In this study, we showed the existence of global, finite energy, weak solutions to a class of compressible Euler equations, describing the dynamics of a quantum fluid interacting with an external magnetic field. To do so, we exploited suitable a priori estimates for weak solutions of the underlying wave-function dynamics, governed by a magnetic semilinear Schrödinger equation.
Citation: Raffaele Scandone. Global, finite energy, weak solutions to a magnetic quantum fluid system[J]. Networks and Heterogeneous Media, 2025, 20(2): 345-355. doi: 10.3934/nhm.2025016
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In this study, we showed the existence of global, finite energy, weak solutions to a class of compressible Euler equations, describing the dynamics of a quantum fluid interacting with an external magnetic field. To do so, we exploited suitable a priori estimates for weak solutions of the underlying wave-function dynamics, governed by a magnetic semilinear Schrödinger equation.
Let
Oε={x=(x∗,xn+1)|x∗=(x1,…,xn)∈Qand0<xn+1<εg(x∗)}, |
where
γ1≤g(x∗)≤γ2,∀x∗∈¯Q. | (1) |
Denote
{dˆuε−Δˆuεdt=(H(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x∈Oε,t>τ,∂ˆuε∂νε=0,x∈∂Oε, | (2) |
with the initial condition
ˆuε(τ,x)=ˆϕε(x),x∈Oε, | (3) |
where
As
{du0−1gn∑i=1(gu0yi)yidt=(H(t,(y∗,0),u0(t))+G(t,(y∗,0)))dt+m∑j=1cju0∘dwj,y∗=(y1,…,yn)∈Q,t>τ,∂u0∂ν0=0,y∗∈∂Q, | (4) |
with the initial condition
u0(τ,y∗)=ϕ0(y∗),y∗∈Q, | (5) |
where
Random attractors have been investigated in [2,5,10,19,9] in the autonomous stochastic case, and in [3,21,22,23] in the non-autonomous stochastic case. Recently, the limiting dynamical behavior of stochastic partial differential equations on thin domain was studied in [16,20,13,14,11,12,17,4]. However, in [17,13], we only investigated the limiting behavior of random attractors in
Let
We organize the paper as follows. In the next section, we establish the existence of a continuous cocycle in
Here we show that there is a continuous cocycle generated by the reaction-diffusion equation defined on
{dˆuε−Δˆuεdt=(H(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x=(x∗,xn+1)∈Oε,t>τ,∂ˆuε∂νε=0,x∈∂Oε, | (6) |
with the initial condition
ˆuετ(x)=ˆϕε(x),x∈Oε, | (7) |
where
H(t,x,s)s≤−λ1|s|p+φ1(t,x), | (8) |
|H(t,x,s)|≤λ2|s|p−1+φ2(t,x), | (9) |
∂H(t,x,s)∂s≤λ3, | (10) |
|∂H(t,x,s)∂x|≤ψ3(t,x), | (11) |
where
Throughout this paper, we fix a positive number
h(t,x,s)=H(t,x,s)+λs | (12) |
for all
h(t,x,s)s≤−α1|s|p+ψ1(t,x), | (13) |
|h(t,x,s)|≤α2|s|p−1+ψ2(t,x), | (14) |
∂h(t,x,s)∂s≤β, | (15) |
|∂h(t,x,s)∂x|≤ψ3(t,x), | (16) |
where
Substituting (12) into (6) we get for
{dˆuε−(Δˆuε−λˆuε)dt=(h(t,x,ˆuε(t))+G(t,x))dt+m∑j=1cjˆuε∘dwj,x=(x∗,xn+1)∈Oε,∂ˆuε∂νε=0,x∈∂Oε, | (17) |
with the initial condition
ˆuετ(x)=ˆϕε(x),x∈Oε. | (18) |
We now transfer problem (17)-(18) into an initial boundary value problem on the fixed domain
