Uniqueness o solu ion o he 2D P imi i e Equa ions wi h
ic ion condi ion on he bo om∗
D. B esch†
, F. Guill´en-Gonz´alez‡
, N. Masmoudi§
, M. A. Rod ´ıguez-Bellido¶
Monog a ´ıas del Semin. Ma em. Ga c´ıa de Galdeano. 27: 135–143, (2003).
Abs ac
Uniqueness o solu ion o he P imi i e Equa ions wi h Di ichle condi ions
on he bo om is an open p oblem e en in 2D domains. In his wo k we p o e
a esul o addi ional egula i y o a weak solu ion o he P imi i e Equa ions
when we eplace Di ichle bounda y condi ions by ic ion condi ions. This allows
o ob ain uniqueness o weak solu ion global in ime, o such a sys em [3]. Indeed,
we show weak egula i y o he e ical de i a i e o he solu ion, ∂z o all ime.
This is because his de i a i e e i ies a linea pde o con ec ion-di usion ype wi h
con ec ion eloci y , and he p essu e belongs o a L2-space in ime wi h alues in
a weigh ed space.
Keywo ds: Bounda y condi ions o ype Na ie , 2D P imi i e Equa ions, uniqueness
AMS Classi ica ion: 35Q30, 35B40, 76D05
1 In oduc ion and mo i a ion.
P imi i e Equa ions a e one o he models used o o ecas he luid eloci y and p essu e
in he ocean. Such equa ions a e ob ained om he dimensionless Na ie -S okes equa ions,
le ing he aspec a io (quo ien be ween e ical dimension and ho izon al dimensions)
go o ze o. The i s esul s abou exis ence o solu ion (weak, in he sense o he Na ie -
S okes equa ions) a e p o ed o bounda y condi ions o Di ichle ype on he bo om o
he domain and wi h wind ac ion on he su ace, in he wo ks by Lions-Temam-Wang,
∗The second and ou h au ho s ha e been inanced by he C.I.C.Y.T p ojec MAR98-0486.
†Labo a oi e de Ma h´ema iques Appliqu´ees CNRS 6620 Uni . Blaise Pascal, 63177 Aubi`e e
(FRANCE), b esc[email p o ec ed]cle mon .
‡Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Fac. Ma em´a icas, Uni . Se illa, C/ Ta ia, s/n
- 41012 Se illa (SPAIN), guillen@nume .us.es
§Cou an Ins i u e o Ma hema ical Science, New Yo k Uni e si y, [email p o ec ed]yu.edu
¶Dp o. Ma em´a ica Aplicada I, E. T. S. de A qui ec u a, Uni . Se illa, A da. Reina Me cedes, s/n -
41012 Se illa (SPAIN), [email p o ec ed]
135
[5, 6], o domains wi h e ical walls and in he wo k o Az´e ad-Guill´en, [1], o domains
wi hou e ical walls. Howe e , uniqueness o solu ion emained as an open p oblem due
o he necessi y o a mo e egula solu ion. In he case o e ical sidewalls, he au ho s
p o ed in [4] he exis ence o a mo e egula solu ion, global in ime o small da a o
local in ime o any da a. In hese cases, uniqueness o solu ion is gua an eed.
Bu , om a physical poin o iew, homogeneous Di ichle bounda y condi ions (on
he bo om) a e only jus i ied when conside ing a molecula iscosi y luid. In many
geophysical luids, he ole o his iscosi y is negligible, being mo e ele an he iscosi y
due o u bulen e ec s. I seems hen logical o use Na ie bounda y condi ions o
he P imi i e Equa ions. Mo eo e , hey p e en he appea ance o a bounda y laye
phenomena on he bo om.
The au ho s ob ained he P imi i e Equa ions model wi h Na ie ype bounda y con-
di ions om he Na ie -S okes equa ions in [2]. He e, we will ocus on he uniqueness
p oblem in he 2D case, see also [3]. We will p esen wha we conside is he i s esul
o uniqueness o weak solu ion o he 2D P imi i e Equa ions.
