Existence and uniqueness of solutions for the magnetohydrodynamic flow of a second grade fluid

Abstract

In this work, we consider the flow of a second grade fluid in a conducting domain of R 3 and in the presence of a magnetic field. When the initial data are of arbitrary size, we prove that the solution of the magnetohydrodynamics problem exists for a small time and is unique. We also show the global existence of solutions for small initial data.where J is the current density, and E and B are the electric field and the magnetic induction satisfying the Maxwell's equations curlB D J,(2)Eliminating J and E between (1) and (2), we find that the magnetic field B satisfies the following equation Á

Existence and regularity of solution for a model in magnetohydrodynamics

“…By using the spectral Galerkin method, Rojas-Medar and Boldrini [15] proved, under smallness of data, global in time existence of strong solutions and gave several estimates for the solution and their approximations. The (MHD) flow of a second grad fluid has been studied by Hamdache and Jaffal-Mourtada [11] where they showed that a unique solution exists for small time and it is actually global in time for small initial data.…”

On the time-dependent grade-two model for the magnetohydrodynamic flow: 2D case

“…One of the first mathematical results for this model type appears in [2], they prove the existence and uniqueness of solutions for a small time and global existence of solutions for small initial data in a conducting domain of R 3 , based on the iterative scheme where discretization is performed in the spatial variables. In this paper we discuss the MHD flow of a second grade fluid, in particular we prove the existence and uniqueness of a weak solution of a time-dependent grade two fluid model in a two-dimensional Lipschitz domain, where we follow the methodology of [3], i.e , we use semi-discretization in time and the work is in a domain of R 2 .…”