Reproductive and time periodic solutions for incompressible fluids

Climent Ezquerra, María Blanca; Guillén González, Francisco Manuel; Rojas Medar, Marko Antonio
Reproductive and time periodic solutions for incompressible fluids

Author: Climent Ezquerra, María Blanca; Guillén González, Francisco Manuel; Rojas Medar, Marko Antonio
Year: 2007
Source: https://idus.us.es/bitstreams/404bd030-0d7e-4263-a6ae-255cce9a1a65/download
REPRODUCTIVE AND TIME PERIODIC
SOLUTIONS FOR INCOMPRESSIBLE
FLUIDS
Blanca Climen Ezque a
F ancisco Guill´
en Gonz´
alez
Ma ko Rojas Meda
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 1/35
Table o con en s
1In oduc ion
2Na ie -S okes equa ions
Main classical esul s o he ini ial-bounda y p oblem
On he ime-pe iodic weak solu ions
Rela ion be ween weak pe iodic solu ions and global
solu ions
3Some a ian s o Na ie -S okes equa ions
Boussinesq equa ions
Mic opola equa ions
4Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em, wi h di usion
depending on empe a u e
Penalized Nema ic liquid c ys al model
5Regula i y o pe iodic solu ions ia egula i y o ep oduc i e
solu ions
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 2/35
Table o con en s
1In oduc ion
2Na ie -S okes equa ions
Main classical esul s o he ini ial-bounda y p oblem
On he ime-pe iodic weak solu ions
Rela ion be ween weak pe iodic solu ions and global
solu ions
3Some a ian s o Na ie -S okes equa ions
Boussinesq equa ions
Mic opola equa ions
4Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em, wi h di usion
depending on empe a u e
Penalized Nema ic liquid c ys al model
5Regula i y o pe iodic solu ions ia egula i y o ep oduc i e
solu ions
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 2/35
Table o con en s
1In oduc ion
2Na ie -S okes equa ions
Main classical esul s o he ini ial-bounda y p oblem
On he ime-pe iodic weak solu ions
Rela ion be ween weak pe iodic solu ions and global
solu ions
3Some a ian s o Na ie -S okes equa ions
Boussinesq equa ions
Mic opola equa ions
4Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em, wi h di usion
depending on empe a u e
Penalized Nema ic liquid c ys al model
5Regula i y o pe iodic solu ions ia egula i y o ep oduc i e
solu ions
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 2/35
Table o con en s
1In oduc ion
2Na ie -S okes equa ions
Main classical esul s o he ini ial-bounda y p oblem
On he ime-pe iodic weak solu ions
Rela ion be ween weak pe iodic solu ions and global
solu ions
3Some a ian s o Na ie -S okes equa ions
Boussinesq equa ions
Mic opola equa ions
4Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em, wi h di usion
depending on empe a u e
Penalized Nema ic liquid c ys al model
5Regula i y o pe iodic solu ions ia egula i y o ep oduc i e
solu ions
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 2/35

