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arXiv · 2207.14797

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

Alex Blumenthal·Sam Punshon-Smith
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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.

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Alex Blumenthal, Sam Punshon-Smith. 2022-07-29. On the norm equivalence of Lyapunov exponents for regularizing linear evolution equations. https://arxiv.org/abs/2207.14797

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