Steady-State Phase Transition States and Large-Time Asymptotics for the Cauchy Problem of Compressible Navier-Stokes Equations with van der Waals Equation of State
Abstract
We study the Cauchy problem for the one-dimensional isothermal compressible Navier--Stokes system with the van der Waals equation of state, and analyze the well-posedness and asymptotic stability of steady-state phase-transition solutions at subcritical temperatures. Working in Lagrangian coordinates, we use variational techniques to construct piecewise smooth steady-state solutions corresponding to two-phase equilibrium states under finite-mass perturbations. Introducing artificial-viscosity regularization, we first establish existence of the global minimizer of the regularized variational problem, unique within each fixed symmetry class; passing to the vanishing artificial-viscosity limit then yields a physically admissible steady-state solution with two phase interfaces. For initial perturbations that are small with respect to energy, piecewise-gradient, total-variation, and integral norms but may exhibit large pointwise jumps across phase boundaries, we derive uniform a priori bounds for weak solutions near the two-phase equilibrium profile. We further prove that these global weak solutions converge to the steady-state two-phase equilibrium in L∞(ℝ) as t→∞, thereby characterizing their large-time asymptotic behavior. Physically, finite mass perturbations break the uniqueness of the uniform steady state and generate infinitely many piecewise constant two-phase equilibria satisfying the Maxwell equal-area rule. Artificial viscosity acts as a selection principle that singles out the physically relevant two-interface steady-state configuration, and this two-phase equilibrium is asymptotically stable under small integral perturbations that allow large-amplitude discontinuities at phase interfaces.
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BibTeX
@misc{chen2026steadystate,
title = {{Steady-State Phase Transition States and Large-Time Asymptotics for the Cauchy Problem of Compressible Navier-Stokes Equations with van der Waals Equation of State}},
author = {Yazhou Chen and Yi Peng and Xiaoding Shi and Xiaoping Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05965},
eprint = {2610.05965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05965},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Steady-State Phase Transition States and Large-Time Asymptotics for the Cauchy Problem of Compressible Navier-Stokes Equations with van der Waals Equation of State
AU - Yazhou Chen
AU - Yi Peng
AU - Xiaoding Shi
AU - Xiaoping Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05965
UR - https://arxiv.org/abs/2610.05965
AB - We study the Cauchy problem for the one-dimensional isothermal compressible Navier--Stokes system with the van der Waals equation of state, and analyze the well-posedness and asymptotic stability of steady-state phase-transition solutions at subcritical temperatures. Working in Lagrangian coordinates, we use variational techniques to construct piecewise smooth steady-state solutions corresponding to two-phase equilibrium states under finite-mass perturbations. Introducing artificial-viscosity regularization, we first establish existence of the global minimizer of the regularized variational problem, unique within each fixed symmetry class; passing to the vanishing artificial-viscosity limit then yields a physically admissible steady-state solution with two phase interfaces. For initial perturbations that are small with respect to energy, piecewise-gradient, total-variation, and integral norms but may exhibit large pointwise jumps across phase boundaries, we derive uniform a priori bounds for weak solutions near the two-phase equilibrium profile. We further prove that these global weak solutions converge to the steady-state two-phase equilibrium in L^∞(ℝ) as t→∞, thereby characterizing their large-time asymptotic behavior. Physically, finite mass perturbations break the uniqueness of the uniform steady state and generate infinitely many piecewise constant two-phase equilibria satisfying the Maxwell equal-area rule. Artificial viscosity acts as a selection principle that singles out the physically relevant two-interface steady-state configuration, and this two-phase equilibrium is asymptotically stable under small integral perturbations that allow large-amplitude discontinuities at phase interfaces.
ER -