Chuyên san Công nghệ thông tin và Truyền thông - Số 02 (4-2013)
EXISTENCE AND UNIQUENESS OF
STRONG SOLUTIONS TO
2D g-NAVIER-STOKES EQUATIONS
Nguyen Thi Thanh Ha1 , Nguyen Van Hong1 and Dao Trong Quyet1
Abstract
We study the existence and uniqueness of strong solutions to the first initial boundary value
problem for the two-dimensional non-autonomous g-Navier-Stokes equations in bounded domain.
Nội dung bài báo nghiên cứu sự tồn tại, duy nhất nghiệm mạnh của bài toán biên ban đầu đối
với hệ phương trình g-Navier-Stokes hai chiều không ô-tô-nôm trên miền bị chặn.
Keywords
Strong solutions, g-Navier-Stokes equations, Galerkin method.
1. Introduction
Let Ω be a bounded domain in R2 with smooth boundary Γ. In this paper we study the
existence and uniqueness of strong solutions to the following two-dimensional (2D) nonautonomous g-Navier-Stokes equations:


 ∂u − ν∆u + (u · ∇)u + ∇p = f (t) in (0, T ) × Ω,



∂t


∇ · (gu)
= 0 in (0, T ) × Ω,
(1.1)


u
= 0 on (0, T ) × Γ,




u(x, 0)
= u0 (x), x ∈ Ω,
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, u0 is the initial velocity.
The g-Navier-Stokes equations is a variation of the standard Navier-Stokes equations. More
precisely, when g ≡ const we get the usual Navier-Stokes equations. The 2D g-Navier-Stokes
equations arise in a natural way when we study the standard 3D problem in thin domains.
We refer the reader to [8] for a derivation of the 2D g-Navier-Stokes equations from the 3D
Navier-Stokes equations and a relationship between them. As mentioned in [7], good properties
of the 2D g-Navier-Stokes equations can lead to an initiate the study of the Navier-Stokes
equations on the thin three dimensional domain Ωg = Ω × (0, g). In the last few years, the
(1), Faculty of Information Technology, Le Quy Don Technical University, Hanoi, Vietnam.
70
Tạp chí Khoa học và Kỹ thuật - Học viện KTQS số 153 (4-2013)
existence and asymptotic behavior of weak solutions to 2D g-Navier-Stokes equations have
been studied extensively in both cases without and with delays (see e.g. [1], [2], [3], [5], [6],
[7], [8]). However, to the best of knowledge, little seems to be known about strong solutions
of the 2D g-Navier-Stokes equations.
In this paper, we will study the existence and uniqueness of strong solutions to the twodimensional non-autonomous g-Navier-Stokes equations. To do this, we make the assumption:
(G) g ∈ W 1,∞ (Ω) such that:
1/2
0 < m0 ≤g(x)≤M0 for all x = (x1 , x2 ) ∈ Ω, and |∇g|∞ < m0 λ1 .
where λ1 > 0 is the first eigenvalue of the g-Stokes operator in Ω (i.e. the operator A
defined in Section 2).
The rest of the paper is organized as follows. In the next section, we recall some auxiliary
results on function spaces and inequalities for the nonlinear terms related to the g-NavierStokes equations. In Section 3, we prove the existence of a strong solution to the problem by
using the Faedo-Galerkin method.
2. Preliminaries
Let L2 (Ω, g) = (L2 (Ω))2 and H01 (Ω, g) = (H01 (Ω))2 be endowed, respectively, with the
inner products
Z
(u, v)g = u.vgdx, u, v ∈ L2 (Ω, g),
Ω
and
Z X
2
((u, v))g =
∇uj ∇vj gdx, u = (u1 , u2 ), v = (v1 , v2 ) ∈ H01 (Ω, g),
Ω j=1
and norms |u|2 = (u, u)g , ||u||2 = ((u, u))g . Thanks to assumption (H1), the norms |.| and
||.|| are equivalent to the usual ones in (L2 (Ω))2 and in (H01 (Ω))2 .
Let
V = {u ∈ (C0∞ (Ω))2 : ∇.(gu) = 0}.
