gif
(3.3)
Similarly, we have
u 2 ( u ) q + G L 1 ( ( 0 , T ) , L 3 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ20_HTML.gif
(3.4)
According to the ADN theory and the theory of linear elliptic equations, we have that
L ( ϕ , p ) = ( P r ( Δ u p ) Δ T L e Δ q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equw_HTML.gif
is a sectorial operator and
L : W 2 , q ( Ω , R 4 ) × W 1 , q ( Ω ) L q ( Ω , R 4 ) , 1 < q < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equx_HTML.gif

Therefore, L generates the analytic semigroup Φ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq74_HTML.gif.

It follows from (3.2), (3.3), and (3.4) that
F ( ϕ ) L 1 ( ( 0 , T ) , L 3 2 ( Ω , R 4 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equy_HTML.gif
Applying Lemma 2.8 yields
ϕ ( x , t ) = Φ ( t ) ϕ 0 + 0 t Φ ( t τ ) P ( F ( ϕ ) ) d τ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equz_HTML.gif

 □

Remark 3.2 The analytic semigroup Φ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq74_HTML.gif generated by L can be read as
Φ ( t ) = ( Φ 1 ( t ) Φ 2 ( t ) Φ 3 ( t ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equaa_HTML.gif
Remark 3.3 The semigroup generated by Eqs. (1.1)-(1.7) can be rewritten as
( u T q ) = ( Φ 1 ( t ) ( ϕ 0 ) + 0 t Φ 1 ( t τ ) P ( F 1 ) d τ Φ 2 ( t ) ( ϕ 0 ) + 0 t Φ 2 ( t τ ) P ( F 2 ) d τ Φ 3 ( t ) ( ϕ 0 ) + 0 t Φ 3 ( t τ ) P ( F 3 ) d τ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equab_HTML.gif

4 Regularity of global solution

Theorem 4.1 If ϕ 0 = ( u 0 , T 0 , q 0 ) H 2 ( Ω , R 4 ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq112_HTML.gif, Q , G L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq107_HTML.gif, then Eqs. (1.1)-(1.7) have a unique solution ( ϕ , p ) = ( u , T , q , p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq113_HTML.gif, and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equac_HTML.gif

for all 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq43_HTML.gif.

Proof Let X 2 = X 1 = H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq114_HTML.gif and X 3 = H 2 ( Ω , R 4 ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq115_HTML.gif. Define F : H 1 H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq116_HTML.gif as
F ϕ , ψ = Ω [ P r u v P r σ u u + P r ( R T R ˜ q ) v 2 ( u ) u v T S + u 2 S ( u ) T S + Q S L e q z + u 2 z ( u ) q z + G z ] d x , for ψ = ( v , S , z ) H 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equad_HTML.gif
Then
F ϕ , ψ = Ω [ P r Δ u P r σ u + P r ( R T R ˜ q ) κ ˜ ( u ) u ] v + [ Δ T + u 2 ( u ) T + Q ] S + [ L e Δ q + u 2 ( u ) q + G ] z d x 1 2 Ω ( | v | 2 + | S | 2 + | z | 2 ) d x + C Ω ( | Δ u | 2 + | T | 2 + | q | 2 + | u D u | 2 + | Δ T | 2 + | u 2 | 2 + | u D T | 2 + | Q | 2 + | Δ q | 2 + | u D q | 2 + | G | 2 ) d x 1 2 ψ H 2 + g ( ϕ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equae_HTML.gif

for any ψ H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq117_HTML.gif and ϕ X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq118_HTML.gif. Then (2.7) holds.