2 The model.
The domain conside ed is de ined by:
Ω = {(x, z)∈R2/ x ∈S, −h(x)<z<0},
whe e S(ocean su ace) is an open in e al and h:¯
S→R+is a nonnega i e con inuous
unc ion de ined on ¯
S ha anishes on ∂S. The bounda y o he domain is ∂Ω = ¯
Γb∪Γs,
whe e he bo om is Γb={(x, z)∈R2:x∈S, z =−h(x)}and he su ace Γs={(x, 0) :
x∈S}. The e o e, he luid eloci y ( , w) and he p essu e psa is y he ollowing
equa ions:
(PE)



































∂ + ∂x +w ∂z −νh∂2
xx −ν ∂2
zz +∂xp= in (0, T)×Ω,
∂zp= 0, w( , x, z) = Z0
z
∂x ( , x, s)ds, h i= 0 in (0, T)×S,
ν ∂z =α| ai |( ai − ) on (0, T)×Γs,
ν ∂z =β(x) on (0, T)×Γb,
| =0 = 0in Ω,
whe e h i( ;x) = Z0
−h(x)
( ;x, z)dz, ai is he ho izon al eloci y o he wind a he
su ace, 0 he ho izon al ini ial eloci y, (νh, ν ) he aniso opic u bulen iscosi y, α∈R
136
a posi i e cons an and β=β(x) a posi i e unc ion de ined on S.
Rema k 2.1 The model o P imi i e Equa ions wi h Na ie condi ions deduced in [2]
was o med by (P E)1,∂zp= 0 and ∂x +∂zw= 0 in (0, T)×Ω,ν ∂z =α( ai − )and
w= 0 on (0, T)×Γs,ν ∂z =β and ( , w)·n= 0 on (0, T)×Γband | =0 = 0in
Ω. The equa ion ∂x +∂zw= 0 and bounda y condi ions o wimply ha w( ;x, z) =
R0
z∂x ( ;x, s)ds and ∂xh i= 0. Finally, as h iis a 1-dimensional unc ion, he hypo hesis
h i= 0 on (0, T )×∂S implies ha h i= 0 on (0, T )×S.
3 De ini ions and p e ious esul s.
Fo he eloci y , we in oduce he ollowing spaces:
V={ϕ∈C∞
s(Ω) : hϕi= 0 in S,}
whe e C∞
s(Ω) is he space o C∞- unc ions ha anish in a neighbou ghood o ∂Γs. We
will deno e by Hand Vi s closu es in he L2(Ω) and H1(Ω)−no ms espec i ely.
De ini ion 3.1 (Weak solu ion) We say ha is a weak solu ion o (PE)in (0, T )
i :
∈L∞(0, T;H)∩L2(0, T;V),
sa is ies he a ia ional o mula ion: ∀ϕ∈C1([0, T ]; V)wi h ϕ(T) = 0,























−ZT
0ZΩ
(∂ ϕ+ ∂xϕ+w∂zϕ) +ZT
0ZΩ
(νh∂x ∂xϕ+ν ∂z ∂zϕ)
+ZT
0ZS
δ(x) |Γbϕ|Γb+ZT
0ZS
α| ai |( |Γs− ai )ϕ|Γs
=ZΩ
0ϕ(0) + ZT
0ZΩ
ϕ +νhZT
0ZS
|Γb∂x[ϕ|Γbh0(x)],
wi h w=R0
z∂x and sa is ies he ollowing ene gy inequali y











1
2k ( )k2
L2(Ω) +νhZ
0k∂x (s)k2
L2(Ω) +ν Z
0k∂z (s)k2
L2(Ω)
+Z
0ZS
γ(x)| |Γb|2+1
2Z
0ZS
α| ai || |Γs|2≤1
2k 0k2
L2(Ω) +1
2Z
0ZS
α| ai |3
wi h δ(x) = β(x)1 + νh
ν |h0(x)|2and γ(x) = δ(x)−νh
2h00(x).
Rema k 3.1 In o de o ensu e ha he sys em is dissipa i e (necessa y p ope y om a
physical poin o iew), we assume ha γ(x)≥0.
