Journal of Differential Equations

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1 J. Differential Equations Contents lists available at ScienceDirect Journal of Differential Equations Existence of the universal attractor for the 3-D viscous rimitive equations of large-scale moist atmoshere Boling Guo a,daiwenhuang a,b, a Institute of Alied Physics and Comutational Mathematics, Beijing 88, China b Deartment of Mathematical Sciences, Tsinghua University, Beijing 8, China article info abstract Article history: Received 5 January 8 Available online 7 May MSC: 35Q3 35Q35 86A Keywords: Primitive equations Navier Stokes equations Global well-osedness Universal attractors In this aer, we consider the initial boundary value roblem for the three-dimensional viscous rimitive equations of largescale moist atmoshere which are used to describe the turbulent behavior of long-term weather rediction and climate changes. By obtaining the existence and uniqueness of global strong solutions for the roblem and studying the long-time behavior of strong solutions, we rove the existence of the universal attractor for the dynamical system generated by the rimitive equations of largescale moist atmoshere. Elsevier Inc. All rights reserved.. Introduction In order to understand the mechanism of long-term weather rediction and climate changes, one can study the mathematical equations and models governing the motion of the atmoshere as the atmoshere is a secific comressible fluid see, e.g., [9,9]. V. Bjerkness, one of the ioneers of meteorology, said that the weather forecasting can be considered as an initial boundary value roblem in mathematical hysics. In 9, Richardson initially introduced the so-called rimitive atmosheric equations which consisted of the hydrodynamic, thermodynamic equations with Coriolis force, cf. [3]. At that time, the rimitive atmosheric equations were too comlicated to be studied theoretically or to be solved numerically. To overcome this difficulty, some simle numerical models were introduced, such as the barotroic model formulated by Neumann etc. in [5] and the quasi-geostrohic model * Corresonding author at: Institute of Alied Physics and Comutational Mathematics, Beijing 88, China. address: hdw55@tom.com D. Huang. -396/$ see front matter Elsevier Inc. All rights reserved. doi:.6/j.jde..5.

2 58 B. Guo, D. Huang / J. Differential Equations introduced by Charney and Philis in [8]. The -D and 3-D quasi-geostrohic models have been the subject of analytical mathematical study, cf., e.g., [,3,6,7,3,,8,37 39] and references therein. Due to the considerable imrovement in comuter caacity and the develoment of atmosheric science, some mathematicians began to consider the rimitive equations of atmoshere in the ast two decades see, e.g., [5 7,3] and references therein. In [5], by introducing viscosity terms and using some technical treatment, Lions, Temam and Wang obtained a new formulation of the rimitive equations of large-scale dry atmoshere which was amenable to mathematical study. In a - coordinate system, the new formulation of the rimitive equations is a little similar to Navier Stokes equations of incomressible fluid. By the methods used to solve Navier Stokes system in [3], they obtained the existence of global weak solutions of the initial boundary value roblem for the new formulation of the rimitive equations. Moreover, under the assumtions that there exists a unique global strong solution for the roblem with vertical viscosity and that H -normofthestrongsolution is bounded uniformly in t, they established some hysically relevant estimates for the Hausdorff and fractal dimensions of the attractor of the rimitive equations with vertical viscosity. Without those assumtions, by the Trajectory Attractors Theory due to Vishik and Cheyzhov cf. [,35], we obtained the existence of trajectory attractors for the large-scale moist atmosheric rimitive equations in [6]. By taking advantage of the geostrohic balance and other geohysical consideration, several intermediate models have been the subject of studying the long-time dynamics and universal attractors in order to describe the mechanism of long-term weather rediction and climate dynamics see, e.g., [9,,,36,37] and references therein. In recent years, there were some mathematicians who considered the existence of strong solutions for the three-dimensional viscous rimitive equations of large-scale atmoshere and ocean see, e.g., [,9,8,,33,3] and references therein. In [8], Guillén-González et al. obtained the global existence of strong solutions to the rimitive equations of large-scale ocean by assuming that the initial data are small enough, and also roved the local existence of strong solutions to the equation for all initial data. In [3], Temam and Ziane considered the local existence of strong solutions for the rimitive equations of the atmoshere, the ocean and the couled atmoshere ocean. The aers [9 ] are devoted to considering the non-dimensional Boussinesq equations or modified models see, e.g., [9,33]. In [9], Cao and Titi considered global well-osedness and finite-dimensional global attractor to a 3-D lanetary geostrohic model. The aer [] is devoted to studying the global wellosedness for the three-dimensional viscous rimitive equations of large-scale ocean. In [], Cao and Titi develoed a beautiful aroach, by which they obtained the fact that L 6 -norm of the fluctuation ṽ of horizontal velocity is bounded uniformly in t. The estimate about L 6 -norm of the fluctuation ṽ is a key roof in []. On the basis of the results of [], we obtain the existence of weak universal attractors for the 3-D viscous rimitive equations of the large-scale ocean in [7]. In the resent aer we are interested in considering the existence of the universal attractor for the dynamical system generated by the rimitive equations of large-scale moist atmoshere. In order to do that, we must study the existence, uniqueness and long-time behavior of global strong solutions to the initial boundary value roblem of the new formulation of large-scale moist atmosheric rimitive equations the roblem is denoted by IBVP and will be given in Section. Our results are Proosition 3., Proosition 3., Proosition 3.3 and Theorem 3., where Theorem 3. is main. First, we obtain the global well-osedness of the roblem IBVP. Second, by studying the long-time behavior of the strong solution, we rove that H -norm of the strong solution is bounded uniformly in t, and also rove that the corresonding semigrou {St} t ossesses a bounded absorbing set B ρ in V the definition of the sace V will be given in Section., by which we construct a weak universal attractor A. Here, the result about the universal attractors in this aer is more stronger than that in [6]. Since the global well-osedness of the 3-D incomressible Navier Stokes system is still oen, by Proosition 3., Proosition 3. Proosition 3.3 and Theorem 3., we rove rigorously in mathematics that the new formulation of large-scale moist atmosheric rimitive equations is simler than the incomressible Navier Stokes system, which is consistent with the hysical oint of view. Insired by the methods used in [], we rove the global well-osedness of IBVP. However, there are three differences between our aer and []. First, our main aim is to study the long-time dynamics and the existence of the weak universal attractors for the large-scale moist atmosheric rimitive equations, while the authors of [] studied the global well-osedness of their considered

