Well-Posedness of Classical Solutions to the Vacuum Free Boundary Problem of the Viscous Saint-Venant System for Shallow Waters in 1D

We establish the local-in-time well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow waters in 1D derived rigorously from incompressible Navier-Stokes system with a moving free surface by Gerbeau and Perthame. Our solutions are uniformly smooth up to the moving boundary, although the depth degenerates as a singularity of the distance to the vacuum boundary. The main proof is built on some elaborate higher-order weighted energy functional and weighted estimates associated to the degenerate structure of the momentum equation.