Chương 1 MỘT SỐ KIẾN THỨC CHUẨN BỊ
2. KIẾN NGHỊ MỘT SỐ VẤN ĐỀ NGHIÊN CỨU TIẾP THE O
Bên cạnh các kết quả đã đạt được trong luận án, một số vấn đề mở cần tiếp tục nghiên cứu như:
• Tiếp tục nghiên cứu tính chất của tập hút ngẫu nhiên của lớp phương trình parabolic suy biến nhận được trong luận án, chẳng hạn đánh giá số chiều Hausdorff và số chiều fractal, nghiên cứu cấu trúc của tập hút ngẫu nhiên, sự phụ thuộc của tập hút vào tham số trong nhiễu ngẫu nhiên.
• Nghiên cứu sự tồn tại và tính ổn định của nghiệm dừng khơng hằng (tức là nghiệm dừng theo nghĩa ngẫu nhiên) của hệ Navier-Stokes-Voigt và hệ Kelvin-Voigt-Brinkmann-Forchheimer ngẫu nhiên.
• Nghiên cứu sự hội tụ của nghiệm đối với hệ Navier-Stokes-Voigt ngẫu nhiên khi tham số α dần đến 0, tức là so sánh nghiệm của hệ Navier-
Stokes-Voigt ngẫu nhiên với nghiệm tương ứng của hệ Navier-Stokes ngẫu nhiên.
DANH MỤC CÁC CƠNG TRÌNH KHOA HỌC CỦA TÁC GIẢ LIÊN QUAN ĐẾN LUẬN ÁN
1. C.T. Anh, T.Q. Bao and N.V. Thanh (2012), Regularity of random at- tractors for stochastic semilinear degenerate parabolic equations, Elect. J. Differential Equations, No. 207, 22 pp.
2. C.T. Anh and N.V. Thanh (2016), Asymptotic behavior of the stochastic Kelvin-Voigt-Brinkman-Forchheimer equations, Stoch. Anal. Appl., 34
no. 3, 441-455
3. C.T. Anh and N.V. Thanh (2016), Stabilization of a class of semilinear degenerate parabolic equations by Ito noise, Random Oper. Stoch. Equ.,
24, no. 3, 147-155.
4. C.T. Anh and N.V. Thanh (2018), On the existence and long-time be- havior of solutions to stochastic three-dimensional Navier-Stokes-Voigt equations, Stochastics, 91 (4), 485-513.
5. N.V. Thanh (2018), Internal stabilization of stochastic 3D Navier-Stokes- Voigt equations with linearly multiplicative Gaussian noise, Random Oper. Stoch. Equ., accepted.
Tài liệu tham khảo
[1] C.T. Anh and T.Q. Bao (2010), Pullback attractors for a non-autonomous semi-linear degenerate parabolic equation, Glasgow Math. J. 52, 537-554. [2] C.T. Anh, T.Q. Bao and L.T. Thuy (2013), Regularity and fractal dimen- sion of pullback attractors for a non-autonomous semilinear degenerate parabolic equations, Glasgow Math. J. 55, 431-448.
[3] C.T. Anh, N.D. Binh and L.T. Thuy (2010), On the global attractors for a class of semilinear degenerate parabolic equations, Ann. Pol. Math. 98, 71-89.
[4] C.T. Anh and P.Q. Hung (2008), Global existence and long-time behavior of solutions to a class of degenerate parabolic equations,Ann. Pol. Math.
93, 217-230.
[5] C.T. Anh and D.T.P. Thanh (2018), Existence and long-time behavior of solutions to Navier-Stokes-Voigt equations with infinite delay, Bull. Korean Math. Soc. 55, 379-403.
[6] C.T. Anh, N.V. Thanh and N.V.Tuan (2017), On the stability of solutions to stochastic 2D g-Navier-Stokes equations with finite delays, Random Oper. Stoch. Equ. 25, 211-224.
[7] C.T. Anh and P.T. Trang (2013), On the 3D Kelvin-Voigt-Brinkman- Forchheimer equations in some unbounded domains, Nonlinear Anal. 89, 36-54.
