Summary of doctoral thesis in mathematics: Stability and stabilization for some evolution equations in fluid mechanics

Purpose of thesis: Resear h thesis on the problem: The stability and stabilization of some evolution equations appear in fluid mechanics.

HA NOI PEDAGOGICAL UNIVERSITY 2



NGUYEN VIET TUAN

STABILITY AND STABILIZATION FOR SOME EVOLUTION

EQUATIONS IN FLUID MECHANICS

Code: 9 46 01 02

SUMMARY OF DOCTORAL THESIS IN MATHEMATICS

The thesis shall be defended at the University level Thesis

Assessment at Ha Noi P University 2

on

The thesis be found in the National Library and the Library

of Ha Noi P University 2

1 MOTIVATION AND HISTORY OF THE PROBLEM

Partial differential evolution equations appear frequently in

the of ph and pro h as heat transfer and

diffusion, pro of wavetransmission influid and

pop-ulation models in biology The study of this equations has

important meaning in and hnology That is why it has

widespread attention

After studying the well-posedness of the problem, it is

im-portant to study the long-time behavior of solutions, as it allows

make the appropriate adjustments to hieve the desired results

An e h is the study of the and stability

of the stationary solutions In the stationary

solu-tions response onds to the stationary state of the pro

and is the solution of the onding problem When

the stationary solutions of the pro is not stability, people try

to stabilize it by using appropriate trols, or using appropriate

random noise

In t years, stability and stabilization issues have been

studied extensively for Navier-Stokes equations and some

of nonlinear parab equations However, the onding

re-sultsforother ofequationsinfluid andparab

systems are still small There are new

be-tween nonlinear terms in the system Therefore, this is a very

t issue and widespread attention from

and international math tists

First, we 3D Navier-Stokes-Voigt (sometimes written

Voight) equations insmooth bounded domains with homogeneous

In the last few years, questions related to 3D

Navier-Stokes-Voigt equations have the attention of a

of solutions in terms of of to the 3D

Navier-Stokes-Voigt equationsindomainsthatareboundedorunbounded

but satisfying the P inequality was investigated extensively

inthe worksofC.T.AnhandP.T.Trang(2013),A.O.Celebi, V.K

Kalantarov and M Polat (2009), J P

Mar½n-Rubio and J Real (2012) The y rate of solutions to the

equations on the whole was studied in the works of C.T

Anh and P.T Trang (2016), C.J he (2016), C Zhao and H

Zhu (2015) The main aim of this thesis to study the exponential

stability and stabilization of strong stationary solutions to

In the past the and long-time behavior of

solutionsinterms of of for 2Dg-Navier-Stokesequations have been studied extensively in both autonomous and

non-autonomous (see e.g C.T Anh and D.T Quyet (2012),

J Jiang, Y Hou and X Wang (2011), J Jiang and X Wang

and therein) However, there are still many open issues

that need to be investigated regarding the system (2) , h as:

1) uniqueness and exponential stability of strong

stationary solutions

2) Stabilization of strong stationary solutions

3) Stabilization of long-time behavior of solutions

Finally,we thefollowingsto 2Dg-Navier-Stokesequations with finite delays

The and stability of stationary solutions to 2D

Navier-Stokes equations with delays have been studied by many authors

in t years, see for 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 and stability of stationary

solutions to the 2D g-Navier-Stokes equations without/with lays have been studied in t works (see C.T Anh and D.T

de-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

h thesis on the problem: Thestability and stabilization

of some evolution equations appear in fluid

• Resear obje The stability and stabilization of some lution equations appear in fluid namely: three-

evo-dimensional Navier-Stokes-Voigtsystem,g-Navier-Stokestem,sto 2D g-Navier-Stokes equations with finite de-lays

sys-• Resear ope:

◦ Content1:Three-dimensional Navier-Stokes-Voigt tem

sys-1) uniqueness andthe exponential stability

of strong stationary solutions

2) Stabilization of strong stationary solutions by

us-ing either an internal k trol with

sup-port large enough or a m e noise of

suf-t intensity

◦ Content 2: Two-dimensional g-Navier-Stokes system.1) uniqueness andthe exponential stability

of strong stationary solutions

2) Stabilization of strong stationary solutions by

us-ing either an internal k trol with

sup-port large enough or finite-dimensional k

trol

3) Stabilization of long-time behavior of solutions

◦ Content 3: Sto 2D g-Navier-Stokes equationswith finite delays

1) and uniqueness of weak stationary

solu-tions to the system

2) The exponential stability in mean square and

al-most sure exponential stability of the weak

solu-tions to the sto equations

• To study the of solutions: Galerkin approximation,the and energy methods.

