|Dept.||Max Planck Institute for Mathematics|
For hyperbolic systems of conservation laws in one space dimension, existence of globally-in-time small-BV solutions is known when the initial data has small BV. Furthermore, it is known that these solutions are unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. One difficulty for the well-posedness theory of conservation laws is that many systems only admit a single entropy condition. In this lecture, I will present a framework for proving well-posedness of solutions to conservation laws using only one entropy. I will use the Burgers equation as a simple example to clarify ideas. Then, I will present a result for 2x2 systems which gives the uniqueness and stability of these globally small-BV solutions, amongst solutions which might have very large data for positive time (and in particular, might not even be BV). This result shows that the Tame Oscillation Condition, or the Bounded Variation Condition are not necessary to ensure the uniqueness of solutions with small BV data. My lecture will not assume strong familiarity with hyperbolic systems. The lecture will be based on joint work with G. Chen, W. Golding, and A. Vasseur.