x∗=y∗,xn+1=εg(y∗)yn+1. |
It follows from [18] that the Laplace operator in the original variable
Δxˆu(x)=|J|divy(|J|−1JJ∗∇yu(y))=1gdivy(Pεu(y)), |
where we denote by
Pεu(y)=(guy1−gy1yn+1uyn+1⋮guyn−gynyn+1uyn+1−n∑i=1yn+1gyiuyi+1ε2g(1+n∑i=1(εyn+1gyi)2)uyn+1). |
In the sequel, we abuse the notation a little bit by writing
Fε(t,y∗,yn+1,s)=F(t,y∗,εg(y∗)yn+1,s),F0(t,y∗,s)=F(t,y∗,0,s), |
where
{duε−(1gdivy(Pεuε)−λuε)dt=(hε(t,y,uε(t))+Gε(t,y))dt+m∑j=1cjuε∘dwj,y=(y∗,yn+1)∈O,Pεuε⋅ν=0,y∈∂O, | (19) |
with the initial condition
uετ(y)=ϕε(y)=ˆϕε∘T−1ε(y),y∈O, | (20) |
where
Given
θ1,t(τ)=τ+t,for allτ∈R. | (21) |
Then
Ω={ω∈C(R,R):ω(0)=0}. |
Let
θtω(⋅)=ω(⋅+t)−ω(t),ω∈Ω,t∈R. | (22) |
Then
dz+αzdt=dw(t), | (23) |
for
Lemma 2.1. There exists a
limt→±∞|ω(t)|t=0for allω∈Ω′, |
and, for such
z∗(ω)=−α∫0−∞eαsω(s)ds |
is well defined. Moreover, for
(t,ω)→z∗(θtω)=−α∫0−∞eαsθtω(s)ds=−α∫0−∞eαsω(t+s)ds+ω(t) |
is a stationary solution of (23) with continuous trajectories. In addition, for
limt→±∞|z∗(θtω)|t=0,limt→±∞1t∫t0z∗(θsω)ds=0, | (24) |
limt→±∞1t∫t0|z∗(θsω)|ds=E|z∗|<∞. | (25) |
Denote by
˜Ω=Ω′1×⋯×Ω′mand F=m⊗j=1Fj, |
Then
Denote by
SCj(t)u=ecjtu,foru∈L2(O), |
and
T(ω):=SC1(z∗1(ω))∘⋯∘SCm(z∗m(ω))=em∑j=1cjz∗j(ω)IdL2(O),ω∈Ω′. |
Then for every
T−1(ω):=SCm(−z∗m(ω))∘⋯∘SC1(−z∗1(ω))=e−m∑j=1cjz∗j(ω)IdL2(O). |
It follows that
On the other hand, since
limt→±∞1t∫t0‖T(θτω)‖2dτ=E‖T‖2=m∏j=1E(e2cjz∗j)<∞, |
and
limt→±∞1t∫t0‖T−1(θτω)‖2dτ=E‖T−1‖2=m∏j=1E(e−2cjz∗j)<∞. |
Remark 1. We now consider
Next, we define a continuous cocycle for system (19)-(20) in
{dvεdt−1gdivy(Pεvε)=(−λ+δ(θtω))vε+T−1(θtω)hε(t,y,T(θtω)vε(t))+T−1(θtω)Gε(t,y),y∈O,t>τ,Pεvε⋅ν=0,y∈∂O, | (26) |
with the initial conditions
vετ(y)=ψε(y),y∈O, | (27) |
where
Since (26) is a deterministic equation, by the Galerkin method, one can show that if
Φε(t,τ,ω,ϕε)=uε(t+τ,τ,θ−τω,ϕε)=T(θt+τω)vε(t+τ,τ,θ−τω,ψε),for all(t,τ,ω,ϕε)∈R+×R×Ω×N. | (28) |
By the properties of
Let
(Rεˆϕε)(y)=ˆϕε(T−1εy),∀ˆϕε∈L2(Oε). |
Given
ˆΦε(t,τ,ω,ˆϕε)=R−1εΦε(t,τ,ω,Rεˆϕε). |
The same change of unknown variable
{dv0dt−n∑i=11g(gv0yi)yi=(−λ+δ(θtω))v0+T−1(θtω)h0(t,y∗,T(θtω)v0(t))+T−1(θtω)G0(t,y∗),y∗∈Q,t>τ,∂v0∂ν0=0,y∗∈∂Q, | (29) |
with the initial conditions
v0τ(y∗)=ψ0(y∗),y∗∈Q, | (30) |
where
The same argument as above allows us to prove that problem (4) and (5) generates a continuous cocycle
Now we want to write equation (26)-(27) as an abstract evolutionary equation. We introduce the inner product
(u,v)Hg(O)=∫Oguvdy,for allu,v∈N |
and denote by
For
aε(u,v)=(J∗∇yu,J∗∇yv)Hg(O), | (31) |
where
J∗∇yu=(uy1−gy1gyn+1uyn+1,…,uyn−gyngyn+1uyn+1,1εguyn+1). |
By introducing on
‖u‖H1ε(O)=(∫O(|∇y∗u|2+|u|2+1ε2u2yn+1)dy)12, | (32) |
we see that there exist positive constants
η1∫O(|∇y∗u|2+1ε2u2yn+1)dy≤aε(u,u)≤η2∫O(∇y∗u|2+1ε2u2yn+1)dy | (33) |
and
η1‖u‖2H1ε(O)≤aε(u,u)+‖u‖2L2(O)≤η2‖u‖2H1ε(O). | (34) |
Denote by
D(Aε)={v∈H2(O),Pεv⋅ν=0on∂O} |
as defined by
