Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

Emmanuel Grenier, Toan T. Nguyen, Frédéric Rousset, Avy Soffer

Research output: Contribution to journalArticle

Abstract

We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel T×R. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order ν−1/3, ν being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schrödinger operators, combined with a hypocoercivity argument to handle the viscous case.

Original languageEnglish (US)
Article number108339
JournalJournal of Functional Analysis
Volume278
Issue number3
DOIs
StatePublished - Feb 1 2020

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Viscous Dissipation
Shear Flow
Damping
Viscosity
Spectral Stability
Mixing Layer
Large Time Behavior
Behavior of Solutions
Operator
Self-adjoint Operator
Euler
Navier-Stokes Equations
Form

All Science Journal Classification (ASJC) codes

  • Analysis

Keywords

  • Conjugate operator method
  • Enhanced dissipation
  • Inviscid damping
  • Linear stability of shear flows

Cite this

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abstract = "We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel T×R. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order ν−1/3, ν being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schr{\"o}dinger operators, combined with a hypocoercivity argument to handle the viscous case.",
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Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method. / Grenier, Emmanuel; Nguyen, Toan T.; Rousset, Frédéric; Soffer, Avy.

In: Journal of Functional Analysis, Vol. 278, No. 3, 108339, 01.02.2020.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

AU - Grenier, Emmanuel

AU - Nguyen, Toan T.

AU - Rousset, Frédéric

AU - Soffer, Avy

PY - 2020/2/1

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N2 - We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel T×R. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order ν−1/3, ν being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schrödinger operators, combined with a hypocoercivity argument to handle the viscous case.

AB - We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel T×R. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order ν−1/3, ν being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schrödinger operators, combined with a hypocoercivity argument to handle the viscous case.

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KW - Enhanced dissipation

KW - Inviscid damping

KW - Linear stability of shear flows

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