On the Regularity of Three-Dimensional Rotating

On the Regularity of Three-Dimensional Rotating Euler-Boussinesq Equations 1

A. Babin1, A. Mahalov2, B. Nicolaenko2

University of California, Irvine, CA 92717, U.S.A. and MIIT, Moscow, Obrazcova 15, Russia 2 Department of Mathematics, Arizona State University, Tempe, AZ 85287

Abstract: The 3D rotating Boussinesq equations (the \primitive" equations of geophysical uid ows) are analyzed in the asymptotic limit of strong stable strati cation. The resolution of resonances and a non-standard small divisor problem are the basis for error estimates for such fast singular oscillating limits. Existence on in nite time intervals of regular solutions to the viscous 3D \primitive" equations is proven for initial data in H  ,   3=4. Existence on a long time interval T  of regular solutions to the 3D inviscid equations is proven for initial data in H  ,  > 5=2 (T  ! 1 as the frequency of gravity waves ! 1). Keywords: Euler-Boussinesq equations, geophysical uid ow.

1 Introduction The turbulent ows that are subject to rotation and strati cation have many important applications in geophysics and engineering (Fernando and Hunt, [16], Pedlosky [27]). An important class of geophysical

ows can be characterized as strongly strati ed with both eects of (stable) strati cation and rotation playing an important role in the dynamics. This is the so called Burger number of order one regimes where the eects of rotation and strati cation enter at the same order in asymptotics (McWilliams [25]). One of the major diculties encountered in understanding dynamics of geophysical ows is the in uence of the oscillations generated by the rotation and strati cation (buoyancy forces). The governing ow equations for rotating stably strati ed uids under the Boussinesq approximation in dimensionless variables are @tU + U
rU + fe3  U = ?rp + 1 e3 + 1
U + F; r
U = 0;

(1.1)

@t1 + U
r1 = ?N 2 U 3 + 2
1 + F4 ;

(1.2)

where rotation and mean strati cation gradient are aligned parallel to the vertical axis x3. Here U = (U 1 ; U 2; U 3) is the velocity eld and 1 is the buoyancy variable (relative density variation); N is the BruntVaisala wave frequency for constant strati cation and is the frequency of background rotation, f = 2

(see [10] on reduction to dimensionless variables and a discussion of related physical questions). Eqs. (1.1) and (1.2) are sometimes called the primitive (non-hydrostatic) equations of geophysical ows. We focus on both inviscid Eqs. (1.1)-(1.2) and with small uniform viscosities, 1  0; 2  0; here 1 and 2 are the kinematic viscosity and the heat conductivity, respectively; the ratio Pr = 1=2 is known as the Prandtl number. The parameter  = f=N plays an important role and it is assumed to be xed as f, N ! 1;  is essentially the \Burger" number of geophysics [25]. We consider periodic boundary conditions in a parallepiped [0; 2a1]  [0; 2a2]  [0; 2a3], as well as stress-free conditions U 3 = 0, @U 1 =@x3 = @U 2=@x3 = 0 at x3 = 0, 2a3. We investigate the fast singular oscillating limits of Eqs. (1.1)-(1.2) as f ! 1, N ! 1,  = f=N xed. In our approach, the collective contribution to the dynamics made by fast \inertio-gravity" waves is accounted for by rigorous estimates of wave resonances and quasi-resonances via small divisors analysis ([1], [2]). This mathematical approach in the context of geophysical ows was initiated in BMN [3], Mahalov and Marcus [23]). In the context of symmetric hyperbolic systems, related singular limits have been investigated by Joly-Metivier-Rauch [21], whose results have been extended by Schochet [30]. 1