137
Rema k 3.2 No ice ha he bounda y condi ion on he bo om is no s anda d because
∂z is no he Neumann condi ion espec o he laplacian ope a o . This ac p oduces
he e m νhRT
0RS |Γb∂x[ϕ|Γbh0(x)] in he a ia ional o mula ion. In o he wo ds, gi ing
a weak solu ion , we can ge an associa e p essu e p h ough he De Rham Lemma (as
a Lag ange mul iplie ) in such a way ha ( , w, p) e i y he di e en ial p oblem (PE)
in he dis ibu ion sense (see [3] o mo e de ails). In pa icula , he ollowing mixed
a ia ional o mula ion can be ob ained: ∀ϕ∈C1([0, T]; C∞
s(Ω)), wi h ϕ(T) = 0, he e
exis s a unc ion ψsmoo h enough, sa is ying (ϕ, ψ)·n|∂Ω= 0 such ha :
−ZT
0ZΩ
(∂ ϕ+ ∂xϕ+w∂zϕ) +ZT
0ZΩ
(νh∂x ∂xϕ+ν ∂z ∂zϕ)
+ZT
0ZS
δ(x) |Γbϕ|Γb+ZT
0ZS
α| ai |( |Γs− ai )ϕ|Γs
=ZΩ
0ϕ(0) + ZT
0ZΩ
ϕ +νhZT
0ZS
|Γs∂x(ϕ|Γbh0) + ZT
0ZΩ
p∇·(ϕ, ψ).
(1)
Theo em 3.2 (See [2] o a p oo o his esul .) Suppose ha h∈H2(S)wi h |h0|>0
on ∂S,β∈L∞(S), ∈L2(0, T ;L2(Ω)), ai ∈L3(0, T ;L3(S)), 0∈Hand γ(x)≥0on
S. Then, he e exis s a weak solu ion o (P E)in (0, T).
De ini ion 3.3 (Weak- o ici y solu ion) We will say ha is a weak- o ici y solu-
ion o (P E)in (0, T)i i is a weak solu ion ha also sa is ies he addi ional egula i y:
∂z ∈L∞(0, T;L2(Ω)) ∩L2(0, T;H1(Ω)).
Rema k 3.3 ∂z can be seen as he o ici y associa ed o he P imi i e Equa ions. In-
deed, i we conside he o ici y o he 2D Na ie -S okes equa ions, ωNS =∂z NS −
∂xwNS, le ing he aspec a io go o ze o we a i e a ∂z .
4 Main esul .
Theo em 4.1 (Uniqueness o weak solu ion) Unde he hypo hesis o Theo em 3.2, i
we also conside ha β∈H1
0(S), ai ∈L∞(0, T;H1
0(S)),∂ ai ∈L2(0, T;L1(S)),∂z ∈
L2(0, T;H−1(Ω)),∂z 0∈L2(Ω) and he dep h unc ion h e i ies |h0|/h ≤c/dis (x, ∂S),
hen he e exis s a unique weak solu ion o (P E). Mo eo e , his solu ion is a weak-
o ici y solu ion.
Ou line o he p oo : He e, we will explain he main ideas ha we ha e ollowed
o p o e Theo em 4.1. Fo a comple e p oo o his esul see [3].
Following he me hod o P. L. Lions, [7], o p o e uniqueness o weak solu ion o
he Na ie -S okes equa ions we obse ed ha addi ional egula i y is necessa y o one
138
o he wo solu ions compa ed. Applying he a gumen o (PE), we obse ed ha his
egula i y should be ∂z ∈L4(0, T;L4(Ω)). In o de o ob ain mo e egula i y o ∂z ,
we sea ch o he p oblem e i ied by ∂z . Fi s , we o mally de i e (PE)1 espec o z,
ob aining ha ∂z sa is ies in D0((0, T)×Ω):
∂ (∂z ) + ∂x(∂z ) + w ∂z(∂z )−νh∂2
xx(∂z )−ν ∂2
zz(∂z ) = ∂z .
Knowing and w, he p e ious equa ion is linea and pa abolic, because he p essu e p
has disappea ed, so we could expec weak egula i y o ∂z . To his end, we need o
s udy a homogeneous sys em, so we conside he auxilia y unc ion ψ=ν ∂z −φ −e
wi h
φ( ;x, z) = −α1 + z
h(x)| ai ( ;x)|− z
h(x)β(x)
and
e( ;x, z) = α| ai ( ;x)| ai ( ;x)1 + z
h(x)
auxilia y unc ions such ha ψ|∂Ω= 0. Then, ψ e i ies he p oblem:
(P)






∂ ψ+ ∂xψ+w ∂zψ−ν ∂2
xxψ−ν ∂2
zzψ=Fin (0, T)×Ω,
ψ= 0 on (0, T)×∂Ω,
ψ| =0 =ν ∂z 0−φ| =0 0−e| =0 in Ω,
whe e F=G(φ, , w, e, ) + φ ∂xp.
