We study the first initial boundary value problem for the nonautonomous 2D gNavierStokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincar´e inequality. We show the existence of a pullback attractor for the process generated by strong solutions to the problem with respect to a large class of nonautonomous forcing terms. To overcome the difficulty caused by the unboundedness of the domain, the proof is based on a pullback asymptotic compactness argument and the use of the enstrophy equation.
Trang 1g-NAVIER-STOKES EQUATIONS IN SOME UNBOUNDED
DOMAINS
CUNG THE ANH \ AND DAO TRONG QUYET
Abstract We study the first initial boundary value problem for the
non-autonomous 2D g-Navier-Stokes equations in an arbitrary (bounded or
un-bounded) domain satisfying the Poincar´ e inequality We show the existence
of a pullback attractor for the process generated by strong solutions to the
problem with respect to a large class of non-autonomous forcing terms To
overcome the difficulty caused by the unboundedness of the domain, the proof
is based on a pullback asymptotic compactness argument and the use of the
enstrophy equation.
1 Introduction Let Ω be a (bounded or unbounded) domain in R2 with smooth boundary ∂Ω
In this paper we study the long-time behavior of strong solutions to the following non-autonomous 2D g-Navier-Stokes equations
ut− ν∆u + (u · ∇)u + ∇p = f, x ∈ Ω, t > τ,
∇ · (gu) = 0, x ∈ Ω, t > τ, u(x, t) = 0, x ∈ ∂Ω, t > τ, u(x, τ ) = u0(x), x ∈ Ω,
(1.1)
where u = u(x, t) = (u1, u2) is the unknown velocity vector, p = p(x, t) is the unknown pressure, ν > 0 is the kinematic viscosity coefficient, f = f (x, t) is a given force field and u0is the initial velocity
The g-Navier-Stokes equations is a variation of the standard Navier-Stokes equa-tions and arises in a natural way when we study the standard 3D problem in the thin domain Ωg= Ω × (0, g) We refer the reader to [16] for a derivation of the 2D g-Navier-Stokes equations from the 3D Navier-Stokes equations and a relationship between them As mentioned in [16], good properties of the 2D g-Navier-Stokes equations can initiate the study of the Navier-Stokes equations on the thin three dimensional domain Ωg In the last few years, the existence and asymptotic be-havior of solutions to g-Navier-Stokes equations have been studied extensively (cf [1, 2, 4, 11, 12, 13, 14, 16]) Very recently, the existence and exponential growth of pullback attractors for strong solutions to 2D g-Navier-Stokes equations in bounded domains were proved in [15]
The aim of this paper is to continue studying the long-time behavior of strong solutions to the non-autonomous 2D g-Navier-Stokes equations in domains that are not necessarily bounded To do this, we also use the theory of pullback at-tractors that has been developed recently and has shown to be very useful in the understanding of the dynamics of non-autonomous dynamical systems (see e.g the
2010 Mathematics Subject Classification 35B41; 35Q30; 35D35.
Key words and phrases g-Navier-Stokes equations; unbounded domain; strong solution; pull-back attractor; pullpull-back asymptotic compactness argument; enstrophy equation.
\ Corresponding author: anhctmath@hnue.edu.vn.