First results on regularity of 3D Euler-Boussinesq and Navier-Stokes systems (1.1)-(1.2) were obtained in BMN[4], BMN [7]. First results on regularity in the context of geophysical ows with both rotation and strati cation were obtained in BMN[5], BMN[10]. It was revealed in these papers the crucial role of parameters ; 2 = 1=a22; 3 = 1=a23 for the properties of the dynamics, in particular for smoothness (we set a1 = 1 using a simple rescaling, in the general case one has to put 2 = a21 =a22; 3 = a21=a23 ). Later, conditions on smoothness of initial data and forcing term were relaxed in Gallagher [17], [18], [19]. In this paper, we relax restrictions to non-resonant domains (non-resonant parameters 2 , 3 ) as well as smoothness conditions of [17], [18], [19]. Namely, we show that global regularity holds not simply for almost all values of parameters ; a2; a3 (as it is stated in [17], [18], [19]), but for triplets ; a2; a3 which do not belong to an explicitly described strictly resonant set. Speci cally, (; a2; a3) do not belong to  = [k;mk;m in the three-dimensional parameter space (; 2 ; 3) where k; m are integer wave vectors and k;m is for every  (2 ; 3 ) where (2 ; 3) = (1=a22; 1=a23). Thus global k; m a smooth analytic surface with equation  = k;m regularity holds for all a2; a3 (all domains) provided that  2=  (a2 ; a3). The small divisor problem for the fast oscillating limits of Eqs. (1.1)-(1.2) is not of the simple type jkj  jmj  jnj = 0, such as in [21] and [30]. We also relax (in viscous case) conditions on time behavior of the forcing term F(t); no conditions is imposed here on @tF in contrast to [5], [7] and [17], [18], [19]. In regularity theorems we impose only an integral regularity condition on forcing term

Z T +1 T

kFk2?1dt  MF 8T  0;

(1.3)

with   3=4. We do not impose (in contrast to Gallagher [17],[18], [19]) conditions on LR 1 norms of F(t). Note also that Gallagher [17, 19] is assuming that F 2 L2 (R+ ; H 0) which implies that T1 kFk0 dt ! 0 as T ! 1, that is F(t) decays; under this kind of condition solutions of viscous Navier-Stokes and Boussinesq equations tend to zero as t ! 1 and their regularity for large times is well known starting from Leray's results on Navier-Stokes system. Therefore in this case regularity is in fact reduced to regularity on nite intervals. We do not assume decay of F(t). We also consider the case when only kinematic viscosity 1 > 0 but 2 = 0; in this case we prove regularity on arbitrary large (but nite) interval when   1. In BMN ([4],[7],[5]) and here (Section 6) we obtain strong convergence results with uniform error estimates on parameter sets of full Lebesgue measure and with initial data being in the Sobolev space H 7 uniformly in 1; 2  0. They cover the physically relevant case of huge atmospheric Reynolds numbers. This is in contrast with the work of Embid and Majda [15] where, following general theorems of Schochet [30], they state a pointwise convergence theorem on a small time interval [0; T] for every value of , a2, a3 without explicit estimate of error; in fact, as it is proven in BMN [7],[8] it is impossible to obtain explicit, uniform estimates if one does not delete almost resonant sets of parameters. Neither are such general problems of uniform convergence addressed in [17]-[20], [30]. Deleting a countable set of strictly resonant  which correspond to 3-wave resonances we obtained global regularity theorems in BMN[5], later similar results were obtained in [18]; if one does not take into account the structure of resonances, only small time regularity is obtained (see Embid and Majda [15], Grenier [20]). An approach based on choosing special sets of `prepared' initial data with in nite codimension is used in Bourgeois and Beale [12] and Chemin [13] to obtain long time existence, but it eectively lters out the nonlinear interactions between inertio-gravity waves and the quasi-geostrophic (potential vorticity) elds. Now we describe the structure of reduced limit equations which will be derived in Sections 2 and 3. From now on we are going to restrict ourselves to  bounded, including  > 1 must be treated separately and it will be published elsewhere. The case  = 1 (f ! 1, N = 0) was the subject of our papers on pure fast rotating limit without strati cation ([3]-[8]). For all parameters a1 , a2 and a3 and all values of the parameter  = f=N in the reduced equations the total velocity splits into the quasi-geostrophic eld wQG (t) satisfying 3DQG (quasi-geostrophic) equations @twQG = B0 (wQG ; wQG ) + QG
wQG + FQG ; 2

(1.4)

and the ageostrophic components satisfy equations of the type: @t wAG = B2 (wQG ; wAG) + B3 (wAG; wAG ) + AG
wAG;

(1.5)