[8] C.T. Anh and P.T. Trang (2013), Pull-back attractors for three- dimensional Navier-Stokes-Voigt equations in some unbounded domains,
Proc. Roy. Soc. Edinburgh Sect. A 143, 223-251.
[9] C.T. Anh and N.V. Tuan (2018), Stabilization of 3D Navier-Stokes-Voigt equations, Georgian Math. J., DOI: 10.1515/gmj-2018-0067.
[10] L. Arnold, H. Crauel and V. Wihstutz (1983), Stabilization of linear sys- tems by noise, SIAM J. Control Optim. 21, 451-461.
[11] L. Arnold (1998), Random Dynamical Systems, Springer-Verlag, Berlin.
[12] V. Barbu (2013), Note on the internal stabilization of stochastic parabolic equations with linearly multiplicative Gaussian noise,ESAIM Control Op- tim. Calc. Var. 19, 1055-1063.
[13] A. Bensoussan (1995), Stochastic Navier-Stokes equations, Acta Appl. Math. 38, 267-304.
[14] V. Barbu and C. Lefter (2003), Internal stabilizability of the Navier-Stokes equations, Systems Control Lett. 48, 161-167.
[15] P.W. Bates, H. Lisei and K. Lu (2006), Attractors for stochastic lattice dynamical system, Stoch. Dyn. 6, 1-21.
[16] P.W. Bates, K. Lu and B. Wang (2009), Random attractors for stochastic reaction-diffusion equations on unbounded domains,J. Differential Equa- tions 246, 845-869.
[17] T.Q. Bao (2014), Dynamics of stochastic three dimensional Navier-Stokes- Voigt equations on unbounded domains, J. Math. Anal. Appl. 419, 583- 605.
[18] N.D. Binh, N.N. Thang and L.T. Thuy (2016), Pullback attractors for a non-autonomous semilinear degenerate parabolic equation on RN, Acta Math. Vietnam. 41, 183-199.
[19] H.I. Breckner (1999), Approximation and Optimal Control of the Stochas- tic Navier-Stokes Equation, PhD dissertation, Martin-Luther University,
Halle-Wittenberg.
[20] Z. Brzezniak and E. Motyl (2013), Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D domains,
J. Differential Equations 254, 1627-1685.
[21] P. Caldiroli and R. Musina (2000), On a variational degenerate elliptic problem, Nonlinear Diff. Equ. Appl. 7, 187-199.
[22] Y. Cao, E.M. Lunasin and E.S. Titi (2006), Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence mod- els, Commun. Math. Sci. 4, 823-848.
[23] T. Caraballo, J.A. Langa and J.C. Robinson (2000), Stability and ran- dom attractors for a reaction-diffusion equation with multiplicative noise,
Discrete Contin. Dynam. Syst. 6, 875-892.
[24] T. Caraballo and P.E. Kloeden (2010), Stabilization of evolution equa- tions by noise, Recent development in stochastic dynamics and stochastic analysis, 43-66, Interdiscip. Math. Sci., 8, World Sci. Publ., Hackensack,
NJ.
[25] T. Caraballo, K. Liu and X. Mao (2001), On stabilization of partial dif- ferential equations by noise, Nagoya Math. J. 161, 155-170.
[26] T. Caraballo, J.A. Langa and T. Taniguchi (2002), The exponential be- haviour and stabilizability of stochastic 2D Navier-Stokes equations, J. Differential Equations 179, 714-737.
[27] T. Caraballo, A.M. Márquez-Durán and J. Real (2006), The asymptotic behaviour of a stochastic 3D LANS-α model,Appl. Math. Optim.53, 141- 161.
[28] T. Caraballo, J. Real and T. Taniguchi (2006), On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian aver- aged Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462, 459-479.
[29] H. Crauel, A. Debussche and F. Flandoli (1997), Random attractors, J. Dynam. Differential Equations 9, 307-341.
[30] H. Crauel and F. Flandoli (1994), Attractors for random dynamical sytems, Probab. Theory Related Fields 100, 365-393.
[31] M. Coti Zelati and C.G. Gal (2015), Singular limits of Voigt models in fluid dynamics, J. Math. Fluid Mech. 17, 233-259.
