Shock Formation for One-and-A-Half-Dimensional Shallow Water Model
摘要
This paper investigates the formation of shocks for the one-dimensional shallow water equations with Coriolis forces, where a system of hyperbolic conservation laws simplified from the two-dimensional shallow water model under the assumption that the solution is independent of the y-direction is considered. By employing modulated self-similar blow-up techniques, we construct solutions that form asymptotically self-similar Burgers-type shock in finite time, starting from a set of smooth initial data. Rather than defining modulation variables by matching function values, we introduce them based on their physical interpretation and hence provide a much more clear understanding of the shock dynamics. For this end, a rigorous proof of the shock formation is demonstrated by using a bootstrap argument and the modulation theory, and tracking the evolution of the self-similar variables to control the shock structure.