3 B. Guo, D. Huang / J. Differential Equations model and did not consider the long-time behavior of strong solutions. By acquiring the uniform, in t, boundedness of L 3 - and L -norm of the temerature T and the fluctuation ṽ of horizontal velocity v, which are most imortant in the roof of our main results see Remark 5.5 for more discussion about the technique of estimates, we obtain the existence of the universal attractor for our considered dynamical system. Second, our ways to rove the global well-osedness are a little different from those in []. We rove the global well-osedness for our considered model by acquiring several a riori estimates about L -norm of T and ṽ. Third, the new formulation of the large-scale moist atmosheric equations is more comlicated than the model studied in []. If we let a =, the model considered in this aer is similar to that in []. The aer is organized as follows. In Section, we ose the rimitive equations of large-scale moist atmoshere. Main results of this aer are formulated in Section 3. In Section, we give our working saces and some reliminaries. In Section 5, we make a riori estimates about the local strong solution and rove that H -norm of local strong solution is bounded uniformly in t, which is the kernel of the roof of our results. We rove main results of our aer in Sections 6, 7.. The three-dimensional viscous rimitive equations of large-scale moist atmoshere The three-dimensional viscous rimitive equations of large-scale moist atmoshere in the ressure coordinate system for details, we refer the reader to [6,,5,6] and references therein is written as v t v v ω v f k v grad Φ v v =, R Re Re. div v ω =,. Φ aqt =,.3 T t v T ω T aqω T T Rt Rt = Q,. q t vq ω q q q Rq Rq = Q,.5 where the unknown functions v, ω, Φ, q, T are: v = v θ, v ϕ the horizontal velocity, ω vertical velocity in -coordinate system, Φ the geootential, q the mixing ratio of water vaor in the air, T temerature, f = cos θ Coriolis arameter, R the Rossby number, k vertical unit vector, Re, Re, Rt, Rt, Rq, Rq Reynolds numbers, P an aroximate value of ressure at the surface of the earth, ressure of the uer atmoshere and >, the variable satisfying = P < P, Q, Q are given functions on S, here we don t consider the condensation of water vaor, a a ositive constant a.68, b a ositive constant. The definitions of v v, v, T, q, v q, v T,divv, gradφ will be given in Section.. Eqs...5 are called the 3-D viscous rimitive equations of the large-scale moist atmoshere. The sace domain of Eqs...5 is = S,, where S is two-dimensional unit shere. The boundary value conditions are given by = = P: v =, ω =, T = α st s T, q = β sq s q,.6

4 6 B. Guo, D. Huang / J. Differential Equations = = : v =, ω =, T =, q =,.7 where α s, β s are ositive constants, T s the given temerature on the surface of the earth, q s the given mixing ratio of water vaor on the surface of the earth. For simlicity and without loss of generality we assume that T s = and q s =. If T s and q s, one can homogenize the boundary value conditions for T, q cf., e.g., [6]. Integrating. and using the boundary conditions.6,.7, we have ωt; θ,ϕ,= W vt; θ,ϕ,= div v t; θ,ϕ, d,.8 div vd =..9 Suose that Φ s obtain is a certain unknown function at the isobaric surface =. Integrating.3, we Φt; θ,ϕ,= Φ s t; θ,ϕ aqt d.. In this article, we assume that the constants Re, Re, Rt, Rt, Rq, Rq are all equal to, which cannot change our results. Then Eqs...5 can be written as v t v v W v v f k v grad Φ s R grad[ ] aqt d v v =,. T t v T W v T aqw v T T = Q,. q t vq W v q q q = Q,.3 div vd =,. where the definitions of grad[ aqt ], gradφ s conditions of Eqs... are given by will be given in Section.. The boundary value = : = : v =, v =, T = α st, T =, q = β sq,.5 q = ;.6

5 B. Guo, D. Huang / J. Differential Equations and the initial value conditions can be given as U t= = v t=, T t=, q t= = U = v, T, q..7 We call..7 the initial boundary value roblem of the new formulation of the 3-D viscous rimitive equations of large-scale moist atmoshere, which is denoted by IBVP. Now we define the fluctuation ṽ of horizontal velocity and find the equations satisfied by ṽ and v. By integrating the momentum equation. with resect to from to and using the boundary value conditions.5 and.6, we get v t v v W v v d f k v grad Φ s R grad[ aqt ] d d v = in S,.8 where v = vd. Denote the fluctuation of the horizontal velocity by ṽ = v v. We notice that ṽ = ṽd =, v =..9 By integration by arts and.9, we have W v v d = v vd = v div vd = ṽ div ṽd,. ṽ ṽd v v.. From.8,. and., we obtain v t v v ṽ div ṽ ṽ ṽ f R k v grad Φ s grad[ aqt ] d d v = in S.. Subtracting. from., we know that the fluctuation ṽ satisfies the following equation and boundary value conditions

6 6 B. Guo, D. Huang / J. Differential Equations ṽ t ṽ ṽ div ṽd ṽ ṽ v v ṽ ṽ div ṽ ṽ ṽ f k ṽ R grad[ aqt ] d grad[ aqt ] d d ṽ ṽ = in,.3 3. Statements of main results = : = : ṽ =,. ṽ =..5 Now we formulate our main results in the resent aer. Proosition 3. Existence of global strong solutions for IBVP. Let Q,Q H, U = v, T, q V. Then for any T > given, there exists a strong solution U of the system..7 on the interval [, T ], where the definition of the sace V will be given in Section., and the definition of strong solutions to the system..7 willbegiveninsection5.. Proosition 3. Uniqueness of global strong solutions for IBVP. Let Q, Q H, U = v, T, q V. Then for any T > given, the strong solution U of the system..7 on the interval [, T ] is unique. Moreover, the strong solution U is deendent continuously on the initial data. Proosition 3.3 Existence of bounded absorbing sets for the dynamical system..6. If Q, Q H, U = v, T, q V, then the global strong solution U of the system..7 satisfies U L, ; V and Ut C U, Q, Q, where C is a ositive constant deendent on U, Q, Q and t. Moreover, the corresonding semigrou {St} t ossesses a bounded absorbing set B ρ in V, i.e., for every bounded set B V, there exists t B> big enough such that StB B ρ, for any t t, where B ρ ={U; U ρ} and ρ is a ositive constant deendent on Q, Q. Theorem 3. Existence of the universal attractor for the system..6. The system..6 ossesses a weak universal attractor A = s t s StB ρ that catures all the trajectories, where the closures are taken with resect to V -weak toology. The weak universal attractor A has the following roerties: i weak comact A is bounded and weakly closed in V ; ii invariant for every t, StA = A;

