非齐次Goldstein-Taylor方程的大时间收敛性
Large time convergence of the non-homogeneous Goldstein-Taylor Equation
摘要：
Goldstein-Taylor方程可以被认为是BGK系统的简化版本，其中速度变量被限制为离散集值。它与湍流运动和电报员方程密切相关。这些方程的解决方案的大时间行为的详细了解已大多实现的情况下，松弛函数，测量的强度，对均匀分布的速度密度的松弛，是恒定的。所提出的工作的目标是提供一个通用的方法来解决问题的收敛到平衡时的松弛函数是不恒定的，并尽可能定量地这样做。在通常的模态分解的方程，这是自然的松弛函数是常数时，我们定义了一个新的李雅普诺夫函数的伪微分性质，一个是由模态分析的动机在常数的情况下，这是能够处理充分的空间依赖性的松弛函数。我们开发的方法是足够强大的，可以将其应用到多速度Goldstein-Taylor模型，并实现显式的收敛速度。然而，我们发现的收敛速度并不是最优的，正如我们通过将我们的结果与Bernard和Salvarani在2013年的工作中发现的结果进行比较所表明的那样。
英文摘要：
The Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher's equation. A detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to the that found in the work of Bernard and Salvarani from 2013.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
