*Title*: **Recent Progress on the Large BV Solution of P-System**

*Abstract*: The large BV existence of p-system is a long standing open problem. We consider two sub-problems:
1, how to find a positive time dependent lower bound on density for BV solutions; 2, how to find a BV estimate for
solutions away from vacuum. The resolution of these two problems will finally provide a large BV existence theory.
In this talk, we will discuss the progress on both of these problems. Especially, a very recent result shows
that Glimm-Lax' existence theory for large BV and small L^\infty solution can be extended to a more general case
when we assume that there are only finite many large shocks on the (x,t)-plane and solution is away from vacuum.
This talk includes joint works with Alberto Bressan and Qingtian Zhang.

CAM Seminar Spring 2018