here QG(1 ; 2) and AG (1; 2) are in general non local zeroth-order pseudodierential operators, whenever 1 6= 2 . The limit Eq. (1.4) results from the \slow" ( wQG ; wQG ; wQG ) triads as well as all resonant (wAG ; wAG ; wQG ) triads (the contribution of the latter is exactly zero in the limit, hence the operator splitting). The limit ageostrophic Eq. (1.5) is derived from both resonant ( wAG ; wQG ; wAG ) and ( wQG ; wAG ; wAG ) triads as well as the 3-wave resonances (wAG ; wAG ; wAG ). Notice that the slow-fastslow (wQG ; wAG ; wQG ) triads are not resonant to the lowest order in 1=N and appear only at the next order in 1=N via 4-wave resonances (see also [11]). For any given parameters a1 ; a2 and a3 there is only a rare non-dense discrete set fj g1 j =1 such that only for  = j there are 3 waves fast-fast-fast resonances; in particular, there is a whole interval of  centered at  = f=N = 1 where there are no 3-wave resonances. Even in the context of  equals resonant j , we demonstrate that the operator B3 in (1.5) generally induces only a nite-dimensional dynamical systems (hence no energy cascades). The quasigeostrophic equations in the inviscid case have a global regular solution according to Bourgeois and Beale [12]; also if 1 > 0 according to a theorem we prove here in Section 3. Ageostrophic equations for general  are very complicated and only local existence theorem can be stated. However, in BMN [4], [5], [7] it is shown that if one deletes a resonant set  of parameters ; 2; 3 , then B3(wAG ; wAG) = 0 and only \catalytic" interactions described by the linear in wAG operator B2 (wQG ; wAG ) rule AG dynamics: @t wAG = B2(wQG ; wAG ) + AG
wAG ; (1.6) where wQG (t) is a solution of 3DQG equations. Here, we demonstrate that only the parameter  need not be resonant (Section 5). The reduction to (1.6) holds for all values of the parameter  except a non-dense set of discrete values fj g1 j =1 ( 6= j ). We prove here that this linear system (1.6) has a global smooth solution even when AG = 0, provided wQG is smooth enough. Further, for all , all a3 when a22 is irrational, B2(wQG ; wAG ) splits in Fourier space into uncoupled, restricted interaction operators on 4 rays families in Fourier space, where  is any rational number: 0 m 1 0 1 0 0 1 0 n 1 @ m12 A =  @ 0 1 0 A @ n12 A : (1.7) m3 0 0 1 n3 Fourier series will be used in this paper to represent physical elds in a parallepiped [0; 2]  [0; 2a2]  [0; 2a3]. We assume that 0 < a30  a3  1,0 < a20  a2  1 We denote k2=a2 by k2 , k3=a3 by k3, k1 = k1, k = (k1 ; k2=a2 ; k3=a3 ), jkj2 = k12 + k22 =a22 + k32 =a23, jk0j2 = k12 + k22 =a22. This paper is organized as follows. In Section 2 we derive the limit equations describing reduced dynamics in framework of the Craya-Herring cyclic basis. In Section 3, we give regularity properties of QG equations with nonlocal limit operators QG > 0, AG > 0 (these are resonant limits whenever both 1, 2 > 0); we also consider the partially inviscid case 2 = 0, QG  0 where now the limit operator QG is only non-negative. In Section 4, we derive the reduced ageostrophic equations and establish their regularity in H s , s  0, in the inviscid case. For s > 0, this is not trivial, as only the energy is conserved (L2 norm). Although the inviscid ageostrophic equations (1.6) with AG = 0, are linear, their coecients involve the time-dependent wQG (t) and the solutions need not be bounded globally for all times in H s norms, s  1. This point is omitted in [18], where existence theorems are in fact conditional upon such regularity. In Section 5, we investigate resonances and quasi-resonances for both 3-waves and 2-waves in detail. We also describe the uniform convergence results. These results require much less dierentiability than in BMN 3

[5]. In fact, approach of BMN [5] can be applied to the case   0; here in the case 1=0    0 with xed 0 > 1 we obtain better and simpler estimates of small divisors which result in milder smoothness restrictions: now only six derivatives on initial data are required for the uniform (in ; a2; a3) convergence results to the QG component, with arbitrarily large ageostrophic initial data. In Section 6, we give new regularity and existence theorems for all times of the viscous rotating Boussinesq equations with \unprepared" initial data in H  ,   3=4, with the weaker condition (1.3) on the force. We establish regularization of Leray's classical weak solutions, for N and f nite, albeit large enough. In Section 7, we establish arbitrarily long-time existence results for the inviscid Euler-Boussinesq equations with initial conditions in H  ,  > 5=2, again with arbitrarily large \unprepared" ageostrophic initial data. Following Metais and Herring [26] we introduce a change of variables 1 = N and combine velocity and buoyancy variable in one variable Uy = (U; ) after which Eqs. (1.1)-(1.2) written in non-dimensional variables take the more symmetric form: @t Uy + U
rUy = ?ryp ? N MUy + 
Uy + Fy; r
U = 0

