Title: Large-Time Asymptotics for Hyperbolic Systems with Non-Symmetric Relaxation: An Algorithmic Approach Show Chinese title

Title: 具有非对称松弛的双曲系统的长时间渐近：一种算法方法

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. Show Chinese abstract

Abstract: 我们研究了一维线性双曲系统的非对称松弛稳定性。引入一个新的与频率相关的Kalman稳定性条件，我们证明了一个抽象的衰减结果，支持一种非齐次的亚 coercivity 形式。与齐次情形相比，衰减率取决于Kalman条件如何满足，并且在大多数情况下会出现导数损失：必须假设初始数据具有额外的正则性假设以确保衰减。在结构假设下，我们通过提供一个广泛适用的构造Lyapunov泛函的算法，细化了我们的抽象结果。这使我们能够系统地为给定系统建立衰减估计，并揭示代数抵消（超出基于Kalman方法的范围），从而减少高频下的导数损失。为了展示我们方法的应用性，我们推导了Sugimoto模型（描述非线性声波传播）和具有记忆的Timoshenko型梁模型的新稳定性结果。