[Paper Review] Linear inviscid damping for the $β$-plane equation

Dongyi Wei, Zhifei Zhang+1 more|arXiv (Cornell University)|Sep 10, 2018

Advanced Mathematical Physics Problems15 references4 citations

TL;DR

This paper establishes linear inviscid damping for the linearized $\beta$-plane equation around monotone shear flows using a novel space-time estimate and vector field method, achieving explicit decay rates. For general flows including the Sinus profile, it proves damping via a refined compactness method, overcoming challenges from additional singularities due to Coriolis effects.

ABSTRACT

In this paper, we study the linear inviscid damping for the linearized $β$-plane equation around shear flows. We develop a new method to give the explicit decay rate of the velocity for a class of monotone shear flows. This method is based on the space-time estimate and the vector field method in sprit of the wave equation. For general shear flows including the Sinus flow, we also prove the linear damping by establishing the limiting absorption principle, which is based on the compactness method introduced by Wei-Zhang-Zhao in \cite{WZZ2}. The main difficulty is that the Rayleigh-Kuo equation has more singular points due to the Coriolis effects so that the compactness argument becomes more involved and delicate.

Motivation & Objective

• To establish linear inviscid damping for the $\beta$-plane equation linearized around monotone shear flows.
• To develop a new analytical method based on space-time estimates and the vector field method to derive explicit decay rates for velocity.
• To extend the analysis to general shear flows, including the Sinus flow, using a limiting absorption principle and compactness arguments.
• To address the increased complexity from Coriolis effects, which introduce additional singular points in the Rayleigh-Kuo equation.
• To prove that the absence of embedding eigenvalues in certain parameter regimes ensures the validity of the compactness argument for damping.

Proposed method

• Applies a space-time estimate and vector field method inspired by wave equation techniques to analyze the linearized $\beta$-plane equation.
• Uses the Fourier transform in the $x$-direction to reduce the PDE to an ODE in $y$, leading to the operator $\mathcal{R}_{\alpha,\beta}$.
• Employs a limiting absorption principle to handle the spectral problem associated with the $\beta$-plane equation.
• Utilizes compactness arguments from Wei-Zhang-Zhao [27] to prove linear damping for general shear flows.
• Analyzes the Rayleigh-Kuo equation for singularities at critical layers ($u = c$) and excludes embedding eigenvalues via energy estimates and Sobolev embedding.
• Applies integration by parts and $L^p$-based estimates to show that nontrivial solutions lead to contradictions, implying $\phi \equiv 0$.

Experimental results

Research questions

• RQ1 Can explicit decay rates for velocity be derived for linear inviscid damping in the $\beta$-plane equation around monotone shear flows?
• RQ2 How does the Coriolis effect ($\beta$) affect the spectral structure and damping behavior in the $\beta$-plane equation compared to the standard Euler equation?
• RQ3 What techniques can be used to prove linear damping for general shear flows like the Sinus profile when the compactness argument is complicated by additional singularities?
• RQ4 Under what conditions are there no embedding eigenvalues in the spectrum of $\mathcal{R}_{\alpha,\beta}$, ensuring the validity of the limiting absorption principle?
• RQ5 Can the vector field method be adapted to the $\beta$-plane setting to yield quantitative decay estimates?

Key findings

• For a class of monotone shear flows, the authors establish explicit decay rates for the velocity via a new space-time estimate and vector field method.
• The method successfully handles the nonlocal term and Coriolis effects, providing a robust framework for linear damping analysis.
• For the Sinus flow, linear inviscid damping is proven using the limiting absorption principle and compactness, even though the Rayleigh-Kuo equation has more singular points.
• The paper proves that $c = 1$ is not an embedding eigenvalue of $\mathcal{R}_{\alpha,\beta}$ for $\beta \in (-\frac{9}{16}\pi^2, \frac{9}{16}\pi^2)$, except on a specific curve $\gamma_4$, which is crucial for the compactness argument.
• For $c \in (0,1)$ with $c \neq \frac{1}{2} - \frac{\beta}{\pi^2}$, the paper shows $c$ is not an embedding eigenvalue in the parameter regime $\alpha > 0$, $\beta \in (-\frac{9}{16}\pi^2, \frac{9}{16}\pi^2)$, ensuring the absence of resonant modes.
• The analysis confirms that the absence of embedding eigenvalues allows the use of the compactness method to prove linear damping, even in the presence of Coriolis-induced singularities.