(1.8)

where ryp = (rp; 0); Fy = (F; F4 ),

      M = (S + R);  = f=N; R = J0 00 ; S = 00 0J ; J = 10 ?01 ;

(1.9)

 = diag(1 ; 1; 1; 2) is the viscosity matrix, 

is xed. We use Fourier series expansions for elds Uy (x) = (U 1 (x); U 2(x); U 3(x); (x)), x = (x1; x2; x3):

Uy(x) =

X n

exp(i(n1 x1 + n2 x2=a2 + n3 x3=a3 ))Uyn =

X n

exp(in
x)Uyn

(1.10)

where Uyn are the (4-component) Fourier coecients, [n1; n2; n3] 2 Z3, n = [n1; n2=a2; n3=a3] are wavenumbers. We introduce the space of functions H s with the norm de ned on Fourier coecients Un as follows (where jn j = (n21 + n22=a22 + n23=a23)1=2 ):

jjUyjj2H s =

X n

jnj2sjUynj2:

(1.11)

We assume that all functions have zero average over the periodic parallepiped. Stress-free boundary conditions at x3 = 0, 2a3 correspond to U 1, U 2 even in x3 and U 3 ,  odd in x3. Sobolev spaces are restricted to such functions. In this paper Rn, Sn will denote the action of R and S on n-th Fourier component,  Rn + S n = Mn . We take into account divergence-free condition by applying the Helmholtz projection Pd onto divergencefree vector elds. The matrix (Pd MPd )n is a real skew-symmetric matrix; the corresponding operator restricted to the 3-dimensional subspace of divergence-free vectors Uyn has one zero eigenvalue and two complex conjugate eigenvalues i!n 6= 0. We introduce the divergence-free vectors (1.13) which form a real cyclic basis for it:

Pdn Mq0n = 0; PdnMq1n = ?!nq2n; PdnMq2n = !nq1n;

(1.12)

q0n = !1 (n p0n + n p2n); q1n = p1n; q2n = !1 (np2n ? n p0n)

(1.13)

where Pdnqjn = qjn, n

n

Here p0n; p1n; p2n form an orthonormal basis of the divergence-free subspace for n-th Fourier mode; the pjn are the Craya-Herring basis for the purely strati ed problem, already used by Riley et al. [29]. 4

 n1 n3 n2 n3 ?n2 ? n2   n2 n1  p0n = ? jn 0j ; jn 0j ; 0; 0 ; p1n = jnj jn 0j ; jnjjn 0 j ; jn1jjn 0j 2 ; 0 ; p2n = e4 = [ 0; 0; 0; 1 ]:

(1.14)

The eigenvalues i!n are given by

0 p !n = 2n + 2n2 ; n = jnn3j ; n = jjnnjj ;  = f=N:

(1.15)

where jn j2 = n21 + n22=a22 + n23=a23 , jn 0j2 = n21 + n22 =a22. We consider the case when the ratio  = f=N is bounded, by a bounded 0 > 1: 1=0   = f=N  0;

(1.16)

1=0  min(1; )  !n  max(1; )  0:

(1.17)

In the case n1 = n2 = 0 (this case corresponds to taking horizontal averages) we choose the basis which is obtained from (1.13) by putting n1 = n2 6= 0 and taking n1 ! 0. In particular, when n1 = n2 = 0 we obtain !n =  and the eigenvectors are (1.18) q0n = (0; 0; 0; 1); q1n = ( p1 ; p1 ; 0; 0); q2n = ( p1 ; ? p1 ; 0; 0) 2 2 2 2 where n = (0; 0; n3) denotes wavenumbers for which n1 = n2 = 0. Any arbitrary divergence-free vector eld Uyn can be written as

Uyn = Vn0q0n + Vn1q1n + Vn2 q2n:

(1.19)

We shall use the variables V to denote vector of coecients corresponding to Uyn : Vn = [Vn0 ; Vn1; Vn2 ] = [Vn0 ; V0n ], V0n = [Vn1; Vn2 ]. Note that the relation between U and V variables is given by

Vn0 = Uyn
q0n; Vn1 = Uyn
q1n; Vn2 = Uyn
q2n: (1.20) Clearly, Vn0 = ?V?0n and Vni = V?i n , i = 1; 2, for real U(x) and (x). We denote by QG n the projection onto q0n and call it as usual the quasi-geostrophic mode: QG Uy(x) =

X n

y 0 Vn0q0nein
x ; QG n Un = Vn q0n:

The projection onto two-dimensional subspace corresponding to i!n is denoted by AG and de nes the ageostrophic component: y 1 2 AG n Un = Vn q1n + Vn q2n: The case when  ! 0 or  ! 1 was discussed in BMN [5]; detailed mathematical consideration of this can be done along the lines of this paper and BMN [7], but requires additional non-trivial considerations; in particular structure of resonant sets and smoothness conditions are dierent from those imposed here. The Boussinesq equations (1.8) in Fourier representation in V variables can be written in the cyclic basis (1.13) as @tVni = ?i 3

X

k+m=n;i1 ;i2

i i V i V i ? N! (M 0 V )i ? (^ jnj2V )i + F i Qikmn n n n n n k m 1 2 3

1

3

2

5

3

3

(1.21)

where i1 ; i2; i3 = 0; 1; 2, M 0 is the matrix M in V -variables given by (1.22); ^ is the viscosity matrix  in the V -basis. Here J, M0n are given by

00 0 0 1   M0n = @ 0 0 ?1 A ; J = 10 ?01 ;

(1.22)

0 1 0

0

12n + 2 n2 2 0 (1 ? 2)n n 1 2 @ ^ = 2 !n (1 ? 02)n n 10 n (2 2 +02 2 1) n n

1 A ;

(1.23)

i i i are determined from the equations using (1.13), (1.24), see BMN[10]. The coecients Qkmn 1 2 3

i i = (qi k
m)(qi m
qi n): Qikmn (1.24) We use notation for the skew-symmetric product n0 ^ m0  n1 m2 ? n2 m1 . To save space, we give formulas only for 0-wave interactions, see BMN [10] for general coecients. !m jm j n 0 ^ m 0 (1.25) Q000 kmn = ! ! jkjjnj k n 1 2 3

1

2

3

i i i j  jm  j. when n = k + m; clearly jQkmn 1 2 3

2 The limit equations describing reduced dynamics We write classical rotating Boussinesq equations using the variables V in the basis (1.13); these equations have the form (see Eqs. (1.21)) @t Vn + N!n MnVn = (B(V; V))n ? An Vn + Fn ; An = ^ jnj2:

(2.1)

V = E(?Nt)v; Vn = exp(?N!n M0nt)vn

(2.2)

E(Nt)[V 0; V0 ]n = exp(N!n tM0n)[V 0 ; V0 ]n = [V 0; exp(N!ntJ)V0 ]:

(2.3)

0

Here M0 is given in cyclic V -variables by Eqs. (1.21)-(1.22). We introduce the linear propagator directly into nonlinearity using the change of variables where v = [v0; v1 ; v2 ] and M0 is de ned by (1.22). The action of the linear propagator on the Fourier components E(Nt) can be written in V -variables corresponding to the Craya-Herring cyclic basis Obviously, E(Nt) represents vector rotation in V 1 ; V 2-plane; orthogonal V 0 component (called QG) along the axis of rotation is not aected. To save space, we always write Vn = [V 0 ; V0 ]n as a row, understanding that it is a column in the matrix multiplication. Eqs. (2.1) written in v variables have the form @t v = B(Nt; v; v) ? E(Nt)AE(?Nt)v + FQG + E(Nt)FAG; B(Nt; v; v) = E(Nt)B(E(?Nt)v; E(?Nt)v) 6

(2.4) (2.5)

where Fy = FQG + FAG . Equations (2.4) are explicitly time-dependent with rapidly varying coecients. The corresponding equations for Fourier coecients have the form, with e0 = (1; 0; 0): X ^ i ;i ;i X ~ i ;i ;i i i Qkmn (Nt)vki vmi ? A~n vni ? A^n(Nt)vni + Qkmn vk vm + @t vni = 1

