Trajectory attractor for a non-autonomous Magnetohydrodynamic equations of Non-Newtonian Fluids

TL;DR

This paper studies the complex dynamics of a nonlinear PDE system modeling magnetohydrodynamics of non-Newtonian fluids, proving existence of solutions and a trajectory attractor, which helps understand long-term behavior.

Contribution

It establishes the existence of weak solutions and a trajectory attractor for a non-autonomous MHD system involving non-Newtonian fluids, advancing mathematical understanding of such models.

Findings

Existence of weak solutions to the PDE system.

Existence of a trajectory attractor for the system.

Structural properties of the attractor analyzed.

Abstract

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.