Qualitative dynamics in the Stefan problem with and without surface tension

本文主要研究相变描述中出现的自由边界问题。一个项目是为存在和不存在表面张力的Stefan问题开发一个统一的适定性框架。在后一种情况下，我们将展示边界的规则性如何自然地与由温度的诺伊曼导数加权的规范联系在一起。在获得（奇异）消失表面张力极限后，我们将理解在没有表面张力的情况下初始小解的长期非线性行为，结合能量方法和harnack型边界来获得Neumann导数时间衰减的尖锐下界。我们还重点讨论了表面张力下Stefan问题中产生的动态不稳定性。特别是，我们将尝试严格描述空腔填充抛物线自由边界流的熔化速率，利用它们与自然标度定律的联系，并开发强大的能量技术来执行非线性分析。此外，我们将研究扩散有限聚集（DLA）的连续体极限行为，这是一个概率晶格模型，产生与外部注入Hele-Shaw问题的病态性质相关的各种形态模式。最后，我们将研究由球对称爱因斯坦-弗拉索夫系统描述的星系的动态稳定性，其中相关的adm -质量将被证明对具有适当小中心红移的星系的测量保持摄动集是强制的，在此基础上我们将研究完全的非线性稳定性。自由边界在各种物理现象的描述中是无处不在的，如熔融、晶体生长、冲击和成核。因此，对这些问题中的不稳定性、形态变化或稳定制度进行详细的现象学理解是至关重要的。对表面张力极限消失的研究将在微观和宏观尺度的Stefan问题之间建立严格的联系，而对解的长期行为的研究将揭示在没有表面张力的情况下抛物线相变的复杂稳定机制。对Stefan和Hele-Shaw问题的融化速率的研究旨在澄清与它们的自然标度不变性之间的联系。此外，受20世纪80年代物理学家Witten和Sander工作的启发，我们将研究上述晶格概率演化与其不适定连续体类比之间的联系。DLA生长不稳定的模式，表现出某些典型晶体生长的统计普遍性，并且它对良好的连续体模型描述提出了挑战。星系动力学的研究力求严格地证实天体物理学家泽尔多维奇等人在20世纪60年代所推测的稳定性情景。

This research focuses on the free boundary problems that arise in the description of phase transitions. One project is to develop a unified well-posedness framework for the Stefan problem in both presence and absence of surface tension. In the latter case, we will show how the regularity of the boundary is naturally tied to the norms weighted by the Neumann derivatives of the temperature. After obtaining the (singular) vanishing surface tension limit, we will understand the long-term non-linear behavior of initially small solutions in the absence of surface tension, combining the energy methods and Harnack-type bounds to obtain sharp lower bounds for the time-decay of Neumann derivatives. We also focus on the dynamic instabilities arising in the Stefan problem with surface tension. In particular, we will try to rigorously describe the melting rates for the cavity-filling parabolic free-boundary flows, exploiting their connection to their natural scaling laws and developing robust energy techniques to perform the non-linear analysis. Moreover, we will study the continuum limit behavior of the Diffusion Limited Aggregation (DLA), a probabilistic lattice model giving rise to various morphological patterns associated with the ill-posed nature of the Hele-Shaw problem with external injection. Finally, we will examine the dynamic stability of galaxies as described by the spherically symmetric Einstein-Vlasov system, where the associated ADM-mass will be shown to be coercive on the set of measure preserving perturbations of galaxies with suitably small central red-shift, upon which we will investigate the full non-linearstability. Free boundaries are ubiquitous in the description of various physical phenomena, such as melting, crystal growth, shocks, and nucleation. A detailed phenomenological understanding of instabilities, morphological changes, or stable regimes in such problems is thus of fundamental importance.The proposed investigation of the vanishing surface tension limit will establish a rigorous link between the micro-and the macro-scale version of the Stefan problem, while the study of long-term behavior of solutions will reveal an intricate stabilizing mechanism for parabolic phase transitions in absence of surface tension. The study of melting rates for the Stefan and Hele-Shaw problem aims at clarifying the link to their natural scaling invariances. Moreover, inspired by the work of physicists Witten and Sander from 1980's, we will investigate the link between the lattice probabilistic evolution mentioned above, and its ill-posed continuum analog. DLA grows unstable patterns, that exhibit certain statistical universality typical of crystal growth---and it presents a challenge for a good continuum model description. The study of galaxy dynamics strives to rigorously confirm the stability scenario conjectured by the astrophysicist Zel'dovitch et al. back in the 1960's.