3

2

3

1

1

2

2

3

1

3

3

2

n=k+m;i1 ;i2

n=k+m;i1 ;i2

Fn0e0 + En(Nt)FAG (2.6) n ;i ;i (Nt) of nonwhere the rst sum consists of resonant terms. In the second sum every matrix element Q^ ikmn resonant part as well as A^n(Nt) equals a sum of terms of the form C exp(iD` Nt) with D` 6= 0. Generally, D` = !n0  !m0  !k0 , ` = 1; ::; 8, where either !n0 = !n or !n0 = 0. When all three !n0 ; !m0 ; !k0 are non-zero we have 3-wave interactions; when exactly two of !n0 ; !m0 ; !k0 are non-zero we have 2-wave interactions; when exactly one of !n0 ; !m0 ; !k0 is non-zero we have 1-wave interactions. When resulting D` = 0 we call these interactions resonant, when D` 6= 0 the interactions are non-resonant; see BMN [7] for more details. We have shown in BMN [5], [10] that for all but a countable non-dense set of , all 3-wave interactions are non-resonant and thus do not contribute to the limit equations; see Section 5 for details. The resonant contribution from viscous term does not coincide with original operator ^
since ^ does not commute with M. Simple computation gives the resonant terms. Let 1 and 2 be the kinematic viscosity and the heat conductivity, respectively. We have in V -basis exp(P d M 0 P d Nt)^ exp(?P d M 0P d Nt) = diag(QG(n); AG(n); AG(n)) + 0 when   max( + 1; 1); Moreover, if 0   < 3=2 j(BQ (u; v)); (
)w)j  CQ C8jjwjj+1jjujj3=4+=2jjvjj3=4+=2   jjwjj2+1=8 + C(1=)jjujj23=4+=2jjvjj23=4+=2: If 0   < 3=2;  3=2 was incorrect, when 2   > 3=2; in proofs of [7] it was used in fact not (6.17), but the following inequality with arbitrary small  > 0

j(B(v; u); A u)j  CB jjvjj2+ jjujj2 : 20

(6.18)

Theorem 6.2 Let   3=4; 1 > 0; 2 > 0,  >  + 2; T0 > 0 and  2=  (2 ; 3) and kUy (0)k  M ; kUy0 (0)k  MF ; Let N  N0 (MF ) Then

Z T +1 T

Z T +1 T

kFyk2?1dt  M 8T  0;

kFyk2?1 dt  MF ;

kUy0(t)k  M0 ; 0  t  T0 ; where M0 depends only on M ; T0 .

ZT

0

0

Z T +1 T

k@tFyk20 dt  MF 8T  0:

kUy0 (t)k2+1 dt  (M0 )2

(6.19) (6.20) (6.21)

Proof It is similar to the proof of Theorem 8.2 given in BMN[7] for Euler-Navier Stokes system with   0. Since proofs are based on energy estimates, one can repeat the proof for Boussinesq almost literally with obvious modi cations. Note that solutions U~ of 2D Euler equations is replaced now by UQG and solutions V of Extended Euler equations correspond now to AG part UAG , large parameter is replaced by large parameter N. We have now according to Theorems 3.2 and 4.2 solutions of the limit QG and AG equations bounded for T0  t  0 in H by M for   3=4. The only essential dierence in proofs is that now thanks to condition  > 0 we can replace condition  > 3=2 imposed in BMN[7] by  > 1=2 (we can do it everywhere, but for global existence of solutions of QG equations we need according to Theorem 3.1   3=4). Note that (6.21) follows from the estimate (6.11) of BMN [7] of the dierence between solution v(t) of (2.4) and the solution w(t) of the limit equation (2.11); this estimate now takes the form jjv(t) ? w(t)jj  (N) 8t 2 [0; T1] (6.22) where (N) ! 0 as N ! 1. Estimate (6.22) follows from (8.14) of BMN[7] which here takes the form jjUy(t)jj  jjUy ? U~ QG ? U~ AG jj + jjU~ QG + U~ AGjj  M =4; 0  t  T: (6.23)

~ AG jj are bounded; they are bounded by Theorems Here  > 1=2; this estimate holds as long as jjU~ QG jj; jjU QG AG 3.1, 4.1 if   3=4. Here U~ ; U~ are solutions of the reduced QG and AG equations, with initial data QG Uy(0), AG Uy (0) and U~ AG = E(?Nt) ~uAG . Inequality (6.23) follows from estimation of the error term y^ in (6.14) of BMN[7]. We recall the following notations adopted from [7]: v^ = R v, w^ = R w. Here R v is the projection of vn onto the Fourier modes with jnj  R (similarly Rw). The truncated elds v^ and w^ satisfy the equations of the same form as v and w but with the extra forcing term gtr = R [B(Nt; vh ; v ? R v) + B(Nt; v ? R v; v^ )]i; (6.24) g~tr = R B~ (v; v ? R v) + B~ (v ? R v; v^ ) : Now we de ne ^r = v^ ? w^ and we also de ne ^r1 through its Fourier coecients: ^r1n = ?i

g^1n = i

X n:r:;l;k+m=n

X

eiNDl (k;m;n)tQkmn;l(^vk ; v^ m )=Dl (k; m; n) ? +1 as it was taken in BMN[7]). The term C(M ; )jjy^jj2 + jjy^jj2+1=2 is obtained from Lemma 6.1. To obtain estimate (6.29) from (6.27) we multiply (6.27) by (?
) y and we apply Lemma 6.1 instead of Lemma 4.3 and Lemma 4.3' of BMN[7]. Taking into account the remarks given, one can follow the proof of Theorem 8.2 BMN[7] to obtain the statement of Theorem 6.2. Theorem 6.3 Let   3=4; 1 > 0; 2 > 0,  2= (2 ; 3 ), (1.3) holds and

kUy (0)k  M;