[32] J. Duan and W. Wang (2014), Effective Dynamics of Stochastic Partial Differential Equations, Elsevier.
[33] R. Dautray and J.L. Lions (1985), Mathematical Analysis and Numerical Methods for Science and Technology, Vol. I: Physical origins and classical
methods, Springer-Verlag, Berlin.
[34] G. Da Prato and J. Zabczyk (2014), Stochastic Equations in Infinite Di- mensions, second edition, Cambridge University Press, Cambridge.
[35] L. Gawarecki and V. Mandrekar (2011),Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations, Springer, Heidelberg.
[36] J. García-Luengo, P. Marín-Rubio and J. Real (2012), Pullback attrac- tors for three-dimensional non-autonomous Navier-Stokes-Voigt equa- tions, Nonlinearity 25, 905-930.
[37] H. Gao and C. Sun (2012), Random dynamics of the 3D stochastic Navier- Stokes-Voight equations,Nonlinear Anal. Real World Appl.13, 1197-1205. [38] R.Z. Has’minskii (1980), Stochastic Stability of Differential Equations, Si-
jthoff and Noordhoff, Netherlands.
[39] M. Holst, E. Lunasin and G. Tsogtgerel (2010), Analysis of a general family of regularized Navier-Stokes and MHD models, J. Nonlinear Sci.
20, 523-567.
[40] N.I. Karachalios and N.B. Zographopoulos (2005), Convergence towards attractors for a degenerate Ginzburg-Landau equation, Z. Angew. Math. Phys. 56, 11-30.
[41] N.I. Karachalios and N.B. Zographopoulos (2006), On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence, Calc. Var. Partial Differential Equations 25, 361-393.
[42] V.K. Kalantarov (1986), Attractors for some nonlinear problems of math- ematical physics,Zap. Nauchn. Sem. Lenigrad. Otdel. Math. Inst. Steklov. (LOMI) 152, 50-54.
[43] V.K. Kalantarov, B. Levant and E.S. Titi, Gevrey regularity for the at- tractor of the 3D Navier-Stoke-Voight equations, J. Nonlinear Sci. 19 (2009), 133-152.
[44] V.K. Kalantarov and E.S. Titi (2009), Global attractor and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser. B 30, 697-714.
[45] V.K. Kalantarov and S. Zelik (2012), Smooth attractors for the Brinkman- Forchheimer equations with fast growing nonlinearities, Comm. Pure Appl. Anal. 11, 2037-2054.
[46] I. Karatzas and S.E. Shreve (1991),Brownian Motion and Stochastic Cal- culus, 2nd edition, Springer-Verlag, New York.
[47] J.A. Langa, J. Real and J. Simon (2003), Existence and regularity of the pressure for the stochastic Navier-Stokes equations, Appl. Math. Optim.
48, 195-210.
[48] D. Li and C. Sun (2016), Attractors for a class of semi-linear degenerate parabolic equations with critical exponent, J. Evol. Equ. 16, 997-1015. [49] X. Li, C. Sun and N. Zhang (2016), Dynamics for a non-autonomous
degenerate parabolic equation in D1
0(Ω, σ), Discrete Contin. Dyn. Syst.
36, 7063-7079.
[50] X. Li, C. Sun and F. Zhou (2016), Pullback attractors for a non- autonomous semilinear degenerate parabolic equation, Topol. Methods Nonlinear Anal. 47, 511-528.
[51] J.L. Lions (1969), Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris.
[52] K. Liu (1997), On stability for a class of semilinear stochastic evolution equations, Stochastic Process. Appl. 70, 219-241.
[53] K. Liu (2006), Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 135. Chapman & Hall/CRC, Boca Raton, FL.
[54] Y. Li and B. Guo (2008), Random attractors for quasi-continuous ran- dom dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations 245, 1775-1800.
[55] J. Li, Y. Li and B. Wang (2010), Random attractors of reaction-diffusion equations with multiplicative noise in Lp, App. Math. Comp. 215, 3399- 3407.
[56] R.S. Liptser and A.N. Shiryayev (1989), Theory of Martingales, Kluwer
Academic Publishers Group, Dordrecht.
