On he uniqueness and egula i y o he P imi i e Equa ions
imposing addi ional aniso opic egula i y.
F. Guill´en-Gonz´alez♣∗†
, M.A. Rod ´ıguez-Bellido♠
♣Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa,
Ap do. 1160, 41080 Se illa, Spain. e-mail: guillen@nume .us.es
♠Dp o. de Ma em´a ica Aplicada I, E. T. S. de A qui ec u a, Uni e sidad de Se illa,
A da. Reina Me cedes, 2, 41012 Se illa, Spain. e-mail: [email p o ec ed]
Abs ac
In his no e, we p o e ha gi en ua weak solu ion o he P imi i e Equa ions, impos-
ing an addi ional condi ion on he e ical de i a i e o he eloci y u(conc e ely ∂zu∈
L∞(0,T;L2(Ω)) ∩L2(0,T;H1(Ω))), hen wo diffe en esul s hold; namely, uniqueness o
weak solu ion (any weak solu ion associa ed o he same da a ha umus coincide wi h u)
and global in ime s ong egula i y o u(wi hou “smallness assump ions” on he da a).
Bo h esul s a e p o ed when ei he Di ichle o Robin ype condi ions on he bo om
a e conside ed. In he las case, a domain wi h a s ic ly bounded om below dep h has o
be imposed, e en o he uniqueness esul .
Key wo ds: Weak-s ong uniqueness, P imi i e Equa ions, aniso opic es ima es, s ong solu-
ion
1 In oduc ion
The P imi i e Equa ions a e ela ed wi h a g ea a ie y o geophysical luids [9, 10, 12].
This sys em can be deduced asymp o ically om he Na ie -S okes equa ions wi h aniso opic
(eddy) iscosi y, when he aspec a io (quo ien be ween e ical and ho izon al cha ac e is ic
dimensions) ends o ze o [1, 2, 3]. The 3D sys em can be w i en as ollows: o ind u:
(0,T)×Ω→R2, he ho izon al eloci y ield, and ps:(0,T)×S→R, a su ace po en ial
unc ion (in ol ing he p essu e), e i ying:
(PE)

















∂ u−νH∆Hu−νz∂2
zzu+αu⊥+
+(u·∇H)u+u3∂zu+∇Hps=Fin (0,T)×Ω,
∇H·�u�=0 in(0,T)×S,
u| =0 =u0in Ω,
νz∂zu|Γs=Υ,u|Γl=0in (0,T),
ei he u|Γb=0o ((νH∇Hu,ν
z∂zu)·n+βu)|Γb=0in (0,T),
whe e u3is he e ical eloci y, ha becomes now a diagnos ic a iable, depending on he
ho izon al eloci y uas ollows:
u3(x,z)=�0
z∇H·u(x,s)ds. (1)
We conside he domain Ω={(x, y, z)=(x,z)∈R3/x∈S, −h(x)<z<0},wi hS⊂R2a
bounded open se ( he su ace) and h:S→Ra non-nega i e con inuous unc ion ( he dep h).
∗Co esponding au ho
†The i s au ho has been pa ially inanced by he p ojec BFM2000-1317.
1
2 THE MAIN RESULTS. 2
I s bounda y is decomposed as ∂Ω=Γ
b∪Γl∪Γswhe e Γs={(x,0) : x∈S}is he su ace,
Γl={(x,z)∈R3:x∈∂S, −h(x)<z<0}a e he side-walls and Γb={(x,z)∈R3:x∈
S, z =−h(x)}is he bo om.
We ha e deno ed by �u�( ;x)=�0
−h(x)u( ;x,z)dz he e ical in eg a ion o u. The ho -
izon al ope a o s ∆Hand ∇H ep esen ∂2
xx +∂2
yy and (∂x,∂
y) espec i ely. The cons an s
νH,ν
z>0 a e he iscosi y coefficien s and nis he ou wa d no mal ec o on he bo om.