Trang 2monograph [8]) The results obtained, in particular, extended the corresponding results in bounded domains for the 2D Navier-Stokes equations [9] and for 2D g-Navier-Stokes equations [15]
In order to study problem (1.1), we assume that the function g satisfies the following hypothesis:
(G) g ∈ W1,∞(Ω) such that
0 < m0≤g(x)≤M0 for all x = (x1, x2) ∈ Ω, and |∇g|∞< m0λ1/21 ,
where λ1> 0 is the constant in the Poincar´e inequality (1.2) below
We also assume that the domain Ω satisfies the Poincar´e inequality:
Z
Ω
φ2gdx ≤ 1
λ1 Z
Ω
|∇φ|2gdx, for all φ ∈ C0∞(Ω) (1.2)
Because the considered domain is unbounded, the compactness of the embed-dings which plays an essential role when proving the existence of pullback attrac-tors in [9, 15] is no longer valid here To overcome this difficulty, we exploit the asymptotic compactness argument introduced the first time by Ball [5] to prove the pullback asymptotic compactness of the process, and as a consequence, we get the existence of a pullback attractor Such an approach has been used recently to prove the existence of pullback attractors for some non-autonomous equations in fluid mechanics in unbounded domains, such as the 2D Navier-Stokes equations [6], the 2D g-Navier-Stokes equations [1], and the 3D Navier-Stokes-Voigt equations [3] In these works, because weak solutions are considered, the pullback asymptotic compactness argument are applied for the energy equations Here because we con-sider strong solutions, we apply the argument to the enstrophy equation instead of the energy equation Note that the nonlinear term of the g-Navier-Stokes equations dissapears in the energy equation due to its antisymmetry, while the corresponding term does not dissappear in the enstrophy equation This introduce some new dif-ficulty In this paper, we deal with this difficulty by a careful analysis using some ideas in [10]
The structure of the paper is as follows In Section 2, for convenience of the reader, we recall some results on functions spaces and inequalities related to g-Navier-Stokes equations and abstract theory of pullback attractors In Section 3,
we prove the existence of a pullback attractor for the process generated by strong solutions to problem (1.1)
2 Preliminaries 2.1 Function spaces and inequalities for the nonlinear terms Let L2(Ω, g) = (L2(Ω))2 and H1(Ω, g) = (H1(Ω))2be endowed, respectively, with the inner prod-ucts
(u, v)g=
Z
Ω
u · vgdx, u, v ∈ L2(Ω, g), and
((u, v))g=
Z
Ω
2
X
j=1
∇uj· ∇vjgdx, u = (u1, u2), v = (v1, v2) ∈ H1(Ω, g),
and norms |u|2= (u, u)g, ||u||2= ((u, u))g Thanks to assumption (G), the norms
|.| and ||.|| are equivalent to the usual ones in (L2(Ω))2 and in (H1(Ω))2
Let
V = {u ∈ (C∞(Ω))2: ∇ · (gu) = 0}
Trang 3Denote by Hg the closure of V in L2(Ω, g), and by Vg the closure of V in H1(Ω, g).
It follows that Vg⊂ Hg≡ H0
g⊂ V0
g, where the injections are dense and continuous
We will use ||.||∗ for the norm in Vg0, and h., i for duality pairing between Vg and
Vg0
We now define the trilinear form b by
b(u, v, w) =
2
X
i,j=1
Z
Ω
ui
∂vj
∂xi
wjgdx,
whenever the integrals make sense It is easy to check that if u, v, w ∈ Vg, then
b(u, v, w) = −b(u, w, v), and b(u, v, v) = 0
Set A : Vg → V0
g by hAu, vi = ((u, v))g, B : Vg× Vg → V0
g by hB(u, v), wi = b(u, v, w), and put Bu = B(u, u) Denote D(A) = {u ∈ Vg : Au ∈ Hg}, then D(A) = H2(Ω, g) ∩ Vg and Au = −Pg∆u, ∀u ∈ D(A), where Pg is the ortho-projector from L2(Ω, g) onto Hg
We have the following results
Lemma 2.1 [1] If n = 2, then
|b(u, v, w)| ≤
c1|u|1/2kuk1/2kvk|w|1/2kwk1/2, ∀u, v, w ∈ Vg,
c2|u|1/2kuk1/2kvk1/2|Av|1/2|w|, ∀u ∈ Vg, v ∈ D(A), w ∈ Hg,
c3|u|1/2|Au|1/2kvk|w|, ∀u ∈ D(A), v ∈ Vg, w ∈ Hg,
c4|u|kvk|w|1/2|Aw|1/2, ∀u ∈ Hg, v ∈ Vg, w ∈ D(A), where ci, i = 1, , 4, are appropriate constants
Lemma 2.2 [2] Let u ∈ L2(0, T ; D(A)) ∩ L∞(0, T ; Vg), then the function Bu defined by
(Bu(t), v)g= b(u(t), u(t), v), ∀v ∈ Hg, a.e t ∈ [0, T ],
belongs to L4(0, T ; Hg), therefore also belongs to L2(0, T ; Hg)
Lemma 2.3 [4] Let u ∈ L2(0, T ; Vg), then the function Cu defined by
(Cu(t), v)g= ((∇g
g .∇)u, v)g= b(
∇g
g , u, v), ∀v ∈ Vg, belongs to L2(0, T ; Hg), and therefore also belongs to L2(0, T ; Vg0) Moreover,
|Cu(t)| ≤ |∇g|∞
m0 .ku(t)k, for a.e t ∈ (0, T ), and
||Cu(t)||∗≤ |∇g|∞
m0λ1/21 ||u(t)||, for a.e t ∈ (0, T )
Since
−1
g(∇.g∇)u = −∆u − (
∇g
g .∇)u,
we have
(−∆u, v)g= ((u, v))g+ ((∇g
g .∇)u, v)g= (Au, v)g+ b(
∇g
g , u, v), ∀u, v ∈ Vg.
