Regularity of global solution to atmospheric circulation equations with humidity effect

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R E S E A R C H

Open Access

Regularity of global solution to atmospheric

circulation equations with humidity effect

Hong Luo

*

*_{Correspondence: [email protected]}

College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, China

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) at (x,t) = (x,x,t)∈×(,∞) (= (, π)×(, ) is a period ofC∞ﬁeld (–∞, +∞)×(, )):

∂u

∂t =Pr(u–∇p–σu) +Pr(RT–Rq)˜ κ– (u· ∇)u, (.) ∂T

∂t =T+u– (u· ∇)T+Q, (.)

∂q

∂t =Leq+u– (u· ∇)q+G, (.)

divu= , (.)

wherePr> ,R> ,R,˜ Le>  are constants,u= (u,u),T,q,pdenote velocity ﬁeld,

tem-perature, humidity and pressure respectively,Q,Gare known functions, andσ is a con-stant matrix

σ=

σ ω ω σ

.

The problems (.)-(.) are supplemented with the following Dirichlet boundary con-dition atx= ,  and the periodic condition forx:

(u,T,q) = , x= , , (.)

(u,T,q)(,x) = (u,T,q)(π,x), (.)

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and initial value conditions

(u,T,q) = (u,T,q), t= . (.)

Partial differential equations (.)-(.) are 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. Atmo-spheric 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 im-proving 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 can-not 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 [–], or an infinite dimensional dynamical system [–]. 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 theT-weakly continuous operator []. In [], the steady state solutions to atmospheric circulation equations with humidity effect are studied. A sufficient condi-tion of the existence of the steady state solucondi-tions to atmospheric circulacondi-tion equacondi-tions 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 (.)-(.). The paper is organized as follows. In Section , we present preliminary results. In Sec-tion , we present the formula of the soluSec-tion to the atmospheric circulaSec-tion equaSec-tions. In Section , we obtain the regularity of the solutions to equations (.)-(.).

· Xdenotes the norm of the spaceX, andC,Ciare variable constants.

2 Preliminaries

We consider the divergence form of the linear elliptic equation

Lu= –Dj(aijDiu) +biDiu+cu=f, (.)

whereaij,bi,c∈L∞(),f ∈L(),aij=aji, (aij) is uniformly elliptic,i.e., there exist

con-stants  <λ≤λsuch that

λ|ξ|≤aij(x)ξiξj≤λ|ξ|, ∀ξ∈Rn,x∈.

The problem (.) is supplemented with the following Dirichlet boundary condition:

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Lemma .[] (Theory of linear elliptic equations) Let⊂Rn_{be a C}_ﬁeld,_a

ij∈C(),

bi,c∈L∞(),f∈Lp(),ϕ∈W,p().If u∈W,pis a solution of Eqs. (.), (.),then u_W,p≤C

uLp+fLp+ϕW,p

,

where C> depends on n,p,λ,and C,α_{-norm or L}∞_{-norm of the coeﬃcient functions.} We consider the Stokes equation

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–μu+∇p=f(x),

divu= , u|∂=ϕ.

(.)

Lemma . [, ] (ADN theory of the Stokes equation) Let f ∈ Wk,p₍_,_Rn_), _ϕ _∈

Wk+,p(,Rn),k≥.If(u,p)∈W,p(,Rn)×W,p() ( <p<∞)is a solution of Eq. (.), then the solution(u,p)∈Wk+,p(,Rn)×Wk+,p(),and

u_Wk+,p+p_Wk+,p≤C

f_Wk,p+(u,p)

Lp+ϕWk+,p

,

where C> depends onμ,n,k,α,.

Let X be a linear space,X, X be two separable reﬂexive Banach spaces, and H be a Hilbert space.X,X, andH are completion spaces of Xunder the respective norm. X,X⊂Hare dense embedding.F:X×(,∞)→X*

is a continuous mapping. We con-sider the abstract equation

⎧ ⎨ ⎩

du

dt =Fu,  <t<∞,

u() =ϕ, (.)

whereϕ∈H,u: [, +∞)→His unknown.

