On the Burgers dynamical system with an external force and its Koopman decomposition

TL;DR

This paper analyzes the Burgers equation with external forcing, proving the existence of a unique steady state and a convergent Koopman Modes decomposition, which aids in understanding fluid instabilities.

Contribution

It demonstrates the Koopman decomposition for a nonlinear forced Burgers flow, despite the inability to linearize via Cole-Hopf transforms, providing new insights into its asymptotic behavior.

Findings

Existence of a unique steady state acting as a sink.

Convergence of Koopman Modes decomposition for orbits near the sink.

Analysis of invariant sets with complete orbits and finite-time blow-up.

Abstract

We prove that the Burgers flow with a steady external forcing has a unique steady state which is a sink. Although this flow cannot be linearized through Cole-Hopf transforms, we prove that it has a convergent Koopman Modes decomposition. This gives an asymptotic formula for solutions of the Burgers equation with an external force. Time dependence and the coefficients of the decomposition are proved to be eigenvalues and eigenfunctionals of the Koopman operator. Convergence of the Koopman decomposition is proved for orbits close to the sink. The analysis of Burgers dynamical system relies on the properties of a nonlinear heat flow, that shows invariant sets with complete orbits, and invariant sets where blow-up in finite time do ocurr. This behaviour helps understanding some instabilities in numeric computing for fluids.