Trang 42.2 Pullback attractors Let X be a Banach space Denote by B(X) the set of all bounded subsets of X and k.k is the corresponding norm For A, B ⊂ X, the Hausdorff semi-distance between A and B is defined by
dist(A, B) = sup
x∈A
inf
y∈Bkx − yk
Let {U (t, τ ) : t ≥ τ, τ ∈ R} be a process in X, i.e., a two-parameter family of mappings U (t, τ ) : X → X such that U (τ, τ ) = Id and U (t, s)U (s, τ ) = U (t, τ ) for all t ≥ s ≥ τ, τ ∈ R
Definition 2.1 The process {U (t, τ )} is said to be pullback asymptotically compact
if for any t ∈ R, any D ∈ B(X), any sequence τn→ −∞, and any sequence xn∈ D, the sequence {U (t, τn)xn} is relatively compact in X
Definition 2.2 A family of bounded sets ˆB = {B(t) : t ∈ R} is called pullback absorbing for the process {U (t, τ )} if for any t ∈ R, any D ∈ B(X), there exists
τ0= τ0(D, t) ≤ t such that
[
τ ≤τ 0
U (t, τ )D ⊂ B(t)
Definition 2.3 A family ˆA = {A(t) : t ∈ R} ⊂ B(X) is said to be a pullback attractor for {U (t, τ )} if
(1) A(t) is compact for all t ∈ R;
(2) ˆA is invariant, i.e.,
U (t, τ )A(τ ) = A(t), for all t ≥ τ ; (3) ˆA is pullback attracting, i.e.,
lim
τ →−∞dist(U (t, τ )D, A(t)) = 0, for all D ∈ B(X), and all t ∈ R;
(4) if {C(t) : t ∈ R} is another family of closed attracting sets then A(t) ⊂ C(t), for all t ∈ R
Theorem 2.1 [6, 7] Let {U (t, τ )} be a continuous process such that {U (t, τ )} is pullback asymptotically compact If there exists a family of pullback absorbing sets ˆ
B = {B(t) : t ∈ R}, then {U(t, τ )} has a unique pullback attractor ˆA = {A(t) : t ∈ R} and
A(t) = \
s≤t
[
τ ≤s
U (t, τ )B(τ )
3 Existence of a pullback attractor
We first recall the definition of strong solutions to problem (1.1)
Definition 3.1 A function u is called a strong solution to problem (1.1) on the interval (τ, T ) if
u ∈ C([τ, T ]; Vg) ∩ L2(τ, T ; D(A)), du/dt ∈ L2(τ, T ; Hg),
d
dtu(t) + νAu(t) + νCu(t) + B(u(t), u(t)) = f (t) in Hg, for a.e t ∈ (τ, T ), u(τ ) = u0
Theorem 3.1 For any T > τ , u0∈ Vg, and f ∈ L2(τ, T ; Hg) given, problem (1.1) has a unique strong solution u on (τ, T ) Moreover, the strong solution depends continuously on the initial data in V
Trang 5Proof The proof is standard and similar to the case of bounded domains (see [2]), except some difficulty arising due to the unboundedness of the domain which can
be overcome by using techniques as for Navier-Stokes equations [17], so we omit
it here However, in what follows we will recall some a priori estimates of strong solutions which will be used later
