Large-time asymptotics for hyperbolic systems with non-symmetric relaxation: An algorithmic approach

Crin-Barat T, Liverani L, Shou Ly, Zuazua Iriondo E (2025)

Publication Language: English

Publication Status: Submitted

Publication Type: Unpublished / Preprint

Future Publication Type: Journal article

Publication year: 2025

URI: https://dcn.nat.fau.eu/wp-content/uploads/CrinBarat_Liverani_Shou_Zuazua_Non_symmetric_dissipation.pdf

Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/CrinBarat_Liverani_Shou_Zuazua_Non_symmetric_dissipation.pdf

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.

Authors with CRIS profile

Timothée Crin-Barat Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur) Lorenzo Liverani Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur) Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Nanjing Normal University (NNU) / 南京师范大学 CN China (CN)

How to cite

APA:

Crin-Barat, T., Liverani, L., Shou, L.-y., & Zuazua Iriondo, E. (2025). Large-time asymptotics for hyperbolic systems with non-symmetric relaxation: An algorithmic approach. (Unpublished, Submitted).

MLA:

Crin-Barat, Timothée, et al. Large-time asymptotics for hyperbolic systems with non-symmetric relaxation: An algorithmic approach. Unpublished, Submitted. 2025.

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