luận án tiến sĩ finite dimensional asymptotic behavior of navier stokes voigt equations

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI NGAN FINITE-DIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIER-STOKES-VOIGT EQUATIONS DOCTORAL DISSERTATION OF MATHEMATICS Hanoi - 2021 MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION NGUYEN THI NGAN FINITE-DIMENSIONAL ASYMPTOTIC BEHAVIOR OF NAVIER-STOKES-VOIGT EQUATIONS Speciality: Differential and Integral Equations Speciality Code: 9.46.01.03 DOCTORAL DISSERTATION OF MATHEMATICS Supervisor: PROF DR CUNG THE ANH Hanoi - 2021 COMMITTAL IN THE DISSERTATION I assure that the scientific results presented in this dissertation are new and original To my knowledge, before I published these results, there had been no such results in any scientific document I take responsibility for my research results in the dissertation The publications in common with other authors have been agreed by the co-authors when put into the dissertation February, 2021 Author Nguyen Thi Ngan i ACKNOWLEDGEMENTS This dissertation was carried out at the Department of Mathematics and Informatics, Hanoi National University of Education It was completed under the supervision of Prof Cung The Anh First and foremost, I would like to express my sincere gratitude to my supervisor, Prof Cung The Anh, for the continuous support of my PhD study, for his carefulness, patience, enthusiasm and immense knowledge His guidance helped me in all the time of research to learn and grow a lot, both professionally and personally Sometimes he set me back on the road when I got lost I would like to say that I am proud to be his student Besides my supervisor, I am greatly grateful to Assoc Prof Tran Dinh Ke for his encouragement during the time I have studied at Department of Mathematics and Informatics, Hanoi National University of Education I am deeply indebted to Dr Vu Manh Toi for his help and many interesting discussions during my first one year I thank all the lecturers and PhD students at the seminar of Division of Mathematical Analysis for their stimulating consultation and valuable comments I would like to thank all my colleagues at University of Education Publisher and Foreign Language Specialized School, VNU, for supporting me to study during the last three years I also thank my friends, who always encourage me to overcome difficulties during my period of study Last but not least, I am greatly thankful to my beloved family for respecting all my decisions and supporting me spiritually throughout my life Hanoi, 2021 Nguyen Thi Ngan ii CONTENTS COMMITTALINTHEDISSERTATION ACKNOWLEDGEMENTS CONTENTS LISTOFSYMBOLS INTRODUCTION Chapter PRELIMINARIES AND AUXILIARY RESULTS 1.1 Function spaces 1.1.1 1.1.2 1.1.3 1.1.4 1.2 The global attractor 1.2.1 1.2.2 1.3 Determining functionals 1.3.1 1.3.2 1.3.3 1.3.4 1.4 The Navier-Stokes-Voigt equations with p 1.5 The Gronwall inequalities Chapter BOUNDS ON THE NUMBER OF DETERMINING NODES FOR 3D NAVIER-STOKES-VOIGT EQUATIONS 2.1 Problem setting 2.2 Preliminaries 2.3 Determining nodes for instationary solutio 2.4 Determining nodes for stationary solution 2.5 Determining nodes for periodic solutions Chapter FEEDBACK CONTROL OF NAVIER-STOKES-VOIGT EQUATIONS iii BY FINITE DETERMINING PARAMETERS 3.1 Problem setting 3.2 Preliminaries 3.3 Stabilization of Navier-Stokes-Voigt equa operator as feedback controllers 3.3.1 Feedback control employing finite v onto Fourier modes as an interpol 3.3.2 Feedback control employing finitely m Chapter ASYMPTOTIC BEHAVIOR OF THREE-DIMENSIONAL NON-HOMOGENEOUS NAVIER-STOKES-VOIGT EQUATIONS 4.1 Problem setting 4.2 Preliminaries 4.3 Existence and uniqueness of weak solutio 4.4 Existence of a global attractor 4.4.1 Existence of an absorbing set 4.4.2 The asymptotic compactness 4.5 Fractal dimension estimate of the global a 4.6 Existence and exponential stability of a st 4.7 Determining projections and functionals fo CONCLUSIONSANDFUTUREWORKS LISTOFPUBLICATIONS REFERENCES iv LIST OF SYMBOLS R Rd A:=B ¯ A (., )X ∥x∥X ∗ X ′ x , x X∗,X X,→Y p L (Ω) L (Ω) ∞ ∞ C0 (Ω) ¯ C(Ω) m,p H (Ω), m Wm (Ω), H0 (Ω) m H − L (Ω) (., ) ((., )) ((., ))1 |.| ∥.∥ ∥.