7 B. Guo, D. Huang / J. Differential Equations ii attracting for every bounded set B in V, the sets StB converge to A with resect to V -weak toology as t, i.e., lim t dw V StB, A =, where the distance d w V is induced by the V -weak toology. Remark 3.5. The weak universal attractor A has the following additional roerties: i By the Rellich Kondrachov Comact Embedding Theorem cf., e.g., [], we know that for any < 6thesetsStB converge to A with resect to the L L L L - norm. ii The weak universal attractor A is unique and is connected with resect to V -weak toology. Remark 3.6. In comarison to the 3-D incomressible Navier Stokes equations, the 3-D viscous rimitive equations of large-scale moist atmoshere have not the time derivative term of the vertical velocity ω = W v. Therefore, we cannot rove that the bounded absorbing set B ρ in V is bounded in H H H H as in the case of 3-D incomressible Navier Stokes equations for the Navier Stokes equations, if there exists a bounded absorbing set B ρ in H H H, then one can rove that B ρ is bounded in H H H, i.e., we cannot rove that the universal attractor A is comact in V.. Preliminaries.. Some function saces Let e θ, e ϕ, e be the unit vectors in θ, ϕ and directions of the sace domain resectively, e θ = θ, e ϕ = sin θ ϕ, e =. The inner roduct and norm on T θ,ϕ, the tangent sace of at the oint θ, ϕ, are given by for X, Y = X Y = X Y X Y X 3 Y 3, X =X, X X = X e θ X e ϕ X 3 e, Y = Y e θ Y e ϕ Y 3 e T θ,ϕ,. L := {h; h : R, h < } with the norm h = h, <. d and S ds are denoted by and S resectively. L T TS is the first two comonents of L vector fields on with the norm v = v θ v ϕ, where v = vθ, v ϕ : TS. C S is the function sace for all smooth functions from S to R. C is the function sace for all smooth functions from to R. C T TS is the first two comonents of smooth vector fields on. H m is the Sobolev sace of functions which are in L, together with all their covariant derivatives with resect to e θ, e ϕ, e of order m, with the norm [ h m = k m i j =,,3; j=,...,k i ik h h ],

8 6 B. Guo, D. Huang / J. Differential Equations where = eθ, = e ϕ, 3 = the definitions of e θ, e ϕ will be given later. Hm T TS = {v; v = v θ, v ϕ : TS, v m m < } is the norm which is similar to that of Hm, thatis,in the above formula of norm, we let h = v θ, v ϕ = v θ e θ v ϕ e ϕ. The horizontal divergence div, the horizontal gradient =grad, the horizontal covariant derivative v and the horizontal Lalace Beltrami oerator for scalar and vector functions are defined by div v = divv θ e θ v ϕ e ϕ = vθ sin θ v ϕ, sin θ θ ϕ. T = grad T = T θ e θ T sin θ ϕ e ϕ,. grad Φ s = Φ s θ e θ sin θ ṽ θ v ṽ = v θ θ v ϕ ṽ θ sin θ ϕ v ϕ ṽ ϕ cot θ e θ Φ s ϕ e ϕ,.3 v θ ṽ ϕ θ v ϕ sin θ ṽ ϕ ϕ v ϕ ṽ θ cot θ e ϕ,. T v T = v θ θ v ϕ T sin θ ϕ,.5 q v q = v θ θ v ϕ q sin θ T = divgrad T = [ sin θ T q = divgrad q = [ sin θ v = v θ cosθ v ϕ sin θ ϕ ϕ,.6 ] T, sin θ θ θ sin θ ϕ.7 sin θ q ] q, θ θ sin θ ϕ.8 v θ sin e θ v ϕ cosθ v θ θ sin θ ϕ v ϕ sin e ϕ, θ.9 where v = v θ e θ v ϕ e ϕ, ṽ = ṽ θ e θ ṽ ϕ e ϕ C T TS, T, q C, Φ s C S. Now we can define our working saces for the roblem IBVP. Let { Ṽ := v; v C T TS, v =, v = =, = { Ṽ := T ; T C, T =, T } = = α s T, = { Ṽ 3 := q; q C, q =, q } = = β s q, = V = the closure of Ṽ with resect to the norm, V = the closure of Ṽ with resect to the norm, V 3 = the closure of Ṽ 3 with resect to the norm, H = the closure of Ṽ with resect to the norm, H = L, } div vd =,

9 B. Guo, D. Huang / J. Differential Equations V = V V V 3, H = H H H. The inner roducts and norms on V, V, V 3 are given by v, v V = eθ v eθ v e ϕ v eϕ v v v v v, v =v, v V, v, v V, T, T V = grad T grad T T T =T, T V, T, T V, q, q V 3 = grad q grad q q T TT q qq q =q, q V 3, q, q V, U, U H = v, v T, T q, q, U, U V = v, v V T, T V q, q V 3, U =U, U V, U = U, U H, U = v, T, q, U = v, T, q V, where, denotes the L inner roducts in H, H... Some lemmas Lemma.. Let u = u θ, u ϕ,u = u θ,u ϕ C T TS,and C S.Then,, in articular, u u = S S div u = u,. S v = for any v V,. eθ u eθ u e ϕ u eϕ u u u.. Proof. We can rove. by using.,. and the Stokes Theorem cf., e.g., [3,36].. is the direct result of. and.. From. and.9, by direct comutation, we can obtain the second art.

