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Abstract In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity.
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Forward-modulated damping estimates and nonlocalized stability of periodic Lugiato–Lefever waves
In an interesting recent analysis, Haragus–Johnson–Perkins–de Rijk have shown modulationalstability under localized perturbations of steady periodic solutions of the Lugiato–Lefeverequation (LLE), in the process pointing out a difficulty in obtaining standard “nonlinear dampingestimates” on modulated perturbation variables to control regularity of solutions. Here, we point outthat in place of standard “inverse-modulated” damping estimates, one can alternatively carry outa damping estimate on the “forward-modulated” perturbation, noting that norms of forward- andinverse-modulated variables are equivalent modulo absorbable errors, thus recovering the classicalargument structure of Johnson–Noble–Rodrigues–Zumbrun for parabolic systems. This observationseems of general use in situations of delicate regularity. Applied in the context of (LLE), it gives thestronger result of stability and asymptotic behavior with respect to nonlocalized perturbations.
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- Award ID(s):
- 2206105
- PAR ID:
- 10580629
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Annales de l'Institut Henri Poincaré C, Analyse non linéaire
- Volume:
- 41
- Issue:
- 2
- ISSN:
- 0294-1449
- Page Range / eLocation ID:
- 497 to 510
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4171/AIHPC/76
