A counterexample concerning regularity properties for systems of conservation laws

25 May 2016
In 1973 D. G. Schaeffer established an interesting regularity result that applies to scalar conservation laws with uniformly convex fluxes. Loosely speaking, it can be formulated as follows: for a generic smooth initial datum, the admissible solution is smooth outside a locally finite number of curves in the time-space plane. Here the term ``generic`` should be interpreted in a suitable technical sense, related to the Baire Category Theorem. Several author improved later his result, also for numerical purposes, while only C. M. Dafermos and X. Cheng extended it in 1991 to a special 2x2 system with coinciding shock and rarefaction curves and which satisfies an assumption that reframes what in the scalar case is the assumption of uniformly convex flux, called `genuine nonlinearity'. My talk will aim at discussing a recent explicit counterexample that shows that for systems of at least three equations, even when the flux satisfies the assumption of genuinely nonlinearity, Schaeffer`s Theorem does not extend because countably many shocks might develop from a ``big`` family of smooth initial data. I will then mention related works in progress.
  • PDE CDT Lunchtime Seminar