First, we have
d
ds|u(s)|2+ 2ν||u(s)||2= 2hf (s), u(s)i − 2ν(Cu(s), u(s))g
Using Lemma 2.3 and Cauchy’s inequality, we have
d
ds|u(s)|2+ 2ν||u(s)||2≤ 2ν||u(s)||2+kf k2
∗
2ν + 2ν
|∇g|∞
m0λ1/21
||u(s)||2,
and hence
d
ds|u(s)|2+ 2ν(γ0− )||u(s)||2≤ kf k
2
∗
where γ0 = 1 − |∇g|∞
m0λ1/21 > 0 and > 0 is chosen such that γ0− > 0 Integrating from τ to t, and applying Cauchy’s inequality we get
|u(t)|2+ 2ν(γ0− )
Z t τ
ku(s)k2ds ≤ |u0|2+ 1
2νkf k2
L 2 (τ,T ;V 0
g ) This implies that u is bounded in L∞(τ, T ; Hg) ∩ L2(τ, T ; Vg) Hence, it is easy to check that Au and Bu are bounded in L2(τ, T ; Vg0)
On the other hand, we have
1
2
d
dtku(t)k2+ ν|Au(t)|2+ ν(Cu(t), Au(t))g+ b(u, u, Au) = hf, Aui
By Lemmas 2.1 and 2.3, we have
1
2
d
dt||u(t)||2+ ν|Au(t)|2
≤ ν
4|Au(t)|2+1
ν|f (t)|2+ c3|u(t)|1/2|Au(t)|3/2||u(t)|| +ν|∇g|∞
m0
||u(t)|||Au(t)| Using Young’s inequality, we obtain
1
2
d
dt||u(t)||2+ ν|Au(t)|2≤ ν
4|Au(t)|2+1
ν|f (t)|2
+ν
4|Au(t)|2+ c03|u(t)|2|||u(t)||4 + ν|∇g|∞
2m0λ1/21 |Au(t)|2+ν|∇g|∞λ
1/2 1
2m0
||u(t)||2 Then, we have
d
dt||u(t)||2+ ν(1 − |∇g|∞
m0λ1/21 )|Au(t)|2
≤ 2
ν|f (t)|2+ 2c03|u(t)|2|||u(t)||4+ν|∇g|∞λ
1/2 1
m0
||u(t)||2
(3.2)
From (3.1) we have
|u(t + 1)|2+ 2ν(γ0− )
Z t+1 t
kuk2ds ≤ |u(t)|2+ 1
2νkf kL2 (t,t+1;V 0
g )
It implies that
Z t+1
|u(s)|2ku(s)k2ds ≤ kuk2L∞ (τ,T ;Hg)
Z t+1
ku(s)k2ds < +∞ for all t ≥ τ
Trang 6So, we can applying the uniform Gronwall inequality to obtain
ku(t)k2≤ C for all t ≥ τ + 1 (3.3) This implies that u is bounded in L∞(τ, T ; Vg)
Integrating (3.2) from τ to t we get
ku(t)k2+ νγ0
Z t τ
|Au(s)|2ds
≤ku0k2+2
ν
Z t τ
|f (s)|2ds + 2c03|u|2
L ∞ (τ,T ;H g )kuk2
L ∞ (τ,T ;V g )
Z t τ
ku(s)k2ds
+ν|∇g|∞λ
1/2 1
m0
Z t τ
||u(t)||2 This implies that u is bounded in L2(τ, T ; D(A)) And then, we also have that Bu
is bounded in L2(τ, T ; Hg)
Now, we prove the boundedness of du
dt We have
Z t
τ
du
ds
2
ds + ν
Z t τ
Z
Ω
∇u∂∇u
∂s dx ds + ν
Z t τ
(Cu, us)gds +
Z t τ
b(u, u, us)ds
=
Z t τ
hf, usids Using Cauchy’s and Ladyzhenskaya’s inequalities, we have
Z t
τ
du
ds
2
ds +ν
2
Z t τ
d
dskuk2ds
=
Z t
τ
hf, usi ds −
Z t τ
b(u, u, us) ds − ν
Z t τ
(Cu, us)gds
≤ kf kL2 (τ,t;H g ).kuskL2 (τ,t;H g )
+
Z t
τ
|u|L4|∇u|L4|us|ds + ν |∇g|∞
m0λ1/21
Z t τ
||u||||us||ds
≤ kf k2
L 2 (τ,T ;Hg)+1
4kusk2
L 2 (τ,t;Hg)
+ c
Z t
τ
|u|1/2|∇u||Au|1/2|us|ds + ν |∇g|∞
m0λ1/21
Z t τ
||u||||us||ds
≤ kf k2
L 2 (τ,T ;H g )+1
4kusk2
L 2 (τ,t;H g )
+ c
Z t
τ
|u||∇u|2|Au|ds
1/2
·
Z t τ
|us|2ds
1/2
+ ν |∇g|∞
m0λ1/21
Z t τ
||u||||us||ds
≤ kf k2L2 (τ,T ;H g )+1
4kusk2L2 (τ,t;H g )+1