Denote by Hg the closure of V in L2 (Ω, g), and by Vg the closure of V in H01 (Ω, g). It follows
that Vg ⊂ Hg ≡ Hg0 ⊂ Vg0 , 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 Z
X
i,j=1
Ω
ui
∂vj
wj gdx,
∂xi
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Chuyên san Công nghệ thông tin và Truyền thông - Số 02 (4-2013)
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).
Hence
b(u, v, v) = 0, ∀u, v ∈ Vg .
Set A : Vg → Vg0 by hAu, vi = ((u, v))g , B : Vg × Vg → Vg0 by hB(u, v), wi = b(u, v, w).
Denote D(A) = {u ∈ Vg : Au ∈ Hg }, then D(A) = H 2 (Ω, g) ∩ Vg and Au = −Pg ∆u, ∀u ∈
D(A), where Pg is the ortho-projector from L2 (Ω, g) onto Hg .
Using the Hölder inequality, the Ladyzhenskaya inequality (when n = 2):
|u|L4 ≤ c|u|1/2 |∇u|1/2 , ∀u ∈ H01 (Ω),
and the interpolation inequalities, as in [9] one can prove the following
Lemma 2.1. If n = 2, then


c1 |u|1/2 kuk1/2 kvk|w|1/2 kwk1/2 , ∀u, v, w ∈ Vg ,




c |u|1/2 kuk1/2 kvk|Aw|1/2 |w|1/2 , ∀u ∈ V , v ∈ D(A), w ∈ H ,
2
g
g
|b(u, v, w)| ≤
1/2
1/2

c3 |u| |Au| kvk|w|, ∀u ∈ D(A), v ∈ Vg , w ∈ Hg ,




c |u|kvk|w|1/2 |Aw|1/2 , ∀u ∈ H , v ∈ V , w ∈ D(A),
4
g
g
where ci , i = 1, . . . , 4, are appropriate constants.
Lemma 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 ).
Proof. By Lemma 2.1, for almost every t ∈ [τ, T ], we have
|Bu(t)| ≤ c3 |u(t)|1/2 |Au(t)|1/2 ||u(t)|| ≤ c03 ||u(t)||3/2 |Au(t)|1/2 ,
Then
Z
T
4
|Bu(t)| dt ≤
c03
Z
0
T
6
2
||u(t)|| |Au(t)| dt ≤
0
c||u||6L∞ (0,T ;Vg )
Z
T
|Au(t)|2 dt ≤ +∞.
0
This completes the proof.
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(
, u, v), ∀v ∈ Vg ,
g
g
belongs to L2 (0, T ; Hg ), and hence also belongs to L2 (0, T ; Vg0 ). Moreover,
|Cu(t)| ≤
72
|∇g|∞
.||u(t)||, for a.e. t ∈ (0, T ),
m0
(2.1)
Tạp chí Khoa học và Kỹ thuật - Học viện KTQS số 153 (4-2013)
and
||Cu(t)||∗ ≤
Since
|∇g|∞
1/2
m0 λ1
.||u(t)||, for a.e. t ∈ (0, T ).
∇g
1
.∇)u,
− (∇.g∇)u = −∆u − (
g
g
we have
(−∆u, v)g = ((u, v))g + ((
∇g
∇g
.∇)u, v)g = (Au, v)g + ((
.∇)u, v)g , ∀u, v ∈ Vg .
g
g
3. Existence and uniqueness of strong solutions
Definition 3.1. Given f ∈ L2 (0, T ; Hg ) and u0 ∈ Vg , a strong solution on the (0, T ) of
problem (1.1) is a function u ∈ L2 (0, T ; D(A)) ∩ L∞ (0, T ; Vg ) with u(0) = u0 , and such that
d
(u(t), v)g + ν((u(t), v))g + ν(Cu(t), v)g + b(u(t), u(t), v) = (f (t), v)g .
dt
for all v ∈ Vg , and for a.e. t ∈ (0, T ).
