发表机构
College of Mathematics and Physics, Beijing University of Chemical Technology; School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（北京化工大学数学与物理学院; 香港中文大学（深圳）理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究含van der Waals状态方程的可压缩Navier-Stokes系统Cauchy问题，通过变分法与人工粘性正则化构造两相稳态解，证明其在亚临界温度下对允许大间断的小扰动具有渐近稳定性。
AI 中文摘要
我们研究一维等温可压缩Navier-Stokes系统（采用van der Waals状态方程）的Cauchy问题，并分析亚临界温度下稳态相变解的存在性与渐近稳定性。在拉格朗日坐标下，我们利用变分技术构造对应于有限质量扰动下两相平衡态的分片光滑稳态解。通过引入人工粘性正则化，我们首先建立正则化变分问题全局极小元的存在性（在每个固定对称类内唯一）；随后令人工粘性趋于零，得到具有两个相界面的物理可容许稳态解。对于在能量、分片梯度、总变差和积分范数意义下小、但可能在相界面上具有大逐点跳跃的初始扰动，我们推导出两相平衡剖面附近弱解的一致先验界。我们进一步证明，当t→∞时，这些全局弱解在L∞(R)中收敛到稳态两相平衡态，从而刻画其大时间渐近行为。物理上，有限质量扰动破坏了均匀稳态的唯一性，并产生满足Maxwell等面积法则的无穷多个分片常数两相平衡态。人工粘性作为一种选择原理，挑选出物理相关的双界面稳态构型，且该两相平衡态在允许相界面处大幅间断的小积分扰动下是渐近稳定的。
英文摘要
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 $L^\infty(\mathbb{R})$ as $t\to\infty$, 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.
Comments41pages, 2 figures
