Two-dimensional solutions of the shallow water equations in the neighbourhood of a shoreline

Two-dimensional shallow water flows when the boundary is smooth at the initial instant of time are considered. Three different flow configurations when a wave reaches the shore: a) with a finite inclination, b) with zero inclination (a long tongue) and c) with a vertical to the shoreline (breaking of the wave) are examined. Theorems in the existence and uniqueness of the solutions of the initial-boundary value problems are proved. The solutions are constructed in the form of series that converge in the neighbourhood of the shoreline. The laws of motion of the shoreline and the instants of time up to which a continuous flow pattern is maintained and after which another flow configuration emerges are found.