Exis ence and Uniqueness Resul s o a Coupled
P oblem Rela ed o he S a iona y Na ie -S okes
Sys em
B. Climen and E. Fe n´
andez-Ca a ∗
June 13, 2007
Abs ac
In his pape , we conside some sys ems which a e close o he s a ion-
a y Na ie -S okes equa ions. The s uc u e o hese sys ems is he ollow-
ing: An N-dimensional equa ion o mo ion, he incomp essibili y condi-
ion and a scala equa ion in ol ing an addi ional unknown, k=k(x).
Among o he hings, hey se e o model he beha io o ce ain u bu-
len lows. Ou main in e es conce ns exis ence and uniqueness. The
main di icul ies a e due o he s uc u e o he scala equa ion; in pa ic-
ula , he igh side is ypically in L1and, u he mo e, he e a e nonlinea
e ms o he kind ∇ · (µ(k)∇k) and ∇ · (B(k)), whe e µand Ba e gene al
con inuous unc ions.
∗Depa men o Di e en ial Equa ions and Nume ical Analysis, Uni e si y o Se illa, Ta ia
s/n, E-41012 Se illa, Spain. Pa ially suppo ed by D.G.I.C.Y.T. (Spain), P oyec o PB92–
0696.
0
No a ion:
•L1=L1(Ω), H1
0=H1
0(Ω), e c.
• | · | ( esp. k·k) deno es he usual no m in L2( esp. H1
0).
•H−1=H−1(Ω) is he dual space o H1
0;k · k∗deno es he usual no m in
H−1.
•z+= max(z, 0) o any eal z.
•TM(s) = si s∈[−M, M] ; TM(s) = Msign so he wise.
•Lnis he piecewise linea e en unc ion sa is ying Ln(s) = 1 i s∈[0, n] ,
Ln(s) = s
n+ 2 i s∈[n, 2n] and Ln(s) = 0 i s > 2n.
•S:D=PN
i,j=1 SijDij o any S={Sij}and D={Dij}.
•N0is he conjuga e exponen o N, i.e. N0=N
N−1.
•Fo each p∈[1,∞] , p∗is he associa ed Sobole embedding exponen :
p∗=Np
N−pi p<N; 1 < p∗<∞is a bi a y i p=Nand p∗=∞
o he wise. In pa icula , (N0)∗=N
N−2i N≥3 .
1
1 In oduc ion. Desc ip ion o he p oblem
This pape is conce ned wi h some nonlinea pa ial di e en ial sys ems s em-
ming om luid mechanics. These a e a ian s o he s a iona y Na ie -S okes
equa ions and ead as ollows:









−∇ · (νDu +kΦ0(Du)) + (u· ∇)u+∇p= ,
∇ · u= 0 ,
−∇ · (µ(k)∇k+B(k)) + u· ∇k=ν0|Du|2+kΦ0(Du) : Du
− |k|1/2kψ0(Du).
(1)
In (1), i is assumed ha Du =∇u+ ∇u. The unc ions D7→ Φ(D) ,
D7→ ψ0(D) , k7→ µ(k) and k7→ B(k) a e gi en. Once an open se Ω ⊂IRN
and he da a ν > 0 , ν0∈[0, ν] and a e ixed, we sea ch o a solu ion {u, p, k}
o (1), oge he wi h app op ia e bounda y alue condi ions.
Sys ems like (1) a e mo i a ed by u bulence modelling. Mo e p ecisely, le
U=U(x, ) and P=P(x, ) be espec i ely he eloci y ield and p essu e
dis ibu ion o a iscous incomp essible luid in u bulen egime. Then, he
couple (U, P) mus sa is y he ins a iona y Na ie -S okes equa ions. Deno ing
by uand p he co esponding ime-a e aged a iables ( ha is o say, u=Uand
p=P) and se ing
U=u+u0, P =p+p0,
i is cus oma y o eplace he sea ch o a solu ion o (1) by he analysis o a
sys em ha should be sa is ied by uand p. A e some compu a ions, one inds:
−∇ · (νDu +R)+(u· ∇)u+∇p= , ∇ · u= 0 ,(2)
whe e is he ime-a e aged ex e nal o ces ield ac ing on he luid pa icles
and Ris he so called Reynolds enso :
R={Rij},wi h Rij =−u0
iu0
j.
Since in (2) we s ill ind he unknown a iables u0
i, i is easonable o in oduce
closing hypo heses ela ing R o u. In he case o usual one-equa ion models,
one imposes he ollowing hypo hesis o he Boussinesq kind:
R=νTDu , whe e νT=F(k) (an algeb aic ela ion). (3)
He e, k=1
2|u0|2is he mean u bulen kine ic ene gy. The p oblem is hus
closed using (2), (3) and an addi ional PDE o k.
Un o una ely, when one ies o deduce an equa ion o k, one inds again
e ms in which he u bulen pe u ba ions u0
i(and k0) appea . Mo e p ecisely,
one has:
−∇ · ν∇k+ (−(p0+k0)u0)+u· ∇k=R:Du −ν
2|Du0|2.(4)
2
Consequen ly, one has o eplace (4) by an app oxima ion. This is made by
in oducing new closing hypo heses:
•The e is gene al ag eemen in he app oxima ion o he dissipa ion e m
ν
2|Du0|2. I is usually eplaced by a cons an imes k3/2.
•O cou se, (3) is used again in o de o app oxima e he p oduc ion e m
R:Du.
•The app oxima ion o −(p0+k0)u0has been achie ed by se e al au ho s
in di e en ways. In mos pape s, his e m is eplaced by cνT∇k, whe e c
is an expe imen al cons an ( o ins ance, see [13], [12] and he e e ences
he ein). In o he s, i is eplaced by a ec o B(k) (see [7]).
Hence, i is clea ha equa ions like (1) can be used o desc ibe he beha io
o ce ain u bulen lows. Ano he mo i a ion o (1) can be ound in non
New onian mechanics. In his se ing, {u, p}a e he ue eloci y ield and
p essu e, kis he empe a u e and i is assumed ha he s ess enso τdepends
on Du and kas ollows:
τ=νDu +kΦ0(Du).(5)
2 The main esul s
In he sequel, we will conside a simpli ied e sion o (1):





