Abstract. We give an upper bound on the number of determining finite volume elements
for 3D Navier-Stokes-Voigt equations in bounded domains with periodic boundary
conditions, which is estimated explicitly in terms of flow parameters, such as viscosity,
smoothing length, forcing and domain size.
Keywords: Navier-Stokes-Voigt equations, weak solutions, determining finite volume
elements.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0024
Natural Science, 2018, Volume 63, Issue 6, pp. 17-22
This paper is available online at
ON THE NUMBER OF DETERMINING FINITE VOLUME ELEMENTS
FOR 3D NAVIER-STOKES-VOIGT EQUATIONS
Nguyen Thi Ngan1, Nguyen Thi Minh Toai1 and Tran Quoc Tuan2
1University of Education Publishing House, Hanoi National University of Education
1Faculty of Mathematics, Hanoi National University of Education
Abstract. We give an upper bound on the number of determining finite volume elements
for 3D Navier-Stokes-Voigt equations in bounded domains with periodic boundary
conditions, which is estimated explicitly in terms of flow parameters, such as viscosity,
smoothing length, forcing and domain size.
Keywords: Navier-Stokes-Voigt equations, weak solutions, determining finite volume
elements.
1. Introduction
Let Ω = (0, L)3, L > 0 be a periodic box in R3. We consider the following 3D
Navier-Stokes-Voigt equations{
ut − α2∆ut − ν∆u+ (u · ∇)u+∇p = f in Ω× (0,∞),
∇ · u = 0 in Ω× (0,∞), (1.1)
subject to the periodic boundary conditions
u(x, t) = u(x+ L, t), x ∈ Ω, t > 0, (1.2)
and the initial condition
u(x, 0) = u0(x), x ∈ Ω. (1.3)
Here u = u(x, t) is the unknown velocity and p = p(x, t) is the unknown pressure, ν > 0 is the
kinematic viscosity coefficient. When α = 0, we formally recover the 3D classical Navier-Stokes
equations.
The system (1.1) was first introduced by Oskolkov in [1] as a model of motion of
certain linear viscoelastic fluids. It was also proposed by Cao, Lunasin and Titi in [2] as a
regularization, for small value of α, of the 3D Navier-Stokes equations for the sake of direct
numerical simulations. The presence of the regularizing term−α2∆ut in (1.1) has many important
consequences. On one hand, it leads to the global well-posedness of (1.1) both forward and
Received July 12, 2018. Revised August 13, 2018. Accepted August 20, 2018.
Contact Nguyen Thi Ngan, e-mail: ngannt.nxb@hnue.edu.vn
17
Nguyen Thi Ngan and Nguyen Thi Minh Toai
backward in time, even in the case of three dimensions. On the other hand, it changes the parabolic
character of the limit Navier-Stokes equations, so the Navier-Stokes-Voigt system behaves like
a damped hyperbolic system. In fact, the Navier-Stokes-Voigt system belongs to the so-called
α-models in fluid mechanics (see e.g. [3]). We also refer the reader to [4] for an interesting
application of the Navier-Stokes-Voigt equations in image inpainting.
The conventional theory of turbulence asserts that turbulent flows are monitored by a
finite number of degrees of freedom. The notions and results for the case of 2D Navier-Stokes
equations on determining modes, determining nodes and determining finite volume elements (see
e.g. [5, 6, 7]) are rigorous attempts to identify those parameters that control turbulent flows.
In the last decade, some results on the number of determining modes and determining nodes
were proved for 3D Navier-Stokes-Voigt equations [8] and some other regularization models of
3D Navier-Stokes equations such as 3D Navier-Stokes-α, 3D Leray-α and 3D Navier-Stokes-ω
equations [9]. However, to the best of our knowledge, there is no existing result on the number of
determining finite volume elements for α-models in fluid mechanics. This is the main motivation
of the present paper.
In this paper, following the general lines of the approach in [7], we give an upper bound
on the number of determining finite volume elements for the 3D Navier-Stokes-Voigt equations.
Here the number of determining finite volume elements is estimated explicitly in terms of flow
parameters, such as viscosity, smoothing length, forcing and domain size, and the obtained
estimate is global as it does not depend on an individual solution.