The ex e nal o ces a e da a deno ed by F:(0,T)×Ω→R2, and αu⊥=α(−u2,u
1) models
he Co iolis o ces, wi h α∈Rdepending on he la i ude. We conside ei he homogeneous
Di ichle o Robin ype bounda y condi ions on he bo om (wi h β:S→Ra non-nega i e
da a unc ion depending on he ugosi y o he bo om) and Neumann bounda y condi ions on
he su ace, whe e Υ:(0,T)×S→R2is a da a unc ion depending on he wind o ce. The
Neumann condi ions on he bo om a e also conside ed aking β= 0.
No ice ha P imi i e Equa ions a e a ian s o he Na ie -S okes equa ions. Now, he
p essu e ield depends only on x(bu no on z). Howe e , he explici o m o u3gi en in (1)
implies ha he sys em is no longe pa abolic espec o (u,u
3) and he egula i y o u3and
∇H·ua e compa able, hence he nonlinea e m co esponding o he e ical con ec ion u3∂zu
is less egula ha in he Na ie -S okes case.
The exis ence o weak solu ion o (PE) we e gi en in [10, 9]. The exis ence o local in ime
s ong solu ion (o global o small enough da a) is p o ed in [7] o he 2D case (whe e Sis a eal
in e al) and in [6] o he 3D case, using s ong egula i y esul s o he s a iona y linea case
gi en in [14]. On he o he hand, some esul s o weak/s ong uniqueness we e gi en in [7, 6],
always imposing addi ional egula i y hypo hesis o e he ho izon al and e ical de i a i es o
u.
In his wo k, we weaken hese addi ional hypo hesis ound in [7, 6] supposing only addi-
ional egula i y o e he e ical de i a i e ∂zu(a oiding he addi ional egula i y o e ∇Hu).
Mo eo e , we will also p o e ha his same addi ional egula i y implies global s ong egula i y
when he da a a e mo e egula bu wi hou smallness assump ions.
We hink ha he aniso opy be ween ho izon al and e ical scales could p oduce aniso opic
egula i y o he solu ion. Indeed, his occu s in he 2D case; exis ence (and uniqueness) o weak
solu ion u o he 2D model such ha ∂zuhas also weak egula i y, i.e. ∂zu∈L∞(0,T;L2(Ω)) ∩
L2(0,T;H1(Ω)), is p o ed in [4] o Robin bounda y condi ions on he bo om and in [5] o
Di ichle condi ions. In his line, he exis ence o weak solu ion o (PE) wi h only weak
egula i y o ∂zu(e en local in ime o global o small enough da a) is an in e es ing open
p oblem, ha we a e going o analyse in a u u e wo k.
2 The main esul s.
Basically, uis a weak solu ion o (PE)in(0,T), i u∈L2(0,T;H1(Ω))2∩L∞(0,T;L2(Ω)2)
and e i ies he es ic ion ∇H·�u�= 0, he Di ichle condi ions in he ace sense and he
momen um equa ions join ly wi h he Neumann and Robin condi ions in a a ia ional sense
([10, 6]). Mo eo e , i u∈L∞(0,T;H1(Ω)2)∩L2(0,T;H2(Ω)2) and ∂ u∈L2(0,T;L2(Ω)2), u
is a s ong solu ion o (PE)in(0,T).
Theo em 2.1 (Uniqueness o solu ion) Le ube a weak solu ion o (PE)in (0,T). I he e
exis s a weak solu ion ¯
uo (PE)in (0,T) e i ying he addi ional egula i y:
∂z¯
u∈L∞(0,T;L2(Ω)2)∩L2(0,T;H1(Ω)2),(2)
hen bo h solu ion coincided in [0,T). When Robin condi ions a e conside ed on he bo om, he
assump ion h≥hmin >0in Shas o be imposed.
Rema k 2.1 No ice ha we ha e educed he hypo heses on ¯
uimposed in [6] o ge ing unique-
ness o weak/s ong solu ion. Conc e ely, in [6] we conside ed
∇H¯
u∈L2(0,T;L∞
zL2
x)and ∂z¯
u∈L∞(0,T;L2(Ω)2)∩L2(0,T;H1(Ω)2)
3 SOME AUXILIARY ANISOTROPIC ESTIMATES. 3
(see he nex sec ion o he de ini ion o he aniso opic space L∞
zL2
x). The e o e, we ha e
emo ed he hypo hesis o ∇H¯
u.