Deﬁnition . Let ϕ ∈H be a given initial value. u ∈Lp_((,_T),_X)_∩_L∞_((,_T_),_H)

( <T<∞) is called a global solution of Eq. (.) ifusatisﬁes

u(t),v_H=

t



Fu,vdt+ (ϕ,v)H, ∀v∈X⊂H.

Deﬁnition . Letun,u∈Lp((,T),X).un→uis called uniformly weak convergence inLp_((,_T_),_X

) if{un} ⊂L∞((,T),H) is bounded, and ⎧

⎨ ⎩

unu, inLp((,T),X), limn→∞

T

 |un–u,vH|

_dt_{= ,} _∀_v_∈_H. (.)

Deﬁnition . A mappingF:X×(,∞)→X*

is calledT-weakly continuous if forp= (p, . . . ,pm),  <T<∞andununiformly weakly converges tou, we have

lim n→∞

T



Fun,vdt=

T



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Lemma .[] Assume F:X×(,∞)→X*

is T -weakly continuous and satisﬁes:

(A) there existsp= (p, . . . ,pm),pi> (≤i≤m),such that

Fu,u ≤–Cup_X_+Cu_H+f(t), ∀u∈X,

whereC,Care constants,f ∈L(,T)( <T<∞), · pX= m

i=| · |

pi

i ,| · |iis a

seminorm ofX, · X=

m

i=| · |i,

(A) there exists <α< for any <h< andu∈C_([,_∞_),_X)_,

t+h

t

Fu,vdt≤Chα_, _∀_v_∈_X_and__≤_t_<_T_,

whereC> depends only onT,vX, t

u

p

Xdt,andsup≤t≤TuH.

Then for anyϕ∈H,Eq. (.)has a global weak solution

u∈L∞(,T),H∩Lp(,T),X,  <T<∞,p in(A).

IfF:X×(,∞)→X*

 is Frechét diﬀerentiable, then the regular solution can be pre-sented under some condition.

We introduce a space sequence

X⊂ ˜H⊂X⊂X⊂H⊂H,

whereX,X,Hare such as in Lemma .,Xis a Banach space,His a Hilbert space, and H⊂Hare compact including. There exist a constantC≥ and a nonnegative function α∈L_(,_T_{) ( <}_T_<_∞_{) such that}

DF(u)v,v≤αv_H–CvH, ∀u,v∈X, (.) ⎧

⎨ ⎩

|Fu,v| ≤ v



H+g(u), v∈X,u∈X,

g:X→[,∞) is a bounded continuous functional. (.)

Lemma . In addition to the assumptions about the existence of a global solution in Lemma.,if F:X×(,∞)→X_*is Frechét diﬀerentiable and satisﬁes(.), (.),then Eq. (.)has a unique global solution

u∈W,∞(,T),H∩W,(,T),H,  <T<∞,

for allϕ∈X.

Lemma .[] Let L be a generator of a strongly continuous semigroup(t).If u(·,ϕ)∈ L_((,_T),_Y₎_{is a weak solution to the equation}

⎧ ⎨ ⎩

du

dt =Lu+F(u),  <t<∞,

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and F(u(·,ϕ))∈L_((,_T),_X),_{then the solution u(t,}_ϕ₎_{can be read as}

u=(t)ϕ+

t



(t–τ)F(u)dτ, ≤t≤T.

Note that we used to assume that the linear operatorLin (.) is a sectorial operator which generates an analytic semigroup(t). It is known that there exists a constantλ≥ such thatL–λIgenerates the fractional power operatorsLα_{and fractional order spaces} Hαforα∈R, whereL= –(L–λI). Without loss of generality, we assume thatLgenerates the fractional power operatorsLα_{and fractional order spaces}_Hα_{as follows:}

Lα_{= (–L)}α_:_Hα_→_H, _α_∈_R_,

whereHα=D(Lα_{) is the domain of}_Lα_{. By the semigroup theory of linear operators (Pazy} []), we know thatHβ⊂Hαis a compact inclusion for anyβ>α.

Lemma .[–] (Imbedding theorem of factional order spaces) Let⊂Rnbe a Lip-schitz ﬁeld,L:Wm,p _→_Lp₍₎_{be a sectorial operator,} _m_≥_,_and__≤_p_<_∞_._{Then for}

≤α≤,the fractional order spaces Hα=D(Lα₎_{satisfy the following relations:}

Hα⊂Wk,q, if k–n

q≤mα– n p,

Hα⊂Ck,β, if≤k+β<mα–n p,

and the inequalities

u_Wk,q ≤CuHα, if k– n

q≤mα– n p,

u_Ck,β≤CuHα, if≤k+β<mα– n p.