10 66 B. Guo, D. Huang / J. Differential Equations Lemma.. For any h C S,v C T TS,wehave S S S v h h div v = divhv =. Proof. From.,.5,., by direct comutation, we can rove Lemma.. Lemma.3. Let v, v V,T V,q V 3.Thenwehave [ v v div vd v ]v =, [ v g div vd g ]g = for g = Torg= q, 3 { grad[ aqt ] d v aqw v T }=. For the details of the roof for Lemma.3, we refer the reader to [6, Lemma 3.]. For convenience, we recall some interolation inequalities for the details of the roof, we refer the reader to see, e.g., [,5,3]. i For u H S for the definitions of H S, L S, cf. [], where L = L S for <, H = u H S. ii For u H, u L c u L u H,.3 u L 6 c u 3 L u 3 H,. u L 8 c u L u H,.5 5. A riori estimates about local strong solutions u c u u 3..6 At first, we recall the local, in time, existence of strong solutions of the 3-D viscous rimitive equations of the large-scale moist atmoshere. Definition 5.. Let U = v, T, q V, and let T be a fixed ositive time. U = v, T, q is called a strong solution of the system..7 on the time interval [, T ] if it satisfies..3 in weak sense such that v C [, T ]; V L, T ; H, T C [, T ]; V L, T ; H, q C [, T ]; V 3 L, T ; H, v t L, T ; L, T t, q t L, T ; L.

11 B. Guo, D. Huang / J. Differential Equations Remark 5.. Since the 3-D viscous rimitive equations of large-scale moist atmoshere have not the time derivative term of the vertical velocity ω = W v, we cannot rove v L, T ; L, t T t, q t L, T ; L. Proosition 5.3. Let Q, Q H, U = v, T, q V. Then there exists T >, T = T U,and there exists a strong solution U of the system..7 on the interval [, T ]. Proof. The roof of Proosition 5.3 is similar to that for the rimitive equations of large-scale ocean given in the aers [8,3]. So we omit the details of the roof here. Remark 5.. In order to rove the global, in time, existence of strong solutions to the system..7 and study the long-time behavior of strong solutions, we should make a riori estimates about H -norm of the local solution Ut obtained in Proosition 5.3. In this section, we should show that H -norm of the strong solution U t is bounded uniformly in t. Remark 5.5. In order to study the long-time behavior of strong solutions, we must make three key estimates. First, we must make estimates about L 3 -normofthetemeraturet and the fluctuation ṽ of horizontal velocity v before we study the long-time behavior of strong solutions by the Uniform Gronwall Lemma, without which we only obtain the global well-osedness of IBVP. Second, on the basis of L 3 estimates of ṽ, T, we ought to make estimates about L -norm of ṽ, T and the mixing ratio of water vaor in the air q. If we only made estimates about L 6 -norm of ṽ, q, T as that in [], we could only rove the global well-osedness for the roblem IBVP, but we could not study long-time behavior of strong solutions and could not obtain a stronger result than the uniqueness of strong solutions to IBVP. Third, since the moist atmosheric equations are more comlicated than the oceanic rimitive equations, we have to make estimates about T, q before we rove that H -norm of v, T, q is bounded. 5.. L estimates of v, T, q Choosing v as a test function in Eq.., we obtain d v = eθ v e ϕ v v v v v W v v f k v grad Φ s R { v grad[ ] } aqt d v. 5. By Lemma., Lemma.3 and f R k v v =, we obtain from 5. d v = { v grad[ ] } aqt d v. 5. eθ v e ϕ v v

12 68 B. Guo, D. Huang / J. Differential Equations Taking the inner roduct of Eq.. with T in L, by Lemma.3, we obtain Similarly to 5.3, we have d T = T T α s T = aqtwv Q T. 5.3 d q q q β s q = = qq. 5. From 5. 5., by Lemma.3, we obtain eθ v e ϕ v v d v T q T T α s T = = Q T qq. q v q β s q = 5.5 By T θ, ϕ,= T d T =, qθ, ϕ,= q d q =, using the Hölder inequality and the Cauchy Schwarz inequality, we have T T T =, q q q =. 5.6 From , by the Young inequality with ε, we obtain d v T q c eθ v e ϕ v v v T T α s T = q q β s q = Q Q. 5.7 In this article, c will denote ositive constant and can be determined in concrete conditions. ε is a small enough ositive constant. By 5.6, 5.7 and thanks to the Gronwall inequality, we have v T q e c t v T q c Q Q E, 5.8

13 B. Guo, D. Huang / J. Differential Equations where c, E are ositive constants and t. By the Minkowski inequality and the Hölder inequality, for any t, we have vt L vt e c t v T q c Q Q E. 5.9 From , we get c tr t [ eθ v e ϕ v v T T T ] q q q T = q = Ut e c t v T q c Q Q r E, 5. where c, E are ositive constants, t, r > tr are given, and t eθ v e ϕ v eθ v e ϕ v, tr ds is denoted by.since t S from 5., we have c tr eθ v e ϕ v v L E, t. 5. t S 5.. L estimates of q By taking the inner roduct of Eq..3 with q q in L, weget d q 3 q q q 3 q β s q = S [ ] = Q q q v q div vd q q q. 5. By Lemma., we have [ ] v q div vd q q q = [ ] v q div vd d q S = v q q div v =. 5.3

14 7 B. Guo, D. Huang / J. Differential Equations Combining 5. with 5.3, we obtain d q 3 q q 3 q q β s q = S = Q q q. 5. Since q θ, ϕ,= q d q =, by using the Hölder inequality and the Cauchy Schwarz inequality, we get q c q q q q =. 5.5 From 5. and 5.5, by the Young inequality, we obtain d q c q c Q, 5.6 where c is a ositive constant. By the Gronwall inequality, we have qt e c t q c Q E, 5.7 where t, E is a ositive constant. From 5. and 5.7, we get c t tr q = E, for any t. 5.8 Before making L 3 and L estimates of T by anisotroic estimates, we need the following lemma. Lemma 5.6. Let v V,T V.Thenwehave div vd L θ,ϕ L div v, { T n c T T, if n =, L θ,ϕ L c T T, if n = 3, where div vd L θ,ϕ L = [ S div vd ] L, T n L θ,ϕ L = S T n L. Proof. By the Hölder inequality, we can get. By.3,. and the Hölder inequality, we can obtain.

