Exponential Stability of Nonlocal Time-Delayed Burgers Equation
Exponential Stability of Nonlocal Time-Delayed Burgers Equation
复制标题
非局部时滞Burgers方程的指数稳定性
DOI：
10.1142/9789814289313_0013
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Yanbin Tang
中科院分区：
文献类型：
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作者：
Yanbin Tang
Burgers方程已经得到了广泛的研究，它已经成为所有非线性偏微分方程中最著名和研究最多的方程之一。Burgers方程是一维Navier-Stokes方程的一种，虽然它不能模拟任何具体的物理流动问题，但它是理解湍流的第一步，而经典的Burgers方程的初边值问题可以通过Cole-Hopf变换精确求解。然而，由于时滞的引入，使得经典的Cole-Hopf变换很难精确求解时滞Burgers方程的初边值问题。这意味着，即使时滞足够小，时滞偏微分方程也会表现出一些完全不同的性质。本文考虑了一类具有时滞的非局部Burgers方程.利用Liapunov函数和一致Gronwall不等式，证明了时滞非局部Burgers方程的解在小时滞下是指数稳定的，并且具有时滞诱导不稳定性.
Burgers equation has been extensively studied and it has become one of the best known and most studied of all nonlinear partial differential equations. Even though Burgers equation, a one-dimensional version of Navier-Stokes, does not model any specific physical flow problem, it would be the first step to understand the turbulence exhibited in a flow.The initial-boundary value problem of classical Burgers equation involving homogeneous Dirichlet data can be solved exactly using the Cole-Hopf transformation. Unfortunately, due to the introduction of time delay, it is very difficult to use the classical Cole-Hopf transformation to solve initial-boundary value problem of time-delay Burgers equation exactly. It means that the partial differential equation with time-delay will exhibit some quite different properties even if the small enough time-delay. In this paper, a nonlocal Burgers equation with a time delay is considered. Using the Liapunov function and uniform Gronwall inequality, the solution of the time-delayed nonlocal Burgers equation is shown to exhibit a delay-induced instability and be exponentially stable under small delays.