Some applications of parabolic Dirac operators to the instationary Navier-Stokes problem on conformally flat cylinders and tori in $\mathbb{R}^3$

TL;DR

This paper surveys how Clifford analysis techniques, involving parabolic Dirac operators and explicit Cauchy kernels, can be applied to model and solve unsteady Navier-Stokes flow problems on conformally flat cylinders and tori.

Contribution

It introduces a method to represent solutions of the instationary Navier-Stokes equations using integral operators derived from Clifford analysis on specific manifolds.

Findings

Explicit integral representations for solutions on cylinders and tori

Application of parabolic Dirac operators to fluid dynamics problems

Use of Cauchy kernels in solving Navier-Stokes equations

Abstract

In this paper we give a survey on how to apply recent techniques of Clifford analysis over conformally flat manifolds to deal with instationary flow problems on cylinders and tori. Solutions are represented in terms of integral operators involving explicit expressions for the Cauchy kernel that are associated to the parabolic Dirac operators acting on spinor sections of these manifolds.