A his poin , we ha e 2 p oblems: ge ing an addi ional egula i y o he p essu e p o
ob ain weak egula i y o ψ, and iden i ying ψ+φ +ewi h ν ∂z . Once hese p oblems
a e sol ed, hen ∂z ∈L2(0, T ;H1(Ω)) ∩L∞(0, T ;L2(Ω)) and in pa icula belongs o
L4(0, T;L4(Ω)), so we will be able o conclude weak uniqueness o (PE).
5 Addi ional egula i y o he p essu e.
Thanks o ∂zp= 0, we can iden i y pwi h a unc ion psonly de ined on S,ps(x) = p(x, z),
h ough he ela ion:
ZΩ
p(x, z)ϕ(x, z)dx dz =ZS
ps(x)hϕi(x)dx ∀ϕ∈L2(Ω).
Theo em 5.1 Assume he hypo hesis o he da a o Theo em 4.1. I ( , p)is a weak
solu ion o (P E), we ha e:
√h ∂xps∈L2(0, T;H−1(S)).
Ou line o he p oo : Fo he equa ions o Na ie -S okes ype, he p essu e egula i y is
no mally ob ained om he egula i y o he emaining e ms o he equa ion. The e m
139

∂ p e en s a L2- egula i y in ime o he p essu e. The ac ha h i= 0 on (0, T )×S
implies ha ∂ h i= 0 on (0, T )×S, so in eg a ing (PE)1in zwe y o imp o e he
egula i y o he p essu e. In a igo ous o m, his e ical in eg a ion co esponds o
ake es unc ions independen om zin he mixed a ia ional o mula ion (1).
On he o he hand, as he p essu e pis independen om z, i s in eg a ion on zonly
adds a ac o h(x) mul iplying p. Mo eo e , o (ϕ, ψ) any es unc ions in (1),
ZΩ
p∇·(ϕ, ψ)dΩ = ZS
ps∂xhϕidx.
Then, we choose ϕ=ζ/√hwi h ζ∈C1
0([0, T]; C∞
0(S)) as a es unc ion (in pa icula ,
his space is dense in L2(0, T ;H1
0(S))). Conc e ely, we ha e o gi e sense o he e m
ZT
0ZS
ps( ;x)∂x(√h ζ)( ;x)dx d .
To his aim, we p o e ha he o he s e ms om he mixed a ia ional o mula ion
a e well-de ined and bounded in unc ion o he L2(0, T ;H1
0(S))-no m o ζ. Addi ional
egula i y equi ed o he da a, hypo hesis |h0|/h ≤c/dis (x, ∂S) join ly wi h Ha dy
inequali ies and he ac ha ∂ h i= 0 le inish he p oo .
6 Iden i ica ion o ψ+φ +ewi h ν ∂z .
Using a Gale kin me hod, he addi ional egula i y o ple us ob ain weak egula -
i y o ψ, so ψ∈L2(0, T ;H1(Ω)) ∩L∞(0, T ;L2(Ω)). To ge ∂z ∈L2(0, T;H1(Ω)) ∩
L∞(0, T;L2(Ω)), we p o e ha ψ+φ +e=ν ∂z .
The i s idea o ge his esul was o use he uniqueness o weak solu ion o p oblem
(P), bu he p oblem was ha we could no assu e he weak egula i y o ∂z (only
∂z ∈L2(0, T;L2(Ω))). Consequen ly, we looked o a new me hod o ou pu pose: We
call a=ψ+φ +eand de ine e ∈L2(0, T;H1
0(Ω)) ∩L∞(0, T;L2(Ω)) such ha ν ∂ze =a
in Ω and he i= 0 on S. In ac , we can choose:
e (x, z) = −1
ν Z0
z
a(x, s)ds +1
ν
1
h(x)Z0
−h(x)Z0
z
a(x, s)dsdz.
The idea is o ob ain uniqueness o bo h eloci ies and e , and hen ∂z =∂ze ∈
L2(0, T;H1(Ω)) ∩L∞(0, T;L2(Ω)).