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

Lemma . Let L:H→H be a sectorial operator which generates an analytic semigroup T(t) =etL_._{If all eigenvalues}_λ_{of L satisfy}_Re_λ_{< –}_λ

for some real numberλ> ,then for

Lα₍_L_{= –L),}_{we have}

() T(t) :H→Hαis bounded for allα∈R_and_t_{> }_,

() T(t)Lα_x₌_Lα_T(t)x_,_∀_x_∈_Hα_,

() for eacht> ,Lα_{T(t) :}_H_→_H_{is bounded and}

_Lα

T(t)≤Cαt–αe–δt,

where someδ> ,Cα> is a constant only depending onα, () theHα-norm can be deﬁned by

xHα=L α

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() ifLis symmetric,for anyα,β∈R_,_{we have}

Lα_u,_v

H=

Lα–β_u,_Lβ_v

H.

3 Formula of global solutions

We introduce the spaces

X=φ= (u,T,q)∈C∞,R|(u,T,q) satisfy (.)-(.), H=φ= (u,T,q)∈L,R|(u,T,q) satisfy (.)-(.),

H=

φ= (u,T,q)∈H,R|(u,T,q) satisfy (.)-(.).

Let

L(φ,p) =

⎛ ⎜ ⎝

L(φ,p) L(φ,p) L(φ,p)

⎞ ⎟ ⎠=

⎛ ⎜ ⎝

Pr(u–∇p)

T Leq

⎞ ⎟ ⎠,

F(φ,p) =

⎛ ⎜ ⎝

F(φ,p) F(φ,p) F(φ,p)

⎞ ⎟ ⎠=

⎛ ⎜ ⎝

–Prσu+Pr(RT–Rq)˜ κ– (u· ∇)u

u– (u· ∇)T+Q u– (u· ∇)q+G

⎞ ⎟ ⎠.

Then Eqs. (.)-(.) can be rewritten as an abstract equation

dφ

dt =Lφ+F(φ).

Theorem . Ifφ= (u,T,q)∈H,Q,G∈L(),then the global solutionφof Eqs. (.)-(.)can be read as

φ(x,t) =(t)φ+

t



(t–τ)PF(φ)dτ, (.)

where(t)is an analytic semigroup generated by L,and P:L₍_,_R₎_→_{H is a Leray} pro-jection.

Proof Asφ= (u,T,q)∈L∞((,T),H)∩L((,T),H) is a weak solution to Eqs. (.)-(.) [], from the Hölder inequality and the Sobolev imbedding theorem, it follows that

T



(u· ∇)u_dx 



dt≤

T



|u||∇_u|_dx 



dt

≤

T



|u|dx

 

|Du|dx

 

dt

≤C

T



|Du|dx dt≤Cu_L_(,_T_;_H₎.

Then (u· ∇)u∈L_((,_T_),_L_₍_{)). Hence,}

–Prσu+Pr(RT–Rq)˜ κ– (u· ∇)u∈L

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From the Hölder inequality and the Sobolev imbedding theorem, we see

T



_(u_{· ∇}_)T _dx

 

dt≤

T



|u||_DT|_dx 



dt

≤

T



|u|dx

 

|DT|dx

 

dt

≤C

T



|Du|dx

 

|DT|dx

 

dt

≤C

T



|Du|dx dt

  T



|DT|dx dt

 

≤CTL_(,_T_;_H₎u_L_(,_T_;_H₎.

Then (u· ∇)T∈L_((,_T_),_L₍_{)). Thus,}

u– (u· ∇)T+Q∈L(,T),L₍₎_. _(.)

Similarly, we have

u– (u· ∇)q+G∈L

(,T),L₍₎_. _(.)

According to the ADN theory and the theory of linear elliptic equations, we have that

L(φ,p) =

⎛ ⎜ ⎝

Pr(u–∇p)

T Leq

⎞ ⎟ ⎠

is a sectorial operator and

L:W,q,R×W,q()→Lq,R, ∀ <q<∞.

