On the Benard Problem with Voight Regularization

Year 2015, Volume: 28 Issue: 3, 523 - 533, 22.05.2015

Meryem Kaya Ahmet Çelebi

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

• 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).
• Cabral, M., Rosa, R. and Temam, R., Existence and dimension of the attractor for Benard problem on channel-like
• Dynamical Systems, vol.10,N.1&2, 89-116 (2004). and
• Continuous [3] Constantin, P. and Foias, C., Navier-Stokes equations, Chicago Lectures in Mathematics, The University of Chicago, (1988).
• 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).
• Galdi, G. P., Lectures in Mathematical Fluid Dynamics, Birkhäuser-Verlag, (2000).
• Hale, J.K., Asymptotic Behaviour of dissipative Systems, Mathematical Survey and Monographs, 25 American Mathematical Society, Providence, RI, (1988).
• Kalantorov, V.K., Attractors for some nonlinear problems of physics, Zap. Nauchn. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152, 50-54, (1986).
• 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).
• 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) .
• 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).
• 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).
• Ladyzhenskaya, O.A., The Mathematical Theory of viscous Incompressible Flow, Gordon and Breach, (1969).
• 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).
• 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).
• Morimoto, H., Non-stationary Boussinesq equations, J. Fac. Sci. Univ. Tokyo, Sect. IA. Math. 39, 61-75 (1992).
• 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).
• 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).