The principal investigator's past research has focused on interface problems arising in a variety of physical phenomena such as phase separation, fluid dynamics, materials science, and continuum limits of nonequilibrium particle systems. The goal of the current project is to gain a better understanding of general properties of free boundary problems using various notions of weak solutions, with the help of maximum-principle-type arguments, harmonic analysis, measure theory, and interacting particle systems. Specific questions under investigation are (i) global-time existence and uniqueness; (ii) uniting notions of weak solutions; (iii) regularity properties; (iv) long-time behavior of solutions and (v) geometric properties of free boundaries. Another project is to study homogenization of the free boundaries in periodic and random media, moving with oscillating normal velocity caused by inhomogeneities in the media. An example is the dynamics of water droplets spreading on an irregular surface. The lower dimensional nature of free boundaries bring major challenges to the homogenization, especially in nonvariational settings. The goal is to prove existence, uniqueness, and stability of the effective free boundary problem in periodic and random media.
Nonlinear interface motions arise naturally in physical applications, and they have been extensively studied in the physics and applied mathematics literature. A classical example is the melting of ice pieces or snow crystals, where the central issue of interest is the evolution of the ice/water/air interface. The highly nonlinear structure and the development, in general, of finite-time singularities on the interface -- such as a stream of water pinching into droplets -- give rise to rather challenging difficulties in the rigorous analysis. It is important, both in theory and in practice, to understand what type of singularities occur and under what circumstances the behavior of the interfaces exhibits stability.
This proposal addresses several aspects of the nonlinear partial differential equations, given in a domain with general geometry. The domain can be given either a priori (fixed boundary problem), or as a part of the problem (free boundary problem). The presence of lower-dimensional structure is ubiquitous in the physical literature, either as a boundary of a domain or a singular part of an evolution. For instance we have investigated a liquid drop sliding on a tilted plane, where the interaction of the droplet boundary in contact with air or with the surface drives the evolution. Another problem we investigated was on the movement of crowd trying to exist a room, where congestion appears and inhibits the movement of the crowd. Besides the nonlinearity of the problem, the difficulty lies in the nonlocality of the problem, in the sense that the behavior of solutions highly depends on the geometric structure of the boundary. Our aim is to develop methods to deal with such difficulties and to gain better understanding on the relation between the boundary structure and the corresponding solutions. The projects resulted in findings in several phenomena in collaboration with Ph.D studens as well as junior and senior researchers in U.S. and in Japan and Korea, and the findings were announced in conference talks and in journal publications.