Therefore,Lgenerates the analytic semigroup(t). It follows from (.), (.), and (.) that

F(φ)∈L(,T),L_,_R_.

Applying Lemma . yields

φ(x,t) =(t)φ+

t



(t–τ)PF(φ)dτ.

Remark . The analytic semigroup(t) generated byLcan be read as

(t) =

⎛ ⎜ ⎝

(t)

(t) (t)

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Remark . The semigroup generated by Eqs. (.)-(.) can be rewritten as

⎛ ⎜ ⎝

u T q

⎞ ⎟ ⎠=

⎛ ⎜ ⎝

(t)(φ) +_t(t–τ)P(F)dτ (t)(φ) +_t(t–τ)P(F)dτ (t)(φ) +_t(t–τ)P(F)dτ

⎞ ⎟ ⎠.

4 Regularity of global solution

Theorem . Ifφ= (u,T,q)∈H(,R)∩H,Q,G∈L(),then Eqs. (.)-(.)have a unique solution(φ,p) = (u,T,q,p),and

φ= (u,T,q)∈L∞(,T),H,R∩H, φt= (u,T,q)t∈L∞

(,T),H∩L(,T),H,

p∈L∞(,T),H()

for all <T<∞.

Proof LetX=X=HandX=H(,R)∩H. DeﬁneF:H→H*as

Fφ,ψ=

–Pr∇u∇v–Prσu·u+Pr(RT–Rq)v˜ – (u· ∇)u·v–∇T∇S

+uS– (u· ∇)TS+QS–Le∇q∇z+uz– (u· ∇)qz+Gz

dx,

for∀ψ= (v,S,z)∈H.

Then

Fφ,ψ=

Pru–Prσu+Pr(RT–Rq)˜ κ˜– (u· ∇)u

v+T+u– (u· ∇)T+QS

+Leq+u– (u· ∇)q+G

z dx

≤ 



|v|+|S|+|z|dx+C

|u|+|T|+|q|+|u·Du|+|T|

+|u|+|u·DT|+|Q|+|q|+|u·Dq|+|G|

dx

≤ 

ψ 

H+g(φ)

for anyψ∈Handφ∈X. Then (.) holds. We prove (.).

DF(φ)ψ,ψ

=

–Pr∇v∇v–Prσv+Pr(RS–Rz)˜ κv– (v· ∇)u·v– (u· ∇)v·v

–∇S∇S+vS– (v· ∇)TS– (u· ∇)SS–Le∇z∇z+vz

– (v· ∇)qz– (u· ∇)zzdx

≤–C

|∇v|+|∇S|+|∇z|dx+C

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+C

_(v_{· ∇}_)u_·_v+(v· ∇)TS+(v· ∇)qzdx

≤–C

|∇v|+|∇S|+|∇z|dx+C

|v|+|S|+|z|dx

+C

|v||Du|+|v||DT||S|+|v||Dq||z|dx

≤–C

|∇v|+|∇S|+|∇z|dx+C

|v|+|S|+|z|dx

+C

|v|dx

 

|Du|dx

 

+

|v|dx

 

|DT|dx

 

+

|S|dx

 

|DT|dx

 

+

|v|dx

 

|Dq|dx

 

+

|z|dx

 

|Dq|dx

 

.

By the interpolation inequality [], we see

vX 

=v

H ≤Cv   Lv

 

H. (.)

By the imbedding theorem of factional order spaces, we have

|v|dx

 

≤CvX 

. (.)

Then it follows from (.) and .) that

DF(φ)ψ,ψ

≤–C

|∇v|+|∇S|+|∇z|dx+C

|v|+|S|+|z|dx

+CvHvHu_H+v_Hv_HT_H+S_HS_HT_H

+vHvHq_H+z_Hz_Hq_H

≤–C 

v_H+S_H+z_H

+Cv_H+S_H+z_H

+Cv_Hu_H+v_HT_H+S_HT_H+v_Hq_H+z_Hq_H

≤–C  ψ



H+Cα(t)ψH.

