For hyperbolic systems of conservation laws in one space dimension endowed with a single convex entropy, it is an open question if it is possible to construct solutions via convex integration. Such solutions, if they exist, would be highly non-unique and exhibit little regularity. In particular, they would not have the strong traces necessary for the nonperturbative L2 stability theory of Vasseur. Whether convex integration is possible is a question about large data, and the global geometric structure of genuine nonlinearity for the underlying PDE.
In this talk, I will discuss recent work which shows the impossibility, for a large class of 2 x 2 systems, of doing convex integration via the use of T_4 configurations. Our work applies to every well-known 2 x 2 hyperbolic system of conservation laws which verifies the "structural Liu entropy condition." This talk is based on joint work with László Székelyhidi