8kusk2L2 (τ,T ;H g )
+ c
Z t
τ
|u||∇u|2|Au|ds +1
8kusk2
L 2 (τ,T ;Hg)+ c
Z t τ
||u(s)||2ds
Hence
Z t
τ
du
ds
2
ds + ν|∇u(t)|2≤ 2kf k2
L 2 (τ,T ;Hg)+ ν|∇u0|2
+ c
Z t
|u||∇u|2|Au|ds + c
Z t
||u(s)||2ds,
Trang 7for all τ ≤ t ≤ T Since u is bounded in L∞(τ, T ; Vg) ∩ L2(τ, T ; D(A)), du
dt is
From now on we assume that f ∈ L2
b(R, Hg), i.e f ∈ L2
loc(R, Hg) and satisfies
kf k2
L 2:= sup
t∈R
Z t+1 t
|f (s)|2ds < +∞
Thanks to Theorem 3.1, we can define a continuous process U (t, τ ) : Vg→ Vg by
U (t, τ )u0= u(t; τ, u0), τ ≤ t, u0∈ Vg, where u(t) = u(t; τ, u0) is the unique strong solution to problem (1.1) with the initial datum u(τ ) = u0
We denote σ = λ1νγ0 with λ1 is the constant in the Poincar´e inequality (1.2), and introduce a new Hilbert norm in D(A) as follows
[[u]]2: = νγ0|Au|2−σ
2kuk2, which is equivalent to the usual norm |Au| in D(A)
We now prove the weak continuity of the process U (t, τ )
Lemma 3.1 Let {u0n} ⊂ Vg be a sequence converging weakly in Vg to an element
u0 in Vg Then
U (t, τ )u0n* U (t, τ )u0 weakly in Vg for all τ ≤ t, (3.4)
U (t, τ )u0n * U (t, τ )u0 weakly in L2(τ, T ; D(A)) for all τ ≤ t (3.5) Proof Let un(t) = U (t, τ, u0n), u(t) = U (t, τ, u0) As in the proof of Theorem 3.1
we have, for all T ≥ τ ,
{un} is bounded in L∞(τ, T ; Vg) ∩ L2(τ, T ; D(A)), (3.6) and
{u0n} is bounded in L2(τ, T ; Hg)
Then, for all v ∈ D(A),
((un(t + a) − un(t), v)) =
Z t+a t
hu0n(s), vids
≤kvkD(A)a1/2ku0
nkL2 (τ,T ,Hg)≤ CTkvkD(A)a1/2,
(3.7)
where CT is positive and independent of n Then, for v = un(t + a) − un(t), which belongs to D(A) for almost every t, from (3.6) we have
kun(t + a) − un(t)k2D(A)≤ CTa1/2kun(t + a) − un(t)kD(A)
Hence
Z T −a
τ
kun(t + a) − un(t)k2D(A)dt ≤ CTa1/2
Z T −a τ
kun(t + a) − un(t)kD(A)dt (3.8) Using Cauchy’s inequality and (3.6), we deduce from (3.8) that
Z T −a τ
kun(t + a) − un(t)k2dt ≤ eCTa1/2, for another positive constant eCT independent of n Therefore
lim
a→0sup
n
Z T −a τ
kun(t + a) − un(t)k2D(A)(Ω
r )dt = 0, (3.9) for all r > 0, where Ωr= {x ∈ Ω : |x| < r} Moreover, from (3.6),
{un|Ω } is bounded in L∞(τ, T ; H1(Ωr, g)) ∩ L2(τ, T ; D(A)(Ωr))
Trang 8for all r > 0 Consider now a truncation function ρ ∈ C1
(R+) with ρ(s) = 1 in [0, 1], and ρ(s) = 0 in [2, +∞) For each r > 0, define vn,r(x, t) = ρ |x|2
r2
un(x, t) for x ∈ Ω√
2r Then, from (3.9), we have
lim
a→0sup
n
Z T −a
τ
kvn,r(t + a) − vn,r(t)k2D(A)(Ω√
2r )dt = 0, for all T > τ, r > 0, while from (3.6) we deduce that vn,r is uniformly bounded in L∞(τ, T ; H1(Ω√