We prove (2.6).
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equaf_HTML.gif
By the interpolation inequality [16], we see
v X 1 4 = v H 1 2 C v L 2 1 2 v H 1 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ21_HTML.gif
(4.1)
By the imbedding theorem of factional order spaces, we have
( Ω | v | 4 d x ) 1 4 C v X 1 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ22_HTML.gif
(4.2)
Then it follows from (4.1) and 4.2) that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equag_HTML.gif
Since ϕ = ( u , T , q ) L 2 ( ( 0 , T ) , H 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq119_HTML.gif is a weak solution to Eqs. (1.1)-(1.7), α ( t ) = C ( u H 1 2 + T H 1 2 + q H 1 2 + 1 ) L 1 ( 0 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq120_HTML.gif, 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq43_HTML.gif. Then (2.6) and (2.7) hold. From Lemma 2.7, we deduce that the solution ϕ is unique and
ϕ t = ( u t , T t , q t ) L ( ( 0 , T ) , H ) L 2 ( ( 0 , T ) , H 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ23_HTML.gif
(4.3)
Multiplying (1.1) by u and integrating over Ω, we get
P r Ω | u | 2 d x + P r Ω σ u u d x = Ω [ P r ( R T R ˜ q ) κ u u t u ] d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equah_HTML.gif
Using the Young inequality, we obtain
P r Ω | u | 2 d x + P r σ ˜ Ω u 2 d x ε Ω | u | 2 d x + C ε 1 Ω [ | T | 2 + | q | 2 + | u t | 2 ] d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equai_HTML.gif
where ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq121_HTML.gif is a real constant satisfying P r σ ˜ ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq122_HTML.gif. Then there exists a constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq27_HTML.gif such that
Ω | u | 2 d x C ( T L 2 2 + q L 2 2 + u t L 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equaj_HTML.gif
Thanks to (4.3), we have
u L ( ( 0 , T ) , H 1 ( Ω , R 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ24_HTML.gif
(4.4)
We consider the Stokes equation
{ Δ u + p = g ( x , t ) , g = σ u + ( R T R ˜ q ) κ 1 P r ( u ) u u t , div u = 0 , u = 0 , x 2 = 0 , 1 , u ( 0 , x 2 ) = u ( 2 π , x 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ25_HTML.gif
(4.5)
From (4.3), (4.4), and the Sobolev imbedding theorem, we find that g ( x , t ) L q ( Ω , R 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq123_HTML.gif, 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq124_HTML.gif. By the ADN theorem, Eq. (4.5) has a solution
u L ( ( 0 , T ) , W 2 , q ( Ω , R 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equak_HTML.gif
Then ( u ) u L 2 ( Ω , R 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq125_HTML.gif and g ( x , t ) L 2 ( Ω , R 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq126_HTML.gif. Using the ADN theorem, we obtain
{ u L ( ( 0 , T ) , H 2 ( Ω , R 2 ) ) , p L ( ( 0 , T ) , H 1 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ26_HTML.gif
(4.6)
Multiplying (1.2) by T and integrating over Ω, we get
Ω | T | 2 d x = Ω ( u 2 T + Q T T t T ) d x ε Ω | T | 2 d x + C ε 1 Ω [ | u | 2 + | Q | 2 + | T t | 2 ] d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equal_HTML.gif
where ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq121_HTML.gif is a constant. Then there exists a constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq27_HTML.gif such that
Ω | T | 2 d x C ( u L 2 2 + Q L 2 2 + T t L 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equam_HTML.gif
Using (4.3), we have
T L ( ( 0 , T ) , H 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ27_HTML.gif
(4.7)
We consider the elliptic equation
{ Δ T = f 1 , f 1 = u 2 ( u ) T + Q T t , T = 0 , x 2 = 0 , 1 , T ( 0 , x 2 ) = T ( 2 π , x 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ28_HTML.gif
(4.8)
It follows from (4.3), (4.7), and the Sobolev imbedding theorem that f 1 ( x , t ) L q ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq127_HTML.gif, 1 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq128_HTML.gif. Using the theory of linear elliptic equations, Eq. (4.8) has a solution
T L ( ( 0 , T ) , W 2 , q ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equan_HTML.gif
Then ( u ) T L 2 ( Ω , R 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq129_HTML.gif and f 1 ( x , t ) L 2 ( Ω , R 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq130_HTML.gif. Using the theory of linear elliptic equations, we have that
T L ( ( 0 , T ) , H 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ29_HTML.gif
(4.9)
Multiplying (1.3) by q and integrating over Ω, we get
Ω | q | 2 d x = Ω ( u 2 q + G q q t ) d x ε Ω | q | 2 d x + C ε 1 Ω [ | u | 2 + | G | 2 + | q t | 2 ] d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equao_HTML.gif
where ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq121_HTML.gif is a constant. Then there exists a constant C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq27_HTML.gif such that
Ω | q | 2 d x C ( u L 2 2 + Q L 2 2 + q t L 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equap_HTML.gif
Using (4.3), we have
q L ( ( 0 , T ) , H 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ30_HTML.gif
(4.10)
We consider the elliptic equation
{ Δ q = f 2 , f 2 = u 2 ( u ) q + G q t , q = 0 , x 2 = 0 , 1 , q ( 2 π , x 2 ) = q ( 0 , x 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ31_HTML.gif
(4.11)
Using the arguments similar to those for (4.8), we get
q L ( ( 0 , T ) , H 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equ32_HTML.gif
(4.12)
It follows from (4.6), (4.9), and (4.12) that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equaq_HTML.gif

