REGULAR TIME-REPRODUCTIVE SOLUTIONS FOR
GENERALIZED BOUSSINESQ MODEL WITH NEUMANN
BOUNDARY CONDITIONS FOR TEMPERATURE
B. Climen -Ezque a, F. Guill´
en-Gonz´
alez
Dp o. EDAN, Uni e sidad de Se illa
Ap do. 1160, 41080 Se illa, SPAIN.
M.A. Rojas-Meda
Dp o. Ma em´a ica Aplicada, IMECC-UNICAMP,
C.P. 6065, 13083-970, Campinas-SP, BRAZIL.
(Communica ed by Aim Sciences)
Abs ac . The aim o his wo k is o p o e exis ence o egula ime ep o-
duc i e solu ions o a gene alized Boussinesq model (wi h nonlinea di usion
o eloci y and empe a u e). The main idea is o ob ain highe egula i y (o
H3 ype) o empe a u e han o eloci y (o H2 ype), using speci ically he
Neumann bounda y condi ion o empe a u e.
1. In oduc ion
Assume a bounded, egula open se Ω in IRN(N= 2 o 3). This pape is con-
ce ned wi h some equa ions go e ning he coupled mass and hea low o a iscous
incomp essible luid in a gene alized Boussinesq app oxima ion by assuming ha
iscosi y and hea conduc i i y a e explici unc ions depending on empe a u e.
The in ol ed equa ions a e
(1)
∂ u−∇·(ν(θ)∇u)+(u· ∇)u−αgθ+∇p= ,
∇ · u= 0,
∂ θ−∇·(k(θ)∇θ)+(u· ∇)θ= 0,
in Ω ×[0,∞), whe e
•u(x, )∈IRNdeno es he eloci y o he luid a poin x∈Ω and ime
∈[0,+∞).
•p(x, )∈IR is he (hyd os a ic) p essu e.
•θ(x, )∈IR is he empe a u e.
•g(x, )∈IRNdeno es he g a i a ional ield and α > 0 is a cons an associ-
a ed o he coe icien o olume expansion.
• (x, )∈IRNdeno es he esul ing o ex e nal o ces.
•ν(·) : IR →IR is he kinema ic iscosi y.
•k(·) : IR →IR is he he mal conduc i i y.
1991 Ma hema ics Subjec Classi ica ion. P ima y: 35Q30; Seconda y: 76D05, 35Q35.
Key wo ds and ph ases. gene alized Boussinesq, egula i y, pe iodic solu ions, ep oduc i e
solu ions.
The au ho s has been pa ially suppo ed by D.G.E.S. and M.C.T. (Spain), P oje BFM2003-
06446. Mo eo e , hi d au ho is also suppo ed by he p oje 301354/03-0 CNPq. (B azil).
1
2 B. CLIMENT, F. GUILL´
EN, M.A. ROJAS
We will sea ch o a iple {u, p, θ} egula ep oduc i e solu ion o (1) in Ω×[0,∞),
oge he he ollows Di ichle -Neumann bounda y condi ions:
(2) u= 0, ∂nθ= 0 on [0,∞)×∂Ω,
and he ime ep oduc i e condi ion:
(3) u(0) = u(T), θ(0) = θ(T) in Ω.
Exis ence and uniqueness o he ini ial alue p oblem ela ed o (1) and wi h
Di ichle ’s bounda y condi ions o eloci y and empe a u e, was p o ed by Lo ca
and Bold ini in [5]. The s a iona y p oblem is s udied in [6] o bounded domains
and in [10] o ex e io domains. On he o he hand, A.C. Mo e i, M.A. Rojas-
Meda and M.D. Rojas-Meda [8] p o ed exis ence o ep oduc i e weak solu ions
in ex e io domains. The classical Boussinesq model, whe e νand ka e posi i es
cons an s, has been analyzed in g ea ex en , see o ins ance, Mo imo o [9], ´
Oeda
[11].
