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Regularity results for the stationary primitive equations of the atmosphere and the ocean
Nonlinear
Analysis,
Theory.
Methods
& Applications,
Vol. 28, No. 2. pp. 289413, 1997 Copyright 0 19% Elsevier Science Ltd Printed in Great Britain. Ail rights reserved 0362-546X/% .$15.00+0.00
0362-546X(95)00154-9
REGULARITY RESULTS FOR THE STATIONARY PRIMITIVE EQUATIONS OF THE ATMOSPHERE AND THE OCEAN MOHAMMED
ZIANE
Department of Mathematics and The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, U.S.A. (Received 16 March 1995; received for publication
17 July 1995)
Key words andphrases: Regularity in domains with corners, Stokes-type systems, primitive equations of the atmosphere, oceanography. 0. INTRODUCTION
The primitive equations of the atmosphere and the large scale equations of the ocean are the equations governing the motion and the state of the atmosphere and the ocean. They also govern the interactions between the atmosphere and the ocean. Recently, Lions et al. started a long range project in order to understand the mechanism of atmospheric and oceanic turbulence and the climate. They derived some mathematical models for the ocean and the atmosphere [l-4]. These models are studied from the numerical view points as well as the mathematical view point (existence and uniqueness of solutions, attractors, etc.). The models studied include, for the atmosphere, the following unknown functions: velocity, pressure and density of the air, temperature and water vapor density; and for the ocean, the unknown functions are: the velocity, pressure and density (with the Boussinesq approximation) of the sea, the temperature, and the salinity. The primitive equations are obtained using the hydrodynamic and thermodynamic equations under Coriolis forces. The vertical momentum equations are approximated with the hydrostatic equations of the atmosphere and the ocean; i.e.
apa
-IL&-= -P%, w
[z
(0.1) = -p”g.
Here, and henceforth, the superscript “a” denotes quantities of the atmosphere, and the superscript “9’ denotes quantities of the ocean. In (O.l), g denotes the acceleration due to gravity, p” and ps denote the pressures, and p’, ps denote the densities. When dealing with the coupled system atmosphere-ocean, phenomenological formulas, near the interface, are used for the momentum transfer flux, heat transfer flux, etc. Lions et al. used (see [3,4]) $9 = p”C;(Va
- uS)(uO - 81”
(0.2)
to represent the horizontal component of the wind force exerted on the surface of the ocean, and, to represent the shear stress of the ocean, they used
a8
shear stress of the ocean = p”vi az. 289
290
The interface condition mechanism
M. ZIANE
is obtained using the well accepted physical law describing the driving shear stress of the ocean = horizontal
wind force.
Here un and us are the horizontal velocities of the air and the sea at the interface. We will give the complete set of the primitive equations after defining the domains and some differential operators. The primary purpose of this paper is to establish the regularity of weak solutions of the stationary primitive equations. The purpose is difficult to obtain for several reasons: (i) the geometry of the domains (presence of corners in the domain occupied by the ocean); (ii) the nonlocal constraint (incompressibility condition) written as an integro-differential equation; (iii) the presence of nonlinearities in the boundary conditions (for the coupled system). In order to understand and overcome the difficulties of the problem, a simplified form of these equations has been studied in [5,6], and the problem of the regularity in that case has been completely solved. We have established regularity results for nonhomogeneous elliptic problems on cylinders [5,6] which we will generalize, in the present paper, to more complicated domains and operators. We also developed a method to handle the nonlocal constraint [6]. We will use it in our present work to establish regularity results for the large scale equations of the ocean and the primitive equations of the coupled system atmosphere-ocean. The regularity of solutions of the linear stationary primitive equations of the atmosphere is studied in [l, lemma 2.81; the domain of the atmosphere has no lateral boundary, i.e. S2 x (0, l), where S2 is the 2D unit sphere. The regularity of solutions to a simplified form of the coupled system atmosphere-ocean is proved in [4, lemma 1.lO], again the domain considered is without lateral boundary, i.e. T2 x (-1, l), where U2 is the 2D-torus (lR2/27rQ2. In both cases the regularity is established using the difference quotient method (see [7]). The regularity results obtained in [l] and [4] can only be used for the atmosphere. The case of the equations of the ocean, for which the domain does have a lateral boundary, remained unsolved up to now. In this paper we prove the regularity of solutions to the linear primitive equations with nonlinear boundary conditions. Various boundary conditions are studied; and in particular the boundary conditions of the large scale equations of the ocean [2] and the coupled system atmosphere-ocean [3,4]. In Section 1 we introduce the primitive equations, following the work of Lions et al. In Section 2 we establish regularity results for systems of elliptic equations in domains with corners and with nonhomogeneous boundary conditions. Finally, Section 3 is devoted to the regularity of solutions of the stationary ocean equations and, also, to the stationary coupled system atmosphere-ocean. 1. THE
PRIMITIVE
EQUATIONS
In this section we give a brief introduction to the primitive equations. We will concentrate on the large scale equations of the ocean; the primitive equations of the atmosphere are obtained in a similar way. In order to obtain the large scale equations of the ocean, Lions et al. started with the following equations:
291
Regularity results for the PEs
the momentum equation, the continuity equation, the equation of diffusion for the temperature T”, the equation of diffusion for salinity S, and the equation of state. They took into account the physical scales and used the following approximations: l Boussinesq approximation: the density differences are neglected except in the Buoyancy term and in the equation of state; l “Linearization” of the domain of the ocean; in spherical coordinates, the metric of the ocean is replaced with dA4” = a2 sin ~9de dp dz, where a is the radius of the earth, and z = r - a is a vertical coordinate denoting the distance of a point to the level of the sea. The change of the metric implies a change of the covariant derivative. Hence, the material derivative is given by d
a
a
ve
--
a
a
V,
A list of differential operators, associated to the new metric, are given below. l The hydrostatic approximation: The vertical momentum equation is replaced by the hydrostatic equation. The approximations above yield the Boussinesq equations, which are not of a CauchyKovalevsky type because of the continuity and the hydrostatic equations. The idea, used by Lions et al. is the integration of these equations with respect to the vertical variable. The integration of the continuity equation gives the vertical components of the velocity in terms of the horizontal one and, also, an incompressibility condition, which is integro-differential. The integration of the hydrostatic equation gives the pressure of the water in terms of the pressure on the surface of the ocean. The primitive equations are au at+Vuv+w~+-kxv+6 a.2
0
f Ro
gradpdz’+gradp,-&Au-&$=% sZ
1
2
aT ;+VnT+w~--&T-&$=0, 2
1
as 1 as -hS--+ V”S + “z + -Rs,, at
i a2s Rszs a7i2= Oy
0
div
vdz,‘=O, -h 0
W; 4 6 2) =
w(v)(t;
8,
vu; 8, 9,~‘) w,
p, Z) = div Z
i p(c 8, q, d =
p,(t;
e,
(p) +
0
z
MC
8, v, 2’) dz’,
p = (1 + /&)j?S - &T + p,S.
2
292
M. ZIANE
Here the parameters Ro, Re,, Re, , RtI, Rt2, RsI and Rs, are Reynolds type numbers; Rossby number: Ro = 1
aZ’
1
P
G=aU’
1
1
av
Re,=zs avT
1
1
PT
Rt,=a
A
1
ah
In the primitive equations, the following notations were used: u is the horizontal velocity, ps is the pressure of the water on the surface of the ocean, T is the temperature, S is the salinity, p is the density, k is the vertical unit vector, f is a forcing term, 6 = gZ/U’, where g is the gravity, Z is a reference for the vertical length, and U is a reference for the horizontal velocity. We will define the domains occupied by the ocean and the atmosphere. Then we give a list of differential operators that will help us to write the primitive equation in a condensed form, and finally, we write the nondimensional primitive equations previously obtained [l-3].
1.1. The geometric domains We denote by M” the subdomain occupied by the atmosphere and MS the subdomain occupied by the sea or the ocean. The domain Ma is defined as follows M” = s2 x (0, l),
(1.1)
where S2 is the unit sphere in Ii?. Islands or continents, Ij (j = 1, . . . , N) are simply connected open subsets of S2 with smooth boundaries such that Jfl& = q5, j,k= l,..., N,j#k, U-2) hereG,j
= 1, . . . , N, denotes the closure of 4 in S2. The water surface M” is given by
MS = S”\(U,“,,Q,
(1.3)
which is an open subset of S2 with a smooth boundary. The depth of the ocean is given by the function h: o, + IR, which is a smooth positive function (at least of class C3) and it is bounded from below by a positive constant do. Now MS is defined as follows
The boundaries of the domains are given as follows (see Fig. 1): l I,, the upper boundary of the atmosphere, is S2 x (1); l I,, the surface of the earth, is U?, 4; a I,, the bottom of the ocean, is h(W); l I,, the lateral boundary of the ocean, is
& = (Bpvaos ((e,9,) X (h(8,9), l
0))
Ii, the interface between the atmosphere and the ocean, is 0’.
(1.5)
Regularity
results
Atmosphere
293
for the PEs
Ma
----t
Fig.
r.
----
1.
The domains Ma and MS are open submanifolds of S2 x m. We denote by A4 any one of the domains Ma or M”. The Riemannian geometry of Mis the same as that of S2 x K?or S2 x (0,l). The tangent space 7;4,n M of A4 at (4, r) E M can be decomposed into the product of TqS2 and IR as follows TcqsE,M = TqS2 x R. (1.6) Therefore,
the Riemannian
metric g, on M is given by
af((q9
0,
(u,
9 w,
vq,
where gsz is the Riemannian gM is given by
(u2
3 w2))
=
w(q;
u19
02)
+
u2 E T2S, wl, 147,E R,
Wl
w2
3
(1.7)
metric on S2. In the spherical coordinates (x1, x2, x3) = (0, q, <),
(gij)
(1.8)
=
where
1.2. Differential
operators
First we recall some basic differential operators for both scalar and vector functions defined on S2, the 2D unit sphere. The horizontal gradient of a scalar function p is gradp = $es The horizontal
+
LaPe
sin8 ap ”
(1.9)
curl of p is I 8~ ap curl p = grad p A e, = sin 0 aq e6 - g,,
where e, and eq are unit vectors in 0 and ~0directions,
respectively.
(1 .lO)
294
M. ZIANE
The horizontal
divergence of a 2D-vector function
i
u = veee + u9ep,
a
(1.11)
div u = - (ve sin 0) + av, . sin ~9ae a The horizontal
curl of u is curlv = div(vAe,)
= 2
i at+ ap ’
+ v,cotgf3 - -sin 0
The horizontal covariant derivatives with respect to v of a 2D vector function function T are V,W
=
Vow
aWe
w and a scalar
a% VP8% t --sinup8 aWe - v,w,cotgO abp 1e, + I vBz + --sin 0 ap + v,w,cotg8 V,,T=
The horizontal
(1.12)
Laplace-Beltrami V2T=
v,aT+
--
VI aT
1
e,;
(1.13) (1.14)
ae sin8 ap’
operators A for scalar and vector functions
divgradT=
i
-sin
a 8 ae
1 a2T sin e ap2’
+ ---
(1.15)
Au = grad div v - curl curl v, and, in spherical coordinates, Au=
t
we can write
V2vg-~$-&]eO+
kvP+G$-&]eP.
(1.16)
In order to write the primitive equations in a condensed form and study them mathematically, we define the second order differential operators L” and L” as follows (1.17) with Ljav = -&AU
-
i a z$&
j = 1,2,3, (1.18) jh where q is the vertical independent variable. Ly , j = 1,2,3 are the dissipative terms for the horizontal velocity, the temperature, and the humidity of the atmosphere, respectively.