Sinceφ= (u,T,q)∈L_((,_T_),_{H) is a weak solution to Eqs. (.)-(.),}_α_{(t) =}_C(_u

H+ T

H+q_H+ )∈L(,T),  <T<∞. Then (.) and (.) hold. From Lemma ., we

deduce that the solutionφis unique and

φt= (ut,Tt,qt)∈L∞

(10)

Multiplying (.) byuand integrating over, we get

Pr

|∇u|dx+Pr

σu·u dx=

Pr(RT–Rq)˜ κu–utu

dx.

Using the Young inequality, we obtain

Pr

|∇u|dx+Prσ˜

udx≤ε

|u|dx+Cε–

|T|+|q|+|ut|

dx,

whereε>  is a real constant satisfyingPrσ˜ ≥ε. Then there exists a constantC>  such

that

|∇u|_dx_≤_C_T

L+q_L+ut_L

.

Thanks to (.), we have

u∈L∞(,T),H,R. (.)

We consider the Stokes equation

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

–u+∇p=g(x,t), g= –σu+ (RT–Rq)˜ κ–_Pr(u· ∇)u–ut, divu= ,

u= , x= , , u(,x) =u(π,x).

(.)

From (.), (.), and the Sobolev imbedding theorem, we ﬁnd thatg(x,t)∈Lq₍_,_R_),

∀ <q< . By the ADN theorem, Eq. (.) has a solution

u∈L∞(,T),W,q,R.

Then (u· ∇)u∈L₍_,_R_{) and}_g(x,_t)_∈_L₍_,_R_{). Using the ADN theorem, we obtain}

⎧ ⎨ ⎩

u∈L∞((,T),H(,R)),

p∈L∞((,T),H₍_)). (.)

Multiplying (.) byT and integrating over, we get

|∇T|dx=

(uT+QT–TtT)dx

≤ε

|T|dx+Cε–

|u|+|Q|+|Tt|

dx,

whereε>  is a constant. Then there exists a constantC>  such that

|∇T|dx≤Cu_L+Q_L+Tt_L

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Using (.), we have

T∈L∞(,T),H. (.)

We consider the elliptic equation

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–T=f, f=u– (u· ∇)T+Q–Tt,

T= , x= , , T(,x) =T(π,x).

(.)

It follows from (.), (.), and the Sobolev imbedding theorem thatf(x,t)∈Lq(),∀ < q< . Using the theory of linear elliptic equations, Eq. (.) has a solution

T∈L∞(,T),W,q().

Then (u· ∇)T∈L₍_,_R_{) and}_f(x,_t)_∈_L₍_,_R_{). Using the theory of linear elliptic} equa-tions, we have that

T∈L∞(,T),H(). (.)

Multiplying (.) byqand integrating over, we get

|∇q|dx=

(uq+Gq–qt)dx

≤ε

|q|dx+Cε–

|u|+|G|+|qt|

dx,

whereε>  is a constant. Then there exists a constantC>  such that

|∇q|dx≤Cu_L+Q_L+qt_L

.

Using (.), we have

q∈L∞(,T),H. (.)

We consider the elliptic equation

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

–q=f, f=u– (u· ∇)q+G–qt,

q= , x= , , q(π,x) =q(,x).

(.)

Using the arguments similar to those for (.), we get

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It follows from (.), (.), and (.) that

φ= (u,T,q)∈L∞(,T),H,R∩H,

p∈L∞(,T),H(). Theorem . Ifφ= (u,T,q)∈Hk+(,R)∩H,Q,G∈Hk(),then Eqs. (.)-(.) have a higher regular solution(φ,p) = (u,T,q,p)and

φ= (u,T,q)∈L∞(,T),Hk+,R∩H∩C[,T],Hk+,R∩H, φt= (u,T,q)t∈L∞

(,T),Hk,R∩H

∩C(,T],Hk,R∩H

,

p∈L∞(,T),Hk+()∩C(,T),Hk+() for∀ <T<∞.

Proof We prove the theorem using mathematical induction.

If k= , φ = (u,T,q)∈ H(,R)∩H, Q,G∈ H(), then φ = (u,T,q) ∈ H₍_,_R₎_∩_H,_Q,_G_∈_L₍_{). Using Theorem ., we ﬁnd that}_φ_{= (u,}_T_,_q)_∈_H₍_,_R_).

