Viscous pressureless flows with free boundary in one space dimension: The constant viscosity case
Xin Liu
TL;DR
The paper analyzes 1D viscous pressureless flows with a free boundary under constant viscosity, proving global well-posedness for arbitrarily large smooth initial data and showing the domain remains bounded with a terminal size determined by the initial data, expressed via $|\Omega(\infty)|=\int_{a(0)}^{b(0)} e^{-\frac{1}{\mu}\int_{a(0)}^{y} ρ_0 u_0 dy'} dy$. It then extends to an almost pressureless Navier–Stokes–Fourier system, establishing global existence for large data when a large parameter $M$ is chosen, and showing the flow domain stays compact, i.e., the NSF flow is a perturbation of the viscous pressureless flow via a Lagrangian energy-method framework. The analysis combines a Lagrangian reformulation with sharp energy and a priori estimates to control the evolution, yielding exponential velocity decay in the viscous case and a closed perturbative scheme in the NSF case. The appendix demonstrates how degenerate (density-dependent) viscosity leads to qualitatively different long-time behavior and self-similar structures, underscoring that constant viscosity is crucial for the established confinement and global regularity results.
Abstract
We establish the global well-posedness of the free boundary problem of the viscous pressureless and almost pressureless heat conductive flows in one space dimension. In both cases, arbitrarily large but smooth initial data is considered, and the evolving fluid domains remain bounded for all time. In the viscous pressureless case, we are able to identify the terminal flow domain in terms of the initial data. In the viscous almost pressureless case, we construct the flow as a perturbation of the viscous pressureless flow, and establish the first result for the Navier-Stokes-Fourier system in the current setting.