时变阻尼系数的等温可压缩欧拉方程组光滑解的爆破研究  

BLOW UP OF SMOOTH SOLUTIONS FOR ISOTHERMAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

作　　者：于慧敏 隋莹 Yu Huimim;Sui Ying(School of Mathematics and Statistics,Shandong Normal University,250358,Jinan,China)

机构地区：[1]山东师范大学数学与统计学院,济南250358

出　　处：《山东师范大学学报（自然科学版）》2021年第4期366-374,共9页Journal of Shandong Normal University(Natural Science)

基　　金：国家自然科学基金资助项目(11671237).

摘　　要：可压缩欧拉方程在物理科学和工程技术等领域有着广泛的应用,它可以用来描述很多出现于流体力学的物理现象,例如:浅水波模型、激波的产生、球对称的多维气体动力学模型等.由于可压缩欧拉方程在物理学中的重要性和在数学中带给人们的挑战性,使得可压缩欧拉方程(组)的研究,成为了非线性偏微分方程(组)中的一个研究热点,引起了许多学者的广泛关注.本文主要研究等温可压缩Euler方程组经典解在有限时间内的爆破问题,并给出了经典解在有限时间内产生激波的充分条件.同时,本文还得到了任何经典解的密度随时间变化的下界估计.值得指出的是:这里保证古典解爆破的条件只与初始数据在某处的取值及导数值有关,与初值在整个空间的分布情况并没有关系.Compressible Euler equations are widely used in many fields,such as physical science and engineering technology.It can be used to describe many physical phenomena in fluid mechanics,for example,shallow water wave model,shock generation,spherically symmetric multi-dimensional gas dynamic model,etc.Due to the importance of compressible Euler equations in physics and the challenge brought by mathematics,the research of compressible Euler equations(systems)has become a research hotspot in nonlinear partial differential equations(systems),and has been widely concerned by many scholars.In this paper,the blow-up problem of classical solutions of isothermal compressible Euler equations in finite time is discussed.A sufficient condition for the classical solution to produce shock wave in finite time is given.At the same time,we obtain the lower bound of the density of any classical solution.It is worth pointing out that the condition to guarantee the blow-up of the classical solution here is only related to the initial data itself and the derivative value at a certain place,and has nothing to do with the distribution of the initial value in the whole space.

关 键 词：爆破 等温 可压缩欧拉方程 时变阻尼 激波 

分 类 号：O175.29[理学—数学]