Notes on Symmetry in Convective Flows

TL;DR

This paper explores how symmetry considerations in convective flow problems can identify invariant solution subspaces, aiding in understanding solution existence and stability in nonlinear fluid dynamics equations.

Contribution

It demonstrates the application of symmetry arguments to classical convection problems for identifying invariant subspaces and analyzing stability.

Findings

Symmetry helps identify invariant subspaces in convective flows.

Invariant subspaces facilitate solution existence proofs.

Symmetry considerations influence stability analysis.

Abstract

In most fluid dynamics problems, the governing equations are nonlinear because of the presence of convective terms. Nevertheless, existence of solutions can be shown by direct sum provided one identifies, in the relevant Banach space of solutions, particular subspaces which are invariant under time evolution. As an example, we consider classical convection problems and show how symmetry arguments can help in identifying such subspaces, as well as the role it can play in some stability issues.