−ν∆u−∇·(kΦ0(∇u)) + (u· ∇)u+∇p= ,
∇ · u= 0 ,
−∇ · (µ(k)∇k+B(k)) + u· ∇k=ν|∇u|2+kΦ0(∇u) : ∇u .
(6)
This is made o con enience only; he esul s in his sec ion also hold o
(1) wi h app op ia e changes. In (6), he i s , second and hi d equa ions will
be espec i ely known as he mo ion equa ion, he incomp essibili y condi ion
and he ene gy equa ion. Ou assump ions a e he ollowing:
•Ω⊂IRNis a bounded, connec ed, open and egula se ; ν > 0 and
∈H−1.
•D7→ Φ(D) is C1, Φ0(0) = 0, |Φ0(D)| ≤ Cons . and D7→ Φ0(D) : Dis con-
ex (consequen ly, i is also locally Lipschi z-con inuous). In pa icula ,
D7→ Φ(D) is con ex and one has (Φ0(D1)−Φ0(D2)) : (D1−D2)≥0 o
all D1and D2.
•k7→ µ(k) and k7→ B(k) a e con inuous unc ions; u he mo e, µ(k)≥
µ0>0 o all k.
3
We wan o sol e (6) oge he wi h Di ichle condi ions o uand k:
u= 0 and k= 0 on ∂Ω.(7)
Ou main in e es conce ns gene al con inuous unc ions µand B. This is o
cou se mo i a ed by he ac ha , in u bulence modelling, an equa ion exac ly
sa is ied by kis unknown. Besides he usual spaces L2,H1
0,V, e c., we will
use he ollowing:
L={ψ∈L1;TM(ψ)∈H1
0∀M > 0,lim
n→∞
1
nZn≤|ψ|≤2n
|∇ψ|2dx = 0 }
(see he No a ion).
Theo em 1–Unde he p e ious assump ions, he e exis s {u, p, k}, wi h
u∈V,p∈L2and k∈ L such ha :
1. The couple {u, p}sol es he i s wo equa ions in (6) in he usual weak o
dis ibu ional sense.
2. k≥0and sol es he hi d equa ion in (6) in he ollowing sense:
(−∇ · (β(k)(µ(k)∇k+B(k))) + β0(k)∇k·(µ(k)∇k+B(k))
+β(k)(u· ∇k) = β(k)ν|∇u|2+kΦ0(∇u) : ∇u(8)
in D0(Ω) o e e y β∈W1,∞(IR) wi h compac suppo .
A iple {u, p, k}as abo e will be called a weak- eno malized solu ion o (6).
Reno malized solu ions o PDE’s seem o ha e been in oduced by R. DiPe na
and P.L. Lions in [8], in he amewo k o he Bol zmann equa ion. They ha e
been used in connec ion wi h a ious nonlinea ellip ic equa ions by P. Benilan
e al. [3], L. Bocca do e al. [6] and P.L. Lions and F. Mu a [10] (see also
[11]). In he analysis o exis ence esul s o p oblems simila o (1) and (6),
weak- eno malized solu ions we e conside ed by R. Lewandowski [9] (see also
[2]). Tha we look o a eno malized solu ion kis mo i a ed by he s uc u e
o he igh side o he ene gy equa ion in (6) ( ypically in L1) and also by ou
in e es in keeping µand Bas gene al as possible.
Le us deno e by ˜µ he ollowing unc ion:
˜µ(s) = Zs
0
µ(σ)dσ ∀s∈IR .
Assume ha , in heo em 1, one has B≡0 . Then i is no di icul o see ha
he solu ion {u, p, k} u nished by heo em 1 sa is ies
˜µ(k)∈
q<N0
W1,q
0,∇˜µ(k) = µ(k)∇k(9)
4

and also he ollowing:


ZΩ
µ(k)∇k· ∇φ+ZΩ
(u· ∇k)φ=ZΩν|∇u|2+kΦ0(∇u) : ∇uφ
∀φ∈ D(Ω) .
(10)
In his case, i will be said ha {u, p, k}is a weak solu ion o (6).
Theo em 2–Assume ha , in heo em 1,B≡0and D→Φ0(D) : Dis
globally Lipschi z-con inuous. Then, he e exis s ν0>0such ha , when ν≥ν0,
he e exis s a mos one weak solu ion {u, p, k} o (6) wi h k≥0.
Be o e gi ing he p oo s o hese esul s, le us make some ema ks:
1. A e y in e es ing ques ion emains: When νis la ge and Bis no ze o,
is i s ill possible o p o e he uniqueness o a eno malized solu ion ?
2. The e a e se e al o he possible condi ions o uniqueness, di e en om
he assump ion ν≥ν0. Fo ins ance, o ixed ν, one can also ob ain a
mos one weak solu ion {u, p, k} o (6) wi h k≥0 i k k∗is su icien ly
small.
3. Resul s simila o hose abo e can be p o ed o he ins a iona y a ian
o (6). This will be analyzed in a o hcoming pape .
4. Se e al mo e o less ob ious gene aliza ions a e possible. In pa icula ,
we ind an in e es ing si ua ion when we simply assume µ(k)≥0 in (6).
This case is a om i ial and will also be he subjec o u u e wo k.
3 The p oo o heo em 1
In his sec ion, Cdeno es a cons an which may depend on N, Ω and he da a
in (6). The p oo o heo em 1 consis s o six s eps:
Fi s s ep: The in oduc ion o a amily o app oxima ions.
Fo each ε > 0, we conside he ollowing app oxima ion o (6):







−ν∆uε− ∇ · (T1
ε(kε)+Φ0(∇uε)) + (uε· ∇)uε+∇pε= ,
∇ · uε= 0 ,
−∇ · T1
ε(µ(kε))∇kε+B(T1
ε(kε)+uε· ∇kε=T1
ε(τε:∇uε).
(11)
He e, we ha e used he ollowing no a ion:
τε=ν∇uε+T1
ε(kε)+Φ0(∇uε).
5
O cou se, hese equa ions a e equi ed o be sa is ied in Ω , oge he wi h ho-
mogeneous Di ichle condi ions o uεand kεon ∂Ω . The exis ence o a iple
{uε, pε, kε}can be es ablished using ( o ins ance) a Gale kin me hod. In ac ,
some non i ial di icul ies a e ound wi h his echnique ha can be sol ed a -
guing as in he ollowing s eps. One inds ha he solu ion belongs o he space
V×L2×H1
0(Ω) and, also, ha kε≥0 .
Second s ep: A p io i es ima es and weak con e gence.
Using uεas a es unc ion in he i s equa ion in (11), one inds:
ZΩ
τε:∇uε≤C . (12)
In pa icula ,
kuεk ≤ C . (13)
In he ene gy equa ion in (11), le us use TM(kε) as es unc ion. This gi es:
kTM(kε)k2≤C·M(14)
On he o he hand, i we choose ξn(kε) = T2n(kε)−Tn(kε) as es unc ion in
he same equa ion, i is no di icul o check ha
1
nZn≤kε≤2n
T1
ε(µ(kε))|∇kε|2≤Zkε≥n
T1
ε(τε:∇uε),(15)
whence 1
nZn≤kε≤2n
|∇kε|2≤C . (16)
¿F om (14) and (16), a guing as in [5] and [11], one deduces he ollowing:
kkεkW1,q
0≤Cq,∀q < N0.(17)
Consequen ly, passing o a subsequence i necessa y, i can be assumed ha
uε→uweakly in V, s ongly in L ∀ < 2∗and a.e.,
kε→kweakly in W1,q
0∀q < N0, s ongly in Lp∀p < (N0)∗and a.e.,
TM(kε)→TM(k) weakly in H1
0∀M > 0.
Ob iously, one has k≥0 .
Thi d s ep: uis, oge he wi h some p, a solu ion o he mo ion equa ion.
Fo each ε > 0 , uεis a solu ion o he ollowing a ia ional inequali y:







νZΩ
∇uε:∇ +ZΩ
(uε· ∇)uε· +ZΩ
T1
ε(kε)Φ(∇ )
≥νZΩ
|∇uε|2+ZΩ
T1
ε(kε)Φ(∇uε) + h , −uεi ∀ ∈V , uε∈V .
6
Taking limi s as ε→0 , one ob ains:
νZΩ
∇u:∇ +ZΩ
(u· ∇)u· +ZΩ
kΦ(∇ )
≥νlim in
ε→0ZΩ
|∇uε|2+ lim in
ε→0ZΩ
T1
ε(kε)Φ(∇uε) + h , −ui
The i s e m in he igh is bounded om below by
νZΩ
|∇u|2.
In wha conce ns he second e m, le us i s no ice ha
ZΩ
T1
ε(kε)Φ(∇uε) = ZΩ
(T1
ε(kε)−k)Φ(∇uε) + ZΩ
kΦ(∇uε).
Thus, aking in o accoun ha he unc ion
7→ ZΩ
kΦ(∇ )
is lowe semicon inous, we ind:
lim in
ε→0ZΩ
T1
ε(kε)Φ(∇uε)≥lim
ε→0ZΩ
(T1
ε(kε)−k)Φ(∇uε) + lim in
ε→0ZΩ
kΦ(∇uε)
≥ZΩ
kΦ(∇u).
Consequen ly, uis a solu ion o he a ia ional inequali y







νZΩ
∇u: (∇ − ∇u) + ZΩ
(u· ∇)u·( −u) + ZΩ
kΦ(∇ )
−ZΩ
kΦ(∇u)≥ h , −ui ∀ ∈V , u ∈V .
(18)
Now, aking in (18) he unc ion o he o m u+ w , whe e w∈Vand ∈IR
and le ing →0 , i is a s anda d ma e o p o e ha usol es, oge he wi h
some p∈L2, he i s wo equa ions in (6) in he usual weak sense.
Fou h s ep: uεcon e ges s ongly in V.
¿F om he mo ion equa ion in (6), i is clea ha
ZΩν|∇u|2+kΦ0(∇u) : ∇u=h , ui.
On he o he hand, choosing uεas es unc ion in he i s equa ion in (11),
one has: ZΩν|∇uε|2+T1
ε(kε)Φ0(∇uε) : ∇uε=h , uεi.
7
Consequen ly,
lim
ε→0ZΩν|∇uε|2+T1
ε(kε)Φ0(∇uε) : ∇uε=ZΩν|∇u|2+kΦ0(∇u) : ∇u,
whence i is also clea ha
lim
ε→0ZΩν|∇uε|2+kΦ0(∇uε) : ∇uε=ZΩν|∇u|2+kΦ0(∇u) : ∇u
( ecall ha Φ0is uni o mly bounded). Hence,
0 = lim
ε→0ZΩν|∇uε|2+kΦ0(∇uε) : ∇uε−ZΩν|∇u|2+kΦ0(∇u) : ∇u
≥lim sup
ε→0νZΩ
|∇(uε−u)|2
+ lim in
ε→0ZΩ
kΦ0(∇uε) : ∇uε−ZΩ
kΦ0(∇u) : ∇u).
He e, he las e m is ≥0 , in iew o he lowe semicon inui y o he unc ion
7→ ZΩ
kΦ0(∇ ) : ∇ .
Thus,
lim
ε→0ZΩ
|∇(uε−u)|2= 0 .
Fi h s ep: Fo all M > 0 , TM(kε) con e ges s ongly in H1
0.
We will use an a gumen due o P.L. Lions and F. Mu a (see [10], [11]). Le us
see ha
T1
ε(µ(kε))1
2∇TM(kε)→µ(k)1
2∇TM(k) s ongly in L2∀M > 0 (19)
(obse e ha µ(k)1
2∇TM(k) has a sense). O cou se, (19) will su ice o ou
pu poses.
I has al eady been p o ed ha
T1
ε(τε:∇uε)→τ:∇us ongly in L1and a.e.
He e, we ha e in oduced τ=ν∇u+kΦ0(∇u) . Choosing TM(kε) as es unc ion
in he ene gy equa ion in (11), one inds:
ZΩ
T1
ε(µ(kε))∇kε· ∇TM(kε) + ZΩ
B(T1
ε(kε)) · ∇TM(kε)
+ZΩ
(uε· ∇kε)TM(kε) = ZΩ
T1
ε(τε:∇uε)TM(kε).
8