Aεv=−1gdivPεv,v∈D(Aε). |
Then we have
aε(u,v)=(Aεu,v)Hg(O),∀u∈D(Aε),∀v∈H1(O). | (35) |
Using
{dvεdt+Aεvε=(−λ+δ(θtω))vε+T−1(θtω)hε(t,y,T(θtω)vε(t))+T−1(θtω)Gε(t,y),y∈O,t>τ,vετ=ψε. | (36) |
To reformulate system (29)-(30), we introduce the inner product
(u,v)Hg(Q)=∫Qguvdy∗,for allu,v∈M, |
and denote by
a0(u,v)=∫Qg▽y∗u⋅▽y∗vdy∗. |
Denote by
D(A0)={v∈H2(Q),∂v∂ν0=0on∂Q} |
as defined by
A0v=−1gn∑i=1(gvyi)yiv∈D(A0). |
Then we have
a0(u,v)=(A0u,v)Hg(Q),∀u∈D(A0),∀v∈H1(Q). |
Using
{dv0dt+A0v0=(−λ+δ(θtω))v0+T−1(θtω)h0(t,y∗,T(θtω)v0(t))+T−1(θtω)G0(t,y∗),y∗∈Q,t>τ,v0τ(s)=ψ0(s),s∈[−ρ,0]. | (37) |
Hereafter, we set
limt→−∞ect‖Bi(τ+t,θtω)‖Xi=0, |
where
Di={Bi={Bi(τ,ω):τ∈R,ω∈Ω}:Bi is tempered in Xi}. |
Our main purpose of the paper is to prove that the cocycle
limε→0supuε∈ˆAεinfu0∈A0ε−1‖uε−u0‖2H1(Oε)=0. | (38) |
To prove (38), we only need to show that the cocycle
limε→0distH(Aε(τ,ω),A0(τ,ω))=0, |
which will be established in the last section of the paper.
Furthermore, we suppose that there exists
¯γΔ=λ0−2E(|δ(ω)|)>0. | (39) |
Let us consider the mapping
γ(ω)=λ0−2|δ(ω)|. | (40) |
By the ergodic theory and (39) we have
limt→±∞1t∫t0γ(θlω)dl=Eγ=¯γ>0. | (41) |
The following condition will be needed when deriving uniform estimates of solutions:
∫τ−∞e12¯γs(‖G(s,⋅)‖2L∞(˜O)+‖φ1(s,⋅)‖2L∞(˜O)+‖ψ3(s,⋅)‖2L∞(˜O))ds<∞,∀τ∈R. | (42) |
When constructing tempered pullback attractors, we will assume
limr→−∞eσr∫0−∞e12¯γs(‖G(s+r,⋅)‖2L∞(˜O)+‖φ1(s+r,⋅)‖2L∞(˜O)+‖ψ3(s+r,⋅)‖2L∞(˜O))ds=0,∀σ>0. | (43) |
Since
∫τ−∞e12¯γs(‖G(s,⋅)‖2L∞(˜O)+‖ψ1(s,⋅)‖L∞(˜O)+‖ψ3(s,⋅)‖2L∞(˜O))ds<∞,∀τ∈R | (44) |
and
limr→−∞eσr∫0−∞e12¯γs(‖G(s+r,⋅)‖2L∞(˜O)+‖ψ1(s+r,⋅)‖2L∞(˜O)+‖ψ3(s+r,⋅)‖2L∞(˜O))ds=0, | (45) |
for any
In this section, we recall and generalize some results in [17] and derive some new uniform estimates of solutions of problem (36) or (19)-(20) which are needed for proving the existence of
Lemma 3.1. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖vε(τ+s,τ−t,θ−τω,ψε)‖2H1ε(O)≤R2(τ,ω), | (46) |
where
R2(τ,ω)=r1(ω)R1(τ,ω)+c∫0−∞e¯γr‖T−1(θrω)‖2(‖G(r+τ,⋅)‖2L∞(˜O)+‖ψ3(r+τ,⋅)‖2L∞(˜O))dr, | (47) |
where
R1(τ,ω)=c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖G(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖ψ1(r+τ,⋅)‖2L∞(˜O)dr, | (48) |
and
Proof. The proof is similar as that of Lemma 3.4 in [17], so we only sketch the proof here. Taking the inner product of (36) with
12ddt‖vε‖2Hg(O)≤−aε(vε,vε)+(−λ0+δ(θtω))‖vε‖2Hg(O)+(T−1(θtω)hε(t,y,T(θtω)vε(t)),vε)Hg(O)+(T−1(θtω)Gε(t,y),vε)Hg(O). | (49) |
By (13), we have
ddt‖vε‖2Hg(O)+2aε(vε,vε)+λ02‖vε‖2Hg(O)+2α1γ1‖T−1(θtω)‖2‖uε‖pLp(O)≤(−λ0+2δ(θtω))‖vε‖2Hg(O)+2λ0γ2|˜O|‖T−1(θtω)‖2‖G(t,⋅)‖2L∞(˜O)+2γ2|˜O|‖T−1(θtω)‖2‖ψ1(t,⋅)‖L∞(˜O). | (50) |
Then, we have for any
e∫στγ(θlω)dl‖vε(σ)‖2Hg(O)+2∫στe∫rτγ(θlω)dlaε(vε(r),vε(r))dr+λ02∫στe∫rτγ(θlω)dl‖vε(r)‖2Hg(O)dr+2α1γ1∫στ‖T−1(θrω)‖2e∫rτγ(θlω)dl‖uε(r)‖pLp(O)dr≤‖vε(τ)‖2Hg(O)+2λ0γ2|˜O|∫στe∫rτγ(θlω)dl‖T−1(θrω)‖2‖G(r,⋅)‖2L∞(˜O)dr+2γ2|˜O|∫στe∫rτγ(θlω)dl‖T−1(θrω)‖2‖ψ1(r,⋅)‖2L∞(˜O)dr, | (51) |