-&v
- --.1 a2v &“R,‘u az2’
j = 1,2,3, (1.19) Jh where Lj, j = 1,2,3 are the dissipative terms for the horizontal velocity, the temperature, and the salinity of the ocean reespectively. Ryh, R$, , R$, j = 1,2,3 are positive (Reynolds type) parameters, E is a small positive parameter, which denotes the ratio between the vertical and horizontal scales of the atmosphere-ocean, and kl is a smooth positive function. Ljsv =
Regularity
results
for the PEs
295
We define the operator CJIZby rl 2 0,
f(s’) dtl’,
(1.20)
fW)W,
q < 0,
for any scalar or vector functions such that (1.20) makes sense. The adjoint operator of 3n, in the L2-sense is given by
II 2 0, (1.21) ?j < 0. Now we introduce the following
operators V,u,
NW
+
div(3n*v) $,
rl > 0, (1.22)
= V,,a, - divLJX*v2 i
A(u) = f
II < 0,
au ’
k x u, 0
yU(P)
= vm(K;T”),
(p”)*(u”)
&(TS)
= V31z(K;(-pTTS));
= -ki
div ‘X*u’,
(1.23)
Pus(S) = vwmm,
where I?,, KS and & are positive constants, denoting physical parameters (see [3]), K; is a smooth positive function of q, and k is the unit vector in the vertical direction q. Now we define the first order differential operators P”, P”, Q” and R
(1.24)
0
K4 -f,'2)divm* -f$') -f3(1) -fi2) divK,"
0
0
-fi3'
-fp' -fi4)
-fi3'
,
(1.25)
and R = (A, 0, 0), wheref,“’ and f3(i) (i = 1, . . . , 4) are given functions of space (and possibly time), and K, is a positive constant (a physical parameter).
2%
M. ZIANE
1.3. The primitive
equations
The prognostic variables are u”, T” and q defined in M”, and us, TS and S defined in MS; they are the unknowns of the model. We introduce the following notations (1.26)
us = us, T”, S).
ua = (u”, T”, q),
Now we are ready to state the nondimensional form of the primitive equations (4.10)-(4.13) from Lions et al. [3] below. In Ma
equations.
We write
(1.27)
1
v” dq = 0. 0
In MS
(1.28)
0
v”dij = 0. -h
Equations (1.27) and (1.28) are supplemented with boundary conditions, and, in the case of the coupled model atmosphere-ocean, we also have interface conditions on Ii. In order to give the boundary conditions, we recall some boundary operators. Let L be the operator defined in a domain M c ll?” by Lu=-~,~,$
where u is a scalar function.
I(
a&)$.
We define au/av,
au = -I ‘,i . 1 aij ~cos(v,x,) avL ,
+ ibiaU+c24 J
>
i=l
(1.29)
axi
as follows = -,il
( jl
a,g),i,
(1.30)
J
where v = (v’ , . . . , v”) is the unit inward normal vector to M. We call the derivative in (1.30) a regular normal derivative associated with the operator L. Now we define the canonical derivatives of us, TS and S on I-j, with respect to the dissipative terms for us, TS and S
f -=avs an,; aT” -= an,;
-=as \ an,;
291
Regularity results for the PEs
The set of interface conditions f
1 m, 1 2
is given by a?Y -=-arj
ER;,
atj
as -= \av
0
onr,:
(onr,:
on r,:
on Iii,
e2
on ri,
all
1 a4 --=-
f
g,
e2Rs JF=T’
aT” -=
E R,,
The set of boundary conditions
C51’2 qsv”)
(1.32) g,
on ri,
E ’
on rim
is given by ava -=o, all avs ---=(I, an,;
aTa
-=o, aq
aT”
= 0, an,; aT”
= 0, avL;
us = 0,
\ f
i ER;,
onr,
4 = 0, (1.33)
au” -
=
if,,
att
1 aT” --= ( CR;” ag
g2y
-- 1 a4 = g,. ERL aq
Fig. 2.
as = 0, an,; - as = 0, avL:
(1.34)
298
M. ZIANE
The right-hand
sides of the interface conditions
g, = g’p’lu” - &fIa(u”
are given by
- cw),
g, = gp + g$‘)(T” - TS) + g$2’lu” - WpyTP i g, = g$“~o’lu”- fwlyq
- TS) + g$3)lua - du81yq - q&J, (1.35)
- qe);
for some cy 1 0, where gI”), g3(‘) and gf) (i = 0, 1,2,3) are smooth positive functions, and qe is a positive given function. The right-hand sides of the boundary conditions on I, are given by g,
=
g$"+alorua,
g, = g$“’ + g$“(T= - T,) + g$“‘I uala(To - T,) + gi3’I u”l(q - q&.), g3
(1.36)
= g~“wI”(q - 4eh
where Te is a positive given function. In the case of the large-scale equations of the ocean, we set u” = 0 in the interface conditions. We will also be interested in the case where us = 0 on aw.
2. FUNCTION
SPACES
AND PRELIMINARY
RESULTS
2.1. Function spaces In order to prove existence results, we need to define some function spaces that will allow the application of the Lax-Milgram theorem, as in the case of the Navier-Stokes equations (see [8]), after establishing a variational form of the primitive equations. Let Cm@) be the function space of all C” functions from ainto IR and C”( TM) be the function space of all Cm-vector fields on ii?. Similarly, let C,“(M) be the function space of all C” functions with compact support in h4, and C,“(TM) be the function space of C” vector fields with compact support in M. Let L2( TM’ I TS2) (resp. L2( TM” / TS’)) be the space of the two dimensional L2-vector fields on M” (resp. MS). The definitions of L2(W), L2(A4”), L2(TM”) and L2(TM”) are classical. The scalar products and norms are denoted (* , *) and I . I, respectively (see [8]). If m is a positive integer, we define the Sobolev spaces EP(Mn), fP(TM’) and H”(TM” I TS2) (resp. Hm(MS), Hm(TM”) and Hm(TMS I TS2)) as the completed spaces of C”(lc;i”), C”(Tw), and C”(TII;I” I TS2) (resp. C”(A?), C”(Tti’), and C”(TA? I TS2)) for the respective norms
II~IIP =
[s(
l/2
l,~sm J, j=Jl
3 IviI***vi,yl2
+
ly12
, 2’ ,...,’ k
where Vi = V8,,,, and in the case of spherical coordinates
Vl = vee9
v, = v, 9
and
a v3=G’
>I dX
’
Regularity
Now we define the following C&J@
for the PEs
299
spaces
( TS2) = (v E C”(P
( TS2) ( u is zero near r, U r,),
C~,
= (h E Cm(@) I h is zero near r,],
CE,(Q’)
= (h E C”(Q)
We define some function equations. We set f ~‘3s =
,
results
spaces similar
I
( h is zero near l?,].
to those defined in the study of Navier-Stokes
us E C;“,,,,(CY 1TS2) 1div
V(; =
ua E C”(i2” 1TS2) 1div pw=oj. l v; = C”(cY), v; = C”(sr),
Now we are able to define the following: f q’ = the closure of VT in the H’-norm,
j = 1,2,3;
5’ = the closure of VJ in the &-norm,
j = 1,2,3;
Hj” = the closure of V; in the L2-norm, j = 1,2,3; ( H;S = the closure of VJ in the L2-norm, j = 1,2,3; vu=
vpxv;xvp,
vs=
HS= H;xH,xH;;
H” = HfxH;xH;, H=
We also define the following
H”xH’,
vl”xv;xx;;
v = v x v.
factor operator
A,u=(AOf.P,AOUS), &u
= 63’2(ua, T”, Nq),
A,&
= (d2us,K2 T”, S).
The factor N is a positive number introduced to establish the semi-coercivity of the bilinear form ar( *, *) defined below. We have the following weak formulation of the linear primitive equations with nonlinear boundary conditions.
300
Problem.
M.
ZIANE
Given u. = (ug, ui> E H, andyE L”(0, t; H). Find u = (u”, u”) satisfying
(uEL2(o,r;V)nLm(o,t;H), T {S 0
a,(u) dt < 00,
$4 .u, ~7) + a(u, 6) + e(u, 6) = (A,j; tl), \ult=o
VizEV,
= uo.
The operators a,, a, and e are defined as follows a@, 22) = aI@, u”) + a2(u, 24,I?),
where
+
+
E-lg3/2
g$“(T=
- T”)@”
- F:“)
dl-.I +
re
ri
a2(u, z&17) = c-1d3’2 s ri +
[g$”
+ N
+
s
+ s
re
(g’“‘Iv= - &fla(D” - c%“) - (0’ - di?)) dri g$2)IVu
-
dVsI”(Ta
-
F’“)
+
gi3’lV”
-
~v~I~(@
g$“‘I vu - 6v”l*(~ - qe)4 dy, + e-1c33’2 ri
-
Q,J](F’”
F’)
(g~“$J=JYY- a”) dr, re
@’ - g$l’T, + g$2+.Pl”(P - T,) + g$3’Ivula(q - qJPdr,
+ iv Ir,Cgi”I ~“1 74 - qeM)dr,.
-
dri
Regularity results for the PEs
ecu, fi) = ([EP” + Q” + R]A,u”,
301
21”) + ([&P”
+ R]A,u”,
ti”),
and a,(u) =
(u’ - cWl”l * [Iv” - W12 + IT” - Tsj2 + 1q12]dT; s ri +
r, (Y~(~ * [(u=(~ + (T=(’ + (q12] dT,. s
Finally, we set a(u, ii) = (A,Au, 21),
e(u, 6) = (A&u,
fi),
It was shown [2, lemma 2.41 that the forms a and a, (i = 1,2, 3) are coercive and continuous; and the “H’-existence” of solutions is established in [4]. Similar arguments can be applied to the atmosphere equations (see [l]) and, also, to the coupled system atmosphere-ocean. 2.2. Regularity results for systems of elliptic equations with nonhomogeneous boundary conditions In previous work [5,6], we studied the regularity of nonhomogeneous elliptic boundary value problems on polyhedral curvilinear domains, using results obtained by Dauge [lo, 1l] and Grisvard [12] in the case of homogeneous boundary conditions. We will generalize the previous results to systems of elliptic equations, which include the elliptic operators involved in the large scale equations of the ocean. Recall that the operators LJS, j = 1,2,3 are given by L,sv = -&lU Jh
I
-
a%
j = 1,2,3,
where A is the Laplace-Beltrami operator for vector functions and u is a 2D vector field defined in MS with values in TS2. We also recall the standard Laplacian operator for a 3D vector field u = u,e, + ueee + u+,e,; written in spherical coordinates, Au = (Au),e, + (Au),e, + (Au),e,, with 2u 2 au0 cot 9 (Au), = V2u, - + - - - 2 7Ue-7L r rs ae r - 7 u0 r, sin 8
(Au), = V2u, + ; s (Au), = V2u, - &
+ &
2 r sin 8
au ap ’
2cose au,
-r2 2
(2.12)
ap’ + -@$ r sin
dUg ,
e a9
where, for a scalar function./‘,
v2f=$+;g+--
i r2 sin 0
a sin oaf + ’ a2f ae ( ae> r2@*
(2.13)
We denote by (xi, x2, x,) the Cartesian coordinates in IR3. The following lemma will allow us to work in the Euclidean geometry instead of the Riemannian geometry, defined above.
302
M. ZIANE
LEMMA 2.2. Let u E N1(MS ( ES’) j E {1,2,3]. Then
be such that LJsv belongs to L’(iW” 1 TS’)
for some (2.14)
where CP = U (m - h(m)e,, m), mE09 where e, is the unit vector in the radial direction. Proof. Let u E iY’(M” o = (0, vg, vcp),and
1 TS2) be such that LJv belongs
to L2(M” 1TS2).