Theo em 2.2 (Global s ong egula i y) Le S⊆R2wi h ∂S∈C3and h∈C3(¯
S)wi h
h≥hmin >0in ¯
S.Suppose ha u0∈H1(Ω)wi h ∇H·�u0�=0(and u0|Γb=0in he case o
Di ichle condi ions on he bo om), F∈L2(0,T;L2(Ω)2)and Υ∈L2(0,T;H1/2+ε
0(Γs)2)∩
L∞(0,T;H−1/2(Γs)2) o some ε>0such ha ∂ Υ∈L2(0,T;H−3/2(Γs)2)wi h Υ(0) ∈
H−1/2(Γs)2.I ∂zu e i ies he addi ional egula i y o (2), hen uis a s ong solu ion o (PE)
in (0,T).
3 Some auxilia y aniso opic es ima es.
Le us o in oduce he aniso opic Lp,q spaces o any exponen s p, q ∈[1,+∞]. I will said
say ha a unc ion belongs o Lq
zLp
x(Ω) i :
(·,z)∈Lp(Sz) and � (·,z)�Lp(Sz)∈Lq(−hmax,0),
whe e hmax = max
S
hand Sz={x∈S:(x,z)∈Ω} o each z∈(−hmax,0).
We will use he ollowing h ee aniso opic esul s, he i s one has al eady been conside ed
and p o ed in [6], and he o he ones a e new in his wo k (see Appendix o he p oo s).
Lemma 3.1 a) Le ∈H1(Ω). Then ∈L2
zL4
x(Ω)and e i ies:
� �L2
zL4
x≤C� �1/2
L2(Ω)� �1/2
H1(Ω)(3)
b) Le ∈L2(Ω)2such ha ∇H· ∈L2(Ω),and 3de ined as in (1). Then, 3∈L∞
zL2
x(Ω)
and
� 3�L∞
zL2
x≤h1/2
max�∇H· �L2(Ω)(4)
Lemma 3.2 Le ∈H1(Ω)such ha ∂z ∈H1(Ω)and |Γb=0. Then ∈L∞
zL4
x(Ω)and
� �L∞
zL4
x≤C� �1/4
L2(Ω)� �1/4
H1(Ω)�∂z �1/4
L2(Ω)�∂z �1/4
H1(Ω)(5)
Lemma 3.3 Assume h≥hmin >0in S.
a) Le ∈L2(Ω)such ha ∂z ∈L2(Ω). Then ∈L∞
zL2
x(Ω)and
hmin� �2
L∞
zL2
x≤� �2
L2(Ω)+2� �L2(Ω)�∂z �L2(Ω).(6)
b) Le ∈H1(Ω)such ha ∂z ∈H1(Ω). Then ∈L∞
zL4
x(Ω)and
h1/2
min� �L∞
zL4
x≤C� �1/4
L2(Ω)� �1/4
H1(Ω)�� �1/4
L2(Ω)� �1/4
H1(Ω)+�∂z �1/4
L2(Ω)�∂z �1/4
H1(Ω)�(7)
4 P oo o he main esul s in he Di ichle case.
P oo o Theo em 2.1:We ollow he di ec me hod o p o e uniqueness, used o ins ance in
[11] o he 3D Na ie -S okes equa ions. Deno ing =u−¯
uand 3=u3−¯u3, one has [6] (see
[13] o mo e de ails):
1
2� ( )�2
L2(Ω)+ν�
0� (s)�2
H1(Ω)ds
≤−
�
0�Ω
[ ·∇H¯
u+ 3∂z¯
u]· dΩds := I1+I2,
(8)
4 PROOF OF THE MAIN RESULTS IN THE DIRICHLET CASE. 4
whe e ν=min{νH,ν
z}. Wi h espec o he p oo o uniqueness done in [6], we will change he
ea men o he e m I1. Now, in eg a ing by pa s and applying (3) and (5) one has (he e,
Di ichle condi ion on Γbis used)
I1=�
0�Ω
[( ·∇H) ·¯
u+(∇H· ) ·¯
u]dΩds
≤C�
0�Ω| ||∇H ||¯
u|dΩds ≤C�
0� �L2
zL4
x�∇H �L2(Ω)�¯
u�L∞
zL4
xds
≤C�
0� �1/2
L2(Ω)�∇H �3/2
L2(Ω)�¯
u�1/4
L2(Ω)�¯
u�1/4
H1(Ω)�∂z¯
u�1/4
L2(Ω)�∂z¯
u�1/4
H1(Ω)ds.