Thanks to the Sobolev imbedding theorem,H₍₎_→_W,₍₎_→_C₍_{) if}_n_{= . We} obtain

_∇(u· ∇)udx

≤C

|∇u|+|u||u|dx

≤C

u_W,+sup

|

u|u_H ≤Cu_H.

Then (u· ∇)u∈H₍_{) and}_F

(φ)∈H(). We have from the formula (.)

ut=L(t)u+

t



L(t–τ)F()dτ+F().

Then there existsαsatisfying  <α<  such that

utH–α =

L(t)u+

t



L(t–τ)F()dτ+F()

H–α

≤L(t)u_H–α+

t



L(t–τ)F()dτ

H–α

+F()_H–α

≤C+

t



L

–α



 (t–τ)F()Ldτ

≤C+

t



L– α



 (t–τ)L

 

F()Ldτ ≤C.

Thus,ut∈H–α(). Theng∈H–α() in Eq. (.). By the ADN theory,u∈H–α(). Thus,

(13)

We have

utH =

L(t)u+

t



L(t–τ)F()dτ+F()

H

≤L(t)u_H+

_tL(t–τ)F()dτ

H

+F()_H

≤C+

t



L

 

(t–τ)F()Ldτ

≤C+

t



_L+α



 (t–τ)L –α



 F()Ldτ≤C,

which implies ut ∈H(). Then g ∈H() in Eq. (.). Using the ADN theory, u∈

H₍_{) and} _p_∈_H₍_{). Thus,}_u_∈_L∞_((,_T),_H₍_,_R_{)) and}_p_∈_L∞_((,_T),_H₍_{)). Then} u∈C([,T],H₍_,_R₎_∩_H

) andut∈C((,T],H(,R)) from the formula (.).

Similarly,

_∇(u· ∇)Tdx≤C

|∇u|+|T||u|dx

≤C

u_W,+sup

|u|T_H

≤Cu_H+u_HT_H

.

Then (u· ∇)T∈H₍_{) and}_F(_φ_,_p)_∈_H₍_). We have from the formula (.)

Tt=L(t)u+ t



L(t–τ)F()dτ+F().

TtH–α =

L(t)T+

t



L(t–τ)F()dτ+F()

H–α

≤L(t)T_H–α+

_tL(t–τ)F()dτ

H–α

+F()_H–α

≤C+

t



L

–α



 (t–τ)F()Ldτ

≤C+

t



L– α



 (t–τ)L

 

F()Ldτ ≤C,

which impliesTt∈H–α(). Thenf∈H–α() in Eq. (.). It follows from the linear

el-liptic equationT∈H–α₍_{). Thus, (u}_{· ∇}_)T_∈_H–α₍_{) and}_F

(φ)∈H–α(). Then

TtH =

L(t)T+

t



L(t–τ)F()dτ+F()

H

≤L(t)T_H+

t



L(t–τ)F()dτ

H

(14)

≤C+

t



L

 

(t–τ)F()Ldτ

≤C+

t



_L+α



 (t–τ)L –α



 F()Ldτ ≤C,

which impliesTt∈H(). We obtain thatf∈H() in Eq. (.). ThenT∈H() from

the theory of linear elliptic equations. Thus,T∈L∞((,T),H₍_,_R_{)). From the formula} (.),T∈C([,T],H₍_{)) and}_T

t∈C((,T],H()).

Similarly,

_∇_(u_{· ∇}_)q dx≤C

|∇u|+|q||u|dx

≤C

u_W,+sup

|u|q_H

≤Cu_H+u_Hq_H

.

Then (u· ∇)q∈H₍_{) and}_F(_φ_,_p)_∈_H₍_). We have from the formula (.)

qt=L(t)q+ t



L(t–τ)F()dτ+F().

qtH–α =

L(t)q+

t



L(t–τ)F()dτ+F()

H–α

≤L(t)u_H–α+

_tL(t–τ)F()dτ

H–α

+F()_H–α

≤C+

t



L

–α



 (t–τ)F()Ldτ

≤C+

t



L– α



 (t–τ)L

 

F()_Ldτ ≤C,

which impliesqt∈H–α(). Thenf∈H–α() in Eq. (.). Thus,q∈H–α() from the

theory of linear elliptic equations. Then (u· ∇)q∈H–α₍_{) and}_F(_φ₎_∈_H–α₍_). Thus,

qtH =

L(t)q+

t



L(t–τ)F()dτ+F()