 □

Theorem 4.2 If ϕ 0 = ( u 0 , T 0 , q 0 ) H k + 2 ( Ω , R 4 ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq131_HTML.gif, Q , G H k ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq132_HTML.gif, then Eqs. (1.1)-(1.7) have a higher regular solution ( ϕ , p ) = ( u , T , q , p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq133_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equar_HTML.gif

for 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq134_HTML.gif.

Proof We prove the theorem using mathematical induction.

If k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq135_HTML.gif, ϕ 0 = ( u 0 , T 0 , q 0 ) H 3 ( Ω , R 4 ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq136_HTML.gif, Q , G H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq137_HTML.gif, then ϕ 0 = ( u 0 , T 0 , q 0 ) H 2 ( Ω , R 4 ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq112_HTML.gif, Q , G L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq107_HTML.gif. Using Theorem 4.1, we find that ϕ = ( u , T , q ) H 2 ( Ω , R 4 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq138_HTML.gif.

Thanks to the Sobolev imbedding theorem, H 2 ( Ω ) W 1 , 4 ( Ω ) C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq139_HTML.gif if n = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq140_HTML.gif. We obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equas_HTML.gif

Then ( u ) u H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq141_HTML.gif and F 1 ( ϕ ) H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq142_HTML.gif.

We have from the formula (3.1)
u t = L 1 Φ 1 ( t ) u 0 + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ + F 1 ( Φ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equat_HTML.gif
Then there exists α satisfying 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq61_HTML.gif such that
u t H 1 α = L 1 Φ 1 ( t ) u 0 + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ + F 1 ( Φ ) H 1 α L 1 Φ 1 ( t ) u 0 H 1 α + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ H 1 α + F 1 ( Φ ) H 1 α C + 0 t L 1 3 α 2 Φ 1 ( t τ ) F 1 ( Φ ) L 2 d τ C + 0 t L 1 1 α 2 Φ 1 ( t τ ) L 1 1 2 F 1 ( Φ ) L 2 d τ C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equau_HTML.gif

Thus, u t H 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq143_HTML.gif. Then g H 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq144_HTML.gif in Eq. (4.5). By the ADN theory, u H 3 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq145_HTML.gif. Thus, ( u ) u H 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq146_HTML.gif and F 1 ( ϕ ) H 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq147_HTML.gif.

We have
u t H 1 = L 1 Φ 1 ( t ) u 0 + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ + F 1 ( Φ ) H 1 L 1 Φ 1 ( t ) u 0 H 1 + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ H 1 + F 1 ( Φ ) H 1 C + 0 t L 1 3 2 Φ 1 ( t τ ) F 1 ( Φ ) L 2 d τ C + 0 t L 1 1 + α 2 Φ 1 ( t τ ) L 1 1 α 2 F 1 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equav_HTML.gif

which implies u t H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq148_HTML.gif. Then g H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq149_HTML.gif in Eq. (4.5). Using the ADN theory, u H 3 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq150_HTML.gif and p H 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq151_HTML.gif. Thus, u L ( ( 0 , T ) , H 3 ( Ω , R 2 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq152_HTML.gif and p L ( ( 0 , T ) , H 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq153_HTML.gif. Then u C ( [ 0 , T ] , H 3 ( Ω , R 2 ) H 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq154_HTML.gif and u t C ( ( 0 , T ] , H 1 ( Ω , R 2 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq155_HTML.gif from the formula (3.1).