where
Thus by the similar arguments as Lemma 3.1 in [17] we get for every
‖vετ(⋅,τ−t,θ−τω,ψ)‖2L2(O)≤c∫0−∞e∫r0γ(θlω)dl‖ψ1(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖G(r+τ,⋅)‖2L∞(˜O)dr+c∫0−∞e∫r0γ(θlω)dl‖T−1(θrω)‖2‖ψ1(r+τ,⋅)‖2L∞(˜O)dr. | (52) |
Moreover, taking the inner product of (36) with
12ddtaε(vε,vε)+‖Aεvε‖2Hg(O)≤(−λ0+δ(θtω))aε(vε,vε)+(T−1(θtω)hε(t,y,T(θtω)vε(t)),Aεvε)Hg(O)+(T−1(θtω)Gε(t,y),Aεvε)Hg(O). | (53) |
By (15)-(16) we have
ddtaε(vε,vε)+‖Aεvε‖2Hg(O)≤(c+2δ(θtω))aε(vε,vε)+c‖T−1(θtω)‖2(‖G(t,⋅)‖2L∞(˜O)+‖ψ3(t,⋅)‖2L∞(˜O)), | (54) |
The left proof is similar of that Lemma 3.4 in [17], so we omit it here.
We are now in a position to establish the uniform estimates for the solution
Lemma 3.2. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖uε(τ+s,τ−t,θ−τω,ϕε)‖2H1ε(O)≤r2(ω)R2(τ,ω), | (55) |
where
Lemma 3.3. Assume that (8)-(11), (39) and (42) hold. Then for every
sup−1≤s≤0‖vε(τ+s,τ−t,θ−τω,ψε)‖pLp(O)+∫ττ−ρ‖vε(r,τ−t,θ−τω,ψε)‖2p−2L2p−2(O)dr≤R3(τ,ω), | (56) |
where
Proof. The proof is similar as that of Lemma 3.6 in [14], so we omit it here.
Lemma 3.4. Assume that (8)-(11), (39) and (42) hold. Then for every
∫0−1eγMp−2s∫{y∈O: vε(s+τ,τ−t,θ−τω,ψε)≥2M}|vε(s+τ,τ−t,θ−τω,ψε)|2p−2dyds≤η, | (57) |
∫0−1eγMp−2s∫{y∈O: vε(s+τ,τ−t,θ−τω,ψε)≤−2M}|vε(s+τ,τ−t,θ−τω,ψε)|2p−2dyds≤η. | (58) |
Proof. Let
1pddt‖(vε−M)+‖pLp(O)+(p−1)∫vε≥M(vε−M)p−2aε(vε,vε)dx≤(δ(θtω)vε,(vε−M)p−1+)+(T−1(θtω)hε(t,y,T(θtω)vε),(vε−M)p−1+)+(T−1(θtω)Gε(t,y),(vε−M)p−1+). | (59) |
For the first term on the right side of (59) we have
|(δ(θtω)vε,(vε−M)p−1+)|≤1p|δ(θrω)|p∫O|vε|pdx+p−1p∫O(vε−M)p+dx. | (60) |
For the second term on the right-hand side of (59), by (8), we obtain, for
hε(t,y,T(θtω)vε) (vε−M)p−1+≤−α1‖T(θtω)‖p−1(vε)p−1(v−M)p−1+ |
+‖T(θtω)‖−1ψ1(t,y∗,εg(y∗)yn+1)(vε)−1(vε−M)p−1+ |
≤−12α1Mp−2‖T(θtω)‖p−1(vε−M)p+−12α1‖T(θtω)‖p−1(vε−M)2p−2+ |
+‖T−1(θtω)‖−1|ψ1(t,y∗,εg(y∗)yn+1)|(vε−M)p−2+ |
which implies
(T−1(θtω)hε(t,y,T(θtω)vε), (vε−M)p−1+) |
≤−12α1Mp−2‖T(θtω)‖p−2∫O(vε−M)p+dx−12α1‖T−1(θtω)‖p−2∫O(vε−M)2p−2+dx |
+‖T(θtω)‖−2∫O|ψ1(t,y∗,εg(y∗)yn+1)|(vε−M)p−2+dx |
≤−12α1Mp−2‖T(θtω)‖p−2∫O(vε−M)p+dx−12α1‖T(θtω)‖p−2∫O(vε−M)2p−2+dx |
+p−2p∫O(vε−M)p+dx+2p‖T(θtω)‖−p∫O|ψ1(t,y∗,εg(y∗)yn+1)|p2dy. | (61) |
The last term in (59) is bounded by
(T−1(θtω)Gε(t,y),(vε−M)p−1+)≤18α1‖T(θtω)‖p−2∫O(vε−M)2p−2+dx+2α1‖T(θtω)‖−p∫vε≥M|Gε(t,y)|2dy. | (62) |
All above estimates yield
ddt‖(vε−M)+‖pLp(O)−(2p−3−12pα1Mp−2‖T(θtω)‖p−2)∫O(vε−M)p+dx+14pα1‖T(θtω)‖p−2∫O(vε−M)2p−2+dx≤|δ(θrω)|p∫O|vε|pdx+2‖T(θtω)‖−p∫O|ψ1(t,y∗,εg(y∗)yn+1)|p2dy+2pα1‖T(θtω)‖−p∫O|Gε(t,y)|2dy. | (63) |
Multiplying (63) by
‖(vε(τ,τ−t,ω,ψε)−M)+‖pLp(O) |
+14pα1∫ττ−1‖T(θζω)‖p−2e−∫ζτ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr |
×∫O(vε(ζ,τ−t,ω,ψε)−M)2p−2+dxdζ |
≤e−∫τ−1τ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖(vε(τ−1,τ−t,ω,ψε)−M)+‖pLp(O) |