(6.31)

let N  N (M ; MF ) Then solution of viscous Boussinesq system is regular for all t,

kUy(t)k  M0 ; 8t  0

(6.32)

If 1 > 0; 2 = 0;   1 then for every T there exists N0 (T; MF ) such that Uy(t) 2 H for 0  t  T when N  N0 .

Proof First, we have solution Uy(t) 2 H  on a small time interval [0; T1], where T1 depends on y jjU (0)jj; ; the proof is similar to proof in Theorem 6.1. Now we consider the case 3=4    1. We have

the energy estimate and



Z T +1 T

jjUy(t)jj  M0 8t  0 jjUy (t)jj21dt  CM02 ; M0 = M0 (jjFy jj?1; jjUy(0)jj0); 22

this implies that on every interval [T1 ; T1 + 1] we have a point t0 for which

jjUy (t0)jj21  CM02(jjFyjj?1; jjUy(0)jj0)=: Therefore we can take U(t1) as new initial data, change t for t ? t1 ; to prove that the solution is bounded in H  for all t  0 it suces to prove that the solution is bounded in H 1 for t 2 [0; 2]. Approximating Fy(t), @t Fy(t), Uy(0) by smooth functions Fy0 (t); @tFy0 (t); U0 (0) in H 1 we obtain (6.33) kW(0)k1 = jjUy(0) ? Uy0 (0)jj1  ; kW(0)k  M();

Z T +1 T

kF0k20 dt  ; jjF0 (t)jj?1  M(); jj@t F0(t)jj?1  M()

(6.34)

for F0 = Fy ? Fy0. We choose  so small that we have by Theorem 6.1 a regular solution on [0; 2] which is bounded in H 1 by 1 when initial data are in H  and functions in coecients are bounded by M0 + 2,  = 1. After that we consider Boussinesq equations with smoothed functions Uy0 (t0 ); Fy0 which satisfy (6.1) for t0  t  t0 + 2. The H  - norms of smooth functions are bounded by (a very large) constant MF depending on this xed . After that we choose N so large that we have (6.21) for solutions Uy0(t) of equations with smoothed data. By (6.21) and (6.5) with C0  1 we have jjUy(t)jj  jjUy0(t)jj + jjUy(t) ? Uy0(t)jj  M10 + 1,  = 1. Using this inequality we easily conclude that Uy (t) is bounded in H  for all t  0. If  > 1 we use uniform in t boundedness in H 1 and after that we apply the smoothing property for solutions of Boussinesq equations and obtain that the solutions are bounded for t  t0 > 0 in H  . Finally, using techniques from BMN [7] we obtain the regularity for all large times for the 3D rotating Boussinesq- Navier Stokes equations (the so called \primitive" equations, not to be confused with equations associated to hydrostatic pressure hypothesis). The Navier-Stokes equations are forced by a force Fy(t). This theorem describes the situation when N is xed, large enough (depending only on magnitude of Fy(t). The situation is that of non-smooth and arbitrary large initial data in H 0. Then weak Leray solutions exist with maybe a blow-up in H 1 .

Theorem 6.4 Let 1 > 0, 2 > 0, (1.3) hold with  = 0, y 2/  (2 ; 3). Let jjUy(0)jj0  M0 . Let y T^ = jjU (0)jj20= , where  = min(1 ; 2); T^ depends only on jjU (0)jj0, 1, 2 . Let N 0 be a number which depends only on M01, 1, 2, a2 , a3. Then for every xed N  N 0 and for any weak solution Uy(t) of the ^ and satis es the classical energy 3D rotating Boussinesq-Navier-Stokes equations which is de ned on [0; T] y ^ inequality on [0; T], the following proposition is true: U (t) can be extended to 0 < t < +1 and it is regular for T^  t < +1; it belongs to H 1 and jjUy(t)jj1  C1(M01 ; 1; 2) for every t  T^. It F is time-independent there exists a global attractor of the \primitive" equations of geophysics, which is bounded in H 1, has a nite fractal dimension, and every weak solution is attracted to the global attractor as t ! +1. Remark 6.2 No \preparation" of initial conditions is needed, contrary to the special restricted results of Chemin [13]. This theorem holds for all a2; a3, 0 < a2; a3  1. This theorem resolves problems of existence of attractors of for the \primitive" equations of geophysics raised by Lions, Temam and Wang [22].

