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.
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Summary of doctoral thesis in mathematics: Stability and stabilization for some evolution equations in fluid mechanics MINISTRY OF EDUCATION AND TRAINING HA NOI PEDAGOGICAL UNIVERSITY 2  NGUYEN VIET TUAN STABILITY AND STABILIZATION FOR SOME EVOLUTION EQUATIONS IN FLUID MECHANICS Spe iality: Mathemati al analysis Code: 9 46 01 02 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Ha Noi - 2019 This thesis has been 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 be defended at the University level Thesis Assessment Coun il at Ha Noi Pedagogi al University 2 on....... The thesis an be 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 appear 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 models in biology. The study of this equations lass has important 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- portant to study the long-time behavior 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 response orresponds to the stationary state of the pro ess, and is the solution of the orresponding ellipti problem. When the stationary solutions of the pro esses is not stability, people 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 orresponding re- sults for other lasses of equations in fluid me hani s and paraboli systems are still small. There are new mathemati al diffi ulties, be ause of the omplexity of the system or the intera tion be- 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 smooth bounded domains with homogeneous 1 Diri hlet boundary onditions:   ut − ν∆u − α2 ∆ut + (u · ∇)u + ∇p = f in O × R+ ,  O × R+ ,  ∇ · u = 0 in (1)   u(x, t) = 0 on ∂O × R+ ,  u(x, 0) = u0 (x) in O.  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 bounded or unbounded 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 exponential stability and stabilization of strong stationary solutions to prob- lem (1). Next, we onsider the following 2D g-Navier-Stokes equations ∂u    − ν∆u + (u · ∇)u = ∇p + f in Ω × R+ ,    ∂t ∇ · (gu) = 0 Ω × R+ ,  in (2) +     u(x, t) = 0 on ∂Ω × R ,  u(x, 0) = u0 (x), in Ω.  In the past de ade, the existen e and long-time behavior of solutions in terms of existen e of attra tors for 2D g-Navier-Stokes equations have been studied extensively in both 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 be investigated regarding the system (2), su h as: 1) Existen e, uniqueness and exponential stability of strong stationary solutions. 2) Stabilization of strong stationary solutions. 3) Stabilization of long-time behavior 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, (3)  u(x, t) = 0, x ∈ ∂O, t > 0,   ...