The a gumen s used in [5] in o de o ob ain egula solu ion (and uniqueness)
a e no alid o ind ep oduc i i y since he ini ial condi ions play a undamen al
ole. Ou con ibu ion in his pape is o ob ain highe o de es ima es o he
empe a u e han in [5]; namely in [5] H2(Ω) egula i y is ob ained o eloci y and
empe a u e, bu now we will a i e a H3(Ω) egula i y o he empe a u e. Con-
sequen ly, a ep oduc i e condi ion o ime de i a i e o empe a u e also holds,
i.e. ∂ θ(0) = ∂ θ(T). In addi ion, he a gumen s used in his pape a e ema kably
simple han he used ones in [5]. By he con a y, now he egula i y ob ained o
he solu ion is no su icien o p o e uniqueness.
No a ion.
•In gene al, he no a ion will be ab idged. We se Lp=Lp(Ω), p≥1,
H1
0=H1
0(Ω), e c. I X=X(Ω) is a space o unc ions de ined in he open
se Ω, we deno e by Lp(X) he Banach space Lp(0, T ;X). Also, bold ace
le e s will be used o ec o ial spaces, o ins ance L2=L2(Ω)N.
•The Lpno m is deno ed by | · |p, 1 ≤p≤ ∞. The Hmno m is deno ed by
k · km
•We se V he space o med by all ields ∈C∞
0(Ω)Nsa is ying ∇ · = 0.
We deno e H( espec i ely V) he closu e o Vin L2( espec i ely H1).
Hand Va e Hilbe spaces o he no ms | · |2and k·k1, espec i ely.
Fu he mo e,
H={u∈L2;∇ · u= 0,u·n= 0 on ∂Ω},
V={u∈H1;∇ · u= 0,u= 0 on ∂Ω}
•I is easy deduce ha d
d ZΩ
θ(x, ) = 0 om he con ec ion-di usion equa-
ion o θ. Then, we can ix ZΩ
θ= 0.The e o e, le us conside he ollowing
spaces
Hk
N=½θ∈Hk;∂θ
∂n = 0 on ∂Ω,ZΩ
θ= 0¾
whe e k= 2,3. Hence, Hk
Nis a closed subspace o Hk. Consequen ly |∆θ|2
is equi alen o kθk2in H2
Nand |∇∆θ|2is equi alen o kθk3in H3
N([12]).
REGULARITY REPRODUCTIVE GENERALIZED BOUSSINESQ 3
Some in e pola ion inequali ies. We will use he ollowing classical in e pola-
ion and Sobole inequali ies ( o 3Ddomains):
| |6≤Ck k1,| |3≤ | |1/2
2k k1/2
1
and
| |∞≤Ck k1/2
1k k1/2
2
In his wo k, i will be use ul o use he ollowing esul (see [5]):
Lemma 1. Le u∈V∩H2and conside he Helmhol z decomposi ion o −∆u, i.e.
−∆u=Au+∇q, whe e q∈H1is aken such ha ZΩ
q dx = 0 and Ais he S okes
ope a o . Then,
kqk1≤C|Au|2.
Mo eo e , o e e y δ > 0 he e exis s a posi i e cons an Cδ(independen o u)
such ha
|q|2≤Cδ|∇u|2+δ|Au|2.
2. The Main Resul
De ini ion 1. I will be said ha (u, p, θ)is a egula solu ion o (1)–(3) in (0, T ),
i
u∈L2(H2)∩L∞(H1)and ∂ u∈L2(L2),
p∈L2(H1),
θ∈L2(H3
N)∩L∞(H2
N)and ∂ θ∈L2(H1
N),
sa is ying (1) a.e. in (0, T)×Ω, bounda y condi ions (2) and ime ep oduc i i y
condi ions (3) in he sense o spaces Vand H2
N espec i ely.
No ice ha we ha e imposed highe egula i y o θ han o u.
