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On the Norm Equivalence of Lyapunov Exponents for Regularizing Linear Evolution Equations

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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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Notes

  1. Despite a wealth of numerical evidence, in the absence of noise it is a notoriously challenging open problem to prove that positivity of Lyapunov exponents for incompressible systems of practical interest. This is already the case for low-dimensional discrete-time toy models [19] of Lagrangian flow such as the Chirikov standard map [16], for which the analogue of (6) is a wide-open problem—see, e.g., the discussion in [10, 21].

  2. When F is white-in-time and satisfies mild nondegeneracy conditions (e.g., those in [41]), the value \(\lambda _1(H^r)\) does not depend on \(u_0\). When F is time-periodic it is possible that \(\lambda _1(H^r)\) depends on \(u_0\). For more details and discussion, see Sect. 4 below.

  3. We call \(\mathcal A\) a T-invariant set if \(T^{-1} \mathcal A \supset \mathcal A\).

  4. That such weak\(^*\) limits exist follows by compactness of \(\mathcal A\). That such limiting measures are T-invariant is straightforward to check: see, e.g., Lemma 2.2.4 of [80]. The above procedure is often referred to as the Krylov-Bogolyubov argument for the existence of T-invariant measures [48].

  5. E.g., Proposition 4.3.2 of [80] and the Krein-Milman Theorem, paragraph I.A.22 in [85].

  6. When B is separable, we say that \(x \mapsto A_x\) is strongly measurable if it is Borel measurable w.r.t. the strong operator topology on L(B), or equivalently, when \(x \mapsto A_x v\) is a Borel measurable mapping for each fixed \(v \in B\). For a summary of alternative measurability requirements for the MET, see, e.g., [79].

  7. Throughout, we consider the space of closed subspaces of B with the Hausdorff metric \(d_{Haus}\) of unit spheres; see (17) for details. Here, we are asserting that \(x \mapsto F_i(x)\) is Borel measurable w.r.t. the topology induced by \(d_H\).

  8. Some authors refer to the property (9) as equivariance.

  9. The measure \({{\,\mathrm{\textrm{Vol}}\,}}_E^B\) is sometimes called the Busemann-Hausdorff measure [13] and appears naturally in Finsler geometry.

  10. A version of this argument is carried out in the proof of Lemma 4.18.

  11. This sufficient condition for passing from discrete to continuous time Lyapunov exponents is classical; see, e.g., [52].

  12. Recall the definition of the minimal angle \(\angle ^{H^s}\) in (13).

  13. When there is no confusion, we will abuse notation somewhat and write \(D_{u_0} \Phi ^t_\omega : \textbf{H}^s \rightarrow \textbf{H}^s\) for the extended operator.

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Acknowledgements

AB was supported by National Science Foundation grant DMS-2009431. SPS was supported by National Science Foundation grant DMS-2205953. SPS is also grateful to the Institute for Advanced Study for their generous support and hospitality during 2021-2022 academic year when this paper was being written.

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  1. Georgia Institute of Technology, Atlanta, USA

    Alex Blumenthal

  2. Tulane University, New Orleans, USA

    Sam Punshon-Smith

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Blumenthal, A., Punshon-Smith, S. On the Norm Equivalence of Lyapunov Exponents for Regularizing Linear Evolution Equations. Arch Rational Mech Anal 247, 97 (2023). https://doi.org/10.1007/s00205-023-01928-y

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