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Viscosity driven instability of shear flows without boundaries

Hui Li, Weiren Zhao

Abstract

In this paper, we study the instability effect of viscous dissipation in a domain without boundaries. We construct a shear flow that is initially spectrally stable but evolves into a spectrally unstable state under the influence of viscous dissipation. To the best of our knowledge, this is the first result of viscosity driven instability that is not caused by boundaries.

Viscosity driven instability of shear flows without boundaries

Abstract

In this paper, we study the instability effect of viscous dissipation in a domain without boundaries. We construct a shear flow that is initially spectrally stable but evolves into a spectrally unstable state under the influence of viscous dissipation. To the best of our knowledge, this is the first result of viscosity driven instability that is not caused by boundaries.

Paper Structure

This paper contains 10 sections, 9 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Main ideas
  4. Stability transition of the shear flow
  5. Existence of neutral mode
  6. Profile of the neutral mode
  7. Proof of Proposition \ref{['lem-neutral']}
  8. Proof of Proposition \ref{['lem-neutral2']}
  9. Quantitative estimate for the unstable eigenvalue
  10. Estimate of solution to homogeneous Rayleigh

Key Result

Theorem 1.1

Given small constant $\gamma>0$, for any $\nu>0$, there exists a shear flow $b(t,y)$ satisfying eq-heat, along with two time points $0<\tilde{T}<T$, such that:

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2: Embedded eigenvalue
  • Remark 1.3: Viscous instability
  • Remark 1.4: Secondary instability
  • Remark 1.5: Nonlinear problem
  • Remark 1.6: Nonlinear problem with arbitrary small force
  • Theorem 1.7
  • Remark 1.8: Bounded monotonic shear flow
  • Remark 2.1
  • Lemma 2.2
  • ...and 10 more