Similarly,
Ω | ( u ) T | 2 d x C Ω | u | 4 + | T | 2 | u | 2 d x C ( u W 1 , 4 4 + sup Ω | u | 2 T H 2 2 ) C ( u H 2 4 + u H 2 2 T H 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equaw_HTML.gif

Then ( u ) T H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq156_HTML.gif and F 2 ( ϕ , p ) H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq157_HTML.gif.

We have from the formula (3.1)
T t = L 2 Φ 2 ( t ) u 0 + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ + F 2 ( Φ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equax_HTML.gif
Then there exists α satisfying 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq61_HTML.gif such that
T t H 1 α = L 2 Φ 2 ( t ) T 0 + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ + F 2 ( Φ ) H 1 α L 2 Φ 2 ( t ) T 0 H 1 α + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ H 1 α + F 2 ( Φ ) H 1 α C + 0 t L 2 3 α 2 Φ 2 ( t τ ) F 2 ( Φ ) L 2 d τ C + 0 t L 2 1 α 2 Φ 2 ( t τ ) L 2 1 2 F 2 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equay_HTML.gif

which implies T t H 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq158_HTML.gif. Then f 1 H 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq159_HTML.gif in Eq. (4.8). It follows from the linear elliptic equation T H 3 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq160_HTML.gif. Thus, ( u ) T H 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq161_HTML.gif and F 2 ( ϕ ) H 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq162_HTML.gif.

Then
T t H 1 = L 2 Φ 2 ( t ) T 0 + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ + F 2 ( Φ ) H 1 L 2 Φ 2 ( t ) T 0 H 1 + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ H 1 + F 2 ( Φ ) H 1 C + 0 t L 2 3 2 Φ 2 ( t τ ) F 2 ( Φ ) L 2 d τ C + 0 t L 2 1 + α 2 Φ 2 ( t τ ) L 2 1 α 2 F 2 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equaz_HTML.gif

which implies T t H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq163_HTML.gif. We obtain that f 1 H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq164_HTML.gif in Eq. (4.8). Then T H 3 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq165_HTML.gif from the theory of linear elliptic equations. Thus, T L ( ( 0 , T ) , H 3 ( Ω , R 2 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq166_HTML.gif. From the formula (3.1), T C ( [ 0 , T ] , H 3 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq167_HTML.gif and T t C ( ( 0 , T ] , H 1 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq168_HTML.gif.

Similarly,
Ω | ( u ) q | 2 d x C Ω | u | 4 + | q | 2 | u | 2 d x C ( u W 1 , 4 4 + sup Ω | u | 2 q H 2 2 ) C ( u H 2 4 + u H 2 2 q H 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equba_HTML.gif

Then ( u ) q H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq169_HTML.gif and F 3 ( ϕ , p ) H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq170_HTML.gif.

We have from the formula (3.1)
q t = L 3 Φ 3 ( t ) q 0 + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ + F 3 ( Φ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbb_HTML.gif
Then there exists α satisfying 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq61_HTML.gif such that
q t H 1 α = L 3 Φ 3 ( t ) q 0 + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ + F 3 ( Φ ) H 1 α L 3 Φ 3 ( t ) u 0 H 1 α + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ H 1 α + F 3 ( Φ ) H 1 α C + 0 t L 3 3 α 2 Φ 3 ( t τ ) F 3 ( Φ ) L 2 d τ C + 0 t L 3 1 α 2 Φ 3 ( t τ ) L 3 1 2 F 3 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbc_HTML.gif

which implies q t H 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq171_HTML.gif. Then f 2 H 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq172_HTML.gif in Eq. (4.11). Thus, q H 3 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq173_HTML.gif from the theory of linear elliptic equations. Then ( u ) q H 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq174_HTML.gif and F 3 ( ϕ ) H 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq175_HTML.gif.

Thus,
q t H 1 = L 3 Φ 3 ( t ) q 0 + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ + F 3 ( Φ ) H 1 L 3 Φ 3 ( t ) q 0 H 1 + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ H 1 + F 3 ( Φ ) H 1 C + 0 t L 3 3 2 Φ 3 ( t τ ) F 3 ( Φ ) L 2 d τ C + 0 t L 3 1 + α 2 Φ 3 ( t τ ) L 3 1 α 2 F 3 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbd_HTML.gif

which implies q t H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq176_HTML.gif. We see f 2 H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq177_HTML.gif in Eq. (4.11). Then q H 3 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq178_HTML.gif from the theory of linear elliptic equations. Thus, q L ( ( 0 , T ) , H 3 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq179_HTML.gif. We have q C ( [ 0 , T ] , H 3 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq180_HTML.gif and q t C ( ( 0 , T ] , H 1 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq181_HTML.gif from the formula (3.1).

