Ola Mæhlen, University of Oslo
Scalar hyperbolic conservation laws are traditionally viewed as transport equations, and the viewpoint pertains to the study of (generalized) characteristics. Here, we instead view these PDEs as continuity equations with an implicitly defined velocity field; this motivates the analysis of the associated particle paths. These paths are implicitly defined as solutions of a family of ODEs. We show that the particle paths of a weak solution of the PDE are well-defined (the ODEs well-posed) if, and only if, the weak solution is the entropy solution. This contrast the ODEs corresponding to characteristics of a solution, which can be ill-posed even for the entropy solution.
We show that the associated flow-map is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.
This is joint work with Ulrik Skre Fjordholm (UiO) and Magnus Christie Ørke (UiO)