We write
r=a+ we have, since ve and vP are in H’(M”),
--1 au,
(2.15)
r ar ’
and also 2cos8 av, 00 +m r2 sin2 0 r sm 19 ap ’ Writing
V,
r2 sin2 e
- $g
gf
E L2(&2”).
(2.16)
(2.16) in the Cartesian coordinates (x1, x2, x3), we obtain (2.17)
Note that the lower order terms that appear in the change of variables are in L2 since v is in H’. Let v E (N’(SY))’ be such that (2.18) If we denote by vr , v2 and v3 the Cartesian components of v and iir,, G2, G,, those of fi, we have qvi=~~$+~)
+-&~=fii~L2(W),
i=
1,2,3.
(2.19)
The same argument of lemma 2.2, applied to the canonical derivatives of us, Ts and S with respect to their respective dissipative terms, yields regular normal derivatives on the domain SY. Hence, the H2-regularity of v reduces to the H2-regularity of the scalar functions vl, v2 and v3. In order to apply the results of regularity of nonhomogeneous elliptic problems on cylinders obtained in [5,6], we will need the following analytic description of the domain as: The set oS is a smooth open submanifold of S2 ; we choose a (?-atlas (&, &)i Sk d N of os such that N
cd=
u co;, k=l
(2.20)
Regularity results for the PEs
and each e.$ is of the form (up to a rotation
i(x, Y9 21,
303
of the sphere) (2.21)
(x9 Y) E Sk, z = dJk(X, YN,
where 5, is a smooth domain in I?, and & is a positive C” function on Sk. Now we prove the following lemma. LEMMA 2.3. Assume that the depth function h is of class C3. Then there exists C3-diffeomorphisms (Ykh s k s N from R3 into R3 such that
(2.22)
ylk@k) = B/cx (0, I), where Bk is a smooth open set in IT?‘, and
(2.23)
Proof. We write Ok in the following
form 0, =
u (x,Y)
(2.24)
&, ~5 Fk
where gk is the geometric line given by d:
_
_
t h(-%
Ys +k(%
Y)@bk/~x)(x,
Y)
y
_
t h(x>
Y,
+k(X,
Y))@~k/@)(x,
Y)
kJ1
+
~v~k~2
J1
’
+
bki2
’
(2.25)
Now let Yk = (YL, Y:, Yz) be given by
y:(x, y, z) = x1 = x + (z - $k) 2
(x, y),
i
I
y&&y,z) =Y’ =Y+ k - ik)!.$ (x,-t’% %xX,Y,Z)
(2.26)
= 2’ = -&y$)d~. 3
3
k
Under the mapping Yk, the line “ek is transformed into ((xl, y’, z’); 0 < z’ C 1). Therefore, Y,@k) = B, X (0, l), where B, iS a smooth (C3) open subset of R3. The fact that Yk iS Of ChSS C3 is obvious. Finally, we are in a position to state the following propositions 2.2 and 2.3, corollaries of propositions 3.1-3.4 of [6].
which are, thanks to lemmas
M. ZIANE
304
PROPOSITION
ub E &-1’2(Ib solution of
2.1. Assume that the function h is of class C3. Let Vi E Ha-1’2(~J 1TS’), 1 ZS2) and uI E Ht-1’2(I, 1 7S2) with 3 5 01 < 2. If 0 E H1(Ms 1 TS’) is a
I
LjSv=f V =
\ where f E L2(M” 1 TS’). Then
in MS, on os,
Vi
(2.27)
u = t+,
on
v = v,
on L
rb,
?.IE H”(i-Y 17x2). Moreover,
(2.28)
if VI E @‘2+e(rl
1 ?‘s2),
pi E H03’2’e(ri
I
TS2)
and
vb E Hi’2+e(rb
17X2),
(2.29)
then u E H2(MS 17’S’). 2.2. Assume that the function h is of class C3. Let tli E H~-3’2(~s 1 7X2), vb E @-3’2(rb 1Ts2), y E Hz-3’2(Ij I 7X2), 3/2 I (Y < 2, and f E L2(M” 1TS2). Then, if u E H1(it4’ I 7S2) is a solution of
PROPOSITION
f Ljv+
v=f
in MS,
a0
on 09,
av
(2.30) onrb,
av
on 6,
\ we have u E wyMe Moreover, if Ui E HJ’2+e(Ii v E H2(MS ) 732). PROPOSITION vb
E &-1’2(rb
v E Hr(kP
I 7X2),
( TS2).
ub E #‘2+e(rb
I 7S2),
(2.31) and
v, E Hi’2+c(I’,
1 KS2), then
2.3. Assume that the function h is of class C3. Let ui E H;-3’2(ws 1 TS2), 1 m2), VI E fp2 (I, I TS’), 3/2 I (Y < 2 and f E L2(Ms I ZS2). Then if I 7S2) is a solution of L,v=f in M”,
a0 5 = ui
on 09, (2.32)
v = vb
on
rb,
v =
on
r,,
q
Regularity
results
for the PEs
305
we have u E H”(kP Moreover, if Ui E Hi”+‘(ri u E H2(A4” ( TS2).
1 2X2),
1 TS2).
U, E Hi’2+E(Ib
(2.33)
1 KS2), and
U, E Hz’2+e(r[ ) 7’S2), then
In the following proposition, we need to make a supplementary assumption on the domain occupied by the ocean. We encountered the assumption when we studied a mixed DirichletNeumann problem ([5], see also [ll]). We will assume that the bottom boundary of the ocean forms a dihedral angle equal to 7r/2 with the lateral boundary of the ocean grad h(0,~) = 0.
(*)
2.4. Assume that the domain MS satisfies the condition (*) and h is of class C3. Let vi E H;-3’2(oS 1 TS2), ub E H;-3’2(r,, 1TS2), and q E H,*-“2(r, 1TS2) with 3/2 5 (Y < 2. If u E H’(M” 1 ES2) is a solution of
PROPOSITION
I
Lju=f
in M”, on wS, (2.34)
au \
on rb
u = VI
onG,
u E H”(kP
1732).
then, for f E L2(Ms I TS’), we have Moreover, if ui E Hi’2+E(w2 I TS2), u E lY2(MS 1TS2). 3. REGULARITY
OF
(2.35)
ub E H;‘2+E(rb I 2X2) and u, E H;‘2+E(rl
SOLUTIONS PRIMITIVE
OF
THE
LINEAR
I 2X2), then
STATIONARY
EQUATIONS
The present section contains our main results. In 3.1 we recall the regularity result for the primitive equations of the atmosphere [l]. In 3.2 we establish the regularity of solutions of the linearized stationary large scale equations of the ocean, and finally, in 3.3 we prove the H2-regularity of solutions of the linear stationary primitive equations of the coupled system atmosphere-ocean. First, we take the linear stationary equations associated to (1.27) and (1.28). We have [EP” + Q” + R]u” + L”u, + VY = f a 1
div
us dq = 0, 0
in Ma, (3.1)
306
M.
ZIANE
and [6&P” + R]d
in MS,
+ &LSuS + VYS = 0
0
div s -We,@ Equations (3.1) are supplemented equations of the atmosphere “alone”)
vSdq = 0,
with the following
v”=0, iiij= aq0 arq = 0 on S x (0) aq
Equations (3.2) are supplemented equations of the ocean “alone”) ai9 az=Vf,
and
(3.3)
-
-&T,
z=
TS = 0,
s=o
us = 0 ,
aTs o a”L; = ’
-=as an,;
onr
(primitive
aTa
1us= v;,
!!Ccvf
conditions
= --cy$T” on S x (1). atl with the following boundary conditions
aT”
We will also consider the following
boundary
on S x (0, l),
i
i
(3.2)
in I-i.
as - 0
z-
(large scale
on ri,
on rb,
o
boundary condition
(3.4)
on r,.
for vs
avs
us=0 onrl. (3.5) -=v; onr,, 84; The existence of solutions to problem (3.1), (3.3), problem (3.2), (3.4) and problem (3.2), (3.5) is obtained thanks to the Lax-Milgram theorem. We have, in each case of boundary conditions vu E J&VP 1TS2), Y” E L2(S2), T”, and q E N’(A4”) (3.6) and us E H1(MS 1TS2), Y’” E L2(ri), TS, and S E H’(AP). (3.7) az
1
is
We note that the differential operators EP’ + Q” + R and 6&P” + R are of order I 1. Therefore, thanks to (3.6) and (3.7), we have (&PO + Q” + R)u”
The regularity problems
E L2(M0 1 TS2)
of the solution
and
(&Ps
E L2(M2 1TS2).
to (3.1), (3.3) reduces to the regularity L’flf
(3.8)
of the following
+ VY” = fp, 1
div and
+ R)d
v” dq = 0,
(3.9)
J0 L; T” = fi”, L’j9 = fP,
(3.9’)
307
Regularity results for the PEs
with the boundary condition (3.3). Also, the regularity of the solution to (3.2), (3.4) (resp. (3.5)) reduces to the regularity of the following problems
(3.10)
0
div
lJ”d.2 = 0, s -h
and
L; TS = f;,
(3.10)
L;s = f;, with the boundary condition 3.1. The primitive
(3.4) (resp. (3.5)).
equations of the atmosphere
The linearized stationary primitive equations of the atmosphere are given by (3.1), (3.3), which have been transformed into equations (3.9), (3.3). The Hz-regularity of T” and q is classical. Using the classical method of difference quotients [7], the Hz-regularity of ua was obtained [l, lemma 2.81, it was shown that if ff E Hk(Ma Therefore,
1TS”), k 1 0, then u’ E Hk+‘(Mo
we have the following
LEMMA 3.1. Letff
(3.1), (3.3) withf”
1TS2),
(3.11)
lemma.
E Hk(Mn 1TS’),ft E Hk(M”),ff = (fp,ft, f;"). Then
ua E Hk+‘(Ma
1 TS’) and ‘P’” E Hk+‘(S2).
Y’” E Hk+‘(S2),
E Hk(M”), T”,
and
k 1 0, and u’ be a solution to q E Hk+2(MO).
(3.12)
3.2. The large-scale equations of the ocean The linearized stationary equations for the ocean are given by (3.2) with either boundary conditions (3.4) or (3.5). We have transformed equations (3.2) into equations (3.10). In the study of regularity of elliptic problems on cyclinders [5,6], we encountered the following angle condition: To obtain H2 regularity, the angle at a corner, where Dirichlet and Neumann boundary conditions are mixed, must be n/2. Therefore, the regularity of T” and S is obtained thanks to propositions 2.3 and 2.4; we need, of course, to make the supplementary condition (*), which is grad h(r3, p) = 0 whenever (0, q) E X,. However, for the regularity of v’, in the case of boundary condition (3.4), we will need h to be only of class C3. For the boundary condition (3.5), we need to make the supplementary condition (*) as in the case of the Laplacian operator. We will concentrate on the boundary condition (3.5) and establish the H2-regularity of v2. The H2 regularity, in the case of the boundary condition (3.4) under the assumption (*), is obtained in a similar fashion. Now we prove the following theorem.
308
M.ZIANE
THEOREM 3.1. , Assume that h E C3(ri) and let II,? E H;-3’2(rj 1ZS2), vi E III,*-“~(~~ 1ZS2) with 3/2 I a > 2, andf: E L2(Ms 17S2). Then if (us, Yy”) E H1(MS 1ZS2) x L2(o”) is a solution of
L;vs + v\ys = j-i”
in MS,
0
div
s -h
vsdz=o
in f+i,
onTi,
us= v;
aUs
-=vf az
(3.13) us=0
Onrb,
on&,
we have
v EHa-l(q).
us E Ha(Ms ( 7x2), Moreover, we have
(3.14)
if f E L2(Ms ) KS’), vf E I-II~‘~+‘(~~ I 2X2) and vg E Hi’2 +e(I’, I TS2), e > 0, then us E H2(MI
TS2)
YS E H’(iw).
and
(3.15)
Proof. First, we reduce the nonhomogeneous boundary condition to a homogeneous one. To this end, we consider the following nonhomogeneous elliptic problem L,?