Table o con en s
1In oduc ion
2Na ie -S okes equa ions
Main classical esul s o he ini ial-bounda y p oblem
On he ime-pe iodic weak solu ions
Rela ion be ween weak pe iodic solu ions and global
solu ions
3Some a ian s o Na ie -S okes equa ions
Boussinesq equa ions
Mic opola equa ions
4Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em, wi h di usion
depending on empe a u e
Penalized Nema ic liquid c ys al model
5Regula i y o pe iodic solu ions ia egula i y o ep oduc i e
solu ions
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 2/35
Table o con en s
1In oduc ion
2Na ie -S okes equa ions
Main classical esul s o he ini ial-bounda y p oblem
On he ime-pe iodic weak solu ions
Rela ion be ween weak pe iodic solu ions and global
solu ions
3Some a ian s o Na ie -S okes equa ions
Boussinesq equa ions
Mic opola equa ions
4Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em, wi h di usion
depending on empe a u e
Penalized Nema ic liquid c ys al model
5Regula i y o pe iodic solu ions ia egula i y o ep oduc i e
solu ions
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 3/35
The i le
Time-condi ions
Ini ial condi ion: u(0) = u0
Condi ion o ep oduc i i y: u(0) = u(T)
Condi ion o ep oduc i i y (o ime-pe iodic condi ion)
⇒Rep oduc i e solu ions
Mo eo e i u( +T) = u( )∀ ∈(0,+∞)
⇒Pe iodic solu ions
Incomp essibili y condi ion
∇ · u=0
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 4/35
The i le
Time-condi ions
Ini ial condi ion: u(0) = u0
Condi ion o ep oduc i i y: u(0) = u(T)
Condi ion o ep oduc i i y (o ime-pe iodic condi ion)
⇒Rep oduc i e solu ions
Mo eo e i u( +T) = u( )∀ ∈(0,+∞)
⇒Pe iodic solu ions
Incomp essibili y condi ion
∇ · u=0
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 4/35
Main classical esul s o he ini ial-bounda y p oblem
Weak solu ion
Fo any u0∈Hand ∈L2(0,T;H−1(Ω)), ini ial-bounda y
p oblem has (a leas ) a weak solu ion.
Regula i y
I u0∈Vand ∈L∞(0,∞;L2(Ω)):
Unique s ong solu ion (u,p)local
I (u0, )a e small enough he s ong solu ion is global.
Weak/s ong uniqueness p ope y
I a solu ion has he s ong egula i y, i coincides wi h any weak
solu ion associa ed wi h he same da a.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 9/35

Main classical esul s o he ini ial-bounda y p oblem
Weak solu ion
Fo any u0∈Hand ∈L2(0,T;H−1(Ω)), ini ial-bounda y
p oblem has (a leas ) a weak solu ion.
Regula i y
I u0∈Vand ∈L∞(0,∞;L2(Ω)):
Unique s ong solu ion (u,p)local
I (u0, )a e small enough he s ong solu ion is global.
Weak/s ong uniqueness p ope y
I a solu ion has he s ong egula i y, i coincides wi h any weak
solu ion associa ed wi h he same da a.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 9/35
Main classical esul s o he ini ial-bounda y p oblem
Weak solu ion
Fo any u0∈Hand ∈L2(0,T;H−1(Ω)), ini ial-bounda y
p oblem has (a leas ) a weak solu ion.
Regula i y
I u0∈Vand ∈L∞(0,∞;L2(Ω)):
Unique s ong solu ion (u,p)local
I (u0, )a e small enough he s ong solu ion is global.
Weak/s ong uniqueness p ope y
I a solu ion has he s ong egula i y, i coincides wi h any weak
solu ion associa ed wi h he same da a.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 9/35
Main classical esul s o he ini ial-bounda y p oblem
N=2
I u0∈Hand ∈L2(0,T;H−1(Ω)) he weak solu ions is
unique.
I u0∈Vand ∈L∞(0,∞;L2(Ω)) he s ong solu ion is
global.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 10/35
On he ime-pe iodic weak solu ions
Theo em
Fo any ∈L2(0,T;H−1(Ω)), he e exis s a weak solu ion o
ep oduc i e p oblem.
Time pe iodic ex ension, e
u, o any weak ep oduc i e solu ion u
o he whole ime in e al (0,+∞)is a pe iodic weak solu ion
co esponding o he da a,e , de ined as he ime pe iodic
ex ension o .
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 11/35
On he ime-pe iodic weak solu ions
Theo em
Fo any ∈L2(0,T;H−1(Ω)), he e exis s a weak solu ion o
ep oduc i e p oblem.
Time pe iodic ex ension, e
u, o any weak ep oduc i e solu ion u
o he whole ime in e al (0,+∞)is a pe iodic weak solu ion
co esponding o he da a,e , de ined as he ime pe iodic
ex ension o .
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 11/35