(3.1)
Remark 3.1. From the above definition, we see that the strong solution u ∈ L2 (0, T ; D(A))
du
= f − νAu − Bu − Cu ∈ L2 (0, T ; Hg ) due to the results of Lemmas 2.2 and 2.3. By
and
dt
Lemma 1.2 in [15], we have u ∈ C([0, T ]; Vg ), which makes the initial condition u(0) = u0
meaningful. It is noticed that if u is a strong solution of (1.1), then u satisfies the following
energy equality:
Z t
Z t
Z t
∇g
2
2
2
, u(r), u(r))dr = |u(s)| + 2 (f (r), u(r))g dr.
|u(t)| + 2ν
ku(r)k dr + 2ν
b(
g
s
s
s
for all 0 ≤ s < t ≤ T.
We now prove some a priori estimates for the (sufficiently regular) strong solutions to
problem (1.1).
Lemma 3.1. For u is strong solution of (1.1), then we have:
Z T
||u(t)||2 dt ≤ K1 , K1 = K1 (u0 , f, ν, T, λ1 , ),
(3.2)
0
sup |u(s)|2 ≤ K2 ,
K2 = K2 (u0 , f, ν, T, λ1 , ).
(3.3)
s∈[0,T ]
Proof. From (3.1), replacing v by u(t) we get
d
(u(t), u(t))g + ν((u(t), u(t)))g + ν(Cu(t), u(t))g + b(u(t), u(t), u(t))
dt
= (f (t), u(t))g .
(3.4)
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Chuyên san Công nghệ thông tin và Truyền thông - Số 02 (4-2013)
Because b(u(t), u(t), u(t)) = 0 and (Cu(t), u(t))g = b(
∇g
, u(t), u(t)), from (3.4) we have
g
d
∇g
(u(t), u(t))g + ν((u(t), u(t)))g + νb(
, u(t), u(t))
dt
g
= (f (t), u(t))g ,
and therefore,
d
∇g
|u(t)|2 + 2ν||u(t)||2 = 2(f (t), u(t))g − 2νb(
, u(t), u(t)).
dt
g
(3.5)
Using Lemma 2.3 and the Cauchy inequality, we get
|f (t)|2
|∇g|∞
d
|u(t)|2 + 2ν||u(t)||2 ≤ 2ν||u(t)||2 +
+ 2ν
||u(t)||2 ,
1/2
dt
2νλ1
m0 λ1
and hence
d
|f (t)|2
|u(t)|2 + 2ν(γ0 − )||u(t)||2 ≤
,
dt
2νλ1
(3.6)
∞
> 0 and > 0 is chosen such that γ0 − > 0.
where γ0 = 1 − |∇g|1/2
m0 λ1
By integration in t from 0 to T, after dropping unnecessary term, we obtain (3.2). Then by
integration in t of (3.10) from 0 to s, 0 < s < T , we obtain
Z s
1
2
|u(s)| ≤ |u0 | +
|f (t)|2 dt
2νλ1 0
Then, we have (3.3).
Lemma 3.2. If u is a sufficiently regular solution of (1.1), then we have:
sup ||u(t)||2 ≤ K3 ,
K3 = K3 (K1 , K2 ),
(3.7)
K4 = K4 (K1 , K2 ).
(3.8)
t∈[0,T ]
Z
T
|Au(t)|2 dt ≤ K4 ,
0
Proof. From (3.1), replacing v by Au(t) we get
d
(u(t), Au(t))g + ν((u(t), Au(t)))g + ν(Cu(t), Au(t))g + b(u(t), u(t), Au(t))
dt
= (f (t), Au(t))g .
Since ((φ, ψ))g = hAφ, ψi ∀ φ, ψ ∈ Vg
This relation can be writen
1d
||u(t)||2 + ν|Au(t)|2 + ν(Cu(t), Au(t))g + b(u(t), u(t), Au(t))
2 dt
ν
1
= (f (t), Au(t))g ≤ |Au(t)|2 + |f (t)|2 .
4
ν
74
(3.9)
(3.10)
Tạp chí Khoa học và Kỹ thuật - Học viện KTQS số 153 (4-2013)
Using Lemma 2.1 and 2.3, (3.10) implies
1d
||u(t)||2 + ν|Au(t)|2
2 dt
ν
1
ν|∇g|∞
≤ |Au(t)|2 + |f (t)|2 + c3 |u(t)|1/2 |Au(t)|3/2 ||u(t)|| +
|u(t)||Au(t)|.