It follows from Eq. (4.5) that
p = Δ u σ u + ( R T R ˜ q ) κ 1 P r ( u ) u u t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Eqube_HTML.gif
Clearly, the right-hand side of the above equality is continuous in H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq182_HTML.gif. Thus,
p C ( ( 0 , T ) , H 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbf_HTML.gif

If k = m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq183_HTML.gif, ϕ 0 = ( u 0 , T 0 , q 0 ) H m + 2 ( Ω , R 4 ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq184_HTML.gif, and Q , G H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq185_HTML.gif, then ϕ 0 = ( u 0 , T 0 , q 0 ) H m + 1 ( Ω , R 4 ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq186_HTML.gif and Q , G H m 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq187_HTML.gif. From the hypothesis of mathematical induction, we see ϕ = ( u , T , q ) H m + 1 ( Ω , R 4 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq188_HTML.gif.

By the Sobolev imbedding theorem, we have H m + 1 ( Ω ) W m , 4 ( Ω ) C m 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq189_HTML.gif if n = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq140_HTML.gif. Then it follows from the Sobolev imbedding theorem and the interpolation inequality that
Ω | m ( u ) u | 2 d x C Ω [ i = 0 m | i u | 2 | m + 1 i u | 2 ] d x C Ω [ i = 1 m | i u | 2 | m + 1 i u | 2 + | u | 2 | m + 1 u | 2 ] d x C [ i = 1 m ( Ω | i u | 4 d x ) 1 2 ( Ω | m + 1 i u | 4 d x ) 1 2 + sup Ω | u | 2 Ω | m + 1 u | 2 d x ] C ( u W 1 , 4 2 u W m , 4 2 + sup Ω | u | 2 u H m + 1 2 ) , C u H m + 1 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbg_HTML.gif

Then ( u ) u H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq190_HTML.gif and F 1 ( ϕ ) H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq191_HTML.gif.

We have from the formula (3.1)
u t = L 1 Φ 1 ( t ) u 0 + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ + F 1 ( Φ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbh_HTML.gif
Then there exists α satisfying 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq61_HTML.gif such that
u t H m α = L 1 Φ 1 ( t ) u 0 + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ + F 1 ( Φ ) H m α L 1 Φ 1 ( t ) u 0 H m α + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ H m α + F 1 ( Φ ) H m α C + 0 t L 1 2 + m α 2 Φ 1 ( t τ ) F 1 ( Φ ) L 2 d τ C + 0 t L 1 1 α 2 Φ 1 ( t τ ) L 1 m 2 F 1 ( Φ ) L 2 d τ C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbi_HTML.gif
Then u t H m α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq192_HTML.gif. We see that g H m α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq193_HTML.gif in Eq. (4.5). Thus, u H m + 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq194_HTML.gif from the ADN theory. Hence, ( u ) u H m + 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq195_HTML.gif and F 1 ( ϕ ) H m + 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq196_HTML.gif. Then
u t H m = L 1 Φ 1 ( t ) u 0 + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ + F 1 ( Φ ) H m L 1 Φ 1 ( t ) u 0 H m + 0 t L 1 Φ 1 ( t τ ) F 1 ( Φ ) d τ H m + F 1 ( Φ ) H m C + 0 t L 1 1 + m 2 Φ 1 ( t τ ) F 1 ( Φ ) L 2 d τ C + 0 t L 1 1 + α 2 Φ 1 ( t τ ) L 1 m + 1 α 2 F 1 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbj_HTML.gif
which implies u t H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq197_HTML.gif. Then g H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq198_HTML.gif in Eq. (4.5). Using the ADN theory, u H m + 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq199_HTML.gif, and p H m + 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq200_HTML.gif, we get
u L ( ( 0 , T ) , H m + 2 ( Ω , R 2 ) ) , p L ( ( 0 , T ) , H m + 1 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbk_HTML.gif
From the formula (3.1), we have
u C ( [ 0 , T ] , H m + 2 ( Ω , R 2 ) ) , u t C ( ( 0 , T ] , H m ( Ω , R 2 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbl_HTML.gif
Similarly,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbm_HTML.gif

Then ( u ) T H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq201_HTML.gif and F 2 ( ϕ ) H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq202_HTML.gif.

