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Linearized Euler equations reveal norm inflation in Gevrey spaces for Couette flow perturbations, identifying echo chains as a linear instability. This analysis refines high-frequency behavior and constructs solutions with diverging Gevrey regularity.

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Area of Science:

  • Fluid dynamics
  • Nonlinear analysis
  • Partial differential equations

Background:

  • Couette flow is a fundamental shear flow.
  • Linearized Euler equations describe perturbations in fluid flow.
  • Gevrey spaces characterize functions with specific smoothness properties.

Purpose of the Study:

  • To investigate the stability of linearized Euler equations around Couette flow.
  • To analyze norm inflation and instability mechanisms.
  • To refine understanding of resonance phenomena and perturbation constraints.

Main Methods:

  • Linearization of Euler equations around Couette flow.
  • Analysis in Gevrey-type function spaces.
  • Asymptotic analysis of solutions in different regularity regimes.

Main Results:

  • Demonstrated norm inflation in Gevrey spaces for low-frequency perturbations.
  • Identified echo chains as a secondary linear instability mechanism.
  • Developed a refined analysis of resonance, modifying the high-frequency exponent and removing prior logarithmic constraints.

Conclusions:

  • Echo chains represent a novel linear instability mechanism in fluid dynamics.
  • The study provides a more precise understanding of perturbation behavior in fluid flows.
  • Constructed solutions exhibit asymptotic convergence in Sobolev regularity while diverging in Gevrey regularity.