Theo em 1. Le T > 0and Ωa bounded domain in IRN(N= 2 o 3) wi h a
bounda y o class C2,1. Le he unc ions ν∈C1(IR) and k∈C2(IR) such ha
0< νmin ≤ν(s)≤νmax 0< kmin ≤k(s)≤kmax in IR,
and ν0, k0, k00 a e bounded in IR (i.e. |ν0(s)| ≤ ν0
max,|k0(s)| ≤ k0
max,|k00(s)| ≤ k00
max).
Assume ha ∈L2(L2)and g∈L∞(L2)and
k kL2(L2)≤δ
o δsmall enough, hen he e exis s a egula (and small) ep oduc i e solu ion o
(1)–(3) in (0, T). Mo eo e , his solu ion also e i ies ∂ θ(0) = ∂ θ(T).
Rema k: The uniqueness o solu ions u nished by p e ious Theo em emains open,
because highe egula i y o he eloci y is necessa y. To ob ain H3 egula i y o
he eloci y seem complica ed because he a gumen made in he p oo o Lemma 3 in
o de o ge H3 egula i y is based in he Neumann condi ion, bu we ha e Di ichle
condi ion o u.
The p oo o his Theo em will be gi en in Sec ion 5. The me hod is based on
he Gale kin app oxima ion wi h spec al basis (de ined in Sec ion 3) and some
di e en ial inequali ies in egula no ms gi en in Sec ion 4.
4 B. CLIMENT, F. GUILL´
EN, M.A. ROJAS
3. The Gale kin Ini ial-Bounda y P oblem
Le {φi}i≥1and {ϕi}i≥1“special” basis o Vand H1
0(Ω), espec i ely, o med
by eigen unc ions o he S okes and he Poisson p oblems ollowing:
(4) ½−∆φi=λiφiin Ω
φi= 0 on ∂Ω½−∆ϕi=µiϕiin Ω
∂nϕi= 0 on ∂Ω,
wi h kφik1= 1, kϕik1= 1 o all iand RΩϕi= 0. Le Vmand Wmbe he
ini e-dimensional subspaces spanned by {φ1, φ2, . . . , φm}and {ϕ1, ϕ2, . . . , ϕm} e-
spec i ely.
Fo each m≥1, gi en u0m∈Vmand θ0m∈Wm, we seek an app oxima e
solu ion (um, θm), wi h um: [0, T ]7→ Vmand θm: [0, T ]7→ Wm, e i ying he
ollowing a ia ional o mula ion a.e. in ∈(0, T ):
(5)
(∂ um( ), m) + ((um( )· ∇)um( ), m)+(ν(θm( ))∇um( ),∇ m)
−(αθm( )g, m)−( , m) = 0 ∀ m∈Vm
(∂ θm( ), em) + ((um( )· ∇)θm( ), em)
+(k(θm( ))∇θm( ),∇em) = 0 ∀em∈Wm
um(0) = u0m, θm(0) = θ0m,
I we pu
um( ) =
m
X
j=1
ξi,m( )φiand θm( ) =
m
X
j=1
ζi,m( )ϕi,
hen (5) can be ew i en as a i s o de o dina y di e en ial sys em (in no mal
o m) associa ed o he unknowns (ξi,m( ), ζi,m( )). Then, one has exis ence o a
maximal solu ion (de ined in some in e al [0, τm)⊂[0, T ]) o he ela ed Cauchy
p oblem. Mo eo e , om a p io i es ima es (independen on m) which will be
ob ained below, in pa icula one has ha τm=T. Finally, using egula i y o he
chosen spec al basis, uniqueness o app oxima e solu ion holds ([2]).
4. Di e en ial inequali ies in egula no ms
In he sequel, δand εwill deno e some cons an s su icien ly small. By Cwe will
deno e di e en cons an s, independen on da a and δand ε.
Lemma 2. Fo each δ, ε > 0su icien ly small, he e exis s a cons an K=
K(δ, ε)>0such ha
d
d ZΩ
(ν(θm) + 1)|∇um|2+νminkumk2
2+|∂ um|2
2≤δk∂ θmk2
1
+εkumk2
2kθmk2+K(kumk6
1+kumk2
1kθmk4
2+kgk2
L∞(L2)kθmk2
2+| |2
2)
P oo .
