二维可压欧拉方程组径向对称解的爆破

BLOWUP OF THE SPHERICALLY SYMMETRIC SOLUTIONS OF THE 2-D FULL COMPRESSIBLE EULER EQUATIONS

本文将考虑二维径向对称完全欧拉方程组经典解的爆破问题.当其初值是一个常状态加上一个具有紧支集的光滑小扰动时,我们建立了精确的生命跨度.对于二维有旋等熵的欧拉方程组,S.Alinhac^([5])建立了其解的生命跨度,本文将针对非等熵情况.我们的主要论证方法是:首先构造一个合适的渐近解,然后利用^([6,7])研究三维欧拉方程组经典解问题时引进的相关范数,并通过细致的分析得到生命跨度的下界,最后再利用常微分方程的技巧证明此下界也同时是上界.

In this paper, we are concerned with the blowup problem of the spherically symmetric solutions to the 2-D full compressible Euler system. When the initial data are of small perturbations with respect to a constant state, we obtain the precise bound of the lifespam With compact simporf for the 2-D isentropic Euler system with the symmetric rotation, S. Alinhac has established the lifespan of the small data symmetric solutions. Our focus is on the non-isentropic case in this paper. The main ingredients are： at first we construct a suitable approximate solution, then by utilizing the norms for studying the 3-D blowup problems and combining some delicate analysis, we obtain the precise lower bound of the lifespan. Finally we show that the lower bound is also the upper bound of the lifespan by exploiting some techniques in ordinary differential equations.