*主持人：袁海荣 教授

*讲座内容简介：
于慧敏，山东师范大学数学与统计学院教授、博导，山东师范大学东岳学者。2008年从中国科学院数学与系统科学研究院获理学博士学位，主要从事偏微分方程的理论与应用研究。在 SIAM J. Math. Anal.、 J. Differential Equations、 Commun. Math. Sci，Z. Angew. Math. Phys. 等国际期刊发表学术论文二十余篇， 一篇文章获首届《中国科学：数学》优秀论文提名奖。

*主讲人简介：
In this talk, the vacuum and shock formation problem will be considered for the compressible Euler equations with general pressure law and time-dependent damping. We investigate the lower bound estimates of density for arbitrary classical solutions for some kind of pressure functions. Precisely, we get the uniform (w.r.t time t) lower bounds of density for Euler equations with an over-damping coefficient; however, if the system is under-damping, the bound tends to zero with a time-dependent speed. Then, several sufficient conditions, under which the classical solutions must break down in finite time, will be shown by delicate analysis of decoupled Riccati type equations. The assumptions on pressure function automatically satisfied for gas dynamics withγ-law. Furthermore, our results have no limits on the size of the solutions or the positive/monotonicity on the initial Riemann invariants.