= 0
in MS,
K = V? on r.IS ’ 1 az Thanks to proposition
P=
V;
fiS=O
onr,,
(3.16) Onrl.
3.3, we have
v EW(iw (TS2),
15a<2.
(3.17)
Let v*’ = us - P. The system of equations satisfied by v*’ is given by L; v*# + VYS = f-i”
in M”,
0
div
0
v*’ dz = -div s -h
-h
i?
az
= 0
on r. 19
3 d2Efza-‘(ri),
u+S=O
Onrb,
v*S=O
(3.18) Onrl.
Second, let V be the unique solution of
1 $AV= Jh v=o
VYs on
on ri,
(3.19)
ar,,
where A is the Laplace-Beltrami operator on S2. Since Ys E L’(r,), Lax-Milgram theorem, V E H& I TS2).
we have, according to the
309
Regularity results for the PEs
Let wS = u*’ - V. Since V is independent
of z, we have
(L,wS = fi” aws -=0 i a2 Thanks to proposition
onTi,
d=-V
w”=O
onr,,
(3.20)
on&.
2.3, we have (3.21)
ws E H3’2(Ms 1 TS2), and 0
div
(3.22)
ws & E fP2(ri). -h
We write -div
ws dz + div
u*‘dz = div(h(8, (p)V),
(3.23)
/3 = min(a, 3/2).
(3.24)
-h
-h
which implies that div V E HB-‘(ri), We rewrite the system of equations (3.24)
satisfied by V with the supplementary
-$w+
VP= 0
result
in ri,
lh
(3.25)
div V E HP-‘(r.) I 3
iv=0
regularity
on Xi.
The regularity result of the Stokes problem on the smooth submanifold (see [Ei]). We have YS E fF(r.) I * U E fP(& 1 TS2),
ri of S2 is well known (3.26)
Hence, V E H[-“(ri 1 7S2) (note that V = 0 on X, and /? - E < t). We return to problem (3.20) and conclude, with proposition 2.3, that ws E Hp+1’2-qMs ( TS2).
(3.27)
Thanks to (3.23) and (3.27), we have div V E HP-1’2-E(ri). Next, we use the Stokes problem
(3.28)
(3.25) with (3.28) and conclude that
V E H13+1’2-e’(ri I TS2)
and
YS E fP-1/2-E'(ru).
(3.29)
Now note that 1 c p 5 3/2, hence, p + $ - E’ > $ for E’ small enough (E’ < /3 - 1). Therefore, au*s av*s E L2(rb ) 7x2). (3.30) E L2(rj I ~32) and a2 rb a2 ri
310
M.
ZIANE
We have shown that II*’ = ws + V, where ws E Hp+1’2-E’(MS
1TS’) and YE HB+1’2-e’(ri
1 TS2).
(3.31)
For the remainder of the proof, we assume that the depth function is constant; the general case can be done by transforming the domain into a right cylinder (see [5,6]). We chose our assumption to avoid complicated formulas but the idea is exactly the same (see [6]). We integrate equations (3.18) with respect to z and obtain, thanks to (3.30), -$
AV’ + 6h V’P E L2(rj 17’S2), lh
(3.32)
div Vs E Ii’( V” = 0
on Xi
According to the regularity of the Stokes problem on ri, we have Ys E H’(ri). Therefore, VY’ E L2(ri / TS2). We move the gradient term to the right-hand side in (3.18) and obtain an elliptic problem for tr*‘. Thanks to prop osition 2.3, we conclude that u*’ E N2(MS 1rS2). Now, with vs = u*’ + ii, we conclude that us E N”(A4’ I 2X2). The last step, in the proof is, to get N2-regularity when the boundary conditions on r, and ri are of the form on ri
aus
= -giuSla an,;
and
and recall that us = 0 on r, . We have, by the Lax-Milgram us I ri E P2(ri
I m2)
and
aus
- = wSlq, anLs,
on r,
theorem, us E N’(M”
uslrb
E
W2(rbI ~2).
(3.33) I 7X2). Hence, (3.34)
The argument given above (with CY= 2 - E’) implies that (3.35)
us E E12-qkfs ) TS2). Therefore, -
aus
an,;
Moreover,
ri
E H312-e’(ri
I m2)
atf
and
an,;
and
an
rb
E H3’2-qrj)
1 TS2).
(3.36)
since us = 0 on rb, we have -
aus
an,;
ar,
= 0
atf L;
ar,
= 0,
(3.37)
we can extend (auS/anLs) Ir. and (a$/an,$ Ir, to S2. The H2-regularity is obtained as in the case of zero boundary cond&ions; we can extend (W/&z,;) Iri to S2 using the extension of uslri and then reduce the problem to zero boundary conditions on ri. We omit the details. The N2-regularity for the other boundary conditions can be obtained in a similar fashion. We state, for the sake of completeness, the following theorem.
Regularity results for the PEs
311
THEOREM 3.2. Assume that h E C3(~i) satisfies grad h(8, p) = 0 for (0, p) E & and f: E L2(M2 ( TS2). Let vs E H’(M 1 TS2) and YysE L2(& 1TS2) satisfy the following system of equations LTV”+ vYs=f; in M”, 0 (3.38) v”&=O in &, div i s -h and the boundary condition
x=O
an,;
0nrUlY b
I9
US= 0 on r1.
(3.39)
Y’” E H’(MS).
(3.40)
Then us E H2(M 1TS2) Remark
3.1. The H2-regularity
3.3. The primitive
and
of TS and S is an easy corollary of propositions
(2.1)-(2.4).
equations of the coupled atmosphere-ocean
In this subsection we establish the H2-regularity associated to (1.27), (1.28); i.e.
of solutions of the linear stationary problem
[&Pa + Q” + R]u4 + L”u” + VYa = f”
in Ma, (3.41)
1
div
v”drj = 0
in S2
s0 and [d&P” + R]uS + ~5L’u’ + d VY’ = fs
in M”, (3.42)
0
div
vSdz = 0 s -h
in
ri.
Equations (3.41) and (3.42) are supplemented with boundary conditions and an interface condition, which makes the coupling between the two equations. The boundary conditions for v’ and us on r, , r, and &, are the same as those considered in Sections 3.1 and 3.2. The interface condition on & is given by (1.32); i.e.
1 ava PI2 a(w) -= -=marl -E2Rs, atl
g, &’
(3.43)
with g, = g~O’Iva - Bvsla(va - BUS), where gi”’ is a smooth function of 8 and p. The boundary condition by (1.35); i.e.
(3.44) for vu on S2\ri = r, is given (3.45)
M. ZIANE
312
We assume that Q = 0; however, the Hz-regularity can be obtained for 0 I 01 5 1 using Sobolev imbedding theorems and the case (Y = 0 (see [6] for more details). It is easy to see, using Lax-Milgram’s theorem that u= E N’(AP
(3.46)
us E H1(MS 1 79).
1 TP),
Hence
Therefore,
(3.47)
V”lri E H1’2(ri 1 TS2)*
?J’lr,, we have (in the case (Y = 0)
(3.48) and 6 II2 atf ---=-((VU-
g:
c2R& aq
cw) EfP2(rj 1TS2).
&
(3.49)
We consider now the equations satisfied by v” and us as decoupled with the Neumann condition on Ii. Thanks to theorem 3.2, we conclude that us E H2(M8 ( TS2).
and
V’ E H2(ri X (0, 1) ( TS’)
(3.50)
We apply theorem 3.1 to u“ in the domain I, x (0, 1) instead to MS and conclude that (3.51)
lf E H2(A4” 1 TS2). Now we have, of course, v E H’(S2) We have proved the following
v EP(r-).
and
(3.52)
theorem.
3.3. Assume that f: E L2(M” I TS2) and f; E L2(Ml TS2), and let (tr”, ‘I’9 (us, ‘P) be such that
THEOREM
v” E H’(M”
\y” E L2(S2),
1TS2),
us E H1(MS 1 TS2),
w” E L2(rJ
and (3.53)
are solutions of Ls,vS+ VYS=f,s
LL;v” + VY = j-p,
0
1
div 1 with interface condition
s0
va dr/ = 0,
div 1
(3.43) and one of the following ( us = 0
va = 0
u”&=O
(3.54)
s -h
boundary conditions
onrrurb, onr,,
1 ava --=g % atl
(3.55) on r,,
313
Regularity results for the PEs
or
x = -Vh vS an,; ’
vs= 0 onr,, ?.I== 0
-_1 avazv'I &R'f, au
on r,,
on I?, (3 56)
on L,
with h E C3(ri). In (3.56), we assume that Vh(B, ~0) = 0 on Xi. Then, we have v“ E H’(M” Remark
1TS’),
‘I”” E H’(S*),
us E H2(MS 1 TS*)
3.2. We also have, under the condition T” E H*(iU”),
Q E H*(M”),
and
‘I’” E H’(Ii).
Vh(8, p) = 0 for (0, ~0)E Xi, the following
S E H*(M”)
and
TS E H*(M”).
Remark 3.3. Thanks to Sobolev imbedding theorem, and an imbedding theorem due to Zolesio [13], which is stated as theorem 6.1 in [6]; we can prove the H*-regularity of the stationary linear primitive equations with nonlinear boundary conditions (with 0 5 (Y 5 1 in the boundary conditions). The proof follows the lines of the proof of theorem 6.3 in [6]. We omit the details. Acknowledgements-The author is deeply thankful to Professor Roger Temam not only for suggesting the problem, but also for the valuable comments pertaining to this work. The appreciation goes, also, to Professor Shouhang Wang for the critical reading of the manuscript. This work was partially supported by the National Science Foundation under Grant NSF-DMS9400615, by the Department of Energy under Grant DOE-DE-FG02-92ER25120, and by the Research Fund of Indiana University. REFERENCES 1. LIONS J.-L., TEMAM R. & WANG S., New formulations of the primitive equations of the atmosphere and applications, Nonlinearity 5, 237-288 (1992). 2. LIONS J.-L., TEMAM R. &WANG S., On the equations of large-scale ocean, Nonlinearity 5, 1007-1053 (1992). 3. LIONS J.-L., TEMAM R. &WANG S., Models of the coupled atmosphere and ocean (CA0 I), in Computational Mechanics Advances (Edited by J. T. Oden), Vol. 1, pp. 5-54. Elsevier, Amsterdam (1993). 4. LIONS J.-L., TEMAM R. & WANG S., Mathematical study of the coupled models of atmosphere and ocean (CA0 III), J. Math. pures uppl. (to appear). 5. ZIANE M. B., Regularity results for Stokes-type systemsrelated to climatology, Appl. Math. Lett. 8,53-58 (1995). 6. ZIANE M. B., Regularity results for Stokes-type systems, Applic. Analysis (to appear). 7. AGMON S., DOUGLIS A. & NIRENBERG L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Communs pure appl. Math 12, 623-723 (1959). 8. TEMAM R., Navier-Stokes Equations, 3rd revised edition, Studies in Mathematics and its Applications, Vol. 2. North-Holland, Amsterdam (1984). 9. AUBIN T., Nonlinear Analysis on Manifolds, Monge-Ampere Equations. Springer, New York (1982). 10. DAUGE M., Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics, Vol. 1341. Springer, Berlin (1988). 11. DAUGE M., Problemes mixtes pour le Laplacien dans des domaines polyedraux courbes, C. r. Acad. Sci. Paris 309, 553-558 (1980). 12. GRISVARD P., Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Vol. 24. Pitman, London. 13. ZOLESIO J. L., Multiplication dans les espaces de Besov, Prov. R. Sot. Edinb. (1977).
