Hyperbolic conservation laws and L2

Abstract

Taking as background the fact that conservation laws in a single space variable are well-posed in the space of functions of bounded variation, while multidimensional systems enjoy short-time well-posedness in Sobolev spaces Hs, we attempt to resolve the discrepancies between these two theories by exploring what can be said about stability of one-dimensional systems in L2. We summarize some positive results for special cases, and also show by a conterexample that there is no straightforward way to resolve the difficulty.