Global-in-time Well-posedness of Classical Solutions to the Vacuum Free Boundary Problem for the Viscous Saint-Venant System with Large Data

Abstract

We establish the global well-posedness of classical solutions to the vacuum free boundary problem of the 1-D viscous Saint-Venant system with large data. Since the depth $\rho$ of the fluid vanishes on the moving boundary, the momentum equations become degenerate both in the time evolution and spatial dissipation, which may lead to singularities for the derivatives of the velocity u of the fluid and then makes it challenging to study classical solutions. By exploiting the intrinsic degenerate-singular structures of this system, we are able to identify two classes of admissible initial depth profile and obtain the global well-posedness theory here: $\rho_0^\alpha\in H^3$ $(\frac{1}{3}<\alpha<1)$ vanishes as the distance to the moving boundary, which satisfies the BD entropy condition; while $\rho_0\in H^3$ vanishes as the distance to the moving boundary, which satisfies the physical vacuum boundary condition, but violates the BD entropy condition. Further, it is shown that for arbitrarily large time, the solutions obtained here are smooth (in Sobolev spaces) all the way up to the moving boundary. One of the key ingredients of the analysis here is to establish some degenerate weighted estimates for the effective velocity $v=u+ (\log\rho)_y$ (y is the Eulerian spatial coordinate) via its transport properties, which enables one to obtain the upper bounds for the first order derivatives of the flow map $\eta$. Then the global regularity uniformly up to the vacuum boundary can be obtained by carrying out a series of weighted energy estimates carefully designed for this system. It is worth pointing out that the result here seems to be the first global existence theory of classical solutions with large data that is independent of the BD entropy for such degenerate systems, and the methodology developed here can be applied to more general degenerate CNS.