2. Preliminaries
Let V be the set of all vector valued trigonometric polynomials u defined in Ω such that
∇ · u = 0 and ∫
Ω
u(x)dx = 0. Denote by H and V the closures of V in L2(Ω)3 and in H1(Ω)3,
respectively. We denote by (·, ·) and | · | the inner product and the norm in H , and by ((·, ·)) =
(∇·,∇·) and ‖ · ‖ = |∇ · | the inner product and norm in V .
Let P be the Leray orthogonal projection in L2(Ω)3 onto the space H . We denote by A =
−P∆ the Stokes operator with domain D(A) = H2(Ω)3 ∩ V . Notice the fact that in the case of
periodic boundary conditions, A = −∆ is a self-adjoint positive operator with compact inverse.
Hence there exists a complete set of eigenfunctions {wj}∞j=1 which is orthonormal in H , and
orthogonal in both V and D(A) such that Awj = λjwj with
(2pi/L)2 = λ1 ≤ λ2 ≤ · · · ≤ λj ∼ j2/3L−2 ≤ · · ·
We have the following Poincaré type inequalities:
‖u‖2 ≥ λ1|u|2 for all u ∈ V,
|Au|2 ≥ λ1‖u‖2 for all u ∈ D(A).
Hence we have the following estimates
‖w‖2 ≥ d0
(|w|2 + α2‖w‖2) ,
|Aw|2 ≥ d0
(‖w‖2 + α2|Aw|2) ,
with
d0 =
λ1
1 + λ1α2
. (2.1)
18
On the number of determining finite volume elements for 3D Navier-Stokes-Voigt equations
We define
B(u, v) = P (u · ∇)v, ∀u, v ∈ V.
From the definition of B we have
〈B(u, v), v〉V ′,V = 0 for all u, v ∈ V. (2.2)
We have the following estimate (see e.g. [10, 11]):
|(B(u, v), w)| ≤ c1|u|1/2‖u‖1/2‖v‖ ‖w‖, ∀u, v, w ∈ V, (2.3)
for the positive constant c1 depending only on Ω.
We apply the Leray projection P to (1.1) to obtain the equivalent system of equations
d
dt
(u+ α2Au) + νAu+B(u, u) = Pf, (2.4)
with the initial datum
u(0) = u0. (2.5)
We first recall the following well-posedness result.
Theorem 2.1. [8, 12] If u0 ∈ V and f ∈ L∞(0,∞;V ′), then problem (2.4)-(2.5) has a unique
global weak solution u ∈ L∞(0,∞;V ) satisfying
lim sup
t→∞
(|u(t)|2 + α2‖u(t)‖2) ≤ ν2λ3/21
d0
Gr2. (2.6)
where d0 is given in (2.1) and Gr is the generalized Grashoff number defined by
Gr =
1
ν2λ
3/4
1
lim sup
t→∞
‖f(t)‖V ′ .
We also need the following generalized Gronwall inequality.
Lemma 2.1. [7] Suppose that φ(t) is an absolutely continuous non-negative function on [0,∞)
that satisfies the following inequality
dφ
dt
+ βφ ≤ γ, a.e. on [0,∞),
where β and γ are locally integrable real-valued functions on [0,∞) that satisfy the following
conditions for some T > 0
lim inf
t→∞
1
T
∫ t+T
t
β(τ)dτ > 0, lim sup
t→∞
1
T
∫ t+T
t
β−(τ)dτ <∞,
and
lim sup
t→∞
1
T
∫ t+T
t
γ+(τ)dτ = 0,
with β− := max{−β, 0}, γ+ := max{γ, 0}. Then it follows that lim
t→∞
φ(t) = 0.
19
Nguyen Thi Ngan and Nguyen Thi Minh Toai
3. Determining finite volume elements
We divide the domain Ω into N equal squares Ωj, j = 1, . . . , N , where Ωj is the j-th cubic
with edge h = L/ 3
√
N , and so the volume of Ωj is |Ωj | = L3/N . The local average of ϕ in Ωj is
defined by
〈ϕ〉
Ωj
=
1
|Ωj|
∫
Ωj
ϕ(x)dx.
To establish the result on the number of determining finite volume elements, we need the
following lemma:
Lemma 3.1. [5] For every w ∈ V , we have
|w|2 ≤ L3χ2(w) + L
2
6N2/3
‖w‖2, (3.1)
where
χ(w) = max
1≤j≤N
∣∣∣〈w〉Ωj
∣∣∣ .
Let f and g belong to L∞(0,∞;V ′) satisfying
lim
t→∞
‖f(t)− g(t)‖V ′ = 0.