H

≤L(t)q_H+

t



L(t–τ)F()dτ

H

+F()_H

≤C+

t



L

 

(t–τ)F()_Ldτ

≤C+

t



L

+α



 (t–τ)L –α_

(15)

which impliesqt∈H(). We seef∈H() in Eq. (.). Thenq∈H() from the theory

of linear elliptic equations. Thus,q∈L∞((,T),H₍_{)). We have}_q_∈_C([,_T_],_H₍_{)) and} qt∈C((,T],H()) from the formula (.).

It follows from Eq. (.) that

∇p=u–σu+ (RT–Rq)˜ κ– 

Pr(u· ∇)u–ut.

Clearly, the right-hand side of the above equality is continuous inH₍_{). Thus,}

p∈C(,T),H().

Ifk=m,φ= (u,T,q)∈Hm+(,R)∩H, andQ,G∈Hm(), thenφ= (u,T,q)∈ Hm+(,R)∩H andQ,G∈Hm–(). From the hypothesis of mathematical induction, we seeφ= (u,T,q)∈Hm+(,R).

By the Sobolev imbedding theorem, we haveHm+()→Wm,()→Cm–() ifn= . Then it follows from the Sobolev imbedding theorem and the interpolation inequality that

_∇m_(u_{· ∇}_)u_dx_≤_C

! _m "

i=

_∇i_u_∇m+–i_u #

dx

≤C

! _m "

i=

_∇i_u_∇m+–i_u₊_|_u_|_∇m+_u

#

dx

≤C

! _m "

i=

_∇i_u_dx 



_∇m+–i_u_dx 



+sup

|u|

_∇m+_u dx

#

≤C

u_W,u_Wm,+sup

|

u|u_Hm+ ,

≤Cu_Hm+.

Then (u· ∇)u∈Hm₍_{) and}_F(_φ₎_∈_Hm₍_).

We have from the formula (.)

ut=L(t)u+

t



L(t–τ)F()dτ+F().

utHm–α =L(t)u+

t



L(t–τ)F()dτ+F()

Hm–α

≤L(t)u_Hm–α+

t



L(t–τ)F()dτ

Hm–α

+F()_Hm–α

≤C+

t



L

+m–α



 (t–τ)F()_Ldτ

≤C+

t



L– α



 (t–τ)L

m



(16)

Thenut∈Hm–α(). We see thatg∈Hm–α() in Eq. (.). Thus,u∈Hm+–α() from the

ADN theory. Hence, (u· ∇)u∈Hm+–α() andF(φ)∈Hm+–α(). Then

utHm =

L(t)u+

t



L(t–τ)F()dτ+F()

Hm

≤L(t)u_Hm+

_tL(t–τ)F()dτ

Hm

+F()_Hm

≤C+

t



L+

m



 (t–τ)F()Ldτ

≤C+

t



L

+α



 (t–τ)L

m+–α



 F()Ldτ≤C,

which implies ut ∈Hm(). Then g∈Hm() in Eq. (.). Using the ADN theory, u∈

Hm+₍_{), and}_p_∈_Hm+₍_{), we get}

u∈L∞(,T),Hm+,R, p∈L∞(,T),Hm+().

From the formula (.), we have

u∈C[,T],Hm+,R, ut∈C

(,T],Hm,R.

Similarly,

_∇m_(u_{· ∇}_)T_dx

≤C

! _m "

i=

_∇i_u_∇m+–i_T

#

dx

≤C

! _m "

i=

_∇i_u

∇m+–i_T

+|u|∇m+T

#

dx

≤C

! _m "

i=

_∇i_u dx

 

_∇m+–i_T dx

 

+sup

|

u|

_∇m+_T dx

#

≤C

_m

"

i=

u_Wi,T_Wm+–i,+sup

|u|T_Hm+

,

≤Cu_Hm+T_Hm+.

Then (u· ∇)T∈Hm₍_{) and}_F

(φ)∈Hm(). We have from the formula (.)

