Cheung Ka Luen
Department of Mathematics and Information Technology, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories Hong Kong.
Springerplus. 2016 Feb 27;5:196. doi: 10.1186/s40064-016-1766-8. eCollection 2016.
The N-dimensional isentropic compressible Euler system with a damping term is one of the most fundamental equations in fluid dynamics. Since it does not have a general solution in a closed form for arbitrary well-posed initial value problems. Constructing exact solutions to the system is a useful way to obtain important information on the properties of its solutions.
In this article, we construct two families of exact solutions for the one-dimensional isentropic compressible Euler equations with damping by the perturbational method. The two families of exact solutions found include the cases [Formula: see text] and [Formula: see text], where [Formula: see text] is the adiabatic constant.
With analysis of the key ordinary differential equation, we show that the classes of solutions include both blowup type and global existence type when the parameters are suitably chosen. Moreover, in the blowup cases, we show that the singularities are of essential type in the sense that they cannot be smoothed by redefining values at the odd points.
The two families of exact solutions obtained in this paper can be useful to study of related numerical methods and algorithms such as the finite difference method, the finite element method and the finite volume method that are applied by scientists to simulate the fluids for applications.
带有阻尼项的N维等熵可压缩欧拉方程组是流体动力学中最基本的方程之一。由于对于任意适定的初值问题，它没有一般的封闭形式解。构造该方程组的精确解是获取其解的性质的重要信息的一种有用方法。
在本文中，我们通过微扰法为带有阻尼的一维等熵可压缩欧拉方程构造了两类精确解。所找到的两类精确解包括情况[公式：见原文]和[公式：见原文]，其中[公式：见原文]是绝热常数。
通过对关键常微分方程的分析，我们表明当参数适当地选取时，解的类别包括爆破解类型和全局存在解类型。此外，在爆破解的情况下，我们表明奇点是本质类型的，即它们不能通过在奇数点重新定义值来平滑。
本文得到的两类精确解对于研究相关数值方法和算法（如科学家用于模拟流体应用的有限差分法、有限元法和有限体积法）可能是有用的。