4
ν
m0
(3.11)
Using Young’s inequality and Cauchy’s inequality, we obtain
1d
||u(t)||2 + ν|Au(t)|2
2 dt
ν
1
≤ |Au(t)|2 + |f (t)|2
4
ν
ν
2
+ |Au(t)| + c03 |u(t)|2 |||u(t)||4
4
1/2
ν|∇g|∞
ν|∇g|∞ λ1
2
+
|u(t)|2 ,
|Au(t)| +
1/2
4m0
m0 λ1
(3.12)
d
|∇g|∞
)|Au(t)|2
||u(t)||2 + ν(1 −
1/2
dt
m0 λ1
2
ν|∇g|∞
≤ |f (t)|2 + 2c03 |u(t)|2 |||u(t)||4 +
||u(t)||2 .
1/2
ν
2m0 λ1
(3.13)
Then, we have
Momentarily dropping the term ν(1 −
|∇g|∞
1/2
m0 λ1
)|Au(t)|2 , we have a differential inequality
y 0 ≤ a + θy
where
y(t) = ||u(t)||2 , a(t) =
2
ν|∇g|∞ |f (t)|2 , θ(t) = 2c03 |u(t)|2 |||u(t)||2 +
,
1/2
ν
2m0 λ1
from which we obtain by the technique of Gronwall’s lemma:
Z t
Z t
d
y(t)exp(−
θ(τ )dτ ) ≤ a(t)exp(−
θ(τ )dτ ),
dt
0
0
Z t
Z t
Z t
y(t) ≤ y(0)exp( θ(τ )dτ ) +
a(s)exp( θ(τ )dτ )ds,
0
0
0
or
2
Z
2
t
2c03 |u(τ )|2 |||u(τ )||2 +
||u(t)|| ≤ ||u0 || exp
0
2
+
ν
Z
t
2
Z
|f (s)| exp
0
0
ν|∇g|∞ dτ
1/2
2m0 λ1
t
2c03 |u(τ )|2 |||u(τ )||2
+
ν|∇g|∞ 1/2
(3.14)
dτ ds.
2m0 λ1
With Lemma 3.1 we have (3.7).
We com back to (3.13), which we integrate from 0 to T , we have (3.8).
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Chuyên san Công nghệ thông tin và Truyền thông - Số 02 (4-2013)
Theorem 3.1. Suppose that f ∈ L2 (0, T ; Hg ) and u0 ∈ Vg are given.Then there exists a
unique strong solution u of (1.1) on (0, T ).
Proof. (i) Uniqueness. Let u, v be two strong solutions of problem (1.1) with the same
initial condition and set w = u − v. Then, using the energy equality, we obtain
Z t
Z t
∇g
2
2
b(
, w(s), w(s))ds
||w(s)|| ds+2ν
|w(t)| + 2ν
g
0
0
Z t
b(w(s), v(s), w(s))ds, ∀ t ∈ (0, T ).
= −2
0
By Lemma 2.1, we have
Z t
Z t
|2
b(w(s), v(s), w(s))ds| ≤ 2c1
|w(s)|||w(s)||||v(s)||ds
0
0
Z t
Z
c21 t
2
||w(s)|| ds +
≤ν
||v(s)||2 |w(s)|2 ds.
ν
0
0
By Lemma 2.3, we have
Z t
Z
∇g
|∇g|∞ t
b(
, w(s), w(s))ds| ≤ 2ν
||w(s)|||w(s)|ds
|2ν
1/2
g
m0 λ1
0
0
Z
Z t
ν|∇g|2∞ t
2
|w(s)|2 ds.
||w(s)|| ds +
≤ν
2
λ
m
1
0
0
0
Thus, one has
Z
Z
c21 t
ν|∇g|2∞ t
2
2
2
|w(t)| ≤
||v(s)|| |w(s)| ds +
|w(s)|2 ds.
ν 0
m20 λ1 0
Hence the Gronwall’s inequality completes the proof.
(ii) Existence. We split the proof of the existence into several steps.
Step 1: A Galerkin scheme.
