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Large-Time Asymptotics for Hyperbolic Systems with Non-Symmetric Relaxation: An Algorithmic Approach

Timothée Crin-Barat, Lorenzo Liverani, Ling-Yun Shou, Enrique Zuazua

Abstract

We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must assume an additional regularity assumption on the initial data to ensure the decay. Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.

Large-Time Asymptotics for Hyperbolic Systems with Non-Symmetric Relaxation: An Algorithmic Approach

Abstract

We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must assume an additional regularity assumption on the initial data to ensure the decay. Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.

Paper Structure

This paper contains 33 sections, 24 theorems, 279 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The symmetric case
  3. The skew-symmetric case -- State of the art
  4. Our contributions
  5. Outline of the paper
  6. The Inhomogeneous Kalman Method
  7. Limitations of Kalman-based analysis
  8. A system with cancellation for specific coefficients
  9. Improving the Kalman approach
  10. Proof of Theorem \ref{['thm:decayKalman']}
  11. Littlewood-Paley decomposition
  12. High-frequency decay estimates
  13. Low-frequency decay estimates
  14. An Algorithm to Improve the Lyapunov Functional
  15. Initialization
  16. ...and 18 more sections

Key Result

Lemma 2.2

Let the inhomogeneous Kalman condition condition:span hold. There exist integers $\alpha\in \mathbb{N}$ and $\beta \in \mathbb{N}$ such that where $c>0$ is a universal constant.

Figures (3)

  • Figure 1: The first few nodes of the tree $\mathsf{T}$.
  • Figure 2: The four possible routes to follow, starting from a node $X_k^i$
  • Figure 3: The paths followed in the three examples.

Theorems & Definitions (57)

  • Definition 2.1: Inhomogeneous Kalman rank condition
  • Lemma 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Example 2.7: Decay estimates for the linear Sugimoto model \ref{['sugimoto_lin']}
  • Theorem 2.8
  • Remark 2.9
  • Lemma 3.1: Bernstein properties
  • ...and 47 more