Well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow waters

TL;DR

This paper proves the local existence and uniqueness of smooth classical solutions to a free boundary problem in shallow water models, handling degeneracy at the vacuum boundary with new energy estimates.

Contribution

It establishes the well-posedness of the viscous Saint-Venant system with a free boundary, including solutions that are smooth up to the vacuum boundary, using novel weighted energy methods.

Findings

Solutions are smooth up to the moving boundary.

The height degenerates as a singularity near the vacuum boundary.

New weighted energy estimates are developed for degeneracy handling.

Abstract

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 derived rigorously from incompressible Navier-Stokes system with a moving free surface by Gerbeau-Perthame. Our solutions (the height and velocity) are smooth (the solutions satisfy the equations point-wisely) all the way to the moving boundary, although the height degenerates as a singularity of the distance to the vacuum boundary. The proof is built on some new higher-order weighted energy functional and weighted estimates associated to the degeneracy near the moving vacuum boundary.