Applications of parabolic Dirac operators to the instationary viscous MHD equations on conformally flat manifolds

TL;DR

This paper develops quaternionic analysis techniques using parabolic Dirac operators to derive analytic solutions for time-dependent viscous MHD equations on conformally flat manifolds like cylinders and tori.

Contribution

It introduces new analytic representation formulas for MHD solutions on conformally flat manifolds using quaternionic analysis and hypercomplex Eisenstein series.

Findings

Representation formulas for MHD solutions on manifolds

Application of hypercomplex Eisenstein series as kernels

Extension of quaternionic analysis to MHD equations

Abstract

In this paper we apply classical and recent techniques from quaternionic analysis using parabolic Dirac type operators and related Teodorescu and Cauchy-Bitzadse type operators to set up some analytic representation formulas for the solutions to the time depedendent incompressible viscous magnetohydrodynamic equations on some conformally flat manifolds, such as cylinders and tori associated with different spinor bundles. Also in this context a special variant of hypercomplex Eisenstein series related to the parabolic Dirac operator serve as kernel functions.