Fi s , aking =Aumas es unc ion in he u-sys em o (5) (Ais he Helmhol z
ope a o men ioned in Lemma 1) one has
(6) (∂ um, Aum)−(∇ · (ν(θm)∇um), Aum) + ((um· ∇)um, Aum)
−α(gθm, Aum) = ( , Aum)
We can w i e he i s e m as
(∂ um, Aum) = 1
2
d
d kumk2
1
REGULARITY REPRODUCTIVE GENERALIZED BOUSSINESQ 5
The second e m o (6) is spli as ollows (using he Helmhol z decomposi ion
∆u=−Au+∇q),
−(∇ · (ν(θm)∇um), Aum) = (ν(θm)Aum, Aum)
+ (ν(θm)∇q, Aum)−(ν0(θm)∇θm∇um, Aum).
Taking in o accoun ha
(ν(θm)∇q, Aum) = −(q, ∇ · (ν(θm)Aum))
=−(q, ν0(θm)∇θmAum)−(q, ν(θm)∇ · Aum)
=−(q, ν0(θm)∇θmAum)
since ∇ · Aum= 0, hence he second e m o (6) emains
−(∇ · (ν(θm)∇um), Aum)=(ν(θm)Aum, Aum)
−(q, ν0(θm)∇θmAum)−(ν0(θm)∇θm∇um, Aum)
Then, (6) can also be w i en as ollows (using ν(θm)≥νmin >0)
(7)
1
2
d
d kumk2
1+νminkumk2
2≤ −((um· ∇)um, Aum)−α(gθm, Aum)
+(q, ν0(θm)∇θmAum) + (ν0(θm)∇θm∇um, Aum)+( , Aum)
:= I1+I2+I3+I4+I5
The i s wo e ms and he las e m on he igh hand side o (15) a e bounded
espec i ely by
I1≤δkumk2
2+Cδkumk6
1, I2≤δkumk2
2+Cδ|g|2
2kθmk2
2
and
I5≤δkumk2
2+Cδ| |2
2.
In o de o es ima e he hi d e m we use he Lemma 1 (and |ν0(θm)| ≤ ν0
max)
|I3|=|(q, ν0(θm)∇θmAum)| ≤ ν0
max|q|3|∇θm|6|Aumk2
≤C|q|1/2
2kqk1/2
1kθmk2kumk2≤C(Cεkumk1/2
1+εkumk1/2
2)kumk3/2
2kθmk2
≤Cεkumk1/2
1kumk3/2
2kθmk2+εkumk2
2kθmk2
≤δkumk2
2+Cε,δkumk2
1kθmk4
2+εkumk2
2kθmk2.
In wha conce ns o he ou h e m,
|I4|=|(ν0(θm)∇θm∇um, Aum)| ≤ ν0
max|∇θm|6|∇umk3|Aum|2
≤Ckθmk2kumk1/2
1kumk3/2
2≤δkumk2
2+Cδkumk2
1kθmk4
2.
Consequen ly, choosing δsmall enough we ob ain ha
(8)
d
d kumk2
1+νminkumk2
2≤Cε(kumk6
1+kumk2
1kθmk4
2+|g|2
2kθmk2
2+| |2
2)
+εkumk2
2kθmk2
On he o he hand, using ∂ umas a es unc ion in he u-sys em o (5), one ob ains
(9) (∂ um, ∂ um)+(ν(θm)∇um, ∂ ∇um) + ((um· ∇)um, ∂ um)
−α(gθm, ∂ um) = ( , ∂ um).