+∫ττ−1|δ(θζω)|pe−∫ζτ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖vε(ζ,τ−t,ω,ψε)‖pLp(O)dζ |
+2|O|∫ττ−1‖T(θζω)‖−pe−∫ζτ+ξ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖ψ1(ζ,⋅)‖p2L∞(˜O)dζ. |
+2p|O|α1∫ττ−1‖T(θζω)‖−pe−∫ζτ+ξ(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖G(ζ,⋅)‖2L∞(˜O)dζ, | (64) |
where
14pα1∫0−1‖T(θζω)‖p−2e−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr |
×∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ |
≤e−∫−10(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖(vε(τ−1,τ−t,θ−τω,ψε)−M)+‖pLp(O) |
+∫0−1|δ(θζ+ξω)|pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖vε(ζ+τ,τ−t,θ−τω,ψε)‖pLp(O)dζ |
+2|O|∫0−1‖T(θζω)‖−pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖ψ1(ζ+τ,⋅)‖p2L∞(˜O)dζ. |
+2p|O|α1∫0−1‖T(θζω)‖−pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖G(ζ+τ,⋅)‖2L∞(˜O)dζ. | (65) |
Since
c1≤12pα1‖T(θrω)‖p−2≤c2 for all r∈[−ρ−1,0]. | (66) |
By (66) we obtain
ec2Mp−2ζ≤e∫ζ+ξξ12pα1Mp−2‖T(θrω)‖p−2dr≤ec1Mp−2ζ for all ζ∈[−1,0]andξ∈[−ρ,0]. | (67) |
For the left-hand side of (65), by (67) we find that there exists
14pα1∫0−1‖T(θζω)‖p−2e−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr |
∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ |
≥c3∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ. | (68) |
For the first term on the right-hand side of (65), by (67) we obtain
e−∫−10(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖(vε(τ−1,τ−t,θ−τω,ψε)−M)+‖pLp(O) |
≤e2p−3e−c1Mp−2‖(vε(τ−1,τ−t,θ−τω,ψε)−M)+‖pLp(O) |
≤e2p−3e−c1Mp−2‖vε(τ−1,τ−t,θ−τω,ψε)‖pLp(O). | (69) |
Similarly, for the second terms on the right-hand side of (65), we have from (67) there exists
∫0−1|δ(θζω)|pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖vε(ζ+τ,τ−t,θ−τω,ψε)‖pLp(O)dζ |
≤c4∫0−1ec1Mp−2ζ‖vε(ζ+τ,τ−t,θ−τω,ψε)‖pLp(O)dζ | (70) |
Since
2|O|∫0−1‖T(θζω)‖−pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖ψ1(ζ+τ,⋅)‖p2L∞(˜O)dζ. |
+2p|O|α1∫0−1‖T(θζω)‖−pe−∫ζ0(2p−3−12pα1Mp−2‖T(θrω)‖p−2)dr‖G(ζ+τ,⋅)‖2L∞(˜O)dζ |
≤c5∫0−1ec1Mp−2ζdζ≤c−11c5M2−p. | (71) |
By (68)-(71) we get from (65) that
c3∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dydζ |
≤e2p−3e−c1Mp−2‖vε(τ−1,τ−t,θ−τω,ψε)‖pLp(O) |
+c4∫0−1ec1Mp−2ζ‖vε(ζ+τ,τ−t,θ−τω,ψε)‖pLp(O)dζ+c−11c5M2−p, |
which together with Lemma 3.2 and Lemma 3.3 implies that there exist
c3∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ |
≤c6e−c1Mp−2+c6∫0−1ec1Mp−2ζdζ+c−11c5M2−p≤c6e−c1Mp−2+c−11(c5+c6)M2−p. | (72) |
Since
∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dydζ≤η. | (73) |
Note that
∫0−1ec2Mp−2ζ∫{y∈O: vε(ζ+τ,τ−t,θ−τω,ψε)≥2M}|vε(ζ+τ,τ−t,θ−τω,ψε)|2p−2dydζ≤22p−2∫0−1ec2Mp−2ζ∫O(vε(ζ+τ,τ−t,θ−τω,ψε)−M)2p−2+dxdζ≤22p−2η. | (74) |
Similarly, one can verify that there exist
∫0−1ec2Mp−2ζ∫{y∈O: vε(ζ+τ,τ−t,θ−τω,ψε)≤−2M}|vε(ζ+τ,τ−t,θ−τω,ψε)|2p−2dydζ≤22p−2η. | (75) |
Then Lemma 3.4 follows from (3) and (75) immediately.
