Delta shock waves in shallow water flow

The shallow-water equations for two-dimensional hydrostatic flow over a bottom bathymetry b(x) areht+(uh)x=0,ut+(gh+u2/2+gb)x=0. It is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally lead to singular solutions featuring Dirac delta distributions. These singular solutions feature a Rankine-Hugoniot deficit, and can readily be understood as generalized weak solutions in the variational context, such as defined in [13], [22]. Complex-valued approximations which become real-valued in the distributional limit are shown to extend the range of possible singular solutions. The method of complex-valued weak asymptotics [22], [23] is used to provide a firm link between the Rankine-Hugoniot deficit and the singular parts of the weak solutions. The interaction of a surface bore (traveling hydraulic jump) with a bottom step is studied, and admissible solutions are found.