• Tostudy thestabilityofstationary and solutions:Energy ratings and Gronwall's inequality

• To study the stabilization problem: The methods of

Mathe-trol theory and Sto analysis

5 RESULTS OF THESIS

The thesis hieves the following main results:

• Proving the uniqueness and exponential stability of strongstationary solutions; proving the for stabilization

of strong stationary solutions by k trol with

sup-port in the domain and by random noise for 3D

Navier-Stokes-Voigt equations in bounded domains These are the

tents of Chapter 2

• Proving the uniqueness and exponential ity of strong stationary solutions; proving the for

stabil-stabilization of strong stationary solutions by using an

in-ternal k trols with support in domain and finite

dimensional k trols; proving the for

stabilization of the long-time behavior under of fast

external for2Dg-Navier-Stokestions in bounded domains These are the tents of

equa-Chapter 3

• Proving the and uniqueness of weak stationary lutions to the system; the exponential mean

so-square stability and almost sure exponential stability of the

weak solution to the sto 2D g-Navier-Stokes tions with finite delays in bounded domains These are the

equa-tents of Chapter 4

Beside Intro Author's worksrelated tothe

thesis and the thesis 4 hapters:

• Chapter 1 Preliminaries

• Chapter 2 Stabilization of 3D Navier-Stokes-Voigt tions

equa-• Chapter 3 Stabilization of 2D g-Navier-Stokes equations

• Chapter 4 The stability of solutions to sto 2D gNavier-Stokes equations with finite delays

In this hapter, we some general and results

about the operators, sto analysis,

inequal-ities for the nonlinear term and some additional results (the usual

inequalities, the methods) to prove the main results

of the thesis in the following hapters

1.1 THE FUNCTION SPACES

In this we repeat some of the results about the

that will be used in the thesis: Sobolev

L p (O), H m (O), H 0 m (O)), L p (0, T ; Y )

and C([0, T ]; Y ) In addition, we also present H

and V related to Navier-Stokes-Voigt equations;

H g and V g related to g-Navier-Stokes equations

1.2 THE OPERATORS

1.2.1 Operators A, B

We define the Stokes operator A : V → V ′ by

(Au, v) = ((u, v)), for all u, v ∈ V.

We also define the operator B : V × V → V ′ by

(B(u, v), w) = b(u, v, w), for all u, v, w ∈ V,

where c are appropriate onstants.

1.2.2 Operators A g, B g and C g

We define the operator A g : V g → V g ′ by

hA g u, vi g = ((u, v)) g , ∀u, v ∈ V g

We denote by η 1 the first eigenvalue of the operator A g

We also define the operator B g : V g × V g → V g ′ by

hB g (u, v), wi g = b g (u, v, w), ∀u, v, w ∈ V g ,

where c i , i = 1, , 4, are appropriate onstants

Lemma 1.3 Let u ∈ L 2 (0, T ; V g ), then the C g u definedby

(C g u(t), v) g = (( ∇g

g · ∇)u, v) g = b g (

∇g

g , u, v), ∀v ∈ Vg,

belongs to L 2 (0, T ; H g ), and e also belongs to L 2 (0, T ; V g ′ ).Moreover,

1.3 RESULTS OF STOCHASTIC ANALYSIS

In this we repeat some of the results about the

probability theory, Brownian motions or Wiener pro and

sto integrals that will be used in the thesis

1.4 RESULTS OF NORMAL USED

Inthis we someoftheprimarybut important

inequalities that are frequently used in the thesis We also present

a number of important propositions and theorems often used to

prove the results of the thesis: Aubin-Lions lemma, the

of the Brouwer fixed point theorem, the T honoff

fixed point theorem

STABILIZATION OF 3D NAVIER-STOKES-VOIGT

EQUATIONS

In this hapter, we 3D Navier-Stokes-Voigt equations

in smooth bounded domains with homogeneous hlet

bound-ary First, we study the and exponential

sta-bility of strong stationary solutions to the problem Then we show

that any unstable strong stationary solution be exponentially

stabilized by using either an internal k trol with

sup-port large enough or a m e Ito noise of t

inten-sity

This hapter is written based on the paper 3

2.1 SETTING OF THE PROBLEM

Let O be a bounded domain in R 3 with smooth boundary ∂O

We the following 3D Navier-Stokes-Voigt equations:

where u = u(x, t) = (u 1 , u 2 , u 3 ) is the unknown velo y v

p = p(x, t) is the unknown pressure, ν > 0 is the

y of the fluid, f = f (x) is a given field and u 0 is theinitial velo y

2.2 UNIQUENESSANDEXPONENTIALSTABILITY OFST

A-TIONARY SOLUTIONS

Definition 2.1 Let f ∈ (L 2 (O)) 3 be given A u ∗ ∈ D(A) is said to be a strong stationary solution to problem (2.1) if

νAu∗ + B(u∗, u∗) = f in (L2(Ω))3. (2.2)

Theorem 2.1 Let f ∈ (L 2 (O)) 3 Then

a) There exists at least one strong stationary solution u ∗ of

2.3 STABILIZATIONOFSTATIONARYSOLUTIONSBY

US-ING AN INTERNAL FEEDBACK CONTROL

We the following trolled 3D Navier-Stokes-Voigt

where 1 ω is the of the subdomain ω ⊂ O

with smooth boundary ∂ω, f ∈ (L2(O))3 and u0 ∈ V are given,

Denote by A ω be the Stokes operator defined on O ω We denote

by λ ∗ 1 (ω) the first eigenvalue of the operator A ω

Consider the k troller

γ ∗ (u ∗ ) := sup {|b(u, u ∗ , u)| : |u| = 1} ≤ γ ku ∗ k H α

We are now in position to state the main result of this

Theorem 2.2 Let u ∗ ∈ V ∩ (H β (O)) 3 , β > 5/2, b any strongstationary solution to (2.1) that

for some η > 0 Here kuk 2 α := |u| 2 + α 2 kuk 2

Remark 2.1 By the P inequality, we have

λ ∗ 1 (ω) ≥ C

 sup

if Oω is tly thin

MULTIPLICATIVE ITO NOISE

We the following sto 3D Navier-Stokes-Voigt

where c 0 is the best onstant in Lemma 1.1, then the solution u ∗ of

problem (2.7) is globallyexponentiallystable More pre there

exists N ⊂ Ω with P(N ) = 0, that for ω / ∈ N there is T (ω)

that for any solution u(t) of problem (2.7) , the followingestimate holds for some ℓ > 0 :

ku(t) − u ∗ k 2 α ≤ ku(0) − u ∗ k 2 α e −ℓt , ∀t ≥ T (ω).

Remark 2.2 Thus, the m e Ito noise stabilizes the

strong stationary solution u ∗ for ν in the interval

Thelargerthe parameterσ,thelongerthestabilityforthesolution

u ∗ Moreover, for any given ν > 0, we always hoose a value

of σ h that (2.8) holds

STABILIZATION OF 2D g-NAVIER-STOKES

EQUATIONS

We theg-Navier-Stokesequationsinatwo-dimensionalsmooth bounded domain O First, we study the andexponential stability of a strong stationary solution under some

we prove that any unstable strong

sta-tionary solution be stabilized by proportional troller with

support in an open subset ω ⊂ O h that O\ω is tly

small" or by using finite-dimensional k trols Finally,

we stabilize the long-time behavior of solutions to 2D gStokes equations under of fast external

-Navier-by showing that in this there exists a unique

time-perio solution and every solution tends to this perio solution

as time goes to infinity

This hapter is written based on the papers 1 and 4

3.1 SETTING OF THE PROBLEM

Let O be a bounded domain in R 2 with smooth boundary ∂O

We the following 2D g-Navier-Stokes equations:

t, u 0 is the initial velo y

We assume that the g satisfies the following tion:

assump-(G1) g ∈ W 1,∞ (O) h that

0 < m 0 ≤g(x)≤M 0 ∀x = (x 1 , x 2 ) ∈ O, v  |∇g| ∞ < m 0 η 1 1/2 ,

where η 1 > 0 is the first eigenvalue of the g-Stokes operator

in O (i.e the operator A g is defined in Chapter 1)

