论文标题
非均匀戈德斯坦 - 泰勒方程的大时间收敛
Large time convergence of the non-homogeneous Goldstein-Taylor Equation
论文作者
论文摘要
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值集。它与动荡的流体运动和电报者方程密切相关。在放松函数（测量放松函数，测量朝向均匀分布的速度密度的强度）的情况下，对这些方程式解决方案的较大时间行为的详细理解大多是实现的。提出的工作的目的是提供一种通用方法，以解决放松函数不恒定并尽可能定量地解决趋于平衡问题。与方程式的通常模态分解相反，方程式是自然的，当松弛函数恒定时，我们定义了一种新的lyapunov伪变异性质的函数，这种函数是由恒定情况下的模态分析激发的，它能够处理宽松函数的全空间依赖性。我们开发的方法足够强大，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，我们发现的收敛速率不是最佳的，正如我们将结果与2013年伯纳德和萨尔瓦拉尼的工作中发现的结果进行了比较。
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.
