MINISTRY OF EDUCATION AND TRAINING
HA NOI PEDAGOGICAL UNIVERSITY 2

NGUYEN VIET TUAN
STABILITY AND STABILIZATION FOR SOME EVOLUTION
EQUATIONS IN FLUID MECHANICS
Sp eiality: Mathematial analysis
Co de: 9 46 01 02
SUMMARY OF DOCTORAL THESIS IN MATHEMATICS
Ha Noi - 2019
This thesis has b een ompleted at the Ha Noi Pedagogial Uni-
versity 2
Sientifi Advisor: Asso .Prof. PhD. Cung The Anh
Referee 1:
Referee 2:
Referee 3:
The thesis shall b e defended at the University level Thesis
Assessment Counil at Ha Noi Pedagogial University 2
on.......
The thesis an b e found in the National Library and the Library
of Ha Noi Pedagogial University 2.
INTRODUCTION
1. MOTIVATION AND HISTORY OF THE PROBLEM
Partial differential evolution equations app ear frequently in
the of physial and biologial pro esses, suh as heat transfer and
diffusion, pro ess of wave transmission in fluid mehanis and pop-
ulation mo dels in biology. The study of this equations lass has
imp ortant meaning in siene and tehnology. That is why it has
attrated widespread attention.
After studying the well-posedness of the problem, it is im-
p ortant to study the long-time b ehavior of solutions, as it allows
us to understand and predit the future dynamis, sine we an
make the appropriate adjustments to ahieve the desired results.
An effetive approah is the study of the existene and stability
of the stationary solutions. In mathematial, the stationary solu-
tions resp onse orresp onds to the stationary state of the pro ess,
and is the solution of the orresp onding ellipti problem. When
the stationary solutions of the pro esses is not stability, p eople try
to stabilize it by using appropriate ontrols, or using appropriate
random noise.
In reent years, stability and stabilization issues have been
studied extensively for Navier-Stokes equations and some lasses
of nonlinear paraboli equations. However, the orresp onding re-
sults for other lasses of equations in fluid mehanis and parab oli
systems are still small. There are new mathematial diffiulties,
b eause of the omplexity of the system or the interation b e-
tween nonlinear terms in the system. Therefore, this is a very
urrent issue and attrated widespread attention from domesti
and international math sientists.
First, we onsider 3D Navier-Stokes-Voigt (sometimes written
Voight) equations in smo oth b ounded domains with homogeneous
1
Dirihlet b oundary onditions:
ut−ν∆u−α2∆ut+ (u· ∇)u+∇p=f
in
O × R+,
∇ · u= 0
in
O × R+,
u(x, t) = 0
on
∂O × R+,
u(x, 0) = u0(x)
in
O.
(1)
In the last few years, mathematial questions related to 3D
Navier-Stokes-Voigt equations have attrated the attention of a
number of mathematiians. The existene and long-time behavior
of solutions in terms of existene of attrators to the 3D Navier-
Stokes-Voigt equations in domains that are b ounded or unb ounded
but satisfying the Poinar² inequality was investigated extensively
in the works of C.T. Anh and P.T. Trang (2013), A.O. Celebi, V.K.
Kalantarov and M. Polat (2009), J. Gar½a-Luengo, P. Mar½n-
Rubio and J. Real (2012). The deay rate of solutions to the
equations on the whole spae was studied in the works of C.T.
Anh and P.T. Trang (2016), C.J. Nihe (2016), C. Zhao and H.
Zhu (2015). The main aim of this thesis to study the exp onential
stability and stabilization of strong stationary solutions to prob-
lem (1).
Next, we onsider the following 2D
g
-Navier-Stokes equations
∂u
∂t −ν∆u+ (u· ∇)u=∇p+f
in
Ω×R+,
∇ · (gu) = 0
in
Ω×R+,
u(x, t) = 0
on
∂Ω×R+,
u(x, 0) = u0(x),
in
Ω.
(2)
In the past deade, the existene and long-time b ehavior of
solutions in terms of existene of attrators for 2D
g
-Navier-Stokes
equations have been studied extensively in b oth autonomous and
non-autonomous ases (see e.g. C.T. Anh and D.T. Quyet (2012),
J. Jiang, Y. Hou and X. Wang (2011), J. Jiang and X. Wang
2
(2013), H. Kwean and J. Roh (2005), D. Wu and J. Tao (2012),
and referenes therein). However, there are still many open issues
that need to b e investigated regarding the system (2), suh as:
1) Existene, uniqueness and exp onential stability of strong
stationary solutions.
2) Stabilization of strong stationary solutions.
3) Stabilization of long-time b ehavior of solutions.
Finally, we onsider the following sto hasti 2D
g
-Navier-Stokes
equations with finite delays
du = [ν∆u−(u· ∇)u− ∇p+f+F(u(t−ρ(t)))]dt
+G(u(t−ρ(t)))dW (t), x ∈ O, t > 0,
∇ · (gu) = 0, x ∈ O, t > 0,
u(x, t) = 0, x ∈∂O, t > 0,
u(x, t) = ϕ(x, t), x ∈ O, t ∈[−τ, 0],
(3)
The existene and stability of stationary solutions to 2D Navier-
Stokes equations with delays have been studied by many authors
in reent years, see for instane, Caraballo and Han (2014, 2015),
Caraballo and Real (2001, 2003), Chen (2012), Garrido-Atienza
and Mar½n-Rubio (2006), Mar½n-Rubio, Real and Valero (2011),
Wan and Zhou (2011). The existene and stability of stationary
solutions to the 2D
g
-Navier-Stokes equations without/with de-
lays have b een studied in reent works (see C.T. Anh and D.T.
Quyet (2012), D.T. Quyet (2014)). However, to the best of our
knowledge, there is no result on the stability of solutions to prob-
lem (3).
2. PURPOSE OF THESIS
Researh thesis on the problem: The stability and stabilization
of some evolution equations app ear in fluid mehanis.
3