6 B. CLIMENT, F. GUILL´
EN, M.A. ROJAS
By aking in o accoun ha he second e m in (9) can be w i en as
(ν(θm)∇um, ∂ ∇um) = 1
2
d
d (ν(θm)∇um,∇um)−1
2(∂ (ν(θm))∇um,∇um),
we deduce om (9) ha
(10)
1
2
d
d ZΩ
ν(θm)|∇um|2+|∂ um|2
2≤ −((um· ∇)um, ∂ um)
+α(gθm, ∂ um) + 1
2(∂ (ν(θm))∇um,∇um)+( , ∂ um)
:= J1+J2+J3+J4
The i s wo e ms and he las e m on he igh side o (10) a e bounded espec-
i ely, by
J1≤δ(|∂ um|2
2+kumk2
2) + Cδkumk6
1, J2≤δ|∂ um|2
2+Cδ|g|2
2kθmk2
2,
and
J4≤δ|∂ um|2
2+Cδ| |2
2.
Las ly, we go in o de ail he hi d e m:
|J3|=|(ν0(θm)∂ (θm)∇um,∇um)| ≤ ν0
max|∂ θm|6|∇umk3|∇umk2
≤Ck∂ θmk1kumk3/2
1kumk1/2
2≤δ(k∂ θmk2
1+kumk2
2) + Cδkumk6
1
Consequen ly, choosing δsmall enough,
(11)
d
d ZΩ
ν(θm)|∇um|2+|∂ um|2
2≤δ(k∂ θmk2
1+kumk2
2)
+Cδ(kumk6
1+|g|2
2kθmk2
2+| |2
2)
Finally, (8) and (11) p o e he Lemma.
Lemma 3. Fo each δ > 0small enough, he e exis s Cδ>0such ha
d
d (kθmk2
2+|∂ θm|2
2) + kmin(kθmk2
3+k∂ θmk2
1)
≤δ|∂ umk2
2+Cδ(kθmk6
2+kθmk4
2|∂ θm|2
2+kθmk2
2kumk4
1)
u
P oo .
Di e en ing espec o he ime he θm-equa ion o (5) and using ∂ θmas es
unc ion, one ob ains
(12) 1
2
d
d |∂ θm|2
2+ (∂ (k(θm)∇θm), ∂ ∇θm)+(∂ um· ∇θm, ∂ θm) = 0
since (um· ∇∂ θm, ∂ θm) = 0.
By aking in o accoun ha he second e m in (12) can be spli as
(∂ (k(θm)∇θm), ∂ ∇θm) = (k0(θm)∂ θm∇θm, ∂ ∇θm)+(k(θm)∂ ∇θm, ∂ ∇θm),
we deduce om (12) ha
(13)
1
2
d
d |∂ θm|2
2+kmin|∂ ∇θm|2
2≤ −(k0(θm)∂ θm∇θm, ∂ ∇θm)
−(∂ um· ∇θm, ∂ θm).
REGULARITY REPRODUCTIVE GENERALIZED BOUSSINESQ 7
Bounding bo h e ms on he igh hand side o (13) (k0
max = max |k0|):
|(k0(θm)∂ θm∇θm, ∂ ∇θm)| ≤ k0
max|∇θm|6|∂ θm|3|∂ ∇θm|2
≤Ckθmk2|∂ θm|1/2
2k∂ θmk3/2
1≤δk∂ θmk2
1+Cδkθmk4
2|∂ θm|2
2
and
|(∂ um· ∇θm, ∂ θm)| ≤ |∂ um|2|∇θm|6|∂ θm|3
≤C|∂ um|2kθmk2|∂ θm|1/2
2k∂ θmk1/2
1
≤δ(k∂ θmk2
1+|∂ um|2
2) + Cδkθmk4
2|∂ θm|2
2
we ob ain, o δsmall enough,
(14) d
d |∂ θm|2
2+kmink∂ θmk2
1≤δ|∂ um|2
2+Cδkθmk4
2|∂ θm|2
2
Now, using ∆2θmas es unc ion (∆2θm∈Wm hanks o he elec ion o spec al
basis) and in eg a ing by pa s in all e ms (bounda y e ms anish since (∇∆θm·
n)|∂Ω= 0), one ob ains:
(15) −(∂ ∇θm,∇∆θm)+(∇[∇·(k(θm)∇θm)],∇∆θm)−(∇(u·∇θm),∇∆θm) = 0.