Analysis,
Theory.
Methods
& Applications,
Vol. 28, No. 2. pp. 289413, 1997 Copyright 0 19% Elsevier Science Ltd Printed in Great Britain. Ail rights reserved 0362-546X/% .$15.00+0.00
0362-546X(95)00154-9
REGULARITY RESULTS FOR THE STATIONARY PRIMITIVE EQUATIONS OF THE ATMOSPHERE AND THE OCEAN MOHAMMED
ZIANE
Department of Mathematics and The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, U.S.A. (Received 16 March 1995; received for publication
17 July 1995)
Key words andphrases: Regularity in domains with corners, Stokes-type systems, primitive equations of the atmosphere, oceanography. 0. INTRODUCTION
The primitive equations of the atmosphere and the large scale equations of the ocean are the equations governing the motion and the state of the atmosphere and the ocean. They also govern the interactions between the atmosphere and the ocean. Recently, Lions et al. started a long range project in order to understand the mechanism of atmospheric and oceanic turbulence and the climate. They derived some mathematical models for the ocean and the atmosphere [l-4]. These models are studied from the numerical view points as well as the mathematical view point (existence and uniqueness of solutions, attractors, etc.). The models studied include, for the atmosphere, the following unknown functions: velocity, pressure and density of the air, temperature and water vapor density; and for the ocean, the unknown functions are: the velocity, pressure and density (with the Boussinesq approximation) of the sea, the temperature, and the salinity. The primitive equations are obtained using the hydrodynamic and thermodynamic equations under Coriolis forces. The vertical momentum equations are approximated with the hydrostatic equations of the atmosphere and the ocean; i.e.
apa
-IL&-= -P%, w
[z
(0.1) = -p”g.
Here, and henceforth, the superscript “a” denotes quantities of the atmosphere, and the superscript “9’ denotes quantities of the ocean. In (O.l), g denotes the acceleration due to gravity, p” and ps denote the pressures, and p’, ps denote the densities. When dealing with the coupled system atmosphere-ocean, phenomenological formulas, near the interface, are used for the momentum transfer flux, heat transfer flux, etc. Lions et al. used (see [3,4]) $9 = p”C;(Va
- uS)(uO - 81”
(0.2)
to represent the horizontal component of the wind force exerted on the surface of the ocean, and, to represent the shear stress of the ocean, they used
a8
shear stress of the ocean = p”vi az. 289
290
The interface condition mechanism
M. ZIANE
is obtained using the well accepted physical law describing the driving shear stress of the ocean = horizontal
wind force.
Here un and us are the horizontal velocities of the air and the sea at the interface. We will give the complete set of the primitive equations after defining the domains and some differential operators. The primary purpose of this paper is to establish the regularity of weak solutions of the stationary primitive equations. The purpose is difficult to obtain for several reasons: (i) the geometry of the domains (presence of corners in the domain occupied by the ocean); (ii) the nonlocal constraint (incompressibility condition) written as an integro-differential equation; (iii) the presence of nonlinearities in the boundary conditions (for the coupled system). In order to understand and overcome the difficulties of the problem, a simplified form of these equations has been studied in [5,6], and the problem of the regularity in that case has been completely solved. We have established regularity results for nonhomogeneous elliptic problems on cylinders [5,6] which we will generalize, in the present paper, to more complicated domains and operators. We also developed a method to handle the nonlocal constraint [6]. We will use it in our present work to establish regularity results for the large scale equations of the ocean and the primitive equations of the coupled system atmosphere-ocean. The regularity of solutions of the linear stationary primitive equations of the atmosphere is studied in [l, lemma 2.81; the domain of the atmosphere has no lateral boundary, i.e. S2 x (0, l), where S2 is the 2D unit sphere. The regularity of solutions to a simplified form of the coupled system atmosphere-ocean is proved in [4, lemma 1.lO], again the domain considered is without lateral boundary, i.e. T2 x (-1, l), where U2 is the 2D-torus (lR2/27rQ2. In both cases the regularity is established using the difference quotient method (see [7]). The regularity results obtained in [l] and [4] can only be used for the atmosphere. The case of the equations of the ocean, for which the domain does have a lateral boundary, remained unsolved up to now. In this paper we prove the regularity of solutions to the linear primitive equations with nonlinear boundary conditions. Various boundary conditions are studied; and in particular the boundary conditions of the large scale equations of the ocean [2] and the coupled system atmosphere-ocean [3,4]. In Section 1 we introduce the primitive equations, following the work of Lions et al. In Section 2 we establish regularity results for systems of elliptic equations in domains with corners and with nonhomogeneous boundary conditions. Finally, Section 3 is devoted to the regularity of solutions of the stationary ocean equations and, also, to the stationary coupled system atmosphere-ocean. 1. THE
PRIMITIVE
EQUATIONS
In this section we give a brief introduction to the primitive equations. We will concentrate on the large scale equations of the ocean; the primitive equations of the atmosphere are obtained in a similar way. In order to obtain the large scale equations of the ocean, Lions et al. started with the following equations:
291
Regularity results for the PEs
the momentum equation, the continuity equation, the equation of diffusion for the temperature T”, the equation of diffusion for salinity S, and the equation of state. They took into account the physical scales and used the following approximations: l Boussinesq approximation: the density differences are neglected except in the Buoyancy term and in the equation of state; l “Linearization” of the domain of the ocean; in spherical coordinates, the metric of the ocean is replaced with dA4” = a2 sin ~9de dp dz, where a is the radius of the earth, and z = r - a is a vertical coordinate denoting the distance of a point to the level of the sea. The change of the metric implies a change of the covariant derivative. Hence, the material derivative is given by d
a
a
ve
--
a
a
V,
A list of differential operators, associated to the new metric, are given below. l The hydrostatic approximation: The vertical momentum equation is replaced by the hydrostatic equation. The approximations above yield the Boussinesq equations, which are not of a CauchyKovalevsky type because of the continuity and the hydrostatic equations. The idea, used by Lions et al. is the integration of these equations with respect to the vertical variable. The integration of the continuity equation gives the vertical components of the velocity in terms of the horizontal one and, also, an incompressibility condition, which is integro-differential. The integration of the hydrostatic equation gives the pressure of the water in terms of the pressure on the surface of the ocean. The primitive equations are au at+Vuv+w~+-kxv+6 a.2
0
f Ro
gradpdz’+gradp,-&Au-&$=% sZ
1
2
aT ;+VnT+w~--&T-&$=0, 2
1
as 1 as -hS--+ V”S + “z + -Rs,, at
i a2s Rszs a7i2= Oy
0
div
vdz,‘=O, -h 0
W; 4 6 2) =
w(v)(t;
8,
vu; 8, 9,~‘) w,
p, Z) = div Z
i p(c 8, q, d =
p,(t;
e,
(p) +
0
z
MC
8, v, 2’) dz’,
p = (1 + /&)j?S - &T + p,S.
2
292
M. ZIANE
Here the parameters Ro, Re,, Re, , RtI, Rt2, RsI and Rs, are Reynolds type numbers; Rossby number: Ro = 1
aZ’
1
P
G=aU’
1
1
av
Re,=zs avT
1
1
PT
Rt,=a
A
1
ah
In the primitive equations, the following notations were used: u is the horizontal velocity, ps is the pressure of the water on the surface of the ocean, T is the temperature, S is the salinity, p is the density, k is the vertical unit vector, f is a forcing term, 6 = gZ/U’, where g is the gravity, Z is a reference for the vertical length, and U is a reference for the horizontal velocity. We will define the domains occupied by the ocean and the atmosphere. Then we give a list of differential operators that will help us to write the primitive equation in a condensed form, and finally, we write the nondimensional primitive equations previously obtained [l-3].
1.1. The geometric domains We denote by M” the subdomain occupied by the atmosphere and MS the subdomain occupied by the sea or the ocean. The domain Ma is defined as follows M” = s2 x (0, l),
(1.1)
where S2 is the unit sphere in Ii?. Islands or continents, Ij (j = 1, . . . , N) are simply connected open subsets of S2 with smooth boundaries such that Jfl& = q5, j,k= l,..., N,j#k, U-2) hereG,j
= 1, . . . , N, denotes the closure of 4 in S2. The water surface M” is given by
MS = S”\(U,“,,Q,
(1.3)
which is an open subset of S2 with a smooth boundary. The depth of the ocean is given by the function h: o, + IR, which is a smooth positive function (at least of class C3) and it is bounded from below by a positive constant do. Now MS is defined as follows
The boundaries of the domains are given as follows (see Fig. 1): l I,, the upper boundary of the atmosphere, is S2 x (1); l I,, the surface of the earth, is U?, 4; a I,, the bottom of the ocean, is h(W); l I,, the lateral boundary of the ocean, is
& = (Bpvaos ((e,9,) X (h(8,9), l
0))
Ii, the interface between the atmosphere and the ocean, is 0’.
(1.5)
Regularity
results
Atmosphere
293
for the PEs
Ma
----t
Fig.
r.
----
1.
The domains Ma and MS are open submanifolds of S2 x m. We denote by A4 any one of the domains Ma or M”. The Riemannian geometry of Mis the same as that of S2 x K?or S2 x (0,l). The tangent space 7;4,n M of A4 at (4, r) E M can be decomposed into the product of TqS2 and IR as follows TcqsE,M = TqS2 x R. (1.6) Therefore,
the Riemannian
metric g, on M is given by
af((q9
0,
(u,
9 w,
vq,
where gsz is the Riemannian gM is given by
(u2
3 w2))
=
w(q;
u19
02)
+
u2 E T2S, wl, 147,E R,
Wl
w2
3
(1.7)
metric on S2. In the spherical coordinates (x1, x2, x3) = (0, q, <),
(gij)
(1.8)
=
where
1.2. Differential
operators
First we recall some basic differential operators for both scalar and vector functions defined on S2, the 2D unit sphere. The horizontal gradient of a scalar function p is gradp = $es The horizontal
+
LaPe
sin8 ap ”
(1.9)
curl of p is I 8~ ap curl p = grad p A e, = sin 0 aq e6 - g,,
where e, and eq are unit vectors in 0 and ~0directions,
respectively.
(1 .lO)
294
M. ZIANE
The horizontal
divergence of a 2D-vector function
i
u = veee + u9ep,
a
(1.11)
div u = - (ve sin 0) + av, . sin ~9ae a The horizontal
curl of u is curlv = div(vAe,)
= 2
i at+ ap ’
+ v,cotgf3 - -sin 0
The horizontal covariant derivatives with respect to v of a 2D vector function function T are V,W
=
Vow
aWe
w and a scalar
a% VP8% t --sinup8 aWe - v,w,cotgO abp 1e, + I vBz + --sin 0 ap + v,w,cotg8 V,,T=
The horizontal
(1.12)
Laplace-Beltrami V2T=
v,aT+
--
VI aT
1
e,;
(1.13) (1.14)
ae sin8 ap’
operators A for scalar and vector functions
divgradT=
i
-sin
a 8 ae
1 a2T sin e ap2’
+ ---
(1.15)
Au = grad div v - curl curl v, and, in spherical coordinates, Au=
t
we can write
V2vg-~$-&]eO+
kvP+G$-&]eP.
(1.16)
In order to write the primitive equations in a condensed form and study them mathematically, we define the second order differential operators L” and L” as follows (1.17) with Ljav = -&AU
-
i a z$&
j = 1,2,3, (1.18) jh where q is the vertical independent variable. Ly , j = 1,2,3 are the dissipative terms for the horizontal velocity, the temperature, and the humidity of the atmosphere, respectively.
