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Seminars

Seminar on PDE

Large Data Solutions to 1-D Hyperbolic Systems, Ill-Posedness, and Convex Integration

2024/01/04 (Thu.) 15:30~16:30
  • Date : 2024/01/04 (Thu.) 15:30~16:30
  • Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
  • Speaker : Sam G. Krupa (Max Planck Institute for Mathematics in the Sciences)
  • Abstract : For hyperbolic systems of conservation laws in one space dimension endowed with a single convex entropy, the well-posedness for large L∞ data is widely open. Even for the famous p-system, uniqueness is not known in this class of data. One possible method of showing ill-posedness is by constructing solutions via convex integration. Such solutions, if they exist, would be highly non-unique and exhibit little regularity. In particular, they would be outside of any known well-posedness results and 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 2x2 systems, of doing convex integration via the use of T4 configurations. Our work applies to every well-known 2x2 hyperbolic system of conservation laws which verifies the Liu entropy condition. This talk is based on joint work with Laszlo Szekelyhidi.
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