15 B. Guo, D. Huang / J. Differential Equations L 3 estimates of T We take the inner roduct of Eq.. with T T in L and obtain 3 d T 3 3 = T T T T α s T = S [ ] Q T T v T div vd T T T div vd T T By the Hölder inequality and Lemma 5.6, we get div vd T T div vd L θ,ϕ L a T L θ,ϕ L 3 div vd q T T. 5.9 c e ϕ v e ϕ v c T T. 5. By the Hölder inequality, the Young inequality and u 6 L 3 have a q c c { S [ S c q T 3 3 c q 7 div vd T T [ q S 3 q T T 3 3 T L ] T T T 3 3 T 3 3 [ } d 5 6 H ] d div v 5 div v c u 3 8 L u 5 8 H,foranyu H S, we div vd L θ,ϕ L eθ v e ϕ v ] ε T T T

16 7 B. Guo, D. Huang / J. Differential Equations By the Young inequality, choosing ε small enough and using an inequality similar to 5.5, we derive from d T 3 3 T T T T α s T = S [ eθ v e ϕ v ] T 3 3 c Q 3 3 c q 7 c q 7 [ 3 eθ v e ϕ v ] c T T. 5. By the Uniform Gronwall Lemma [3,. 9], 5., 5.7 and T 3 3 c T 3 T 3,weobtain T t r 3 3 E 3, 5.3 where E 3 = E 3 U, Q, Q > and t. By the Gronwall inequality, we rove where C = C U, Q, Q >. 5.. L estimates of T T t 3 3 C, for any t < r, 5. We take the inner roduct of Eq.. with T T in L and obtain d T = 3 T T 3 T T α s T = S [ ] Q T T v T div vd T T T div vd T T a Similarly to 5., by the Hölder inequality and Lemma 5.6, we get div vd q T T. 5.5 div vd T T c e ϕ v e ϕ v c T T. 5.6 Similarly to 5., by the Hölder inequality, the Young inequality and u L 6 S c u 3 L S u 3 H S, for any u H S, we have

17 a q c q T B. Guo, D. Huang / J. Differential Equations div vd T T eθ v e ϕ v ε T T T. 5.7 By the Young inequality, 5.7, choosing ε small enough and using an inequality similar to 5.5, we derive from , d T 3 T T 3 T T α s T = S [ c T eθ v e ϕ v ] T c Q c v. 5.8 By the Uniform Gronwall Lemma, 5., 5.3 and T c T 3 T, we obtain T t r E, 5.9 where E = E U, Q, Q > and t. By the Gronwall inequality, from 5.8 we rove T t C, 5.3 where C = C U, Q, Q > and t < r. From 5.8 and 5.9, we get c t3r tr T = E E, for any t L 3 estimates of ṽ We take the inner roduct of Eq..3 with ṽ ṽ in L and obtain 3 d ṽ 3 3 = [ eθ ṽ eϕ ṽ ṽ eθ ṽ 3 ] e ϕ 9 9 ṽ 3 ṽ 3 ṽ ṽ ṽ 3 9 [ ] ṽ ṽ div ṽd ṽ { grad[ aqt ] d ṽ ṽ v ṽ ṽ ṽ ṽ ṽ ṽ v grad[ ] } aqt d d ṽ ṽ

18 7 B. Guo, D. Huang / J. Differential Equations ṽ div ṽ ṽ ṽ ṽ ṽ f k ṽ ṽ ṽ, 5.3 R where ṽ = ṽ. Similarly to 5.3, by Lemma. and integration by arts, we have [ ] ṽ ṽ div ṽd ṽ ṽ ṽ = By Lemma. and.9, we get v ṽ ṽ ṽ = 3 v ṽ 3 = 3 ṽ 3 div v =. 5.3 Using Lemma., we have = [ div ṽ ṽ v ] ṽ = = ṽ ṽ ṽ v ṽ ṽ v div ṽ [ ] ṽ ṽ ṽ v v ṽ ṽ ṽ ṽ ṽ v div ṽ. So ṽ ṽ ṽ v = v ṽ ṽ ṽ ṽ ṽ v div ṽ Using integration by arts, we obtain [ ṽ div ṽ ṽ ṽd ] ṽ ṽ = ṽ eθ ṽd eθ ṽ ṽ ṽ e ϕ ṽd e ϕ ṽ ṽ From 5.3 to 5.36, by f R k ṽ ṽ ṽ = and Lemma., we get 3 d ṽ 3 3 = [ eθ ṽ eϕ ṽ ṽ eθ ṽ 3 ] e ϕ 9 9 ṽ 3 ṽ 3 ṽ 3 ṽ ṽ 9 [ ] v ṽ ṽ ṽ ṽ ṽ v div ṽ

19 [ [ B. Guo, D. Huang / J. Differential Equations ṽ θ ṽd eθ ṽ ṽ ṽ ϕ ṽd aqt d ] e ϕ ṽ ṽ ] aqt d d div ṽ ṽ By the Hölder inequality, we derive from d ṽ 3 3 c [ eθ ṽ eϕ ṽ ṽ eθ ṽ 3 ] e ϕ 9 9 ṽ 3 ṽ 3 ṽ ṽ ṽ 3 9 v ṽ eθ ṽ e ϕ ṽ d S c [ ṽ d ṽ eθ ṽ e ϕ ṽ d ] S [ c T S [ c qt ṽ eθ ṽ e ϕ ṽ d ṽ eθ ṽ e ϕ ṽ d ] ] S c v L c [ [ c T L ṽ ṽ [ eθ ṽ e ϕ ṽ ] ṽ eθ ṽ e ϕ ṽ ] c qt L ṽ [ [ S S ṽ eθ ṽ e ϕ ṽ ] ṽ 3 d ṽ d ] 5 ṽ eθ ṽ e ϕ ṽ ] By the Minkowski inequality, the Hölder inequality and.3, we have

20 76 B. Guo, D. Huang / J. Differential Equations [ ṽ 3 d ] [ ] 3 ṽ d S S c ṽ 3 3 [ ṽ 3 ] ṽ 3 L L d By the Minkowski inequality, the Hölder inequality and u L 5 c u 3 5 u 5,foranyu H S,we L 3 H get 5 [ ṽ d ] ṽ d c ṽ ṽ S S By the Minkowski inequality,.3,.5 and the Hölder inequality, we get qt [ L = T [ L = S S qt d T d ] T, 5. ] c q q T T. 5. By the Young inequality,.3, 5.7, 5.9 we obtain from d ṽ 3 3 [ eθ ṽ eϕ ṽ ṽ eθ ṽ 3 ] e ϕ 9 9 ṽ 3 ṽ 3 ṽ ṽ ṽ 3 9 c v v ṽ ṽ 3 L H 3 c T c q ṽ c. 5.3 By the Uniform Gronwall Lemma, and ṽ 3 3 ṽ 3 ṽ 3,weobtain ṽt 3r 3 3 E 5, 5. where E 5 = E 5 U, Q, Q > and t L estimates of ṽ Taking the inner roduct of Eq..3 with ṽ ṽ in L, similarly to 5.37, we obtain d ṽ [ eθ ṽ eϕ ṽ ṽ eθ ṽ ] e ϕ ṽ ṽ