-&v
- --.1 a2v &“R,‘u az2’
j = 1,2,3, (1.19) Jh where Lj, j = 1,2,3 are the dissipative terms for the horizontal velocity, the temperature, and the salinity of the ocean reespectively. Ryh, R$, , R$, j = 1,2,3 are positive (Reynolds type) parameters, E is a small positive parameter, which denotes the ratio between the vertical and horizontal scales of the atmosphere-ocean, and kl is a smooth positive function. Ljsv =
Regularity
results
for the PEs
295
We define the operator CJIZby rl 2 0,
f(s’) dtl’,
(1.20)
fW)W,
q < 0,
for any scalar or vector functions such that (1.20) makes sense. The adjoint operator of 3n, in the L2-sense is given by
II 2 0, (1.21) ?j < 0. Now we introduce the following
operators V,u,
NW
+
div(3n*v) $,
rl > 0, (1.22)
= V,,a, - divLJX*v2 i
A(u) = f
II < 0,
au ’
k x u, 0
yU(P)
= vm(K;T”),
(p”)*(u”)
&(TS)
= V31z(K;(-pTTS));
= -ki
div ‘X*u’,
(1.23)
Pus(S) = vwmm,
where I?,, KS and & are positive constants, denoting physical parameters (see [3]), K; is a smooth positive function of q, and k is the unit vector in the vertical direction q. Now we define the first order differential operators P”, P”, Q” and R
(1.24)
0
K4 -f,'2)divm* -f$') -f3(1) -fi2) divK,"
0
0
-fi3'
-fp' -fi4)
-fi3'
,
(1.25)
and R = (A, 0, 0), wheref,“’ and f3(i) (i = 1, . . . , 4) are given functions of space (and possibly time), and K, is a positive constant (a physical parameter).
2%
M. ZIANE
1.3. The primitive
equations
The prognostic variables are u”, T” and q defined in M”, and us, TS and S defined in MS; they are the unknowns of the model. We introduce the following notations (1.26)
us = us, T”, S).
ua = (u”, T”, q),
Now we are ready to state the nondimensional form of the primitive equations (4.10)-(4.13) from Lions et al. [3] below. In Ma
equations.
We write
(1.27)
1
v” dq = 0. 0
In MS
(1.28)
0
v”dij = 0. -h
Equations (1.27) and (1.28) are supplemented with boundary conditions, and, in the case of the coupled model atmosphere-ocean, we also have interface conditions on Ii. In order to give the boundary conditions, we recall some boundary operators. Let L be the operator defined in a domain M c ll?” by Lu=-~,~,$
where u is a scalar function.
I(
a&)$.
We define au/av,
au = -I ‘,i . 1 aij ~cos(v,x,) avL ,
+ ibiaU+c24 J
>
i=l
(1.29)
axi
as follows = -,il
( jl
a,g),i,
(1.30)
J
where v = (v’ , . . . , v”) is the unit inward normal vector to M. We call the derivative in (1.30) a regular normal derivative associated with the operator L. Now we define the canonical derivatives of us, TS and S on I-j, with respect to the dissipative terms for us, TS and S
f -=avs an,; aT” -= an,;
-=as \ an,;
291
Regularity results for the PEs
The set of interface conditions f
1 m, 1 2
is given by a?Y -=-arj
ER;,
atj
as -= \av
0
onr,:
(onr,:
on r,:
on Iii,
e2
on ri,
all
1 a4 --=-
f
g,
e2Rs JF=T’
aT” -=
E R,,
The set of boundary conditions
C51’2 qsv”)
(1.32) g,
on ri,
E ’
on rim
is given by ava -=o, all avs ---=(I, an,;
aTa
-=o, aq
aT”
= 0, an,; aT”
= 0, avL;
us = 0,
\ f
i ER;,
onr,
4 = 0, (1.33)
au” -
=
if,,
att
1 aT” --= ( CR;” ag
g2y
-- 1 a4 = g,. ERL aq
Fig. 2.
as = 0, an,; - as = 0, avL:
(1.34)
298
M. ZIANE
The right-hand
sides of the interface conditions
g, = g’p’lu” - &fIa(u”
are given by
- cw),
g, = gp + g$‘)(T” - TS) + g$2’lu” - WpyTP i g, = g$“~o’lu”- fwlyq
- TS) + g$3)lua - du81yq - q&J, (1.35)
- qe);
for some cy 1 0, where gI”), g3(‘) and gf) (i = 0, 1,2,3) are smooth positive functions, and qe is a positive given function. The right-hand sides of the boundary conditions on I, are given by g,
=
g$"+alorua,
g, = g$“’ + g$“(T= - T,) + g$“‘I uala(To - T,) + gi3’I u”l(q - q&.), g3
(1.36)
= g~“wI”(q - 4eh
where Te is a positive given function. In the case of the large-scale equations of the ocean, we set u” = 0 in the interface conditions. We will also be interested in the case where us = 0 on aw.
2. FUNCTION
SPACES
AND PRELIMINARY
RESULTS
2.1. Function spaces In order to prove existence results, we need to define some function spaces that will allow the application of the Lax-Milgram theorem, as in the case of the Navier-Stokes equations (see [8]), after establishing a variational form of the primitive equations. Let Cm@) be the function space of all C” functions from ainto IR and C”( TM) be the function space of all Cm-vector fields on ii?. Similarly, let C,“(M) be the function space of all C” functions with compact support in h4, and C,“(TM) be the function space of C” vector fields with compact support in M. Let L2( TM’ I TS2) (resp. L2( TM” / TS’)) be the space of the two dimensional L2-vector fields on M” (resp. MS). The definitions of L2(W), L2(A4”), L2(TM”) and L2(TM”) are classical. The scalar products and norms are denoted (* , *) and I . I, respectively (see [8]). If m is a positive integer, we define the Sobolev spaces EP(Mn), fP(TM’) and H”(TM” I TS2) (resp. Hm(MS), Hm(TM”) and Hm(TMS I TS2)) as the completed spaces of C”(lc;i”), C”(Tw), and C”(TII;I” I TS2) (resp. C”(A?), C”(Tti’), and C”(TA? I TS2)) for the respective norms
II~IIP =
[s(
l/2
l,~sm J, j=Jl
3 IviI***vi,yl2
+
ly12
, 2’ ,...,’ k
where Vi = V8,,,, and in the case of spherical coordinates
Vl = vee9
v, = v, 9
and
a v3=G’
>I dX
’
Regularity
Now we define the following C&J@
for the PEs
299
spaces
( TS2) = (v E C”(P
( TS2) ( u is zero near r, U r,),
C~,
= (h E Cm(@) I h is zero near r,],
CE,(Q’)
= (h E C”(Q)
We define some function equations. We set f ~‘3s =
,
results
spaces similar
I
( h is zero near l?,].
to those defined in the study of Navier-Stokes
us E C;“,,,,(CY 1TS2) 1div
V(; =
ua E C”(i2” 1TS2) 1div pw=oj. l v; = C”(cY), v; = C”(sr),
Now we are able to define the following: f q’ = the closure of VT in the H’-norm,
j = 1,2,3;
5’ = the closure of VJ in the &-norm,
j = 1,2,3;
Hj” = the closure of V; in the L2-norm, j = 1,2,3; ( H;S = the closure of VJ in the L2-norm, j = 1,2,3; vu=
vpxv;xvp,
vs=
HS= H;xH,xH;;
H” = HfxH;xH;, H=
We also define the following
H”xH’,
vl”xv;xx;;
v = v x v.
factor operator
A,u=(AOf.P,AOUS), &u
= 63’2(ua, T”, Nq),
A,&
= (d2us,K2 T”, S).
The factor N is a positive number introduced to establish the semi-coercivity of the bilinear form ar( *, *) defined below. We have the following weak formulation of the linear primitive equations with nonlinear boundary conditions.
300
Problem.
M.
ZIANE
Given u. = (ug, ui> E H, andyE L”(0, t; H). Find u = (u”, u”) satisfying
(uEL2(o,r;V)nLm(o,t;H), T {S 0
a,(u) dt < 00,
$4 .u, ~7) + a(u, 6) + e(u, 6) = (A,j; tl), \ult=o
VizEV,
= uo.
The operators a,, a, and e are defined as follows a@, 22) = aI@, u”) + a2(u, 24,I?),
where
+
+
E-lg3/2
g$“(T=
- T”)@”
- F:“)
dl-.I +
re
ri
a2(u, z&17) = c-1d3’2 s ri +
[g$”
+ N
+
s
+ s
re
(g’“‘Iv= - &fla(D” - c%“) - (0’ - di?)) dri g$2)IVu
-
dVsI”(Ta
-
F’“)
+
gi3’lV”
-
~v~I~(@
g$“‘I vu - 6v”l*(~ - qe)4 dy, + e-1c33’2 ri
-
Q,J](F’”
F’)
(g~“$J=JYY- a”) dr, re
@’ - g$l’T, + g$2+.Pl”(P - T,) + g$3’Ivula(q - qJPdr,
+ iv Ir,Cgi”I ~“1 74 - qeM)dr,.
-
dri
Regularity results for the PEs
ecu, fi) = ([EP” + Q” + R]A,u”,
301
21”) + ([&P”
+ R]A,u”,
ti”),
and a,(u) =
(u’ - cWl”l * [Iv” - W12 + IT” - Tsj2 + 1q12]dT; s ri +
r, (Y~(~ * [(u=(~ + (T=(’ + (q12] dT,. s
Finally, we set a(u, ii) = (A,Au, 21),
e(u, 6) = (A&u,
fi),
It was shown [2, lemma 2.41 that the forms a and a, (i = 1,2, 3) are coercive and continuous; and the “H’-existence” of solutions is established in [4]. Similar arguments can be applied to the atmosphere equations (see [l]) and, also, to the coupled system atmosphere-ocean. 2.2. Regularity results for systems of elliptic equations with nonhomogeneous boundary conditions In previous work [5,6], we studied the regularity of nonhomogeneous elliptic boundary value problems on polyhedral curvilinear domains, using results obtained by Dauge [lo, 1l] and Grisvard [12] in the case of homogeneous boundary conditions. We will generalize the previous results to systems of elliptic equations, which include the elliptic operators involved in the large scale equations of the ocean. Recall that the operators LJS, j = 1,2,3 are given by L,sv = -&lU Jh
I
-
a%
j = 1,2,3,
where A is the Laplace-Beltrami operator for vector functions and u is a 2D vector field defined in MS with values in TS2. We also recall the standard Laplacian operator for a 3D vector field u = u,e, + ueee + u+,e,; written in spherical coordinates, Au = (Au),e, + (Au),e, + (Au),e,, with 2u 2 au0 cot 9 (Au), = V2u, - + - - - 2 7Ue-7L r rs ae r - 7 u0 r, sin 8
(Au), = V2u, + ; s (Au), = V2u, - &
+ &
2 r sin 8
au ap ’
2cose au,
-r2 2
(2.12)
ap’ + -@$ r sin
dUg ,
e a9
where, for a scalar function./‘,
v2f=$+;g+--
i r2 sin 0
a sin oaf + ’ a2f ae ( ae> r2@*
(2.13)
We denote by (xi, x2, x,) the Cartesian coordinates in IR3. The following lemma will allow us to work in the Euclidean geometry instead of the Riemannian geometry, defined above.
302
M. ZIANE
LEMMA 2.2. Let u E N1(MS ( ES’) j E {1,2,3]. Then
be such that LJsv belongs to L’(iW” 1 TS’)
for some (2.14)
where CP = U (m - h(m)e,, m), mE09 where e, is the unit vector in the radial direction. Proof. Let u E iY’(M” o = (0, vg, vcp),and
1 TS2) be such that LJv belongs
to L2(M” 1TS2).
We write
r=a+ we have, since ve and vP are in H’(M”),
--1 au,
(2.15)
r ar ’
and also 2cos8 av, 00 +m r2 sin2 0 r sm 19 ap ’ Writing
V,
r2 sin2 e
- $g
gf
E L2(&2”).