Note that
0≤λε1≤λε2≤…≤λεn≤⋯→+∞, |
and their associated eigenfunctions
It follows from Corollary 9.7 in [8] that the eigenvalues and the eigenfunctions of
Next, we introduce the spectral projections. We use
Pεn(u)=m∑i=1(u,ϖεi)Yεϖεiforu∈Yε. |
We use
aε(u,u)=(Aεu,u)Hg(O)≤λεn(u,u)Hg(O),∀u∈PεnD(A1/2ε). | (76) |
and
aε(u,u)=(Aεu,u)Hg(O)≥λεm+1(u,u)Hg(O),u∈QεmD(A1/2ε). | (77) |
Let
Lemma 3.5. Assume that (8)-(11), (39) and (42) hold. Then for every
‖uε2(τ,τ−t,θ−τω,ϕε)‖H1(O)≤η. |
Proof. Taking the inner product (36) with
12ddtaε(vε2,vε2)+‖Aεvε2‖2≤(δ(θtω)vε2,Aεvε2)+(QεnT−1(θtω)hε(t,y,T(θtω)vε),Aεvε2)+(QεnT−1(θtω)Gε(t,y),Aεvε2). | (78) |
For the first term on the right-hand side of (78), we have
(δ(θtω)vε2,Aεvε2)≤18‖Aεvε2‖2+2|δ(θtω)|2‖vε2‖2. | (79) |
For the superlinear term, we have from (9) that
(QεnT−1(θtω)hε(t,y,T(θtω)vε),Aεvε2)≤18‖Aεvε2‖2+2‖T−1(θtω)‖2∫O|hε(t,y,T(θtω)vε)|2dy≤18‖Aεvε2‖2+2α2‖T−1(θtω)‖2∫O(|T(θtω)vε|p−1+ψ2(t,y∗,εg(y∗)yn+1))2dy≤18‖Aεvε2‖2+4α2‖T(θtω)‖2p−4‖v‖2p−22p−2+4α2|O|‖T−1(θtω)‖2‖ψ2(t,⋅)‖2L∞(˜O). | (80) |
For the last term on the right-hand side of (78), we have
(QεnT−1(θtω)Gε(t,y),Aεvε2)≤18‖Aεvε2‖2+2|O|‖T−1(θtω)‖2‖G(t,⋅)‖2L∞(˜O) | (81) |
Noting that
ddtaε(vε2,vε2)+λεn+1aε(vε2,vε2)≤4δ2(θtω)‖vε2‖2+8α2‖T(θtω)‖2p−4‖vε‖2p−22p−2+c‖T−1(θtω)‖2(‖ψ2(t,⋅)‖2L∞(˜O)+‖G(t,⋅)‖2L∞(˜O)). | (82) |
Taking
aε(vε2(τ,τ−t,θ−τω,ψε),vε2(τ,τ−t,θ−τω,ψε))≤∫ττ−1eλεn+1(r−τ)aε(vε2(r,τ−t,θ−τω,ψε),vε2(r,τ−t,θ−τω,ψε))dr+4δ2∫ττ−1eλεn+1(r−τ)δ2(θr−τω)aε(vε2(r,τ−t,θ−τω,ψε),vε2(r,τ−t,θ−τω,ψε))dr+8α2∫ττ−1eλεn+1(r−τ)‖T(θr−τω)‖2p−4‖vε(r,τ−t,θ−τω,ψε)‖2p−22p−2dr+c∫ττ−1eλεn+1(r−τ)‖T−1(θr−τω)‖2(‖ψ2(r,⋅)‖2L∞(˜O))dr+c∫ττ−1eλεn+1(r−τ)‖T−1(θr−τω)‖2‖G(r,⋅)‖2L∞(˜O)dr. | (83) |
Since
aε(vε2(τ,τ−t,θ−τω,ψε),vε2(τ,τ−t,θ−τω,ψε))≤c∫0−1eλεn+1r‖vε(r+τ,τ−t,θ−τω,ψε)‖2p−22p−2dr+c∫0−1eλεn+1raε(vε(r+τ,τ−t,θ−τω,ψε),vε(r+τ,τ−t,θ−τω,ψε))dr |