Using he Young inequali y o he indexes (4/3,4), we ha e:
I1≤ε�
0� �2
H1(Ω)ds +Cε�
0�¯
u�L2(Ω)�¯
u�H1(Ω)�∂z¯
u�L2(Ω)�∂z¯
u�H1(Ω)� �2
L2(Ω)ds.
We bound I2as in [6] (using (3) and (4)), ob aining
I2=�
0�Ω
3∂z¯
u· dΩds ≤�
0� 3�L∞
zL2
x�∂z¯
u�L2
zL4
x� �L2
zL4
xds
≤ε�
0� �2
H1(Ω)ds +Cε�
0�∂z¯
u�2
L2(Ω)�∂z¯
u�2
H1(Ω)� �2
L2(Ω)ds
Using he p e ious bounds in (8), one has:
� ( )�2
L2(Ω)+ν�
0� (s)�2
H1(Ω)ds ≤C�
0
a(s)� (s)�2
L2(Ω)ds (9)
whe e a=�¯
u�L2(Ω)�¯
u�H1(Ω)�∂z¯
u�L2(Ω)�∂z¯
u�H1(Ω)+�∂z¯
u�2
L2(Ω)�∂z¯
u�2
H1(Ω).Sincea∈L1(0,T)
( hanks o he egula i y hypo hesis o ¯
uand ∂z¯
u), we a e in he hypo hesis o G onwall Lemma,
hence he uniqueness is deduced.
P oo o Theo em 2.2:Fi s , we li he bounda y da a Υusing an adequa e (s ong)
solu ion (e,q
s) o a s a iona y hyd os a ic S okes sys em. Obse e ha hypo hesis ∂ Υ∈
L2(0,T;H−3/2(Γs)2) implies ha ∂ e∈L2(0,T;L2(Ω)2) (see [8]). Then, we eason o e he
homogeneous a iables ( ,
3,π
s)=(u−e,u
3−e3,p
s−qs), e i ying:







∂ −νH∆H −νz∂2
zz +u·∇H + 3∂zu+∇Hπs=Gin (0,T)×Ω,
∇H·� �=0in (0,T)×S,
| =0 = 0in Ω,
νz∂z |Γs=0, |Γb=0, |Γl=0in (0,T),
(10)
whe e 0=u0−e(0) and G=F−∂ e−u·∇He+e3∂zu.Thanks o he addi ional egula i y o
∂zuand he s ong egula i y o e, one has ha G∈L2(0,T;L2(Ω)2). Indeed, in he con ec i e
e ms, we ha e p oduc s o u(and e3) belonging o L4
L∞
zL4
x,by∂zu(and ∇He) belonging o
L4
L2
zL4
x(acco dingly Lemmas 3.1 and 3.2). Using he same a gumen han in [6], we apply a
Gale kin me hod, using A mas es unc ions, whe e Ais he hyd os a ic S okes ope a o and
mi s eigen unc ions. In o de o bound he con ec ion e ms, we use he inequali ies (3) and
(5) as ollows ( o simplici y, we d op he m-indexes):
�Ω
u·∇H ·A dΩ≤�u�L∞
zL4
x�∇H �L2
zL4
x�A �L2(Ω)
≤C�u�1/4
L2(Ω)�u�1/4
H1(Ω)�∂zu�1/4
L2(Ω)�∂zu�1/4
H1(Ω)�∇H �1/2
L2(Ω)�A �3/2
L2(Ω)
≤ε�A �2
L2(Ω)+a( )�∇H �2
L2(Ω),
(11)
5 PROOFS FOR ROBIN CONDITIONS ON THE BOTTOM 5
whe e a( )=C�u�L∞(0,T;L2(Ω))�u( )�H1(Ω)�∂zu�L∞(0,T;L2(Ω))�∂zu( )�H1(Ω), and
�Ω
3∂zu·A dΩ≤� 3�L∞
zL4
x�∂zu�L2
zL4
x�A �L2(Ω)
≤C�∇H· �1/2
L2(Ω)�∂zu�1/2
L2(Ω)�∂zu�1/2
H1(Ω)�A �3/2
L2(Ω)≤ε�A �2
L2(Ω)+b( )�∇H �2
L2(Ω),
(12)
whe e b( )=C�∂zu�2
L∞(0,T;L2(Ω))�∂zu( )�2
H1(Ω). The addi ional egula i y o ∂zugua an ees
ha a,b∈L1(0,T). Then,
1
2
d
d �∇ �2
L2(Ω)+�A �2
L2(Ω)≤(a( )+b( ))�∇ �2
L2(Ω)+�G( )�2
L2(Ω).
