On the Incompressible Viscous Stationary MHD Equations and Explicit Solution Formulas for Some Three-dimensional Radially Symmetric Domai

Abstract

The quaternionic calculus is a powerful tool to treat many complicated systems of linear and non-linear PDEs in higher dimensions. In this paper we apply these new techniques to treat the stationary incompressible viscous magnetohydrodynamic equations. For the highly viscous case, in which the convective terms are negligibly small we present explicit analytic representation formulas for some three-dimensional radially symmetric domains. Then we look at the fully non-linear case for which we propose a fixed point algorithm. In this more complicated context, the solutions of the simpler linear problems treated in the first part of the paper need to be used to solving the corresponding equations in each step of the proposed iteration.