Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.10175

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

Xin Zhouping·Zhang Jiawen·Zhu Shengguo
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xin Zhouping, Zhang Jiawen, Zhu Shengguo. 2025-04-14. Global-in-time Well-posedness of Classical Solutions to the Vacuum Free Boundary Problem for the Viscous Saint-Venant System with Large Data. https://arxiv.org/abs/2504.10175

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Multilayered fluid-structure interactions: existence of weak solutions for time-periodic and initial-value problems

We establish the existence of weak solutions for a class of fully coupled multilayered fluid-structure interaction systems in a three-dimensional spatial setting. The model consists of an incompressible viscous fluid interacting with a thin elastic shell, which is in turn coupled to a three-dimensional elastic solid, yielding a nonstandard $3D/2D/3D$ coupling configuration. The system is driven by time-periodic boundary forcing through Bernoulli-type pressure conditions. For sufficiently small forcing in $L^2$, we prove the existence of at least one time-periodic weak solution. A central analytical difficulty stems from the strong nonlinear coupling across interfaces of different dimensionality and the absence of classical compactness mechanisms. This challenge is overcome through a carefully designed energy framework combined with and new $L^{2}$ compactness arguments adapted to the multilayered geometry. A key structural assumption is the viscoelasticity of the three-dimensional solid, which yields additional diffusion estimates and ensures energy stability. In the purely elastic case, we establish the global-in-time existence of weak solutions to the corresponding initial-value problem, provided that no degeneration (self-contact) of the fluid domain occurs. These results extend existing theories for two-dimensional and reduced-dimensional configurations to a genuinely three-dimensional multilayered setting, providing new analytical insight into complex coupled PDE systems arising in fluid-structure interaction.

math.AP↗
arXiv2026

A linear test approach to global controllability of third- and fifth-order nonlinear dispersive equations

We investigate third- and fifth-order nonlinear dispersive equations of KdV type on the torus and establishes approximate controllability by a fixed four-dimensional control; rather than relying solely on the saturation machinery, the analysis exploits the finite-dimensional controllability of the inviscid Burgers equation linearized around a carefully constructed return trajectory, with the trajectory itself obtained from an observable family. This ``linear test" strategy, yields more information about the structure of the control than the standard approach. In particular, the constructed control is shown to depend continuously on the initial and target states, a property that is by no means automatic in nonlinear control problems, and to decompose as a bounded linear operator applied to the data plus a fixed remainder, with the operator part interestingly independent of the order of dispersion.

math.AP↗
arXiv2026

Global in-time rough large data solution to complex-valued semilinear damped evolution equations

We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-Δ)^σu+(-Δ)^δ\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with $δ\in[0,σ]$, $σ\in\mathbb{R}_+$ and $p\in\mathbb{N}_+\backslash\{1\}$, where the initial data belong to the rough space $E^α_s$ endowed with the norm \begin{align*} \|f\|_{E^α_s}=\big\|\langleξ\rangle^s\,2^{α|ξ|}\widehat{f}(ξ)\big\|_{L^2}\ \ \mbox{with}\ \ α<0, \ s\in\mathbb{R}. \end{align*} Concerning $(u_0,u_1)\in E^α_{s+\barκ}\times E^α_s$ when $s\geqslant\frac{n}{2}-\frac{2κ+\barκ-2δ}{p-1}-\barκ$ with $κ=\min\{2δ,σ\}$ and $\barκ=\max\{2δ,σ\}$ whose Fourier transforms are supported in a suitable subset of first octant, we prove a global in-time existence result without requiring the smallness of rough initial data.

math.AP↗