21 = B. Guo, D. Huang / J. Differential Equations ṽ ṽ ṽ [ v ṽ ṽ ṽ ṽ ṽ v ] div ṽ [ [ ṽ θ ṽd ṽ ṽ eθ aqt d ṽ ϕ ṽd aqt d d ] e ϕ ṽ ṽ ] div ṽ ṽ. 5.5 Similarly to 5.38, by the Hölder inequality, we derive from 5.5 d ṽ c v L c [ eθ ṽ eϕ ṽ ṽ eθ ṽ ] e ϕ ṽ ṽ ṽ ṽ ṽ [ [ c T L c qt L [ ṽ eθ ṽ e ϕ ṽ ] [ ṽ eθ ṽ e ϕ ṽ ] [ [ S S [ ṽ eθ ṽ e ϕ ṽ ] ṽ d ṽ d S [ ṽ eθ ṽ e ϕ ṽ ] S By the Minkowski inequality, the Hölder inequality and.3, we have 3 ] ] ṽ d ṽ d ] ]. 5.6 [ ṽ d ] [ ] ṽ d S S c ṽ ṽ L ṽ L d. 5.7 By the Minkowski inequality, the Hölder inequality and., we get

22 78 B. Guo, D. Huang / J. Differential Equations S ṽ d 3 [ S ] ṽ d c ṽ ṽ. 5.8 By the Young inequality,.3, 5.7, 5.9, 5.3, 5. and 5., we obtain from d ṽ [ eθ ṽ eϕ ṽ ṽ eθ ṽ ] e ϕ ṽ ṽ ṽ ṽ ṽ c v L v H ṽ q ṽ c T c. 5.9 By the Uniform Gronwall Lemma, , 5., 5.9 and ṽ ṽ 3 ṽ, we obtain ṽt r E 6, 5.5 where E 6 = E 6 U, Q, Q > and t. From 5.9 and 5.5, we have t5r tr { [ eθ ṽ eϕ ṽ ṽ eθ ṽ ] e ϕ ṽ ṽ ṽ ṽ ṽ } E 6 E 6 = E By the Gronwall inequality, from 5.9 we obtain ṽt C, 5.5 where C = C U, Q, Q > and t < r H estimates of v Taking the inner roduct of Eq.. with v in L,weget d v H v L [ ] = v v ṽ div ṽ ṽ ṽ d v S S { grad Φ s f k v R grad[ ] } aqt d d v By the Hölder inequality,.3 and the Young inequality, we have

23 B. Guo, D. Huang / J. Differential Equations [ v v v c v L S S c v L v H [ {[ S ] eθ v e ϕ v v L S eθ v e ϕ v ] eθ v e ϕ v ] v L } v L S c v L v H v L v H v H ε v L. 5.5 By the Hölder inequality and the Minkowski inequality, we obtain S ṽ div ṽ ṽ ṽ d v c ṽ eθ ṽ e ϕ ṽ ε v L From , by f R k v v = and Lemma., choosing ε small enough, we obtain d v H v L c v v v H L H v H c ṽ eθ ṽ e ϕ ṽ By the Uniform Gronwall Lemma, and 5.5, we get vt 5r H E 8, 5.57 where E 8 = E 8 U, Q, Q >. By the Gronwall inequality, from 5.56 we obtain vt H C 3, 5.58 where C 3 = C 3 U, Q, Q > and t < 5r L estimates of v Taking the derivative, with resect to, of Eq.., then taking the inner roduct of equation obtained with v in L,weobtain d v = [ v v eθ v e ϕ v v f k v R div vd v v ] [ v v v v div v v ] v grad[ aqt ] v. 5.59

24 8 B. Guo, D. Huang / J. Differential Equations By integration by arts, the Hölder inequality,.6 and the Young inequality, we have [ v v div v v ] v c v v eθ v e ϕ v c v v v 3 [ eθ v e ϕ v ] ε v c v 8 v. 5.6 By Lemma., the Hölder inequality and the Young inequality, we obtain grad[ ] aqt v = aqt div v c T c q T ε v. 5.6 By integration by arts, Lemma. and f R k v v =, choosing ε small enough, we derive from , d v eθ v e ϕ v v c v 8 H ṽ 8 v c T c q c T. 5.6 v By the Uniform Gronwall Lemma, 5., 5.7, 5.9, 5.5, 5.57 and 5.6, we get v t 6r E 9, 5.63 where E 9 = E 9 U, Q, Q > and t. From 5.6 and 5.63, we have c t7r t6r v E 9 E 9 = E. 5.6 By the Gronwall inequality, from 5.6, we obtain v t C, 5.65 where C = C U, Q, Q > and t < 6r L estimates of T,q Taking the derivative, with resect to, of Eqs.., then taking the inner roduct of equation obtained with T in L, weobtain

25 d T = B. Guo, D. Huang / J. Differential Equations T [ v T W v T T T = T = S ] [ T v T div v T ] T Q T [ aqdiv v P ] aqw v aq W v T By integration by arts, the Hölder inequality,.6, the Poincaré inequality and the Young inequality, we obtain [ v T divv T ] T [ c eθ v e ϕ v T T v T T v T T ] c eθ v e ϕ v ε T c T T c v T c v T ε [ T T c v eθ v e ϕ v ] c T 8 v c T 8 v 8 T By integrating by arts, the Hölder inequality, the Minkowski inequality, the Poincaré inequality, the Young inequality and.6, we obtain [ aqdiv v P aqw v a ] q W v T [ q vt ] [ ] aqv T P aqw v T [ ] a q v T a q v T ε q T c q c v c v q T c v q q c T ε T T ε q q c v 8 q 8 T c v 8 q c q v c v q From.5, by taking the trace on = of Eq.., we get