2r, g))∩
L2(τ, T ; D(A)(Ω√
2r)) for all T > τ , r > 0 Thus, by applying Theorem 13.3 and Remark 13.1 in [18], we obtain
{vn,r} is relatively compact in L2(τ, T ; H01(Ω√
2r, g)), for all T > τ, r > 0
It follows that
{un|Ωr} is relatively compact in L2(τ, T ; H01(Ω√
2r, g)), for all T > τ, r > 0 Then, by a diagonal process, we can extract a subsequence {un 0} such that
un 0 →eu weakly-* in L2loc(R; D(A)),
un 0 →eu strongly in L2loc(R; H1(Ωr, g)), r > 0, (3.10) for some eu ∈ L∞loc(R, Ω) The convergences (3.10) allows us to pass to the limit in the equation for un 0to find thatu is a strong solution of (1.1) withe u(τ ) = ue 0 Since the uniqueness of the strong solution, we must haveeu = u Then by a contradiction argument we deduce that the whole sequence {un} converges to u in the sense of (3.10) This proves (3.5)
Now, from the strong convergence in (3.10) we also have that un(t) converges strongly in H1(Ωr, g) to u(t) for a.e t ≥ τ and all r > 0 Hence for all v ∈ V,
((un(t), v))g→ ((u(t), v))g for a.e t ∈ R
Moreover, from (3.6) and (3.7), we see that {(un(t), v)g} is equibounded and equicontinuous on [τ, T ], for all T > τ Therefore
((un(t), v))g → ((u(t), v))g, ∀ t ∈ R, ∀ v ∈ V
Finally, (3.4) follows from the fact that V is dense in Vg Lemma 3.2 Let {u0n} ⊂ Hgbe a sequence converging strongly in Hgto an element
u0 in Hg Suppose u(t) = U (t, τ )u0, un(t) = U (t, τ )u0n Then, for all T > τ ,
un→ u strongly in L2(τ, T ; Vg)
Proof Suppose un and u are solutions to (1.1) with initial conditions u0n and u0,
we have
1
2
d
ds|un− u|2+ νkun− uk2
= −b(un, un, un− u) + b(u, u, un− u) − ν(C(un− u), un− u)g
= −b(un− u, u, un− u) − ν(C(un− u), un− u)g
≤ c|un− u|kun− ukkuk + ν |∇g|∞
m0λ1/21 ||un− u||2, or
d
ds|un− u|2+ 2νγ0kun− uk2≤ 2c|un− u|kun− ukkuk
≤ νγ ku − uk2+ ckuk2|u − u|2
Trang 9d
ds|un− u|2+ νγ0kun− uk2≤ ckuk2|un− u|2 Therefore,
νγ0
Z T
τ
kun(s) − u(s)k2ds ≤ |u0n− u0|2+ c
Z T τ
ku(s)k2|un(s) − u(s)|2ds
Noting that u0n→ u0strongly in Hg, we have |un(t) − u(t)|2→ 0 for all t ∈ (τ, T )
By Lebesgue’s dominant convergence theorem, we have
lim
n→∞
Z T τ
kun(s) − u(s)k2ds = 0,
Theorem 3.2 Suppose that f ∈ L2b(R; Hg) Then, there exists a unique pullback attractor ˆA = {A(t) : t ∈ R} for the process U(t, τ )
Proof Let τ ∈ R, u0∈ Vg be fixed, and denote
u(t) = u(t; τ, u0) = U (t, τ )u0 for all t ≥ τ
We will check the two conditions in Theorem 2.1
i) The process U (t, τ ) has a family ˆB of pullback absorbing sets
From
dt, v + ν((u, v))g+ ν(Cu, v)g+ b(u, u, v) = hf, vi, (3.11) choosing v = eσsu(s), we have