∥1 x·y (Ω) ∇ ∇y y·∇ ∇ · y, div y V H, V p L (0, T ; X), < p < ∞ the space of functions f : [0, T ] → X such ∞ L 1,p W (0, T ; X) C([0, T]; X) {xk} xk → x xk ⇀ x ∗ xk ⇀ x i.e a.e p 2D 3D (0,T;X) where β = 87 Applying Lemma 1.5.2, we obtain 2 2 2 lim (|z(t)| + α ∥z(t)∥ ) = lim (|ue(t) − ve(t)| + α ∥ue(t) − ve(t)∥ ) = t→∞ t→∞ Since u = ue + ψ and v = ve + ψ, we have 2 lim (|u(t) − v(t)| + α ∥u(t) − v(t)∥ ) = t→∞ This completes the proof Remark 4.7.3 It is worthy noticing that when ψ ≡ 0, from Theorem 4.7.2 we can get the bounds on some determining functionals such as determining modes, nodes and finite volumes for 3D NavierStokes-Voigt equations with homogeneous Dirichlet boundary conditions, which are similar to those in the case of periodic boundary conditions obtained recently in [29, 36] The existence of global solutions and of a finite-dimensional global attractor to three-dimensional Navier-Stokes-Voigt equations with non-homogeneous boundary conditions in this chapter can be seen as an extension of previous results in the case of homogeneous boundary con-ditions (i.e φ ≡ 0) obtained in [17, 36] Conclusion of Chapter In this chapter, we have studied the existence and long-time behavior of solutions to the three-dimensional Navier-Stokes-Voigt equations with non-homogeneous Dirichlet boundary conditions We have achieved the following results: 1) The existence and uniqueness of global weak solutions (Theorem 4.3.3); 2) The existence of a global attractor for the semigroup generated by weak solutions (see Remark 4.1.1 and Theorem 4.4.4); 3) The finiteness of the fractal dimension of the global attractor (see Remark 4.1.1 and Theorem 4.7.2); 4) A sufficient condition for exponential stability of weak stationary so-lutions (Theorem 4.6.2); 88 5) The existence of determining projections for weak solutions (Theorem 4.7.2) These are the first results about long-time behavior of solutions to the three-dimensional Navier-Stokes-Voigt equations with nonhomogeneous Dirichlet boundary conditions 89 CONCLUSIONS AND FUTURE WORKS Conclusion In this dissertation, we have been investigated the long-time behavior of solutions to three-dimensional Navier-Stokes-Voigt equations by some approaches: the global attractor, determining nodes and stabilization of stationary solutions by finite-dimensional feedback controls The main contributions of the dissertation are to the following: 1) Give bounds on the number of determining nodes for solutions to the three-dimensional Navier-Stokes-Voigt equations with periodic boundary conditions 2) Prove the stabilization of stationary solutions to the NavierStokes-Voigt equations with periodic boundary conditions by finitedimensional feedback controls 3) Prove the existence and long-time behavior of solutions to the three-dimensional Navier-Stokes-Voigt equations with nonhomogeneous Dirichlet boundary conditions in terms of the existence of a finite-dimensional global attractor and the existence of determining projec-tions The results obtained in the dissertation are meaningful contributions to the Navier-Stokes-Voigt equations as well as the theory of infinite-dimensional dissipative dynamical systems Future works Some suggestions for potential future works are proposed below: 1) Existence of approximate internal manifold for the threedimensional Navier-Stokes-Voigt equations (see the survey article [56] for related results on Navier-Stokes equations) 2) Data assimilation for three-dimensional Navier-Stokes-Voigt equations (see [9, 19] for results on Navier-Stokes equations) 90 LIST OF PUBLICATIONS Published papers [CT1 ] V.M Toi and N.T Ngan (2020), Upper bounds on the number of determining nodes for 3D Navier-Stokes-Voigt equations, Ann Pol Math 125, no 1, 83-99 [CT2 ] N.T Ngan and V.M Toi (2020), Feedback control of NavierStokes-Voigt equations by finite determining parameters Acta Math Viet-nam 45 (2020), no 4, 917-930 Submitted papers [CT3 ] C.T Anh and N.T Ngan (2020), Asymptotic behavior of threedimensional non-homogeneous Navier-Stokes-Voigt equations, sub-mitted to Acta Applicandae Mathematicae 91 REFERENCES [1] R.A Adams (1975), Sobolev Spaces, Academic Press, New York [2] D.A.F Albanez, H.J Nussenzveig-Lopes and E.S Titi (2016), Con-tinuous data assimilation for the three-dimensional Navier-Stokes-α model, Asymptot Anal 97, 139-164 [3] C.T Anh, N.T.M Toai and V.M Toi (2020), Upper bounds on the number of determining modes, nodes, and volume elements for a 3D magenetohydrodynamic-α model, J Appl Anal Comput 10, 624-648 [4] C.T Anh and V.M Toi (2017), Stabilizing the long-time behavior of the Navier-Stokes-Voigt equations by fast oscillating-in-time forces, Bull Polish Acad Sci Math 65, 177185 [5] C.T Anh and P.T Trang (2013), Pull-back attractors for three-dimensional Navier-Stokes-Voigt equations in some unbounded do-mains, Proc Roy Soc 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mathematicians The existence and long-time behavior of solutions in terms of existence of attractors to the Navier- Stokes- Voigt. .. the three -dimensional NavierStokes -Voigt equations with periodic boundary conditions (P2) The stabilization of stationary solutions of the Navier- StokesVoigt equations in both cases of dimension