高维欧拉方程正则解的有限时间爆破

Finite-Time Blow-Up of the Regular Solution in the Multi-Dimensional Euler Equations

带阻尼的欧拉方程是流体力学基本模型之一。欧拉方程解的爆破在理论研究和数值模拟上都有重要意义,尤其是带真空状态的欧拉方程在工程应用中有着重要作用且具有一定的难度。本文讨论了带真空状态的阻尼欧拉方程正则解的相关理论。通过特征线方法讨论了带阻尼的高维欧拉方程正则解的相关估计,并构造了一个加权质量的泛函来证明当初始密度有紧支集但并不恒为零时,带阻尼高维欧拉方程正则解的有限时间爆破。

Euler equations with damping is one of the basic models in Fluid Mechanics.The blow-up of the smooth solu-tions of the Euler equations plays an important role not only in the theoretical study but also in numerical simulations.The the-ory about the solutions including vacuum states,having important applications in engineering,but is difficult to analysis.In this paper,the regular solution including the vacuum states of the damped high dimensional Euler equations is discussed.The method of characteristics is used to obtain the estimation of the regular solutions.When the initial density is non zero and has a compact supported,the finite time blow-up of the regular solutions is proved based on the analysis of these estimates on a mass weighted functional.