Trajectory attractor for a non-autonomous magnetohydrodynamic equation of non-Newtonian fluids

In this article we initiate the mathematical study of the dynamics of a system of nonlinear partial differential equations modelling the motion of incompressible, isothermal and conducting modified bipolar fluids in presence of magnetic field. We mainly prove the existence of weak solutions to the model. We also prove the existence of a trajectory attractor to the translation semigroup acting on the trajectories of the set of weak solutions and that of external forces. Some results concerning the structure of this trajectory attractor are also given. The results from this paper may be useful in the investigation of some system of PDEs arising from the coupling of incompressible fluids of p-structure and the Maxwell equations.

Keywords

non-Newtonian fluids, bipolar fluids, shear thinning fluids, shear thickening fluids, MHD, magnetohydrodynamics, weak solution, asymptotic behavior, long-time behavior, trajectory attractor

2010 Mathematics Subject Classification

35B40, 35B41, 35D30, 35K55, 76W05

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Published 14 September 2012