+c∫0−1eλεn+1rdr≤c∫0−1e(λ0n+1−1)r‖vε(r+τ,τ−t,θ−τω,ψε)‖2p−22p−2dr+c∫0−1e(λ0n+1−1)raε(vε(r+τ,τ−t,θ−τω,ψε),vε(r+τ+s,τ−t,θ−τω,ψε))dr+c∫0−1e(λ0n+1−1)r‖vε(r+τ−ρ0(r+τ+s),τ−t,θ−τω,ψε)‖2dr+c∫0−1e(λ0n+1−1)rdr. | (84) |
Given
c∫0−1e(λ0n+1−1)r‖vε(r+τ,τ−t,θ−τω,ψε)‖2p−22p−2dr≤c∫0−1e(λ0n+1−1)r∫{y∈O:|vε|≥2M}|vε(r+τ,τ−t,θ−τω,ψε)|2p−2dydr+c∫0−1e(λ0n+1−1)r∫{y∈O:|vε|<2M}|vε(r+τ,τ−t,θ−τω,ψε)|2p−2dydr≤c∫0−1eγMp−2r∫{y∈O:|vε|≥2M}|vε(r+τ,τ−t,θ−τω,ψε)|2p−2dydr+c∫0−1e(λ0n+1−1)r∫{y∈O:|vε|<2M}|vε(r+τ,τ−t,θ−τω,ψε)|2p−2dydr≤η+c22p−2M2p−2|O|∫0−1e(λ0n+1−1)rdr≤η+c22p−2M2p−2|O|1λ0n+1−1. | (85) |
For the last three terms on the right-hand side of (84), by Lemma 3.1, we find that there exist
c∫0−1e(λ0n+1−1)raε(vε(r+τ,τ−t,θ−τω,ψε),vε(r+τ,τ−t,θ−τω,ψε))dr+c∫0−1e(λ0n+1−1)rdr≤c1∫0−1e(λ0n+1−1)rdr≤c11λ0n+1−1. | (86) |
Since
aε(vε2(τ+s,τ−t,θ−τω,ψε),vε2(τ+s,τ−t,θ−τω,ψε))≤2η, |
which together
In this subsection, we establish the existence of
Lemma 4.1. Suppose (8)-(11), (39) and (43) hold. Then the cocycle
Proof. We first notice that, by Lemma 3.2,
K(τ,ω)={u∈H1(O):‖u‖2H1(O)≤L(τ,ω)}, | (87) |
where
Φε(t,τ−t,θ−tω,D(τ−t,θ−tω))⊆K(τ,ω). |
Thus we find that
Lemma 4.2. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
Proof. We will show that for every
‖uε2(τ,τ−tn,θ−τω,ϕε)‖H1(O)=‖Qm0uε(τ,τ−tn,θ−τω,ϕε)‖H1(O)<η4. | (88) |
On the other hand, by Lemma 3.2 we find that the sequence
Theorem 4.3. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
Proof. First, we know from Lemma 4.1 that
Analogous results also hold for the solution of (4)-(5). In particular, we have:
Theorem 4.4. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle
The following estimates are needed when we derive the convergence of pullback attractors. By the similar proof of that of Theorem 5.1 in [14], we get the following lemma.