(2.16)
(2.16) in the Cartesian coordinates (x1, x2, x3), we obtain (2.17)
Note that the lower order terms that appear in the change of variables are in L2 since v is in H’. Let v E (N’(SY))’ be such that (2.18) If we denote by vr , v2 and v3 the Cartesian components of v and iir,, G2, G,, those of fi, we have qvi=~~$+~)
+-&~=fii~L2(W),
i=
1,2,3.
(2.19)
The same argument of lemma 2.2, applied to the canonical derivatives of us, Ts and S with respect to their respective dissipative terms, yields regular normal derivatives on the domain SY. Hence, the H2-regularity of v reduces to the H2-regularity of the scalar functions vl, v2 and v3. In order to apply the results of regularity of nonhomogeneous elliptic problems on cylinders obtained in [5,6], we will need the following analytic description of the domain as: The set oS is a smooth open submanifold of S2 ; we choose a (?-atlas (&, &)i Sk d N of os such that N
cd=
u co;, k=l
(2.20)
Regularity results for the PEs
and each e.$ is of the form (up to a rotation
i(x, Y9 21,
303
of the sphere) (2.21)
(x9 Y) E Sk, z = dJk(X, YN,
where 5, is a smooth domain in I?, and & is a positive C” function on Sk. Now we prove the following lemma. LEMMA 2.3. Assume that the depth function h is of class C3. Then there exists C3-diffeomorphisms (Ykh s k s N from R3 into R3 such that
(2.22)
ylk@k) = B/cx (0, I), where Bk is a smooth open set in IT?‘, and
(2.23)
Proof. We write Ok in the following
form 0, =
u (x,Y)
(2.24)
&, ~5 Fk
where gk is the geometric line given by d:
_
_
t h(-%
Ys +k(%
Y)@bk/~x)(x,
Y)
y
_
t h(x>
Y,
+k(X,
Y))@~k/@)(x,
Y)
kJ1
+
~v~k~2
J1
’
+
bki2
’
(2.25)
Now let Yk = (YL, Y:, Yz) be given by
y:(x, y, z) = x1 = x + (z - $k) 2
(x, y),
i
I
y&&y,z) =Y’ =Y+ k - ik)!.$ (x,-t’% %xX,Y,Z)
(2.26)
= 2’ = -&y$)d~. 3
3
k
Under the mapping Yk, the line “ek is transformed into ((xl, y’, z’); 0 < z’ C 1). Therefore, Y,@k) = B, X (0, l), where B, iS a smooth (C3) open subset of R3. The fact that Yk iS Of ChSS C3 is obvious. Finally, we are in a position to state the following propositions 2.2 and 2.3, corollaries of propositions 3.1-3.4 of [6].
which are, thanks to lemmas
M. ZIANE
304
PROPOSITION
ub E &-1’2(Ib solution of
2.1. Assume that the function h is of class C3. Let Vi E Ha-1’2(~J 1TS’), 1 ZS2) and uI E Ht-1’2(I, 1 7S2) with 3 5 01 < 2. If 0 E H1(Ms 1 TS’) is a
I
LjSv=f V =
\ where f E L2(M” 1 TS’). Then
in MS, on os,
Vi
(2.27)
u = t+,
on
v = v,
on L
rb,
?.IE H”(i-Y 17x2). Moreover,
(2.28)
if VI E @‘2+e(rl
1 ?‘s2),
pi E H03’2’e(ri
I
TS2)
and
vb E Hi’2+e(rb
17X2),
(2.29)
then u E H2(MS 17’S’). 2.2. Assume that the function h is of class C3. Let tli E H~-3’2(~s 1 7X2), vb E @-3’2(rb 1Ts2), y E Hz-3’2(Ij I 7X2), 3/2 I (Y < 2, and f E L2(M” 1TS2). Then, if u E H1(it4’ I 7S2) is a solution of
PROPOSITION
f Ljv+
v=f
in MS,
a0
on 09,
av
(2.30) onrb,
av
on 6,
\ we have u E wyMe Moreover, if Ui E HJ’2+e(Ii v E H2(MS ) 732). PROPOSITION vb
E &-1’2(rb
v E Hr(kP
I 7X2),
( TS2).
ub E #‘2+e(rb
I 7S2),
(2.31) and
v, E Hi’2+c(I’,
1 KS2), then
2.3. Assume that the function h is of class C3. Let ui E H;-3’2(ws 1 TS2), 1 m2), VI E fp2 (I, I TS’), 3/2 I (Y < 2 and f E L2(Ms I ZS2). Then if I 7S2) is a solution of L,v=f in M”,
a0 5 = ui
on 09, (2.32)
v = vb
on
rb,
v =
on
r,,
q
Regularity
results
for the PEs
305
we have u E H”(kP Moreover, if Ui E Hi”+‘(ri u E H2(A4” ( TS2).
1 2X2),
1 TS2).
U, E Hi’2+E(Ib
(2.33)
1 KS2), and
U, E Hz’2+e(r[ ) 7’S2), then
In the following proposition, we need to make a supplementary assumption on the domain occupied by the ocean. We encountered the assumption when we studied a mixed DirichletNeumann problem ([5], see also [ll]). We will assume that the bottom boundary of the ocean forms a dihedral angle equal to 7r/2 with the lateral boundary of the ocean grad h(0,~) = 0.
(*)
2.4. Assume that the domain MS satisfies the condition (*) and h is of class C3. Let vi E H;-3’2(oS 1 TS2), ub E H;-3’2(r,, 1TS2), and q E H,*-“2(r, 1TS2) with 3/2 5 (Y < 2. If u E H’(M” 1 ES2) is a solution of
PROPOSITION
I
Lju=f
in M”, on wS, (2.34)
au \
on rb
u = VI
onG,
u E H”(kP
1732).
then, for f E L2(Ms I TS’), we have Moreover, if ui E Hi’2+E(w2 I TS2), u E lY2(MS 1TS2). 3. REGULARITY
OF
(2.35)
ub E H;‘2+E(rb I 2X2) and u, E H;‘2+E(rl
SOLUTIONS PRIMITIVE
OF
THE
LINEAR
I 2X2), then
STATIONARY
EQUATIONS
The present section contains our main results. In 3.1 we recall the regularity result for the primitive equations of the atmosphere [l]. In 3.2 we establish the regularity of solutions of the linearized stationary large scale equations of the ocean, and finally, in 3.3 we prove the H2-regularity of solutions of the linear stationary primitive equations of the coupled system atmosphere-ocean. First, we take the linear stationary equations associated to (1.27) and (1.28). We have [EP” + Q” + R]u” + L”u, + VY = f a 1
div
us dq = 0, 0
in Ma, (3.1)
306
M.
ZIANE
and [6&P” + R]d
in MS,
+ &LSuS + VYS = 0
0
div s -We,@ Equations (3.1) are supplemented equations of the atmosphere “alone”)
vSdq = 0,
with the following
v”=0, iiij= aq0 arq = 0 on S x (0) aq
Equations (3.2) are supplemented equations of the ocean “alone”) ai9 az=Vf,
and
(3.3)
-
-&T,
z=
TS = 0,
s=o
us = 0 ,
aTs o a”L; = ’
-=as an,;
onr
(primitive
aTa
1us= v;,
!!Ccvf
conditions
= --cy$T” on S x (1). atl with the following boundary conditions
aT”
We will also consider the following
boundary
on S x (0, l),
i
i
(3.2)
in I-i.
as - 0
z-
(large scale
on ri,
on rb,
o
boundary condition
(3.4)
on r,.
for vs
avs
us=0 onrl. (3.5) -=v; onr,, 84; The existence of solutions to problem (3.1), (3.3), problem (3.2), (3.4) and problem (3.2), (3.5) is obtained thanks to the Lax-Milgram theorem. We have, in each case of boundary conditions vu E J&VP 1TS2), Y” E L2(S2), T”, and q E N’(A4”) (3.6) and us E H1(MS 1TS2), Y’” E L2(ri), TS, and S E H’(AP). (3.7) az
1
is
We note that the differential operators EP’ + Q” + R and 6&P” + R are of order I 1. Therefore, thanks to (3.6) and (3.7), we have (&PO + Q” + R)u”
The regularity problems
E L2(M0 1 TS2)
of the solution
and
(&Ps
E L2(M2 1TS2).
to (3.1), (3.3) reduces to the regularity L’flf
(3.8)
of the following
+ VY” = fp, 1
div and
+ R)d
v” dq = 0,
(3.9)
J0 L; T” = fi”, L’j9 = fP,
(3.9’)
307
Regularity results for the PEs
with the boundary condition (3.3). Also, the regularity of the solution to (3.2), (3.4) (resp. (3.5)) reduces to the regularity of the following problems
(3.10)
0
div
lJ”d.2 = 0, s -h
and
L; TS = f;,
(3.10)
L;s = f;, with the boundary condition 3.1. The primitive
(3.4) (resp. (3.5)).
equations of the atmosphere
The linearized stationary primitive equations of the atmosphere are given by (3.1), (3.3), which have been transformed into equations (3.9), (3.3). The Hz-regularity of T” and q is classical. Using the classical method of difference quotients [7], the Hz-regularity of ua was obtained [l, lemma 2.81, it was shown that if ff E Hk(Ma Therefore,
1TS”), k 1 0, then u’ E Hk+‘(Mo
we have the following
LEMMA 3.1. Letff
(3.1), (3.3) withf”
1TS2),
(3.11)
lemma.
E Hk(Mn 1TS’),ft E Hk(M”),ff = (fp,ft, f;"). Then
ua E Hk+‘(Ma
1 TS’) and ‘P’” E Hk+‘(S2).
Y’” E Hk+‘(S2),
E Hk(M”), T”,
and
k 1 0, and u’ be a solution to q E Hk+2(MO).
(3.12)
3.2. The large-scale equations of the ocean The linearized stationary equations for the ocean are given by (3.2) with either boundary conditions (3.4) or (3.5). We have transformed equations (3.2) into equations (3.10). In the study of regularity of elliptic problems on cyclinders [5,6], we encountered the following angle condition: To obtain H2 regularity, the angle at a corner, where Dirichlet and Neumann boundary conditions are mixed, must be n/2. Therefore, the regularity of T” and S is obtained thanks to propositions 2.3 and 2.4; we need, of course, to make the supplementary condition (*), which is grad h(r3, p) = 0 whenever (0, q) E X,. However, for the regularity of v’, in the case of boundary condition (3.4), we will need h to be only of class C3. For the boundary condition (3.5), we need to make the supplementary condition (*) as in the case of the Laplacian operator. We will concentrate on the boundary condition (3.5) and establish the H2-regularity of v2. The H2 regularity, in the case of the boundary condition (3.4) under the assumption (*), is obtained in a similar fashion. Now we prove the following theorem.
308
M.ZIANE
THEOREM 3.1. , Assume that h E C3(ri) and let II,? E H;-3’2(rj 1ZS2), vi E III,*-“~(~~ 1ZS2) with 3/2 I a > 2, andf: E L2(Ms 17S2). Then if (us, Yy”) E H1(MS 1ZS2) x L2(o”) is a solution of
L;vs + v\ys = j-i”
in MS,
0
div
s -h
vsdz=o
in f+i,
onTi,
us= v;
aUs
-=vf az
(3.13) us=0
Onrb,
on&,
we have
v EHa-l(q).
us E Ha(Ms ( 7x2), Moreover, we have
(3.14)
if f E L2(Ms ) KS’), vf E I-II~‘~+‘(~~ I 2X2) and vg E Hi’2 +e(I’, I TS2), e > 0, then us E H2(MI
TS2)
YS E H’(iw).
and
(3.15)
Proof. First, we reduce the nonhomogeneous boundary condition to a homogeneous one. To this end, we consider the following nonhomogeneous elliptic problem L,?