No ice ha i Di ichle bounda y condi ion is imposed o he empe a u e θ, he
bounda y e ms do no anish in he in eg a ion by pa s and we can no ob ain
he p e ious inequali ies.
In eg a ing by pa s he i s e m o (15) (again he bounda y e m anishes
since (∂ ∇θm·n)|∂Ω= 0), he e m emains 1
2
d
d |∆θm|2
2. The second e m is
(∇[∇ · (k(θm)∇θm)],∇∆θm) = (k00(θm)(∇θm)3,∇∆θm)
+2(k0(θm)∇2θm∇θm,∇∆θm)
+(k0(θm)∇θm∆θm,∇∆θm) + (k(θm)∇∆θm,∇∆θm).
Hence, we deduce o (15) ha (|k00(θm)| ≤ k00
max = max |k00|)
1
2
d
d |∆θm|2
2+k0|∇∆θm|2
2≤k00
max|((∇θm)3,∇∆θm)|
+2k0
max|(∇2θm∇θm,∇∆θm)|+k0
max|(∇θm∆θm,∇∆θm)|
+|(∇um∇θm,∇∆θm)|+|(um∇2θm,∇∆θm)|
:= L1+L2+L3+L4+L5.
Replacing in he abo e inequali y he ollowing es ima ions
L1≤C|∇θm|3
6|∇∆θm|2≤δkθmk2
3+Cδkθmk6
2
L2≤C|∇2θm|3|∇θm|6|∇∆θm|2≤Ckθmk3/2
2kθmk3/2
3≤δkθmk2
3+Cδkθmk6
2
L3≤C|∇θm|6|∆θm|3|∇∆θm|2≤Ckθmk3/2
2kθmk3/2
3≤δkθmk2
3+Cδkθmk6
2
L4≤C|∇um|2|∇θm|∞|∇∆θm|2≤Ckumk1kθmk1/2
2kθmk3/2
3
≤δkθmk2
3+Cδkumk4
1kθmk2
2
L5≤C|um|6|∇2θm|3|∇∆θm|2≤Ckumk1kθmk1/2
2kθmk3/2
3
≤δkθmk2
3+Cδkumk4
1kθmk2
2
8 B. CLIMENT, F. GUILL´
EN, M.A. ROJAS
we ge , aking δsmall enough,
(16) d
d kθmk2
2+kminkθmk2
3≤Cδ(kθmk6
2+kumk4
1kθmk2
2)
Finally, (14) added o (16) p o es he Lemma.
5. P oo o Theo em 2.2
I we deno e
Φm( ) = ZΩ
(ν(θm) + 1)|∇um|2+kθmk2
2+|∂ θm|2
2
Ψm( ) = kumk2
2+|∂ um|2
2+kθmk2
3+k∂ θmk2
1
aking an adequa e balance be ween inequali ies om Lemmas 2 and 3 in o de o
anish he e m ||g||2
L∞(L2)||θm||2
2a he igh hand-side, one has
(17) (Φ0
m+CΨm≤εΨmΦ1/2
m+C0( ) + DΦ3
m
Φm(0) = Φm0
whe e C, D > 0 a e cons an and C0( ) is a posi i e unc ion depending on da a .
Conc e ely, C0( ) = C0| |2
2.
Le Φm(0) ≤δ o δa small enough cons an ( ha we will speci y la e ).
Fi s s ep: I Φm(0) ≤δand k kL2(L2)≤δ, hen Φm( )<2δ∀ ∈[0, T ].
Indeed, by an absu d a gumen , le T∗be he i s alue in [0, T ] such ha
Φm(T∗) = 2δ, hence
Φm(T∗) = 2δand Φm(s)<2δ∀s∈[0, T ∗).
Mo eo e , he e exis s a Poinca ´e cons an Cp>0 such ha Φm( )≤CpΨm( ).
Then o εsmall enough we ha e
CΨm−εΨmΦ1/2
m≥CΨm−εΨm(2δ)1/2≥¯
CΨm≥¯
C
Cp
Φm≡˜
CΦm.