Let v1 , v2 , ...., be a basis of Vg consisting of eigenfunctions of the operator A, which is orP
thonormal in Hg . Denote Vm = span{v1 , ..., vm } and consider the projector Pm u = m
j=1 (u, vj )vj .
Define also
m
X
m
u (t) =
αm,j (t)vj ,
j=1
where the coefficients αm,j are required to satisfy the following system
d m
(u (t), vj )g + ν(Aum (t), vj )g + ν(Cum (t),vj )g + b(um (t), um (t), vj )
dt
= (f (t), vj )g , ∀j = 1, ..., m,
(3.15)
and the initial condition is um (0) = Pm u0 . This system of ordinary differential equations in
the unknown (αm,1 (t), ..., αm,m (t)) fulfills the Peano theorem, so the approximate solutions
um exist.
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Tạp chí Khoa học và Kỹ thuật - Học viện KTQS số 153 (4-2013)
Step 2: A priori estimates.
From (3.15), replacing vj by Aum (t) we get
d m
(u (t), Aum (t))g +ν((um (t), Aum (t)))g + ν(Cum (t), Aum (t))g
dt
+ b(um (t), um (t), Aum (t)) = (f (t), Aum (t))g .
(3.16)
This relation is similar to (3.9). Exactly as in Lemma 3.2, for um , with u0 replace by um
0 .
Since um
0 = Pm u0 and Pm is an orthogonal projector in Vg , then
||um
0 || = ||Pm u0 || ≤ ||u0 ||,
and we then find for um exactly the same bounds as for u in (3.7) and (3.8), or
um remains in a bounded set of L2 (0, T ; D(A)) ∩ L∞ (0, T ; Vg ).
(3.17)
Now, observe that (3.15) is equivalent to
dum
= −νAum − νCum − Pm B(um , um ) + Pm f (t)
dt
Hence, since Lemma 2.2 we have
{(um )0 } is bounded in L2 (0, T ; Hg ),
Step 3: Passage to the limit.
From the above estimates, we conclude that there exist u ∈ L2 (0, T ; D(A)) ∩ L∞ (0, T ; Vg )
with u0 ∈ L2 (0, T ; Hg ), and a subsequence of {um }, relabelled the same, such that
{um } converges weakly-star to u in L∞ (0, T ; Vg ),
{um } converges weakly to u in L2 (0, T ; D(A)),
(3.18)
0
{(um ) } converges weakly to u0 in L2 (0, T ; Hg ).
Since Ω is bounded, we can use the Compactness Lemma (see [11, Chapter III. Theorem
2.1]) to deduce the existence of a subsequence (still denoted by) um which converges strong
to u in L2 (0, T ; Vg ), and therefore also in L2 (0, T ; Hg ).
Then we can pass to the limit in the nonlinearity b thanks to the following lemma whose
proof is exactly the proof of Lemma 3.2 in [9, Chapter III].
Lemma 3.3. If um converges to uu in L2 (0, T ; Hg ) strongly, then for any vector function w
with components belong to C 1 ([0, T ] × (Ω)), we have
Z T
Z T
m
m
b(u (t), u (t), w(t))dt →
b(u(t), u(t), w(t))dt.
0
0
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Chuyên san Công nghệ thông tin và Truyền thông - Số 02 (4-2013)
Let ψ be a continuously differentiable function on [0, T ] with ψ(T ) = 0. We multiply (3.15)
by ψ(t) and then integrate by parts. This leads to the equation:
Z T
Z T
0
m
(Aum (t), wj ψ(t))g dt
(u (t), ψ (t)wj )g dt + ν
−
0
0
Z T
Z T
m
(Cu (t), wj ψ(t))g dt +
b(um (t), um (t), wj ψ(t))dt
+ν
0
0
Z T
(f (t), wj ψ(t))g dt.
= (um (0), wj )g ψ(0) +
0
Passing to the limit, we have:
Z T
Z T
0
−
(u(t), vψ (t))g dt + ν
(Au(t), vψ(t))g dt
0
Z
0
T
+ν
T
Z
(Cu(t), vψ(t))g dt +
b(u(t), u(t), vψ(t))dt
0
0
T
Z
(f (t), vψ(t))g dt.