S a ing om he a ia ional o mula ion o ψ, aking χ=Z0
z
η(x, s)ds as es unc-
ions, whe e η∈ D(Ω) wi h hηi= 0 and aking in o accoun ha
ν ∂ze =α| ai |( ai − ) on Γsand ν ∂ze =β on Γb,
140
we can easily deduce ha e e i ies he ollowing a ia ional o mula ion (g
FV ): ∀η∈
C1([0, T]; V),


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































Z
0h∂ e , ηiΩ+Z
0ZΩ
( ∂xe +w ∂ze )η
+Z
0ZΩ
(νh∂xe ∂xη+ν ∂ze ∂zη) + Z
0ZS
α| ai |( |Γs− ai )η|Γs
+Z
0ZS
δ(x) |Γbη|Γb=Z
0ZΩ
η
+Z
0ZΩ ∂xe +Z0
z
∂x( ∂ze )(x, s)dsη+ν Z
0ZS
|Γb∂x[η|Γbh0(x)] .
On he o he hand, we know ha sa is ies he ollowing a ia ional o mula ion
(FV ): ∀ϕ∈C1([0, T]; V),



































h ( ), ϕ( )iΩ−Z
0ZΩ
(∂ ϕ+ ∂xϕ+w∂zϕ)
+Z
0ZΩ
(νh∂x ∂xϕ+ν ∂z ∂zϕ)
+Z
0ZS
α| ai |( |Γs− ai )ϕ|Γs+Z
0ZS
δ(x) |Γbϕ|Γb
=ZΩ
0ϕ(0) + νhZ
0ZS
|Γb∂x[ϕ|Γbh0(x)] + Z
0ZΩ
ϕ,
Taking in o accoun he weak egula i y o e and ∂ze and a guing by densi y, we can
ake e as a es unc ion in (F V ) and as a es unc ion in (g
FV ). Sub ac ing bo h
exp essions o he ene gy equali y o e and he ene gy inequali y o , we a i e a ([3]):
a. e. ∈(0, T ),
1
2k ( )−e ( )k2
L2(Ω) +Z
0νhk∂x( −e ) (s)k2
L2(Ω) +ν k∂z( −e ) (s)k2
L2(Ω)ds
≤Z
0ZΩ ∂xe +Z0
z
∂x( ∂ze ) (x, s)ds(e − )dΩds
+νh
2Z
0ZS|e |Γb− |Γb|2h00(x)dxds ≡I+J.
(2)
No ice ha i e = , hen I= 0 and J= 0. In eg a ing by pa s espec o z, we
141
ew i e Ias:
I=Z
0ZΩ{∂ze ∂x( −e )−∂xe ∂z( −e )}Z0
z
( −e )(x, s)dsdΩds
≤min{νh, ν }
4Z
0k −e k2
H1(Ω)ds
+C(νh, ν )Z
0k∂xe k2
L2(Ω) +k∂x(∂ze )k4/3
L2(Ω)k −e k2
L2(Ω)ds.
We bound Jusing he T ace and In e pola ion Theo y in Hs(Ω)-spaces wi h s∈R:
J≤CZ
0kh00kL2(S)k( −e )|Γbk2
L4(S)ds
≤CZ
0kh00kL2(S)k −e k2
H3/4(Ω)ds
≤CZ
0kh00kL2(S)k −e k1/2
L2(Ω)k −e k3/2
H1(Ω)ds
≤min{νh, ν }
4Z
0k −e k2
H1(Ω)ds +C(νh, ν )Z
0kh00k4
L2(S)k −e k2
L2(Ω)ds
Then, (2) becomes:
k ( )−e ( )k2
L2(Ω) +Z
0νhk∂x( −e ) (s)k2
L2(Ω) +ν k∂z( −e ) (s)k2
L2(Ω)ds
≤C(νh, ν )Z
0k∂ze kL2(Ω)k∂ze kH1(Ω) +k∂xe k2
L2(Ω)
+k∂x(∂ze )k4/3
L2(Ω) +kh00k4
L2(S)k −e k2
L2(Ω)ds.
Since ∂ze has weak egula i y, we can use he G onwall Lemma and deduce ha e = .
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 ap-
p oxima ion in he P imi i e Equa ions o Geophysical luid dynamics. To appea in
Siam J. Ma h. Anal. , Vol. 33, No. 4, 847-859.
[2] 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. Submi ed o
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