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-linear stability.
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.
