Variational principles in quaternionic analysis with applications to the stationary MHD equations

TL;DR

This paper develops new methods combining variational calculus and quaternionic analysis to establish global existence, uniqueness, and computational algorithms for stationary magnetohydrodynamic equations, applicable to large data scenarios.

Contribution

It introduces a quaternionic variational framework applying fixed point theorems for existence and uniqueness, and provides explicit iterative algorithms for solving stationary MHD equations.

Findings

Existence of weak solutions via mountain pass theorem.

Applicability of Schauder's fixed point theorem to large data.

Development of iterative algorithms for numerical solutions.

Abstract

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder's fixed point theorem. The advantage of the approach using Schauder's fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These consideration will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.