G onwall’s Lemma allows us o conclude ha ∈L∞(0,T;H1(Ω)2)∩L2(0,T;H2(Ω)2) and,
hanks o he s ong egula i y o e, one has he same egula i y o u. Regula i y o ∂ uis
ollowed by a s anda d way.
5 P oo s o Robin condi ions on he bo om
P oo o Theo em 2.1:In his case, one a i es a (8) wi h he supplemen a y non-nega i e
e m �
0�Γbβ| |2dσds in he le hand-side. Now, i is necessa y o change he bound o he
e m I1in (8). Indeed, using di ec ly (8) (wi hou by pa s in eg a ion), we ge :
I1≤�
0� �2
L2
zL4
x�∇H¯
u�L∞
zL2
x≤�
0� �L2(Ω)� �H1(Ω)�∇H¯
u�L∞
zL2
x(13)
In o de o bound ∇H¯
u, we canno use he ollowing inequali y (p o ed in [6])
�∇H¯
u�L∞
zL2
x≤C�∇H¯
u�1/2
L2(Ω)�∇H¯
u�1/2
H1(Ω)
because ∇H¯
u�∈ H1(Ω). Ins ead o his, we will use Lemma 3.3 a). Indeed, applying (6) o
=∇H¯
uin (13), we a i e a
I1≤ε�
0� �2
H1(Ω)ds +Cε
hmin �
0��∇H¯
u�2
L2(Ω)+�∇H¯
u�L2(Ω)�∂z(∇H¯
u)�L2(Ω)�� �2
L2(Ω)ds
Adding his exp ession o he es ima e o I2, one can also p o e uniqueness o solu ion.
P oo o Theo em 2.2:The main diffe ence in he p oo is he use o inequali y (7) ins ead
o (5) o Lemma 3.2 in o de o bound he con ec i e e ms.
A Appendix
P oo o Lemma 3.2.We will use he ollowing inequali y, p o ed in [6]; o any p, q ∈[1,+∞]
wi h q>p,
� �Lq
xLp
z≤� �Lp
zLq
x(14)
Wi h he same a gumen s one can change he in eg a ion o de , i.e.,
� �Lq
zLp
x≤C� �Lp
xLq
z(15)
Le in he hypo hesis o Lemma 3.2. Since |Γb= 0, we ha e
(x,z)4=�2�z
−h(x)
(x,s)∂z (x,s)ds�2
≤4� (x,·)�2
L2
z�∂z (x,·)�2
L2
z

REFERENCES 6
Consequen ly,
� (x,·)�L∞
z≤√2� (x,·)�1/2
L2
z�∂z (x,·)�1/2
L2
z.(16)
Taking L4
x-no m,
� �L4
xL∞
z≤√2� �1/2
L4
xL2
z�∂z �1/2
L4
xL2
z.
Now using (14) and (3), we ob ain:
� �L4
xL∞
z≤√2� �1/2
L2
zL4
x�∂z �1/2
L2
zL4
x≤C�u�1/4
L2(Ω)� �1/4
H1(Ω)�∂z �1/4
L2(Ω)�∂z �1/4
H1(Ω).
