On the Benard Problem with Voight Regularization

Abstract

In this paper we consider the Benard problem involving some regularizing terms. Using maximum principle which is given by Foias, Manley and Temam in [4] we prove the existence-uniqueness of weak solution and the global attractor has a Önite fractal dimension.

Keywords

• Benard Problem
• weak solution
• global attractor

References

1. Birnir, B and Svanstedt, N., Existence theory and strong attractors for the Rayleigh-Benard problem with a large aspect ratio, Discrete and Continuous Dynamical Systems,Vol.10 no1&2, 53-74, (2004).
2. Cabral, M., Rosa, R. and Temam, R., Existence and dimension of the attractor for Benard problem on channel-like
3. Dynamical Systems, vol.10,N.1&2, 89-116 (2004). and
4. Continuous [3] Constantin, P. and Foias, C., Navier-Stokes equations, Chicago Lectures in Mathematics, The University of Chicago, (1988).
5. Foias, C., Manley, O. and Temam, R., Attractors for the Benard problem: existence and physical bounds on their fractal dimension, Nonlinear Analysis Theory, Methods and Applications, Vol.11.No.8, 939-967 (1987).
6. Galdi, G. P., Lectures in Mathematical Fluid Dynamics, Birkhäuser-Verlag, (2000).
7. Hale, J.K., Asymptotic Behaviour of dissipative Systems, Mathematical Survey and Monographs, 25 American Mathematical Society, Providence, RI, (1988).
8. Kalantorov, V.K., Attractors for some nonlinear problems of physics, Zap. Nauchn. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152, 50-54, (1986).

9. 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).
10. 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) .
11. Kapusyan, O.V., Melnik, V.S., and Valero, J., A Weak Attractor and Properties of solutions for the three-dimensional Benard Problem, Discrete and Continuous Dynamical Systems, Vol.18, N 2&3, 449- 481, (2007).
12. Khouider, B. and Titi, E.S. An Inviscid regularization for the surface quasi geostrophic equation, Comm. Pure Appl. Math., 61 No 10, 1331- 1346, (2008).
13. Ladyzhenskaya, O.A., The Mathematical Theory of viscous Incompressible Flow, Gordon and Breach, (1969).
14. Ladyhenskaya, O.A., On the determination of minimal global attractors for the Navier Stokes and other partial differential equations, Uspekhi Math. Nauk, 42:6, 25-60, (1987), Russian Math. Survey, 42:6, 27-73 (1987).
15. Ladyhenskaya, O.A., Finite dimensionality of bounded invariant sets for Navier-Stokes systems and other dissipative systems, Zap. Nauchn. Sem. LOMI 115, 137-155, (1982), J.Soviet Math., 28, 714-726, (1985).
16. Morimoto, H., Non-stationary Boussinesq equations, J. Fac. Sci. Univ. Tokyo, Sect. IA. Math. 39, 61-75 (1992).
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] Robinson, J.C., Infinite-dimensional Dynamical Systems, Cambridge University Press "Text in Applied Mathematics" Series, (2001).
18. Temam, R., Navier-Stokes equations: Theory and numerical analysis, North-Holland-Amsterdam, (1984). [19] Temam, R., Infinite Dimensional Dynamical System in Mechanic and Physics, Springer Verlag, (1997).