d
ds(e
σs|u(s)|2) + 2νeσsku(s)k2= σeσs|u(s)|2+ 2eσshf (s), u(s)i − 2eσsν(Cu(s), u(s))g
≤ σ
λ1
eσsku(s)k2+ 2eσs|f (s)||u(s)| + 2eσsν |∇g|∞
m0λ1/21
||u(s)||2, or
d
ds(e
σs|u(s)|2) + 2νγ0eσsku(s)k2≤ σ
λ1e
σsku(s)k2+ 2eσs|f (s)||u(s)|
≤ νγ0eσsku(s)k2+ 1
σe
σs
|f (s)|2+ σeσs|u(s)|2 Hence
d
dse
σs|u(s)|2≤ 1
σe
σs|f (s)|2 Integrating from τ to t we get
eσt|u(t)|2≤ eστ|u(τ )|2+ 1
σ
Z t τ
eσs|f (s)|2ds
Hence it follows that
|U (t, τ, u0)|2≤ eσ(τ −t)|u0|2+e
−σt
σ
eσt
Z t t−1
|f (s)|2ds + eσ(t−1)
Z t−1 t−2
|f (s)|2ds + · · ·
≤ eσ(τ −t)|u0|2+e
−σt
σ kf k2
L 2
eσt+ eσ(t−1)+ · · ·
≤ eσ(τ −t)|u0|2+e
−σt
σ
eσt
1 − e−σkf k2
L 2
≤ eσ(τ −t)|u0|2+1 + σ
σ2 kf kL2
Trang 10BHg =v ∈ Hg : |v|2≤ R2Hg := 2(1 + σ)
σ2 kf k2L2 Then, for a given bounded set B ⊂ BHg there exists τ0(B) such that
|U (t, τ )u0|2≤ R2
Hg for all τ ≤ τ0(B)
On the other hand, from (3.11) choosing v(s) = u(s) and applying Cauchy’s in-equality, we obtain
d
ds|u(s)|2+ νγ0ku(s)k2≤ 1
σ|f (s)|2 Integrating from t to t + 1
|u(t + 1)|2+ νγ0
Z t+1 t
|∇u|2ds ≤ |u(t)|2+ 1
σ
Z t+1 t
|f (s)|2ds
≤2(1 + σ)
σ2 kf k2
L 2+ 1
σkf k2
L 2= 2 + 3σ
σ2 kf k2
L 2
for all t ≥ T0
On the other hand, from (3.11) choosing v(s) = Au(s) we have
1
2
d
ds|∇u(s)|2+ ν|Au(s)|2+ ν(Cu(s), Au(s))g+ b(u, u, Au) = hf, Aui
It implies that
d
ds|∇u(s)|2+ 2ν|Au(s)|2= − 2b(u, u, Au) + 2hf, Aui − 2ν(Cu(s), Au(s))g
≤c3|u|1/2kuk|Au|3/2+ 2|f ||Au| + 2ν|∇g|∞
m0 ||u|||Au|
≤c3|u|1/2kuk|Au|3/2+ 2|f ||Au|
+ 2ν|∇g|∞
m0λ1/21
|Au|2+ ν|∇g|∞λ1/21
2m0 ||u||2 Then, we have
d
ds|∇u(s)|2+ 2νγ0|Au(s)|2≤c3|u|1/2kuk|Au|3/2+ 2|f ||Au| + ν|∇g|∞λ1/21
2m0
||u||2
≤νγ0(|Au|3/2)4/3+ c(|u|1/2kuk)4
+ νγ0|Au|2+ 1
νγ0
|f |2+ ν|∇g|∞λ1/21
2m0
||u||2, or
d
dsku(s)k2≤ 1
νγ0
|f (s)|2+ (c|u|2kuk2+ ν|∇g|∞λ1/21
2m0
)||u||2 Applying the uniform Gronwall inequality with
y(s) = ku(s)k2; a(s) = (c|u|2kuk2+ ν|∇g|∞λ1/21
2m0 ); b(s) =
1
νγ0|f (s)|2, for all u0∈ B ⊂ B(Vg), t ≥ τ0(t, B) ≥ τ , we obtain
kU (t, τ, u0)k2≤ 2 + 3σ
σ2νγ0
kf k2
L 2+ 1
νγ0
kf k2
L 2
e(
2c(1+σ)(2+3σ) νγ0σ 4 kf k 4
L2b+ν|∇g|∞λ
1/2 1
= 2 + 3σ + σ
2
σ2νγ kf k2L2e(
2c(1+σ)(2+3σ) νγ0σ 4 kf k 4
L2b+ν|∇g|∞λ
1/2 1 2m0 ):= R2pVg,
... converges strongly in H1(Ωr, g) to u(t) for a.e t ≥ τ and all r > Hence for all v ∈ V,((un(t), v))g< /small>→ ((u(t), v))g< /small> for. ..
Finally, (3.4) follows from the fact that V is dense in Vg< /small> Lemma 3.2 Let {u0n} ⊂ Hg< /sub>be a sequence converging strongly in Hg< /sub>to...
Trang 7for all τ ≤ t ≤ T Since u is bounded in L∞(τ, T ; Vg< /small>) ∩ L2(τ,