To inish, i suffices o conside (15) in he le hand side, ge ing (5).
P oo o Lemma 3.3.Fo any unc ion g=g(z)de inedinz∈(−h(x),0) wi h x∈S,
we w i e g2(z)=g2(z�)+2�z
z�g(s)∂zg(s)ds. In eg a ing in z�∈(−h(x),0), h(x)g2(z)≤
�g�2
L2
z+2�g�L2
z�∂zg�L2
z, hush(x)1/2�g�L∞
z≤�g�L2
z+√2�g�1/2
L2
z�∂zg�1/2
L2
z. Applying he p e ious
inequali y o g= (x,·) and bounding om below h(x)≥hmin, we ge
h1/2
min� (x,·)�L∞
z≤� (x,·)�L2
z+√2� (x,·)�1/2
L2
z�∂z (x,·)�1/2
L2
z.(17)
In o de o p o e es ima e (7), we ollow he same a gumen ha in he p oo o Lemma 3.2,
eplacing (16) by (17) and adap ing he calculus he ein. Finally, (6) ollows di ec ly aking
L2
x-no m in (17).
Re e ences
[1] P. Az´e ad & F. Guill´en-Gonz´alez, Ma hema ical jus i ica ion o he hyd os a ic app oxima ion in
he P imi i e Equa ions o Geophysical luid dynamics. Siam J. Ma h. Anal. ,Vol. 33, No. 4, 847-859
(2001).
[2] O. Besson & M. R. Laydi, Some Es ima es o he Aniso opic Na ie -S okes Equa ions and o he
Hyd os a ic App oxima ion, M2AN-Mod. Ma h. Ana. Nume., Vol. 7, 855-865 (1992).
[3] D. B esch, F. Guill´en-Gonz´alez, N. Masmoudi & M. A. Rod ´ıguez-Bellido, Asymp o ic de i a ion o
a Na ie condi ion o he P imi i e Equa ions, o appea in Asymp o ic Analysis.
[4] D. B esch, F. Guill´en-Gonz´alez, N. Masmoudi & M. A. Rod ´ıguez-Bellido, On he uniqueness o weak
solu ions o he wo-dimensional p imi i e equa ions. Diff. and In eg al Eqs. ,Volume 16, Numbe 1,
77-94 (2003).
[5] D. B esch, A. Kazhikho & J. Lemoine, On he wo-dimensional hyd os a ic equa ions. Submi ed
o A ch. Ra . Mech. Anal., (2001).
[6] F. Guill´en-Gonz´alez, N. Masmoudi & M. A. Rod ´ıguez-Bellido, Aniso opic es ima es and s ong
solu ions o he P imi i e Equa ions. Diff. and In eg al Eqs. ,Volume 14, Numbe 11, 1381-1408
(2001).
[7] F. Guill´en-Gonz´alez & M.A. Rod ´ıguez-Bellido. On he s ong solu ions o he P imi i e Equa ions
in 2D domains, Nonlinea Analysis 50, 621-646 (2002).
[8] F. Guill´en-Gonz´alez, M. A. Rod ´ıguez-Bellido & M. A. Rojas-Meda , Hyd os a ic S okes equa ions
wi h non-smoo h Neumann da a. Submi ed.
[9] R. Lewandowski, Analyse Ma h´ema ique e Oc´eanog aphie, Masson, (1997).
[10] J. L. Lions, R. Teman & S. Wang,On he equa ions o he la ge scale Ocean. Nonlinea i y, 5, 1007-
1053 (1992).
[11] P. L. Lions, Ma hema ical Topics in Fluids Mechanics, Vol.1, Incomp essible Models, Uni . Pa is-
Dauphine & ´
Ecole Poly echnique, Ox o d Uni . P ess. Inc., NY, (1996).
[12] J. Pedlosky, Geophysical luid dynamics, Sp inge -Ve lag, (1987).
[13] M. A. Rod ´ıguez Bellido, Ph. Thesis, Uni e si y o Se ille (2001).
[14] M. Ziane, Regula i y Resul s o S okes Type Sys ems. Applicable Analysis, Vol. 58, 263-292 (1995).