On the “Vacuum” Dam-Break Problem: Exact Solutions and Their Long Time Asymptotics

The classical dam-break problem for the shallow water system with a dry/vacuum downstream state is revisited in the context of exact solutions which generalize the Riemann setup of a Heaviside jump between constant states to continuous initial data. Two main setups are considered, chosen to illustrate how local properties of the dependent variables at the vacuum point influence the evolution over different time scales. For the first case, the elevation (density) variable is initially continuous but not differentiable at the dry (vacuum) point: no gradient catastrophe develops in this variable, with the time evolution eventually merging into a single (shifted) Stoker wave. Conversely, for the second case, the elevation joins the dry state with vanishing first derivative and a curvature jump: for an instant in time a global gradient catastrophe at the fixed contact point forms and immediately evolves as a Stoker parabolic simple wave, allowing the contact point to split into two moving points, one at the dry bed and one at a fixed elevation, where curvature singularities persist at all times. Although in both cases shocks develop for the velocity field, these are nongeneric, in that, in contrast to the usual case, infinitely many conservation laws are satisfied at all times. Long time evolution is further analyzed with the help of new stretched ``unfolding"" variables to extract the details of the asymptotic approach to a rarefaction Stoker-like wave.