[57] Q.F. Ma, S.H. Wang and C.K. Zhong (2002), Necessary and sufficient conditions for the existence of global attractor for semigroups and appli- cations, Indiana University Math. J. 51, 1541-1559.
[58] T. Medjo (2011), The exponential behavior of the stochastic three- dimensional primitive equations with multiplicative noise,Nonlinear Anal. Real World Appl. 12, 799-810.
[59] X. Mao (1997), Stochastic Differential Equations and Applications, Hor-
wood, Chichester.
[60] E. Pardoux (1975), Équations aux Dérivées Partielles Stochastiques Non Linéaires Monotones, Thèse, Université Paris XI.
[61] M. Scheutzow (1993), Stabilization and destabilization by noise in the plane, Stoch. Anal. Appl. 11, 97-113.
[62] A.P. Oskolkov (1973), The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers, Zap. Nauchn. Sem. Leningrad. Otdel. Math. Inst. Steklov. (LOMI) 38, 98-136.
[63] P.A. Razafimandimby and M. Sango (2012), On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids,Appl. Anal.91, 2217-2233.
[64] P.A. Razafimandimby and M. Sango (2012), Strong solution for a stochas- tic model of two-dimensional second grade fluids: existence, uniqueness and asymptotic behavior, Nonlinear Anal. 75, 4251-4270.
[65] P.A. Razafimandimby and M. Sango (2015), Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equa- tions, Z. Angew. Math. Phys. 66, 2197-2235.
[66] B. Schmalfuss (1997), Qualitative properties for the stochastic Navier- Stokes equation, Nonlinear Anal. 28, 1545-1563.
[67] C. Sun and H. Gao (2010), Hausdorff dimension of random attractor for stochastic Navier-Stokes-Voight equations and primitive equations, Dyn. Partial Differ. Equ. 7, 307-326.
[68] R. Temam (1979), Navier-Stokes Equations: Theory and Numerical Anal- ysis, 2nd edition, Amsterdam: North-Holland.
[69] R. Temam (1995), Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, Philadelphia.
[70] Y. Wang and C.K. Zhong (2008), On the existence of pullback attractors for non-autonomous reaction diffusion, Dyn. Syst. 23, 1-16.
[71] Z. Wang and S. Zhou (2011), Random attractors for stochastic reaction- diffusion equation with multiplicative noise on unbounded domains, J. Math. Anal. Appl. 384, 160-172.
[72] M. Yang and P.E. Kloeden (2011), Random attractors for stochastic semi- linear degenerate parabolic equations, Nonlinear Anal. Real World Appl.
12, 2811-2821.
[73] J. Yin, Y. Li and H. Zhao (2013), Random attractors for stochastic semi- linear degenerate parabolic equations with additive noise in Lq, Appl. Math. Comput. 225, 526-540.
[74] G. Yue and C.K. Zhong (2011), Attractors for autonomous and nonau- tonomous 3D Navier-Stokes-Voight equations,Discrete. Cont. Dyna. Syst. Ser. B 16, 985-1002.
[75] W. Zhao and Y. Li (2012), (L2, Lp)-random attractors for stochastic
reaction-diffusion equation on unbounded domains, Nonlinear Anal. 75, 485-502.
[76] C.K. Zhong, M.H. Yang and C.Y. Sun (2006), The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations 223, 367-399.
[77] W. Zhao (2014), Regularity of random attractors for a degenerate parabolic equations driven by additive noises, Appl. Math. Comput. 239, 358-374.
[78] W. Zhao (2016), Regularity of random attractors for a stochastic degen- erate parabolic equations driven by multiplicative noise, Acta Math. Sci. Ser. B (Engl. Ed.) 36, 409-427.
[79] W. Zhao (2018), Random dynamics of non-autonomous semi-linear degen- erate parabolic equations on RN driven by an unbounded additive noise,
Discrete Contin. Dyn. Syst. Ser. B 23, 2499-2526.
[80] W. Zhao and Y. Li (2014), Random attractors for stochastic semi-linear degenerate parabolic equations with additive noises, Dyn. Partial Differ. Equ. 11, 269-298.