Lemma 5.1. Assume that (8)-(11) and (39) hold. Then for every
∫tτ‖vε(r,τ,ω,ψε)‖2H1ε(O)dr≤c‖ψε‖2N+c∫τ+Tτ(‖G(r,⋅)‖2L∞(˜O)+‖ψ1(r,⋅)‖2L∞(˜O))dr, |
where
Similarly, one can prove
Lemma 5.2. Assume that (8)-(11) and (39) hold. Then for every
∫tτ‖v0(r,τ,ω,ψ0)‖2H1(Q)dr≤c‖ψ0‖2M+c∫τ+Tτ(‖G(r,⋅)‖2L∞(˜O)+‖ψ1(r,⋅)‖2L∞(˜O))dr, |
where
In the sequel, we further assume the functions
‖Gε(t,⋅)−G0(t,⋅)‖L2(O)≤κ1(t)ε | (89) |
and
‖Hε(t,⋅,s)−H0(t,⋅,s)‖L2(O)≤κ2(t)ε, | (90) |
where
By (12) and (90) we have, for all
‖hε(t,⋅,s)−h0(t,⋅,s)‖L2(O)≤κ2(t)ε. | (91) |
Since
Theorem 5.3. Suppose (8)-(11), (39), and (89)-(90) hold. Given
limn→∞‖Φεn(t,τ,ω,ϕεn)−Φ0(t,τ,ω,ϕ0)‖N=0. |
Proof. Since
‖vεn(t)−v0(t)‖2N≤c‖ϕεn−ϕ0‖2N+cmaxν∈[τ,t]ξ(θνω)∫tτ‖vεn(s)−v0(s)‖2Nds |
+cεnmaxν∈[τ,t]‖T−1(θνω)‖∫tτ(‖vεn(s)‖2H1εn(O)+‖v0(s)‖2H1(Q))ds+cεnmaxν∈[τ,t]‖T−1(θνω)‖∫tτ(κ21(s)+κ22(s))ds+cεn∫tτ(‖vεn(s)‖2H1εn(O)+‖v0(s)‖2H1(Q))ds, | (92) |
where
‖vεn(t)−v0(t)‖2N≤ec(1+maxν∈[τ,τ+T]ξ(θνω))T‖ϕεn−ϕ0‖2N+ϱεnec(1+maxν∈[τ,τ+T]ξ(θνω))T[‖ψ0‖2M+‖ψεn‖2N+∫τ+Tτ(κ21(s)+κ22(s))ds+∫τ+Tτ(‖G(s,⋅)‖2L∞(˜O)+‖ψ1(s,⋅)‖2L∞(˜O))ds]. | (93) |
Notice that, for all
‖uεn(t,τ,ω,ϕε)−u0(t,τ,ω,ϕ0)‖2N≤maxν∈[τ,τ+T]‖T(θνω)‖2‖vεn(t,τ,ω,T−1(θτω)ϕε)−v0(t,τ,ω,T−1(θτω)ϕ0‖2N, |
which together with (93) implies the desired results.
The next result is concerned with uniform compactness of attractors with respect to
Lemma 5.4. Assume that (8)-(11), (39) and (43) hold. If
limn→∞‖uεn−u‖H1(O)=0. |
Proof. Take a sequence
uεn=Φεn(tn,τ−tn,θ−tnω,ϕεn). | (94) |
By Lemma 4.1, we have
‖QεnN1uεn(τ,τ−tn,θ−τω,ϕεn)‖H1(O)≤η. | (95) |
By Lemma 3.2, we have
‖PεnN1uεn(τ,τ−tn,θ−τω,ϕεn)‖H1(O)<M. | (96) |
It follows from (95) and (96) that
limn→∞‖uεn−u‖H1(O)=0. | (97) |
This completes the proof.
Now we are in a position to prove the main result of this paper.
Theorem 5.5. Assume that (8)-(11), (39), (43), and (89)-(90) hold. The attractors
limε→0distH1(O)(Aε(τ,ω),A0(τ,ω))=0. |
Proof. Given
‖u‖2H1ε(O)≤L(τ,ω)for all 0<ε<ε0 and u∈Aε(τ,ω), | (98) |
where
distH1(O)(zn,A0(τ,ω))≥δfor alln∈N. | (99) |
By Lemma 5.4 there exists
limn→∞‖zn−z∗‖H1(O)=0. | (100) |
By the invariance property of the attractor
zn=Φεn(t,τ−t,θ−tω,ytn). | (101) |
By Lemma 5.4 again there exists
limn→∞‖ytn−yt∗‖H1(O)=0. | (102) |
It follows from Theorem 5.3 that for every
limn→∞Φεn(t,τ−t,θ−tω,ytn)=Φ0(t,τ−t,θ−tω,yt∗)inN. | (103) |
By (100), (101), (103) and uniqueness of limits we obtain
z∗=Φ0(t,τ−t,θ−tω,yt∗)inH1(O). | (104) |
Notice that
limsupn→∞‖ytn‖H1(O)≤‖K(τ−t,θ−tω)‖H1(O)≤L(τ−t,θ−tω). | (105) |
By (102) and (105) we get, for every
‖yt∗‖H1(Q)≤L(τ−t,θ−tω). | (106) |
By
distH1(Q)(z∗,A0(τ,ω))=distH1(Q)(Φ0(t,τ−t,θ−tω,yt∗),A0(τ,ω))≤distH1(Q)(Φ0(t,τ−t,θ−tω,K0(τ−t,θ−tω)),A0(τ,ω))→0,ast→∞. | (107) |
This implies that
distH1(O)(zn,A0(τ,ω))≤distH1(O)(zn,z∗)→0, |
a contradiction with (99). This completes the proof.
The authors would like to thank the anonymous referee for the useful suggestions and comments.
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