We have from the formula (3.1)
T t = L 2 Φ 2 ( t ) T 0 + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ + F 2 ( Φ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbn_HTML.gif
Then there exists α satisfying 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq61_HTML.gif such that
T t H m α = L 2 Φ 2 ( t ) T 0 + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ + F 2 ( Φ ) H m α L 2 Φ 2 ( t ) T 0 H m α + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ H m α + F 2 ( Φ ) H m α C + 0 t L 2 2 + m α 2 Φ 2 ( t τ ) F 2 ( Φ ) L 2 d τ C + 0 t L 2 1 α 2 Φ 2 ( t τ ) L 2 m 2 F 2 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbo_HTML.gif
which implies T t H m α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq203_HTML.gif. Then f 1 H m α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq204_HTML.gif in Eq. (4.8). It follows from the linear elliptic equation T H m + 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq205_HTML.gif that ( u ) T H m + 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq206_HTML.gif and F 2 ( ϕ ) H m + 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq207_HTML.gif. We obtain
T t H m = L 2 Φ 2 ( t ) T 0 + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ + F 2 ( Φ ) H m L 2 Φ 2 ( t ) u 0 H m + 0 t L 2 Φ 2 ( t τ ) F 2 ( Φ ) d τ H m + F 2 ( Φ ) H m C + 0 t L 2 1 + m 2 Φ 2 ( t τ ) F 2 ( Φ ) L 2 d τ C + 0 t L 2 1 + α 2 Φ 2 ( t τ ) L 2 m + 1 α 2 F 2 ( Φ ) L 2 d τ C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbp_HTML.gif
Then T t H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq208_HTML.gif. We have f 1 H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq209_HTML.gif in Eq. (4.8). Then T H m + 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq210_HTML.gif from the theory of linear elliptic equations. Thus,
T L ( ( 0 , T ) , H m + 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbq_HTML.gif
From the formula (3.1), we induce
T C ( [ 0 , T ] , H m + 2 ( Ω ) ) , u t C ( ( 0 , T ] , H m ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbr_HTML.gif
Similarly,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbs_HTML.gif

Then ( u ) q H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq211_HTML.gif and F 3 ( ϕ ) H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq212_HTML.gif.

We have from the formula (3.1)
q t = L 3 Φ 3 ( t ) q 0 + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ + F 3 ( Φ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbt_HTML.gif
Then there exists α satisfying 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq61_HTML.gif such that
q t H m α = L 3 Φ 3 ( t ) q 0 + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ + F 3 ( Φ ) H m α L 3 Φ 3 ( t ) u 0 H m α + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ H m α + F 3 ( Φ ) H m α C + 0 t L 3 2 + m α 2 Φ 3 ( t τ ) F 3 ( Φ ) L 2 d τ C + 0 t L 3 1 α 2 Φ 3 ( t τ ) L 3 m 2 F 3 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbu_HTML.gif
which implies q t H m α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq213_HTML.gif. Then f 2 H m α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq214_HTML.gif in Eq. (4.11). Thus, q H m + 2 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq215_HTML.gif from the theory of linear elliptic equations. Then ( u ) q H m + 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq216_HTML.gif and F 3 ( ϕ ) H m + 1 α ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq217_HTML.gif. Thus,
q t H m = L 3 Φ 3 ( t ) q 0 + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ + F 3 ( Φ ) H m L 3 Φ 3 ( t ) q 0 H m + 0 t L 3 Φ 3 ( t τ ) F 3 ( Φ ) d τ H m + F 3 ( Φ ) H m C + 0 t L 3 1 + m 2 Φ 3 ( t τ ) F 3 ( Φ ) L 2 d τ C + 0 t L 3 1 + α 2 Φ 3 ( t τ ) L 3 m + 1 α 2 F 3 ( Φ ) L 2 d τ C , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbv_HTML.gif
which implies q t H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq218_HTML.gif. We find f 2 H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq219_HTML.gif in Eq. (4.11). Then q H m + 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq220_HTML.gif from the theory of linear elliptic equations. We have
q L ( ( 0 , T ) , H m + 2 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbw_HTML.gif
From the formula (3.1), we see
q C ( [ 0 , T ] , H m + 2 ( Ω ) ) , q t C ( ( 0 , T ] , H m ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbx_HTML.gif
It follows from Eq. (4.5) that
p = Δ u σ u + ( R T R ˜ q ) κ 1 P r ( u ) u u t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equby_HTML.gif
Clearly, the right-hand side of the above equality is continuous in H m ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq221_HTML.gif. Then
p C ( ( 0 , T ) , H m + 1 ( Ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equbz_HTML.gif

