Evolution of Derivative Singularities in Hyperbolic Quasilinear Systems of Conservation Laws

Motivated by problems arising in the piecewise construction of physically relevant solutions to models of shallow water fluid flows, we study the initial value problem for quasilinear hyperbolic systems of conservation laws in dimensions when the initial data are continuous with “corners,” i.e., derivative discontinuities. While it is well known that generically such discontinuities propagate along characteristics, under which conditions the initial corner points may fission into several ones, and which characteristics they end up following during their time evolution, seems to be less understood; this study aims at filling this knowledge gap. To this end, a distributional approach to moving singularities is constructed, and criteria for selecting the corner-propagating characteristics are identified. The extreme case of initial corners occurring with at least a one-sided infinite derivative is special. Generically, these gradient catastrophe initial conditions for hyperbolic systems (or their parabolic limits) can be expected to evolve instantaneously into either shock discontinuities or rarefaction waves. It is shown that when genuine nonlinearity does not hold uniformly and fails at such singular points, the solutions’ continuity along with their infinite derivatives persist for finite times. All the results are demonstrated in the context of explicit solutions of problems emerging from applications to fluid flows.