26 8 B. Guo, D. Huang / J. Differential Equations T = T = S [ ] T = = α s T = v T = T = Q = t S d T = [ ] = α s T = α s T = v T = Q = S By Lemma., we have [ ] α s T = v T = Q = S = α s v T = α s T = Q = S S = α s T = div v = α s T = Q = S S = α s T = div v d div v α s T = Q = S S c T = c v c v c T = c Q =. 5.7 By Lemma.3, the Young inequality, 5.7, 5.9, 5.3, 5.5, 5.5, 5.57, 5.58, 5.63 and 5.65, we derive from d T α s T = T T α s T = ε T T ε q q c T c q c v c v c q c T = c T = c Q = c Q c Q c. 5.7 Similarly to 5.7, we have d q β s q = q q β s q = ε q q c q c v c v c v c q = c q = c Q = c Q c Q. 5.7 From 5.7 and 5.7, choosing ε small enough, we obtain

27 B. Guo, D. Huang / J. Differential Equations d T q β s q = α s T = α s T = q T q β s q = T c c T q c v c v c q c T = c T = c q = c q = c Q = Q = c Q Q By the Uniform Gronwall Lemma, 5., 5.8, 5.6 and 5.73, we get T t 7r q t 7r E, 5.7 where E = E U, Q, Q >. From 5.73 and 5.7, we have c t8r t7r T q T = q = E E E = E By the Gronwall inequality, from 5.73 we obtain T t q t C 5, 5.76 where C 5 = C 5 U, Q, Q > and t < 7r. 5.. H estimates of v, T, q Taking the inner roduct of Eq.. with v in L,weget d e θ v e ϕ v v v eθ v e ϕ v v { [ ] = v v W vv v grad[ ] } aqt d v f k v grad Φ s v R By the Hölder inequality,.6 and the Young inequality, we have v v v v eθ v e ϕ v v

28 8 B. Guo, D. Huang / J. Differential Equations c v [ eθ v e ϕ v ] [ eθ v e ϕ v eθ v e ϕ v v ] 3 ε v c v 8 v eθ v e ϕ v ε [ v eθ v e ϕ v ] By the Hölder inequality, the Minkowski inequality,.3 and the Young inequality, we obtain W vv v S c c [ { { eθ v e ϕ v d S v v d v [ ] } eθ v e ϕ v d S S [ [ eθ v e ϕ v ] S ] d ε v eθ v e ϕ v v ] d { [ } c v eθ v e ϕ v v ] d ε v S S [ ε v c v v v eθ v e ϕ v ] eθ v e ϕ v By the Hölder inequality, the Young inequality, the Minkowski inequality, and.3, we have c grad[ ] aqt d v [ c [ q d T d ] [ ] [ T d q d ] ] c T ε v }

29 B. Guo, D. Huang / J. Differential Equations c q c T [ T L T L T L ] d [ q L q L q L ] d c T ε v c q T c T q c q T c T q c T ε v ε T ε q. 5.8 By Lemma., f R k v v =, 5.7, 5.9, 5.3, 5.5, 5.5, 5.57, 5.58, 5.63 and 5.65, we derive from d e θ v e ϕ v v v eθ v e ϕ v v c v eθ v e ϕ v c T c q ε Similarly to 5.8, we obtain eθ v e ϕ v 5ε v ε T ε q. 5.8 d T T T α s T = 5ε T ε T ε v c T eθ v e ϕ v c T c Q, 5.8 d q q q β s q = ε q ε q ε v c q eθ v e ϕ v c q c Q From , choosing ε small enough, we obtain d[ e θ v e ϕ v v T q ] v T q eθ v e ϕ v v T α s T = q β s q =

30 86 B. Guo, D. Huang / J. Differential Equations c v [ T q eθ v e ϕ v T q By the Uniform Gronwall Lemma, 5., 5.6, 5.75 and 5.8, we get ] c Q c Q. 5.8 eθ vt 8r e ϕ vt 8r T t 8r qt 8r E 3, 5.85 where E 3 = E 3 U, Q, Q >. By the Gronwall inequality, from 5.8 we obtain eθ vt e ϕ vt T t qt C 6, 5.86 where C 6 = C 6 U, Q, Q > and t < 8r. 6. The existence and uniqueness of global strong solutions 6.. The existence of global strong solutions Proof of Proosition 3.. By Proosition 5.3, we can use the method of contradiction to rove Proosition 3.. Indeed, let U be a strong solution to the system..7 on the maximal interval [, T ]. If T <, then lim su U =, t T which is imossible from 5., 5.63, 5.65, 5.7, 5.76, 5.85, The roof is comlete. 6.. The uniqueness of global strong solutions Proof of Proosition 3.. Let v, T, q and v, T, q be two strong solutions of..7 on the time interval[, T ] with corresonding geootentials Φ s, Φ s, and initial data v,t, q, v,t,q, resectively. Define v = v v, T = T T, q = q q, Φ s = Φ s Φ s. Then v, T, q, Φ s satisfy the following system v t v v v v v v W v v W v v f R k v grad Φ s grad Td a gradq T d a gradqt d =, 6. T t T T v T v T W v T W v T W v a q W v a qwv =, 6. q t q q v q v q W v q W v q =, 6.3

31 B. Guo, D. Huang / J. Differential Equations v t=, T t=, q t= = v v,t T,q q, 6. = : = : v =, v =, T = α st, T =, q = β sq, 6.5 q =. 6.6 We take the inner roduct of Eq. 6. with v in L and obtain d v = eθ v e ϕ v v [ v v W v v f [ R k v grad Φ s ] v v ] a gradq T d v v v v v [ W v v v grad Td v Using Lemma., the Hölder inequality, the Young inequality and.6, we get ] a gradqt d v. 6.7 v v v = v v v v v div v c ε v v eθ v e ϕ v eθ v e ϕ v c v v v 3 ε v c v 8 v. 6.8 By the Hölder inequality, the Young inequality, the Minkowski inequality and.3, we obtain W v v v ε ε eθ v e ϕ v c eθ v e ϕ v c [ ] v d S S [ v L S v d eθ v e ϕ v ] d