The proof is completed. □

Since the differentiability of time and of space can be transformed into each other, we obtain

Remark 4.3 If ϕ 0 = ( u 0 , T 0 , q 0 ) H k + 2 ( Ω , R 4 ) H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq131_HTML.gif, Q , G H k ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq132_HTML.gif, then Eqs. (1.1)-(1.7) have a higher regular solution ( ϕ , p ) = ( u , T , q , p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq133_HTML.gif, and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_Equca_HTML.gif

for 0 < T < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq134_HTML.gif, where l, r, α, β are positive integers satisfying 2 l + r = k + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq222_HTML.gif and 2 α + β = k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-143/MediaObjects/13661_2012_Article_257_IEq223_HTML.gif.

Declarations

Acknowledgements

The author is very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. The project is supported by the National Natural Science Foundation of China (11271271), the NSF of Sichuan Science and Technology Department of China (2010JY0057) and the NSF of Sichuan Education Department of China (11ZA102).

Authors’ Affiliations

(1)
College of Mathematics and Software Science, Sichuan Normal University

References

  1. Luo, H: Global solution of atmospheric circulation equations with humidity effect (submitted)
  2. Ma T, Wang SH: Phase Transition Dynamics in Nonlinear Sciences. Springer, New York; 2012.
  3. Ma T: Theories and Methods in Partial Differential Equations. Science Press, China; 2011. in Chinese
  4. Phillips NA: The general circulation of the atmosphere: a numerical experiment. Q. J. R. Meteorol. Soc. 1956, 82: 123-164. 10.1002/qj.49708235202View Article
  5. Rossby CG: On the solution of problems of atmospheric motion by means of model experiment. Mon. Weather Rev. 1926, 54: 237-240. 10.1175/1520-0493(1926)54<237:OTSOPO>2.0.CO;2View Article
  6. Lions JL, Temam R, Wang SH: New formulations of the primitive equations of atmosphere and applications. Nonlinearity 1992, 5(2):237-288. 10.1088/0951-7715/5/2/001MathSciNetView Article
  7. Lions JL, Temam R, Wang SH: On the equations of the large-scale ocean. Nonlinearity 1992, 5(5):1007-1053. 10.1088/0951-7715/5/5/002MathSciNetView Article
  8. Lions JL, Temam R, Wang SH: Models for the coupled atmosphere and ocean. (CAO I,II). Comput. Mech. Adv. 1993, 1(1):5-54.MathSciNet
  9. Luo H: Steady state solution to atmospheric circulation equations with humidity effect. J. Appl. Math. 2012. doi:10.1155/2012/867310
  10. Evens LC: Partial Differential Equations. Am. Math. Soc., Providence; 1998.
  11. Temam R CBMS-NSF Regional Conference Series in Applied Mathematics. In Navier-Stokes Equation and Nonlinear Functional Analysis. SIAM, Philadelphia; 1983.
  12. Temam R: Navier-Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam; 1979.
  13. Pazy A Appl. Math. Sci. 44. In Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin; 2006.
  14. Ma T, Wang SH: Stability and Bifurcation of Nonlinear Evolution Equations. Science Press, China; 2007. in Chinese
  15. Ma T, Wang SH Nonlinear Science Ser. A 53. In Bifurcation Theory and Applications. World Scientific, Singapore; 2005.
  16. Temam R Applied Mathematical Science 68. In Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd edition. Springer, New York; 1997.View Article

Copyright

© Luo; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.