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

TL;DR

The paper proves global-in-time well-posedness of classical solutions to the 1-D vacuum free boundary problem for the viscous Saint-Venant system with large data, addressing the double degeneracy that occurs as the depth vanishes at the moving boundary. By reformulating in Lagrangian coordinates and introducing a degenerate weighted framework with the effective velocity $V=U+(H_x)/\rho_0$, the authors derive global weighted energy estimates that persist up to the vacuum boundary, even when BD entropy is not available (e.g., in the physical vacuum case). The analysis handles two admissible initial profiles—one BD-entropy compliant with $\rho_0^\alpha\in H^3$ for $\frac{1}{3}<\alpha<1$, and one physical-vacuum type with $\rho_0\in H^3$—and proves local well-posedness via a modified Galerkin-Picard scheme and global-in-time estimates through transport-structure and cross-derivative embedding techniques. A key contribution is a BD-entropy-free global theory for degenerate CNS with vacuum, offering a robust method that can extend to broader degenerate fluid systems. The results also show smoothness up to the moving boundary and provide insights into the behavior of the Eulerian velocity, which may not decay to zero as time grows. Overall, this work advances the mathematical understanding of degenerate free-boundary fluid models with large data and paves the way for applying the approach to more general degenerate CNS systems.

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 $ρ$ 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: $ρ_0^α\in H^3$ $(\frac{1}{3}<α<1)$ vanishes as the distance to the moving boundary, which satisfies the BD entropy condition; while $ρ_0\in H^3$ 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 $v=u+ (\logρ)_y$ (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 $η$. 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.

Theorems & Definitions (110)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.1
• Remark 1.2
• Remark 1.3