AI 中文总结
本文证明了任意维度下 Stefan 问题两相和单相弱解自由边界的稳定性与连续性，通过能量守恒原理和新的下密度界控制边界位移，并展示了弱解渐近行为的定性与定量结果。
AI 中文摘要
本文证明了任意维度下 Stefan 问题两相和单相弱解的自由边界的稳定性与连续性。我们的证明包含两个主要要素。第一个要素是弱解的能量守恒原理，我们在单相和两相情形下对一般区域推导了该原理；这一能量守恒原理提供了自由边界位移的连续性模量，并且是分析解的长时间行为的灵活工具。控制自由边界位移的第二个关键要素是，在熔融区域中以最大速度移动的自由边界点上，我们建立了一个新的下密度界。我们还讨论了一些应用，展示了弱解渐近行为的一系列定性和定量结果。
英文摘要
In this paper we prove the stability and the continuity of the free boundaries of two-phase and one-phase weak solutions to the Stefan problem in any dimension. Our proof has two main ingredients. The first one is an energy conservation principle for weak solutions that we derive for general domains in both the one-phase and the two-phase settings; this energy conservation principle provides the modulus of continuity for the displacement of the free boundary and is a flexible tool for analyzing long-time behavior of solutions. The second key ingredient for the control of the displacement of the free boundary is a new lower density bound for the melting region at free boundary points that move with maximal velocity. We also discuss some applications, showing a series of qualitative and quantitative results on the asymptotic behavior of weak solutions.
Comments32 pages, 6 figures