7 Regularity results for inviscid case

Now we give inviscid versions of the above theorems. We again consider (6.3), now 1 = 2 = 0; Fy = Fy0 = 0. Theorem 7.1 Let  > 5=2;  = 0; Fy = 0,

kUy0(t)k+1  M0 ; 0  t  T 23

(7.1)

let   0: Then regular W(t) exists and kW(t)k  C0 M0 ; 0  t  T:

 Proof The proof is similar to proof of Theorem 6.1. Multiplying the inviscid form of (6.3) by 2(?
) W we now obtain by using Lemma 4.3' of BMN [7] in 3D case @t kW(t)k2  C1kUy0(t)k+1 kWy(t)k2 + C2kWy (t)k3: (7.2) This equation implies @tkW(t)k2  C2kW(t)k3 + C1 M0 kW(t)k2; kW(0)k   (7.3) where C1; C2 depend on . We easily obtain an inequality similar to (6.7) and (6.8). Theorem 7.2 Let  > 5=2; 1 = 2 = 0,  >  + 3,  2=  (2 ; 3) and kUy0 (0)k  M ; k Uy(0)k  M : (7.4) Let solution W0 (t) of QG system be bounded in H  for 0  t  T ; let N  N0 (M ). Then (7.5) kUy (t)k  M0 ; 0 for 0  t  T where M depends only on M . Proof It is given in the proof of Theorem 8.1 in BMN [7], the estimate (7.5) follows from (8.6) of BMN [7]. One has to replace global regularity of 2D Euler equation by global regularity of QG equations (regularity for QG equation is proved in Bourgeois and Beale for   3, but the same proof works for  > 5=2); one also has to replace global regularity of Extended Euler equations by global regularity of AG equations (Theorem 4.1). One also can use a theorem similar to Theorem 7.1. Theorem 7.3 Let  > 5=2; 1 = 2 = 0,  2=  (2; 3 ), F = 0, kUy (0)k  M; (7.6) let T > 0, N  N (M ; T): Then solution of inviscid Boussinesq system is regular for t  T , kUy(t)k  M0 ; 0  t  T: (7.7)

Solutions of the limit equations are bounded for all 0  t  T in H  by M . The bound for QG equation is proved in Bourgeois and Beale for   3, but the same proof works for  > 5=2. After that we proceed like in proof of Theorem 6.3, but now we restrict from the very beginning all considerations to a xed interval [0; T] and do not use smoothing arguments. Now we give a uniform theorem on regularity; such theorem requires more smoothness of initial data. This theorem improves the result of BMN[5] using more precise small divisors estimates given in Section 5. Proof

Theorem 7.4 Let  2= 3 (2; 3), 2 2/ 2 . Let 1 = 2 = 0. Let  > 19=2 , and M > 0, T  > 0 be y    arbitrarily large. Then there exists N = N (M ; T ; 3; 2) such that for jjU (0)jj  M and N  N  , there exists a unique regular solution Uy (t) of the 3D rotating Euler-Boussinesq equations which belongs to H  for 0  t  T  . For M xed, T  ! +1 as N  ! +1 with explicit uniform dependence of T  on M , 3 , 2 , N  . Simultaneously, we can take arbitrary large (but bounded) set of initial data: M ! +1 if N  ! +1, for xed T  . The proof is similar to the proof of Theorem 6.2 in BMN [7], BMN [5]. Acknowledgements 3

2

The authors wish to thank for their support the AFOSR (grant F49620-96-0-0165) and the ASU Center for Environmental Fluid Dynamics. The hospitality of the Newton Institute (Cambridge) under the special programme \Mathematics of Atmosphere and Ocean Dynamics" is gratefully acknowledged as well as the hospitality of the Ecoles Normales of Paris and Cachan. We would like to thank Professors C.W. Bardos, P. Bartello, Y. Brenier, P. Constantin, M.J.P. Cullen, M. Farge, C. Foias, F. Golse, J.C.R. Hunt and H.K. Moatt for very useful discussions. 24

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