Initial-boundary value problems on the sphere

Abstract

In this chapter, we consider classes of fluid flow problems on the sphere and in ball shells with given initial and boundary value conditions. We focus our attention on the corresponding Navier-Stokes equations and their linearizations – the socalled forecasting equations. Shallow water equations are rather similar to this set of equations, and we shall discuss them as well. The physical background of such type of equations is described, for instance, in the book “Turbulence in fluids” by M. LESIEUR [195]. For a better understanding, we will give a brief introduction to the corresponding physical problems. The main aim of the section is to construct a quaternionic operator calculus tailored to the above mentioned applications.