Then one can check that the generalized Grashoff numbers Gr corresponding to the external forces
f and g are the same.
A set of volume elements is said to be determining if for any two solutions u and v of (2.4)
corresponding to the above external forces f and g, from the condition
lim
t→∞
(
〈u〉
Ωj
− 〈v〉
Ωj
)
= 0,
it implies that
lim
t→∞
(|u(t)− v(t)|2 + α2‖u(t) − v(t)‖2) = 0.
The following theorem is the main result of the paper.
Theorem 3.1. Suppose that
lim
t→∞
(
〈u〉
Ωj
− 〈v〉
Ωj
)
= 0,
for j = 1, . . . , N . Then the volume elements are determining provided that
N >
(
9c41λ
3
1L
3Gr4
4α4d20
)3/2
.
Proof. Let u and v be two solutions of (2.4). Then w = u− v satisfies the following system
d
dt
(
w + α2Aw
)
+ νAw +B(w, u) +B(v,w) = f − g.
Multiplying both sides by w we get
1
2
d
dt
(|w|2 + α2‖w‖2)+ ν‖w‖2 + (B(w, u), w) = 〈f − g,w〉, (3.2)
20
On the number of determining finite volume elements for 3D Navier-Stokes-Voigt equations
where we have used property (2.2). Applying the Cauchy inequality, we have
〈f − g,w〉 ≤ 1
ν
‖f − g‖2V ′ +
ν
4
‖w‖2. (3.3)
Now, using (2.3) and the Young inequality, we have
(B(w, u), w) ≤ c1|w|1/2‖w‖1/2‖u‖ ‖w‖
= c1|w|1/2‖u‖ ‖w‖3/2
≤ 27c
4
1
4ν3
‖u‖4|w|2 + ν
4
‖w‖2.
(3.4)
Combining (3.3) and (3.4) then (3.2) becomes
d
dt
(|w|2 + α2‖w‖2)+ ν‖w‖2 ≤ 27c41
2ν3
‖u‖4|w|2 + 2
ν
‖f − g‖2V ′ .
Hence, using (3.1), we obtain that
d
dt
(|w|2 + α2‖w‖2)+ (ν − 9c41L3
4ν3N2/3
‖u‖4
)
‖w‖2 ≤ 27
2ν3
L3χ2(w) +
2
ν
‖f − g‖2V ′ . (3.5)
Using the estimate (2.6), there exists a time t1 > 0 such that
‖u(t)‖4 ≤ ν
4λ31Gr
4
α4d20
, ∀t ≥ t1.
So, we get from (3.5) that
d
dt
(|w|2 + α2‖w‖2)+ (ν − 9c41νλ31L3Gr4
4α4d20N
2/3
)
‖w‖2
≤ 27
2ν3
L3χ2(w) +
2
ν
‖f − g‖2V ′ , ∀t ≥ t1.
(3.6)
Now, for N is chosen large enough such that
N ≥
(
9c41λ
3
1L
3Gr4
4α4d20
)3/2
,
we deduce from (3.6) that
d
dt
(|w|2 + α2‖w‖2)+ β(t) (|w|2 + α2‖w‖2) ≤ γ(t), ∀t ≥ t1
where
β(t) = d0
(
ν − 9c
4
1νλ
3
1L
3Gr4
4α4d20N
2/3
)
,
and
γ(t) =
27
2ν3
L3χ2(w) +
2
ν
‖f(t)− g(t)‖2V ′ .
21
Nguyen Thi Ngan and Nguyen Thi Minh Toai
To complete the proof, we will show that β(t) and γ(t) fulfill the requirements of Lemma 2.1.
First, since the assumption of f, g and χ one can check that for any fixed T > 0,
lim sup
t→∞
1
T
∫ t+T
t
γ+(τ)dτ = 0. (3.7)
Moreover,
lim sup
t→∞
1
T
∫ t+T
t
β−(τ)dτ <∞ and lim inf
t→∞
1
T
∫ t+T
t
β(τ)dτ > 0 (3.8)
hold provided
N >
(
9c41λ
3
1L
3Gr4
4α4d20
)3/2
.
From eqs. (3.7) and (3.8), applying Lemma 2.2 we arrive at
lim
t→∞
(|w(t)|2 + α2‖w(t)‖2) = 0.
This completes the proof.
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