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arXiv · 2202.06340

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

Hai-Liang Li·Yuexun Wang·Zhouping Xin
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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.

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Hai-Liang Li, Yuexun Wang, Zhouping Xin. 2022-02-13. Well-posedness of classical solutions to the vacuum free boundary problem of the viscous Saint-Venant system for shallow waters. https://doi.org/10.1142/s0218202526500120

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