Main ideas p oo exis ence ep oduc i e solu ions
Le uk he unique app oxima e solu ion o he Gale kin ini ial
bounda y p oblem o Na ie -S okes in he ini e-dimensional
subspace Vk, spanned by he i s kelemen s o he “spec al”
basis o V(o hogonal in Vand o hono mal in H), associa ed
o a ini ial disc e e da a uk
0∈Vk.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 12/35
Main ideas p oo exis ence ep oduc i e solu ions
Ene gy inequali y + Poinca ´
e inequali y + in eg a ing [0,T]⇒
ec1Tkuk(T)k2
L2≤ kuk(0)k2
L2+CZT
0ec1 k ( )k2
H−1d .(1)
We de ine he ope a o Lk: [0,T]→Rk,
Lk( ) = (ck
1( ), ..., ck
k( )) whe e ck
i( ),i=1, . . . , k,a e he
coe icien s o he expansion o uk( )in Vk.
No e ha
kLk( )kRk=kuk( )kL2,(2)
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 13/35
Main ideas p oo exis ence ep oduc i e solu ions
We de ine he ope a o Φk:Rk→Rkas ollows: Gi en
Lk
0∈Rk,Φk(Lk
0) = Lk(T), whe e Lk( )a e he coe icien s o he
Gale kin solu ion wi h ini ial alue wi h coe icien s Lk
0.
Le ay-Schaude Theo em
Fo all λ∈[0,1], he possible solu ions o he equa ion
Lk
0(λ) = λΦk(Lk
0(λ)),a e bounded independen ly o λ?
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 14/35
Main ideas p oo exis ence ep oduc i e solu ions
We de ine he ope a o Φk:Rk→Rkas ollows: Gi en
Lk
0∈Rk,Φk(Lk
0) = Lk(T), whe e Lk( )a e he coe icien s o he
Gale kin solu ion wi h ini ial alue wi h coe icien s Lk
0.
Le ay-Schaude Theo em
Fo all λ∈[0,1], he possible solu ions o he equa ion
Lk
0(λ) = λΦk(Lk
0(λ)),a e bounded independen ly o λ?
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 14/35
Table o con en s
1In oduc ion
2Na ie -S okes equa ions
Main classical esul s o he ini ial-bounda y p oblem
On he ime-pe iodic weak solu ions
Rela ion be ween weak pe iodic solu ions and global
solu ions
3Some a ian s o Na ie -S okes equa ions
Boussinesq equa ions
Mic opola equa ions
4Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em, wi h di usion
depending on empe a u e
Penalized Nema ic liquid c ys al model
5Regula i y o pe iodic solu ions ia egula i y o ep oduc i e
solu ions
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 21/35

Rep oduc i i y and maximum p inciple
Gi en u:Q→R3such ha ∇ · u=0 in Qand u·n=0 on ∂Ω,
we conside he ( ep oduc i e) con ec ion-di usion p oblem o
he unknown c:Q→R(a concen a ion):
∂ c−∆c+u· ∇c=0,c|Σ=cΣ,c(0) = c(T),
whe e 0 <c≤cΣ≤con Σ, o some cons an s cand c.
Any ep oduc i e solu ion sa is ies he maximum p inciple.
In pa icula ,
∂ (c−c)−∆(c−c)+(u· ∇)(c−c) = 0 in Q.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 22/35
Rep oduc i i y and maximum p inciple
Gi en u:Q→R3such ha ∇ · u=0 in Qand u·n=0 on ∂Ω,
we conside he ( ep oduc i e) con ec ion-di usion p oblem o
he unknown c:Q→R(a concen a ion):
∂ c−∆c+u· ∇c=0,c|Σ=cΣ,c(0) = c(T),
whe e 0 <c≤cΣ≤con Σ, o some cons an s cand c.
Any ep oduc i e solu ion sa is ies he maximum p inciple.
In pa icula ,
∂ (c−c)−∆(c−c)+(u· ∇)(c−c) = 0 in Q.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 22/35
Rep oduc i i y and maximum p inciple
Mul iplying by (c−c)+and in eg a ing in Ω:
d
d ZΩ
|(c−c)+|2+ZΩ
|∇(c−c)+|2≤0.
In eg a ing in ∈(0,T)and using he pe iodic condi ion
c(0) = c(T):ZT
0k∇(c−c)+k2
L2=0.
Hence c≤cin Qhold. Simila ly c≥cin Qhold.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 23/35
Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em