The abo e inequali y oge he (17) lead:
(18) (Φ0
m+˜
CΦm≤C0( ) + DΦ3
m
Φm(0) = Φm0.
in [0, T∗]. Then, Φ0
m+˜
CΦm≤C0( )+4δ2DΦmin [0, T∗]. We can ind δsuch ha
˜
C−4δ2D≥¯
Cbeing ¯
Ca posi i e cons an . The e o e,
Φ0
m+¯
CΦm≤C0( ) in [0, T∗]
hence
(19) (e¯
C Φm)0≤e¯
C C0( ) in [0, T ∗].
In eg a ing in [0, T ∗] one inds:
e¯
CT ∗Φm(T∗)≤Φm(0) + ZT∗
0
e¯
C C0( ),
hence
Φm(T∗)≤δe−¯
CT ∗+ZT∗
0
C0( ).
REGULARITY REPRODUCTIVE GENERALIZED BOUSSINESQ 9
We can choose ZT∗
0
C0( ) small enough such ha he igh side is smalle ha 2δ
( o example ZT∗
0
C0( )≤δ). Thus, we a i e a a con adic ion.
Second s ep: I Φm(0) and k kL2(L2)a e small enough, hen Φm(T)≤Φm(0)
Now, as Φm( )<2δ∀ ∈[0, T ], we can epea he abo e a gumen and o ob ain
(18) in [0, T]. The e o e, in eg a ing (19) in [0, T ] we a i e a
Φm(T)≤Φm(0)e−¯
CT +ZT
0
C0(s).
Again, o Z
0
C0( ) small enough ( o example ZT∗
0
C0( )≤δ(1 −e−¯
CT )) one
ob ains ha Φm(T)≤Φm(0).
Thi d s ep: Exis ence o app oxima e ep oduc i e solu ion
Gi en (um0, θm0)∈Vm×Wm, we de ine he map
Lm: [0, T]7→ IRm×IRm
7→ (ξ1m( ), ..., ξmm( ), ζ1m( ), ..., ζmm( ))
whe e (ξ1m( ), ..., ξmm( )) and (ζ1m( ), ..., ζmm( )) a e coe icien s o um( ) and θm( )
espec o Vmand Wm espec i ely, being (um( ), θm( )) he (unique) app oxima e
solu ion o (5) co esponding o he ini ial da a (um0, θm0).
Now, a ying he ini ial da a (um0, θm0), we a e going o de ine a new map
Rm:¯
B⊂IRm×IRm7→ IRm×IRm
as ollows: gi en Lm
0∈IRm×IRm, we de ine Rm(Lm
0) = Lm(T), whe e Lm( ) is
ela ed o he solu ion o p oblem (5) wi h ini ial da a Lm
0(= Lm(0)) and
¯
B={(ξ1m, ..., ξmm, ζ1m, ..., ζmm) := Lm
0: Φm(0) ≤δ}.
By uniqueness o app oxima e solu ion o p oblem (5), his map is well-de ined.
Mo eo e , using egula i y o he co esponding o dina y di e en ial sys em (equi -
alen o (5)), his map is con inuous. By he second s ep, Rmapply ¯
Bin o ¯
Band
¯
Bis a closed, con ex and compac se . Consequen ly, B ouwe Theo em implies
he exis ence o ixed poin o Rm, which gi e us exis ence o ep oduc i e Gale kin
solu ion.
Fou s ep: Pass o he limi in ep oduc i e app oxima e solu ions
I he da a o he p oblem a e small, hanks o he i s s ep we ha e
Φm( ) = ZΩ
(ν(θm) + 1)|∇um|2+kθmk2
2+|∂ θm|2
2≤2δ
and (19). The e o e, he ollowing uni o mly bounds hold:
(um) in L∞(H1)∩L2(H2),
(θm) in L∞(H2
N)∩L2(H3
N),
(∂ um) in L2(L2),
(∂ θm) in L∞(L2)∩L2(H1).
