Structural stability for the Benard problem with Voight regularization

Abstract

In this paper we consider the Bénard problem involving Voight regularizing terms. The continuous dependence of solutions of the given problem on the coefficients of the Voight regularizing terms is established.

Keywords

• Bénard Problem,structural stability

References

1. [1] Adams R.A.,Sobolev Spaces New York (NY): Academic Press; (1975).
2. [2] Ames K.A., Straughan B.,Non-Standart and Improperly Posed problems, in: Mathematics in Science and Engineering Series, Vol.194, Academic Press, San Diego, (1997).
3. [3] Birnir, B and Svanstedt, N., Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio, Discrete and Continuous Dynamical Systems,Vol.10 no1&2, 53-74, (2004).
4. [4] Cabral, M., Rosa, R. and Temam, R., Existence and dimension of the attractor for Bénard problem on channel-like domains, Discrete and Continuous Dynamical Systems, vol.10,N.1&2, 89-116 (2004).
5. [5] Constantin, P. and Foias, C., Navier-Stokes equations, Chicago Lectures in Mathematics, The University of Chicago, (1988).
6. [6] Çelebi A. O.,Kalantarov V.K., and Uğurlu D., Continuous dependence for the convective Brinkman - Forchheimer equations. Applicable Analysis. 84:9, 877-888. (2005).
7. [7] Çelebi A. O.,Kalantarov V.K Structural stability for the double diffusive convective Brinkman equations. Appl. Anal. 87, 933-942 (2008).
8. [ 8] Çelebi A. O., Gür Ş., Kalantarov V.K., Structural stability and decay estimate for Marine Riser Equations, Mathematical and Computer Modelling, 54 3182-3188, (2011).

9. [9] Foias, C., Manley, O. and Temam, R., Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Analysis Theory, Methods and Applications, Vol.11.No.8, 939-967 (1987).
10. [10] Kalantorov, V.K., Attractors for some nonlinear problems of physics, Zap. Nauchn. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152, 50-54, (1986).
11. [11] Kalantarov, V.K., Levant, B. and Titi, E. S., Gevrey regularity for the attractor of 3D Navier-Stokes-Voight equations, Journal of Nonlinear Science,.19, 133-152, (2009).
12. [12] Kalantarov, V.K. and Titi, E.S., Global attractors and determining modes for the 3D Navier-Stokes-Voight equations, Chin. Ann. Math. Ser.B 30, no 6, 697-714 (2009) .
13. [13] Kapusyan, O.V., Melnik, V.S., and Valero, J., A Weak Attractor and Properties of solutions for the three-dimensional Bénard Problem, Discrete and Continuous Dynamical Systems, Vol.18, N 2&3, 449-481, (2007).
14. [14] Kaya M., Çelebi A.O., On the Bénard problem with Voight regularization, GUJSci,28(3):523-533 (2015). [15] Ladyzhenskaya, O.A., The Mathematical Theory of viscous Incompressible Flow, Gordon and Breach, (1969).
15. [16] Morimoto, H., Non-stationary Boussinesq equations, J. Fac. Sci. Univ. Tokyo, Sect. IA. Math. 39, 61-75 (1992).
16. [17] Oskolkov, A.P., The uniqueness and Global Solvability of boundary-value problems for the equations of motion for aqueous solutions of polymers, Journal Soviet Mathematics, 8, No 4, 427-455, (1977).
17. [18] Özlük M., Kaya M., On the strong solutions and structural stability of the g-Bénard problem, Numerical Functional Analysis and Optimization, 39, NO. 4, 383–397 (2018)
18. [19] Robinson, J.C., Infinite-dimensional Dynamical Systems, Cambridge University Press "Text in Applied Mathematics" Series, (2001).
19. [20] Temam, R., Navier-Stokes equations: Theory and numerical analysis, North-Holland-Amsterdam, (1984).
20. [21] Temam, R., Infinite Dimensional Dynamical System in Mechanic and Physics, Springer Verlag, (1997).