How to distinguish a local semigroup from a global semigroup

TL;DR

The paper presents a linear eigenvalue test to distinguish between local and global semigroups generated by autonomous systems, with applications to Navier-Stokes equations and numerical demonstrations.

Contribution

It introduces a novel eigenvalue-based method to identify whether a system generates a local or global semigroup, providing a practical test for such classification.

Findings

Eigenvalue test effectively distinguishes local from global semigroups.

Numerical examples validate the proposed method.

Potential application to Navier-Stokes problems demonstrated.

Abstract

For a given autonomous time-dependent system that generates either a global, in time, semigroup or else only a local, in time, semigroup, a test involving a linear eigenvalue problem is given which determines which of 'global' or 'local' holds. Numerical examples are given. A linear transformation A is defined so that one has 'global' or 'local' depending on whether A does not or does have a positive eigenvalue. There is a possible application to Navier-Stokes problems.