Global Stability Analysis of Fluid Flows using Sum-of-Squares

TL;DR

This paper presents a novel approach using sum-of-squares optimization to construct polynomial Lyapunov functions for proving the global stability of fluid flows, extending finite-dimensional methods to infinite-dimensional Navier-Stokes systems.

Contribution

It introduces a new SOS-based method for stability analysis of fluid flows that surpasses classical energy methods, applicable to infinite-dimensional systems.

Findings

SOS methods can construct better Lyapunov functions than classical energy methods.

The approach extends finite-dimensional stability analysis to infinite-dimensional Navier-Stokes systems.

Polynomial Lyapunov functions can efficiently determine stability bounds for fluid flows.

Abstract

This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed optimization methods based on sum-of-squares decomposition to construct a polynomial Lyapunov function. We then show how these methods can be extended to infinite dimensional Navier-Stokes systems using robust optimization techniques. Crucially, this extension requires only the solution of infinite-dimensional linear eigenvalue problems and finite-dimensional sum-of-squares optimization problems. We further show that subject to minor technical constraints, a general polynomial Lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower-bound on the maximum Reynolds number for which a flow is globally stable, if the flow does remain globally stable for Reynolds numbers at least slightly beyond the energy stability limit. Such polynomial functions can be searched for efficiently using the SOS technique we propose.