= (u0 , v)g ψ(0) +
(3.19)
0
holds for any v ∈ Vg0 , we see that u satisfies (3.1) in the distribution sense.
Finally, it remains to prove that u satisfies u(0) = 0. For this we multiply (3.15) by ψ(t),
and integrate. After integrating the first term by parts, we get:
Z T
Z T
0
−
(u(t), vψ (t))g dt + ν
(Au(t), vψ(t))g dt
0
Z
0
T
T
Z
b(u(t), u(t), vψ(t))dt
(Cu(t), vψ(t))g dt +
+ν
0
0
Z
= (u(0), v)g ψ(0) +
T
(f (t), vψ(t))g dt.
(3.20)
0
By comparison with (3.19), we have:
(u(0) − u0 , v)ψ(0) = 0,
We can choose ψ with ψ(0) 6= 0, thus u(0) = u0 . This completes the proof.
Acknowledgements. This work has been supported by a grant from the Le Quy Don Technical
University.
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Tạp chí Khoa học và Kỹ thuật - Học viện KTQS số 153 (4-2013)
References
[1] C.T. Anh and D.T. Quyet, Long-time behavior for 2D non-autonomous g-Navier-Stokes equations, Ann. Pol.
Math, 103 (2012), 277-302.
[2] C.T. Anh and D.T. Quyet, g-Navier-Stokes equations with infinite delays, Viet. J. Math, 40 (2012), 57-78.
[3] C.T. Anh, D.T. Quyet and D.T. Tinh, Existence and finite time approximation of strong solutions of the 2D
g-Navier-Stokes equations, Acta. Math. Viet. 38 (2013), to appear.
[4] H. Bae and J. Roh, Existence of solutions of the g-Navier-Stokes equations, Taiwanese J. Math. 8 (2004), 85-102.
[5] J. Jiang and Y. Hou, The global attractor of g-Navier-Stokes equations with linear dampness on R2 , Appl. Math.
Comp. 215 (2009), 1068-1076.
[6] M. Kwak, H. Kwean and J. Roh, The dimension of attractor of the 2D g-Navier-Stokes equations, J. Math. Anal.
Appl. 315 (2006), 436-461.
[7] H. Kwean and J. Roh, The global attractor of the 2D g-Navier-Stokes equations on some unbounded domains,
Commun. Korean Math. Soc. 20 (2005), 731-749.
[8] J. Roh, Dynamics of the g-Navier-Stokes equations, J. Differential Equations 211 (2005), 452-484.
[9] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, 2nd edition, Amsterdam: North-Holland,
1979.
[10] R. Temam, Navier-Stokes Equations and Nonlinear functional Analysis, 2nd edition, Philadelphia, 1995.
Ngày nhận bài 29-01-2013; ngày gửi phản biện 6-2-2013.
Ngày chấp nhận đăng 5-6-2013.
Đào Trọng Quyết sinh ngày 10/07/1980 tại Hà Nam. Tốt nghiệp ĐH Sư Phạm Hà Nội năm 2002,
nhận bằng Thạc sỹ chuyên ngành Toán học tính toán tại ĐH KHTN-ĐHQG Hà Nội năm 2008. Công
tác tại Bộ môn Toán, Khoa CNTT- Học viện KTQS và đang làm NCS tại Học viện KTQS chuyên
ngành Toán ứng dụng. Hướng nghiên cứu: Toán ứng dụng, Phương trình vi phân và tích phân.
Email: dtq100780@gmail.com
Nguyễn Thị Thanh Hà sinh ngày 07/11/1972. Tốt nghiệp ĐHSP Vinh năm 1993, nhận bằng Thạc sĩ
năm 1997 tại ĐHSP Hà Nội. Hiện đang công tác tại: Bộ môn Toán, Khoa CNTT, Học viện KTQS.
Email: hanguyen117@yahoo.com
Nguyễn Văn Hồng sinh ngày 21/10/1966. Tốt nghiệp ĐHSP Hà Nội ngành Toán, năm 1988. Tốt
nghiệp Thạc sĩ năm 1997 tại ĐHSP Hà Nội. Hiện đang công tác Bộ môn Toán, khoa CNTT Học
Viện KTQS.
Email: hongktqs@gmail.com
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