On the norm equivalence of Lyapunov exponents for regularizing linear evolution equations

TL;DR

This paper proves that for certain dissipative linear evolution equations on infinite-dimensional spaces, the top Lyapunov exponent remains invariant under different norm choices, with applications to fluid mechanics.

Contribution

It establishes the norm invariance of Lyapunov exponents for cocycles of compact operators in infinite-dimensional spaces, extending finite-dimensional results.

Findings

Lyapunov exponent is independent of the norm in broad settings

Application to enhanced dissipation in advection-diffusion equations

Application to Lyapunov exponents in 2D Navier-Stokes equations

Abstract

We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of nonequivalent spaces mitigating, e.g., integrability ($L^p$) or differentiability ($W^{s, p}$). In contrast to finite dimensions, the Lyapunov exponent could apriori depend on the choice of norm used. In this paper we show that under quite general conditions, the Lyapunov exponent of a cocycle of compact linear operators is independent of the norm used. We apply this result to two important problems from fluid mechanics: the enhanced dissipation rate for the advection diffusion equation with ergodic velocity field; and the Lyapunov exponent for the 2d Navier-Stokes equations with stochastic or periodic forcing.