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 those found in [8].

可以将Goldstein-Taylor方程视为BGK系统的简化版本，其中速度变量被限制为一组离散值。它与湍流运动和电报机方程密切相关。在测量弛豫强度朝着均匀分布的速度密度的弛豫函数恒定的情况下，对于这些方程的解的长时间行为的详细了解已得到大部分理解。提出的工作的目的是提供一种通用方法，以解决松弛函数不恒定时收敛到平衡的问题，并尽可能定量地进行。与通常的等式的模态分解相反，当松弛函数为常数时，这很自然，我们定义了一种新的具有伪微分性质的Lyapunov函数，该函数由恒定情况下的模态分析驱动，能够处理松弛函数的完整空间依赖性。我们开发的方法足够健壮，可以将其应用于多速度Goldstein-Taylor模型，并实现明确的收敛速度。但是，正如我们通过将结果与[8]中的结果进行比较所显示的那样，我们发现收敛速度不是最佳的。