Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids

TL;DR

This paper investigates the long-term behavior of nonlinear PDEs modeling bipolar fluid flow under magnetic fields, proving the existence of a finite-dimensional global attractor with explicit bounds.

Contribution

It establishes the existence, finite-dimensionality, and differentiability of the global attractor for the magnetohydrodynamic equations of bipolar fluids, with explicit dimension bounds.

Findings

Existence of a global attractor for the system.

The attractor is finite-dimensional.

Explicit bounds for Hausdorff and fractal dimensions.

Abstract

In this article we study the long-time behaviour of a system of nonlinear Partial Differential Equations (PDEs) modelling the motion of incompressible, isothermal and conducting modified bipolar fluids in presence of magnetic field. We mainly prove the existence of a global attractor denoted by $\A$ for the nonlinear semigroup associated to the aforementioned systems of nonlinear PDEs. We also show that this nonlinear semigroup is uniformly differentiable on $\A$. This fact enables us to go further and prove that the attractor $\A$ is of finite-dimensional and we give an explicit bounds for its Hausdorff and fractal dimensions.