3.2 EXISTENCE,UNIQUENESSAND EXPONENTIAL ST

A-BILITY OF STATIONARY SOLUTIONS

Definition 3.1 Let f ∈ L 2 (Ω, g) be given A strong stationarysolution to problem (3.1) is an element u ∗ ∈ D(A g ) h that

3.3 STABILIZATIONOFSTATIONARYSOLUTIONSBY

US-ING AN INTERNAL FEEDBACK CONTROL

We the following trolled 2D g-Navier-Stokes tions:

where 1 ω is the of the subset ω ⊂ O withsmooth boundary ∂ω, f ∈ L 2 (O, g) and u 0 ∈ H g are given, h g =

Let A gω be the g-Stokes operator defined on O ω We denote by

η 1 ∗ (ω) the first eigenvalue of the operator A gω

Consider the k troller

γ g ∗ (u ∗ ) = sup {|b g (u, u, u ∗ )| : |u| g = 1} ≤ γ g ku ∗ k D(A g )

Theorem 3.2 Let u ∗ ∈ D(A g ) b any strong stationary solution

Then for e u 0 ∈ H g and k ≥ k 0 large but

inde-pendent of u 0, there is a weak solution u ∈ C([0, +∞); H g ) ∩

L 2 loc (0, +∞; V g ) to (3.5) that

|u(t) − u ∗ | g ≤ e −δt |u 0 − u ∗ | g , ∀t ≥ 0,

for some δ > 0

η 1 ∗ (ω) ≥ C

 sup

if O ω is tly thin"

3.4 STABILIZATIONOFSTATIONARYSOLUTIONSBY

US-ING FINITE-DIMENSIONAL FEEDBACK CONTROLS

We the following trolled 2D g-Navier-Stokes tions with interpolant operator Ih:

Then for e u 0 ∈ H g given, there exists a unique weak solution

u to system (3.7) that for any T > 0,

3.5 STABILIZATION BY USING FAST OSCILLA

TING-IN-TIME EXTERNAL FORCES

In this we the following system

We give the following assumption on the external

(F1) For any positive onstant ω 0 > 0, we assume that the for e

term F (x, ω 0 t) is a time perio with period T per

having the following e: There exists a time perio

h(x, ω0t) with period Tper that

F ∈ L ∞ (0, T per ; D(A g )) and kF k L ∞

(0,T per ;D(A g )) with upperbound is independent of ω 0

Moreover, we assume thath ∈ L ∞ (0, T per ; D(A g )) andthereexists a positive onstant L h independent of ω 0 that

khk 2 L ∞

(0,T per ;D(A g )) ≤ L h kF k 2 L ∞

(0,T per ;D(A g )) (3.13)Theorem 3.4 Let hypothesis (F1) hold Then there exists ω 0 ′ > 0

2c 1 1 −

|∇g| ∞

m 0 η 1 1/2

! , ∀t ∈ [0, T per ], (3.14)

where c 1 , c 3 are the onstants in Lemma 1.2

Theorem 3.5 Let hypothesis (F1) hold and let u0 ∈ Vg b given.Then any solution u(·) to system (3.11) with initial datum u0

THE STABILITY OF SOLUTIONS TO STOCHASTIC

2D g-NAVIER-STOKES EQUATIONS WITH FINITE

DELAYS

Inthis hapter, we the sto 2D g-Navier-Stokesequations with finite delays First, we study the of weak

stationary solutions to the system onding to

thesto system byusingthe method,andwhen

the y is large enough, we show that this weak stationary

solution is unique we study the exponential stability in

mean square and almost sure exponential stability of the weak

so-lutions to the sto 2D g-Navier-Stokes equations with finitedelays

This hapter is written based on the paper 2

4.1 SETTING OF THE PROBLEM

Let O be a bounded domain in R 2 with smooth boundary ∂O

We the following sto 2D g-Navier-Stokesequationswith finite delays:

where u = u(x, t) = (u 1 , u 2 ) is the unknown velo y v

p = p(x, t) is the unknown pressure, ν > 0 is the

y t, f = f (x) is a time-independent externalfield without delay, F (·) is the external field withdelay, G(u(t − ρ(t)))dW (t) is the random field with de-lay, W (t) is an infinite-dimensional Wiener pro the