32 88 B. Guo, D. Huang / J. Differential Equations [ v L eθ v e ϕ v v ] d S [ v ε v c eθ v e ϕ v ] v v. 6.9 By Lemma., the Hölder inequality, the Young inequality, the Minkowski inequality and.3, we have [ ] a gradqt d v [ ] c q d T S S c T T q ε q ε d ε eθ v e ϕ v By Lemma., Lemma.3 and f R k v v =, we derive from d v eθ v e ϕ v v v eθ v e ϕ v. 6. 5ε v ε q c T T q [ c v 8 v v eθ v e ϕ v ] v grad Td v [ ] a gradq T d v. 6. Similarly to 6., we get d T T T α s T = [ 3ε v 3ε T c T 8 T v c T ] T T T W vt a q a W vt qt Wv, 6. d q q q β s q = 3ε v H 3ε q H c[ q ] q q q c q 8 q v. 6.3

33 B. Guo, D. Huang / J. Differential Equations Similarly to 5.8, we have a qwv T = a gradqt d v c v q T c v T q c v 8 q T ε q ε T. 6. By integration by arts, from 6. 6., by using 5.6 and choosing ε small enough, we obtain d v T q T eθ v e ϕ v v v T α s T = q q β s q = [ c v 8 T 8 q 8 v v eθ v e ϕ v ] v [ c v 8 T 8 T T ] T T c [ v 8 T T q 8 q q q ] q. 6.5 By the Gronwall inequality, Proosition 3. and 6.5, we rove Proosition 3.. In fact, we have established the following result which is stronger than Proosition 3.. Proosition 6. The uniqueness of strong/weak solutions. Let U be a weak solution to the system..7. If there exists a weak solution U of the system..7 on the interval [, T ] with the same initial conditions, such that U L 8, T ; L, U L, T ; L L, T ; H, then the solutions U,U coincide on [, T ]. 7. The existence of universal attractors Proof of Proosition 3.3. From 5.8, 5.63, 5.65, 5.7, 5.76, 5.85, 5.86, we know U L, ; V and Ut C U, Q, Q, where C is a ositive constant deendent on U, Q, Q and t. ByProosition3. and Proosition 3., we can define the semigrou {St} t corresonding to the system..6 where St : V V, StU = Ut. By 5.8, 5.63, 5.65, 5.7, 5.76, 5.85, 5.86, we rove that the corresonding semigrou {St} t ossesses a bounded absorbing set B ρ in V, i.e., for any U V,thereexistst big enough such that StU B ρ, for any t t, where B ρ ={U; U V, U ρ} and ρ is a ositive constant deendent on Q, Q.

34 9 B. Guo, D. Huang / J. Differential Equations In order to rove Theorem 3., we need the following roerty about the semigrou {St} t. Proosition 7.. For every t, the maing St is weakly continuous from V to V. Proof. Let {U n } be a sequence in V such that U n U weakly in V.Then{U n } is bounded in V. By the a riori estimates in Section 5, we know that, for every t, {StU n } is bounded in V. So we extract a subsequence {StU nk } such that StU nk u weakly in V. Since the embedding V L L L L is comact, U nk U strongly in L L L L. By 6.5, we obtain that StU nk StU strongly in L L L L. Then u = StU. Therefore, the sequence {StU n } satisfies: StU nk StU weakly in V.Proosition7. is roved. Proof of Theorem 3.. With Proosition 3.3 and Proosition 7., we know that Theorem 3. follows directly from Theorem I.. in [3] since this is a result for a dynamical system on a general metric sace. So the details of the roof for Theorem 3. is omitted here. Acknowledgments The authors would like to exress their heartful thanks to the referee and Prof. Songying Li for the useful comments and suggestions. The work was suorted in art by the NSF of China grants No. 959, No. 87 and the National Basic Research Program of China 973 Program No. 7CB88. References [] R.A. Adams, Sobolev Sace, Academic Press, New York, 975. [] A.J. Bourgeois, J.T. Beale, Validity of the quasigeostrohic model for large-scale flow in the atmoshere and ocean, SIAM J. Math. Anal [3] D. Cordoba, Nonexistence of simle hyerbolic blow-u for the quasi-geostrohic equation, Ann. of Math [] F. Charve, Global well osedness and asymtotics for a geohysical fluid system, Comm. Partial Differential Equations [5] J.G. Charney, R. Fjortaft, J. Von Neumann, Numerical integration of the barotroic vorticity equation, Tellus [6] P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the -D quasigeostrohic thermal active scalar, Nonlinearity [7] P. Constantin, A. Majda, E. Tabak, Singular front formation in a model for quasigeostrohic flow, Phys. Fluids [8] J.G. Charney, N.A. Philis, Numerical integration of the quasi-geostrohic equations for barotroic simle baroclinic flows, J. Meteor [9] C. Cao, E.S. Titi, Global well-osedness and finite dimensional global attractor for a 3-D lanetary geostrohic viscous model, Comm. Pure Al. Math [] C. Cao, E.S. Titi, M. Ziane, A horizontal hyer-diffusion 3-D thermocline lanetary geostrohic model: well-osedness and long-time behavior, Nonlinearity [] C. Cao, E.S. Titi, Global well-osedness of the three-dimensional viscous rimitive equations of large-scale ocean and atmoshere dynamics, Ann. of Math [] V.V. Cheyzhov, M.I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Al [3] P. Constantin, J. Wu, Behavior of solutions of D quasi-geostrohic equations, SIAM J. Math. Anal [] P.F. Embid, A.J. Majda, Averaging over fast gravity waves for geohysical flows with arbitrary otential vorticity, Comm. Partial Differential Equations [5] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier Stokes Equations, vol. I, Sringer-Verlag, 99. [6] B. Guo, D. Huang, Existence of weak solutions and trajectory attractors for the moist atmosheric equations in geohysics, J. Math. Phys [7] B. Guo, D. Huang, Existence of the universal attractor for the 3-D viscous rimitive equations of the large-scale ocean, rerint. [8] F. Guillén-González, N. Masmoudi, M.A. Rodríguez-Bellido, Anisotroic estimates and strong solutions for the rimitive equations, Differential Integral Equations [9] J.R. Holton, An Introduction to Dynamic Meteorology, third edition, Academic Press, 99. [] C. Hu, R. Temam, M. Ziane, The rimitive equations of the large scale ocean under the small deth hyothesis, Discrete Contin. Dyn. Syst [] J. Li, J. Chou, Existence of atmoshere attractors, Sci. China Ser. D

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