Tt=L(t)T+ t



(17)

TtHm–α =L(t)T+

t



L(t–τ)F()dτ+F()

Hm–α

≤L(t)T_Hm–α+

t



L(t–τ)F()dτ

Hm–α

+F()_Hm–α

≤C+

t



L

+m–α



 (t–τ)F()_Ldτ

≤C+

t



L– α



 (t–τ)L

m



 F()Ldτ≤C,

which impliesTt ∈Hm–α(). Then f∈Hm–α() in Eq. (.). It follows from the linear

elliptic equationT∈Hm+–α₍_{) that (u}_{· ∇}_)T _∈_Hm+–α₍_{) and}_F(_φ₎_∈_Hm+–α₍_{). We} obtain

TtHm =L(t)T+

t



L(t–τ)F()dτ+F()

Hm

≤L(t)u_Hm+

_tL(t–τ)F()dτ

Hm

+F()_Hm

≤C+

t



L+

m



 (t–τ)F()Ldτ

≤C+

t



L

+α



 (t–τ)L

m+–α



 F()_Ldτ≤C.

ThenTt∈Hm(). We havef∈Hm() in Eq. (.). ThenT∈Hm+() from the theory of

linear elliptic equations. Thus,

T∈L∞(,T),Hm+(). From the formula (.), we induce

T∈C[,T],Hm+(), ut∈C

(,T],Hm().

Similarly,

_∇m_(u_{· ∇}_)q dx

≤C

! _m "

i=

_∇i_u_∇m+–i_q #

dx

≤C

! _m "

i=

_∇i_u_∇m+–i_q₊_|_u_|_∇m+_q

#

dx

≤C

! _m "

i=

_∇i_u_dx 



_∇m+–i_q_dx 



+sup

|u|

_∇m+_q dx

(18)

≤C

_m

"

i=

u_Wi,q_Wm+–i,+sup

|

u|q_Hm+

,

≤Cu_Hm+q_Hm+.

Then (u· ∇)q∈Hm() andF(φ)∈Hm(). We have from the formula (.)

qt=L(t)q+

t



L(t–τ)F()dτ+F().

qtHm–α =

L(t)q+

t



L(t–τ)F()dτ+F()

Hm–α

≤L(t)u_Hm–α+

_tL(t–τ)F()dτ

Hm–α

+F()_Hm–α

≤C+

t



L

+m–α



 (t–τ)F()Ldτ

≤C+

t



_L–α



 (t–τ)L

m



 F()Ldτ≤C,

which impliesqt∈Hm–α(). Thenf∈Hm–α() in Eq. (.). Thus,q∈Hm+–α() from

the theory of linear elliptic equations. Then (u· ∇)q∈Hm+–α₍_{) and}_F(_φ₎_∈_Hm+–α₍_). Thus,

qtHm =L(t)q+

t



L(t–τ)F()dτ+F()

Hm

≤L(t)q_Hm+

t



L(t–τ)F()dτ

Hm

+F()_Hm

≤C+

t



L+

m



 (t–τ)F()_Ldτ

≤C+

t



L

+α



 (t–τ)L

m+–α



 F()Ldτ≤C,

which impliesqt∈Hm(). We ﬁndf∈Hm() in Eq. (.). Thenq∈Hm+() from the

theory of linear elliptic equations. We have

q∈L∞(,T),Hm+().

From the formula (.), we see

q∈C[,T],Hm+(), qt∈C

(,T],Hm().

It follows from Eq. (.) that

∇p=u–σu+ (RT–Rq)˜ κ– 

(19)

Clearly, the right-hand side of the above equality is continuous inHm₍_{). Then}

p∈C(,T),Hm+().

The proof is completed.

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

Remark . Ifφ= (u,T,q)∈Hk+(,R)∩H, Q,G∈Hk(), then Eqs. (.)-(.) have a higher regular solution (φ,p) = (u,T,q,p), and

φ= (u,T,q)∈Cl(,T),Hr,R∩H,

p∈Cα(,T),Hβ()

for∀ <T<∞, wherel,r,α,βare positive integers satisfying l+r=k+  and α+β= k+ .

Competing interests

The author declares that he has no competing interests.

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).

Received: 1 June 2012 Accepted: 14 November 2012 Published: 5 December 2012

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Cite this article as:Luo:Regularity of global solution to atmospheric circulation equations with humidity effect.