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Uniform Linear Inviscid Damping and Enhanced Dissipation Near Monotonic Shear Flows in High Reynolds Number Regime (I): The Whole Space Case

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Abstract

We study the dynamics of the two dimensional Navier Stokes equations linearized around a strictly monotonic shear flow on \({\mathbb {T}}\times {\mathbb {R}}\). The main task is to understand the associated Rayleigh and Orr–Sommerfeld equations, under the natural assumption that the linearized operator around the monotonic shear flow in the inviscid case has no discrete eigenvalues. We obtain precise control of solutions to the Orr–Sommerfeld equations in the high Reynolds number limit, using the perspective that the nonlocal term can be viewed as a compact perturbation with respect to the main part that includes the small diffusion term. As a corollary, we give a detailed description of the linearized flow in Gevrey spaces (linear inviscid damping) that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to accurately capture the behavior of the solution to Orr–Sommerfeld equations in the critical layer. In this paper we consider the case of shear flows on \({\mathbb {T}}\times {\mathbb {R}}\). The case of bounded channels poses significant additional difficulties, due to the presence of boundary layers, and will be addressed elsewhere.

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Notes

  1. We should mention that there are many important nonlinear stability results for fluid equations in Lyapunov or orbital sense based on variational argument, see Arnold [1] and recent works [9, 17] for more references. Our focus is dynamic stability, or asymptotic stability, which requires a more precise understanding on the evolution of solutions.

  2. The study of the Orr–Sommerfeld equation has a long history. We refer to [20, 21] for an introduction to other important aspects of the equation.

  3. Strictly speaking the test functions need to be at least Gevrey-2 smooth due to the strong growth of the symbol.

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Acknowledgements

We are grateful to Alexandru Ionescu for invaluable discussions during the project. We also thank Bernard Helffer for comments which improved the presentation of the paper.

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  1. University of Minnesota, Twin Cities, USA

    Hao Jia

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  1. Hao Jia
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Supported in part by NSF Grant DMS-1945179. This article is part of the Topical collection Ladyzhenskaya Centennial Anniversary edited by Gregory Seregin, Konstantinas Pileckas and Lev Kapitanski.

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Jia, H. Uniform Linear Inviscid Damping and Enhanced Dissipation Near Monotonic Shear Flows in High Reynolds Number Regime (I): The Whole Space Case. J. Math. Fluid Mech. 25, 42 (2023). https://doi.org/10.1007/s00021-023-00794-8

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