∂ u− ∇ · (ν(θ)∇u)+(u· ∇)u+∇p=αθg+ ,
∇ · u=0,
∂ θ− ∇ · (k(θ)∇θ)+(u· ∇)θ=0,
u=0, θ =θ∂Ωon ∂Ω×[0,T),
u(0) = u(T), θ(0) = θ(T)in Ω.
ν:IR →IR+and k:IR →IR+a e con inuous unc ions.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 24/35
Rep oduc i i y and maximum p inciple
θm´ın =m´ınθ∂Ωθm´
ax =m´
axθ∂Ω
Maximum p inciple =⇒θm´ın ≤θ≤θm´
ax.
Then
∃νm´ın >0,km´ın >0, νm´
ax >0,km´
ax >0
such ha
νm´ın ≤ν(s)≤νm´
ax,km´ın ≤k(s)≤km´
ax,∀s∈[θm´ın, θm´
ax].
Changing νby eνand kby e
k, whe e eνand e
ka e bounded
unc ions, he same way ha in he Na ie -S okes case.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 25/35

Rep oduc i i y and maximum p inciple
θm´ın =m´ınθ∂Ωθm´
ax =m´
axθ∂Ω
Maximum p inciple =⇒θm´ın ≤θ≤θm´
ax.
Then
∃νm´ın >0,km´ın >0, νm´
ax >0,km´
ax >0
such ha
νm´ın ≤ν(s)≤νm´
ax,km´ın ≤k(s)≤km´
ax,∀s∈[θm´ın, θm´
ax].
Changing νby eνand kby e
k, whe e eνand e
ka e bounded
unc ions, he same way ha in he Na ie -S okes case.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 25/35
Rep oduc i i y and maximum p inciple
Penalized Nema ic liquid c ys al model







∂ u+ (u· ∇)u−µ∆u+∇p=−λ∇ · (∇d ∇d),
∇ · u=0,
∂ d+ (u· ∇)d=γ(∆d− ε(d)).
u=0,d=hon ∂Ω×(0,T)
u(0) = u(T),d(0) = d(T)in Ω.
ε(d) = ε−2(|d|2−1)d
Assuming |h| ≤ 1, we can apply he maximum p inciple
a gumen ob aining |d| ≤ 1.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 26/35
Rep oduc i i y and maximum p inciple
We conside a equi alen p oblem changing εbye ε, he
auxilia y unc ion
e ε(d) = ( ε(d)i |d| ≤ 1,
0 i |d|>1.
The key is ha |e ε(d)| ≤ 1
ε2∀d∈R3. Then, exis ence o weak
ep oduc i e solu ion o his model can be p o ed.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 27/35
Table o con en s
1In oduc ion
2Na ie -S okes equa ions
Main classical esul s o he ini ial-bounda y p oblem
On he ime-pe iodic weak solu ions
Rela ion be ween weak pe iodic solu ions and global
solu ions
3Some a ian s o Na ie -S okes equa ions
Boussinesq equa ions
Mic opola equa ions
4Rep oduc i i y and maximum p inciple
Gene alized Boussinesq sys em, wi h di usion
depending on empe a u e
Penalized Nema ic liquid c ys al model
5Regula i y o pe iodic solu ions ia egula i y o ep oduc i e
solu ions
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 28/35
Regula i y o pe iodic solu ions
Gene alized Boussinesq model
The egula i y o a ime-pe iodic solu ion o he gene alized
Boussinesq model is an open p oblem.
Gene alized Boussinesq model wi h Neumann bounda y
condi ions o empe a u e
Assuming small enough, ep oduc i e solu ion has
H2(Ω)- eloci y and H3(Ω)- empe a u e egula i y.
When Di ichle bounda y condi ions o uand θa e assumed, i
is no clea how o ob ain app op ia e di e en ial inequali ies in
H2 o eloci y and H3 o empe a u e.
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 34/35

Gene alized Boussinesq model
Pe iodic solu ions
Uniqueness o egula ime pe iodic solu ions emains
open: H3 egula i y o he eloci y ↔Di ichle condi ion o
eloci y !!
A gumen o egula ime pe iodic solu ion small da a ⇒
ku( ?)k2
H1+kθ( ?)k2
H1is small bu kθ( ?)k2
H2?
Blanca Climen Ezque a. Uni e sidad de Se illa. Rep od. and ime pe iodic solu ions o incomp essible luids 35/35