= 0
in MS,
K = V? on r.IS ’ 1 az Thanks to proposition
P=
V;
fiS=O
onr,,
(3.16) Onrl.
3.3, we have
v EW(iw (TS2),
15a<2.
(3.17)
Let v*’ = us - P. The system of equations satisfied by v*’ is given by L; v*# + VYS = f-i”
in M”,
0
div
0
v*’ dz = -div s -h
-h
i?
az
= 0
on r. 19
3 d2Efza-‘(ri),
u+S=O
Onrb,
v*S=O
(3.18) Onrl.
Second, let V be the unique solution of
1 $AV= Jh v=o
VYs on
on ri,
(3.19)
ar,,
where A is the Laplace-Beltrami operator on S2. Since Ys E L’(r,), Lax-Milgram theorem, V E H& I TS2).
we have, according to the
309
Regularity results for the PEs
Let wS = u*’ - V. Since V is independent
of z, we have
(L,wS = fi” aws -=0 i a2 Thanks to proposition
onTi,
d=-V
w”=O
onr,,
(3.20)
on&.
2.3, we have (3.21)
ws E H3’2(Ms 1 TS2), and 0
div
(3.22)
ws & E fP2(ri). -h
We write -div
ws dz + div
u*‘dz = div(h(8, (p)V),
(3.23)
/3 = min(a, 3/2).
(3.24)
-h
-h
which implies that div V E HB-‘(ri), We rewrite the system of equations (3.24)
satisfied by V with the supplementary
-$w+
VP= 0
result
in ri,
lh
(3.25)
div V E HP-‘(r.) I 3
iv=0
regularity
on Xi.
The regularity result of the Stokes problem on the smooth submanifold (see [Ei]). We have YS E fF(r.) I * U E fP(& 1 TS2),
ri of S2 is well known (3.26)
Hence, V E H[-“(ri 1 7S2) (note that V = 0 on X, and /? - E < t). We return to problem (3.20) and conclude, with proposition 2.3, that ws E Hp+1’2-qMs ( TS2).
(3.27)
Thanks to (3.23) and (3.27), we have div V E HP-1’2-E(ri). Next, we use the Stokes problem
(3.28)
(3.25) with (3.28) and conclude that
V E H13+1’2-e’(ri I TS2)
and
YS E fP-1/2-E'(ru).
(3.29)
Now note that 1 c p 5 3/2, hence, p + $ - E’ > $ for E’ small enough (E’ < /3 - 1). Therefore, au*s av*s E L2(rb ) 7x2). (3.30) E L2(rj I ~32) and a2 rb a2 ri
310
M.
ZIANE
We have shown that II*’ = ws + V, where ws E Hp+1’2-E’(MS
1TS’) and YE HB+1’2-e’(ri
1 TS2).
(3.31)
For the remainder of the proof, we assume that the depth function is constant; the general case can be done by transforming the domain into a right cylinder (see [5,6]). We chose our assumption to avoid complicated formulas but the idea is exactly the same (see [6]). We integrate equations (3.18) with respect to z and obtain, thanks to (3.30), -$
AV’ + 6h V’P E L2(rj 17’S2), lh
(3.32)
div Vs E Ii’( V” = 0
on Xi
According to the regularity of the Stokes problem on ri, we have Ys E H’(ri). Therefore, VY’ E L2(ri / TS2). We move the gradient term to the right-hand side in (3.18) and obtain an elliptic problem for tr*‘. Thanks to prop osition 2.3, we conclude that u*’ E N2(MS 1rS2). Now, with vs = u*’ + ii, we conclude that us E N”(A4’ I 2X2). The last step, in the proof is, to get N2-regularity when the boundary conditions on r, and ri are of the form on ri
aus
= -giuSla an,;
and
and recall that us = 0 on r, . We have, by the Lax-Milgram us I ri E P2(ri
I m2)
and
aus
- = wSlq, anLs,
on r,
theorem, us E N’(M”
uslrb
E
W2(rbI ~2).
(3.33) I 7X2). Hence, (3.34)
The argument given above (with CY= 2 - E’) implies that (3.35)
us E E12-qkfs ) TS2). Therefore, -
aus
an,;
Moreover,
ri
E H312-e’(ri
I m2)
atf
and
an,;
and
an
rb
E H3’2-qrj)
1 TS2).
(3.36)
since us = 0 on rb, we have -
aus
an,;
ar,
= 0
atf L;
ar,
= 0,
(3.37)
we can extend (auS/anLs) Ir. and (a$/an,$ Ir, to S2. The H2-regularity is obtained as in the case of zero boundary cond&ions; we can extend (W/&z,;) Iri to S2 using the extension of uslri and then reduce the problem to zero boundary conditions on ri. We omit the details. The N2-regularity for the other boundary conditions can be obtained in a similar fashion. We state, for the sake of completeness, the following theorem.
Regularity results for the PEs
311
THEOREM 3.2. Assume that h E C3(~i) satisfies grad h(8, p) = 0 for (0, p) E & and f: E L2(M2 ( TS2). Let vs E H’(M 1 TS2) and YysE L2(& 1TS2) satisfy the following system of equations LTV”+ vYs=f; in M”, 0 (3.38) v”&=O in &, div i s -h and the boundary condition
x=O
an,;
0nrUlY b
I9
US= 0 on r1.
(3.39)
Y’” E H’(MS).
(3.40)
Then us E H2(M 1TS2) Remark
3.1. The H2-regularity
3.3. The primitive
and
of TS and S is an easy corollary of propositions
(2.1)-(2.4).
equations of the coupled atmosphere-ocean
In this subsection we establish the H2-regularity associated to (1.27), (1.28); i.e.
of solutions of the linear stationary problem
[&Pa + Q” + R]u4 + L”u” + VYa = f”
in Ma, (3.41)
1
div
v”drj = 0
in S2
s0 and [d&P” + R]uS + ~5L’u’ + d VY’ = fs
in M”, (3.42)
0
div
vSdz = 0 s -h
in
ri.
Equations (3.41) and (3.42) are supplemented with boundary conditions and an interface condition, which makes the coupling between the two equations. The boundary conditions for v’ and us on r, , r, and &, are the same as those considered in Sections 3.1 and 3.2. The interface condition on & is given by (1.32); i.e.
1 ava PI2 a(w) -= -=marl -E2Rs, atl
g, &’
(3.43)
with g, = g~O’Iva - Bvsla(va - BUS), where gi”’ is a smooth function of 8 and p. The boundary condition by (1.35); i.e.
(3.44) for vu on S2\ri = r, is given (3.45)
M. ZIANE
312
We assume that Q = 0; however, the Hz-regularity can be obtained for 0 I 01 5 1 using Sobolev imbedding theorems and the case (Y = 0 (see [6] for more details). It is easy to see, using Lax-Milgram’s theorem that u= E N’(AP
(3.46)
us E H1(MS 1 79).
1 TP),
Hence
Therefore,
(3.47)
V”lri E H1’2(ri 1 TS2)*
?J’lr,, we have (in the case (Y = 0)
(3.48) and 6 II2 atf ---=-((VU-
g:
c2R& aq
cw) EfP2(rj 1TS2).
&
(3.49)
We consider now the equations satisfied by v” and us as decoupled with the Neumann condition on Ii. Thanks to theorem 3.2, we conclude that us E H2(M8 ( TS2).
and
V’ E H2(ri X (0, 1) ( TS’)
(3.50)
We apply theorem 3.1 to u“ in the domain I, x (0, 1) instead to MS and conclude that (3.51)
lf E H2(A4” 1 TS2). Now we have, of course, v E H’(S2) We have proved the following
v EP(r-).
and
(3.52)
theorem.
3.3. Assume that f: E L2(M” I TS2) and f; E L2(Ml TS2), and let (tr”, ‘I’9 (us, ‘P) be such that
THEOREM
v” E H’(M”
\y” E L2(S2),
1TS2),
us E H1(MS 1 TS2),
w” E L2(rJ
and (3.53)
are solutions of Ls,vS+ VYS=f,s
LL;v” + VY = j-p,
0
1
div 1 with interface condition
s0
va dr/ = 0,
div 1
(3.43) and one of the following ( us = 0
va = 0
u”&=O
(3.54)
s -h
boundary conditions
onrrurb, onr,,
1 ava --=g % atl
(3.55) on r,,
313
Regularity results for the PEs
or
x = -Vh vS an,; ’
vs= 0 onr,, ?.I== 0
-_1 avazv'I &R'f, au
on r,,
on I?, (3 56)
on L,
with h E C3(ri). In (3.56), we assume that Vh(B, ~0) = 0 on Xi. Then, we have v“ E H’(M” Remark
1TS’),
‘I”” E H’(S*),
us E H2(MS 1 TS*)
3.2. We also have, under the condition T” E H*(iU”),
Q E H*(M”),
and
‘I’” E H’(Ii).
Vh(8, p) = 0 for (0, ~0)E Xi, the following
S E H*(M”)
and
TS E H*(M”).
Remark 3.3. Thanks to Sobolev imbedding theorem, and an imbedding theorem due to Zolesio [13], which is stated as theorem 6.1 in [6]; we can prove the H*-regularity of the stationary linear primitive equations with nonlinear boundary conditions (with 0 5 (Y 5 1 in the boundary conditions). The proof follows the lines of the proof of theorem 6.3 in [6]. We omit the details. Acknowledgements-The author is deeply thankful to Professor Roger Temam not only for suggesting the problem, but also for the valuable comments pertaining to this work. The appreciation goes, also, to Professor Shouhang Wang for the critical reading of the manuscript. This work was partially supported by the National Science Foundation under Grant NSF-DMS9400615, by the Department of Energy under Grant DOE-DE-FG02-92ER25120, and by the Research Fund of Indiana University. REFERENCES 1. LIONS J.-L., TEMAM R. & WANG S., New formulations of the primitive equations of the atmosphere and applications, Nonlinearity 5, 237-288 (1992). 2. LIONS J.-L., TEMAM R. &WANG S., On the equations of large-scale ocean, Nonlinearity 5, 1007-1053 (1992). 3. LIONS J.-L., TEMAM R. &WANG S., Models of the coupled atmosphere and ocean (CA0 I), in Computational Mechanics Advances (Edited by J. T. Oden), Vol. 1, pp. 5-54. Elsevier, Amsterdam (1993). 4. LIONS J.-L., TEMAM R. & WANG S., Mathematical study of the coupled models of atmosphere and ocean (CA0 III), J. Math. pures uppl. (to appear). 5. ZIANE M. B., Regularity results for Stokes-type systemsrelated to climatology, Appl. Math. Lett. 8,53-58 (1995). 6. ZIANE M. B., Regularity results for Stokes-type systems, Applic. Analysis (to appear). 7. AGMON S., DOUGLIS A. & NIRENBERG L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Communs pure appl. Math 12, 623-723 (1959). 8. TEMAM R., Navier-Stokes Equations, 3rd revised edition, Studies in Mathematics and its Applications, Vol. 2. North-Holland, Amsterdam (1984). 9. AUBIN T., Nonlinear Analysis on Manifolds, Monge-Ampere Equations. Springer, New York (1982). 10. DAUGE M., Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics, Vol. 1341. Springer, Berlin (1988). 11. DAUGE M., Problemes mixtes pour le Laplacien dans des domaines polyedraux courbes, C. r. Acad. Sci. Paris 309, 553-558 (1980). 12. GRISVARD P., Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Vol. 24. Pitman, London. 13. ZOLESIO J. L., Multiplication dans les espaces de Besov, Prov. R. Sot. Edinb. (1977).