Dynamic Free-Boundary Problems

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A "dynamic free-boundary problem" is the problem of finding the solution of a partial differential equation in a domain whose evolution is a priori unknown. One example is the problem of modeling melting ice, where the interface of ice and water is determined dynamically by the distribution of temperature (i.e., the solution of the heat equation) in the water region. This project addresses some fundamental questions concerning free-boundary problems, such as the existence and long-time behavior of solutions. This is an important topic, since without a proper mathematical theory it is difficult to develop accurate and trustworthy numerical methods for dealing with concrete physical problems. A good example is the problem of liquid drops sliding on a tilted surface. In this case the shape of the drop can change drastically when the velocity is increased, and a singularity (corner) develops at the rear of the drop when the velocity exceeds a certain critical value. At higher speeds the tail of the drop may break into another component (pearling). Accurate modeling of the motion of drops is therefore a highly complex question in fluid mechanics, with many important applications in engineering. Another example is in the crowd motion of individuals or cars in congested areas exiting through part of the boundary of the confining domain (e.g., a room or a highway); in these examples, the exit pattern is heavily influenced by the shape of the domain as well as the location of the exit. The project aims towards a better understanding of the properties of these problems and seeks to provide a framework for developing accurate computer-based numerical simulations of the processes. The problems studied in this project arise in a variety of physical phenomena, including the phase change between liquid and solid, the motion of capillary drops, congested crowd motion, and tumor growth. Particular focus will be on the asymptotic behavior of solutions starting with general initial data, either in the context of homogenization and long-time behavior or in the "stiff-pressure" limit of nonlinear diffusion. In addition to standard methods in partial differential equations, such as integral estimates, it will often be necessary to introduce geometric methods to understand the pointwise behavior of the moving interface. The first part of the project concerns volume-preserving geometric motions and their large-time behavior. The challenge lies in possible topological changes of the interface caused by the merging and splitting of fronts. For this reason most results in this area hold only for convex surfaces. The principal investigator will introduce a modified version of the moving-planes method to investigate a more general class of interfaces and to study their convergence to equilibrium. The second subproject concerns the evolution of capillary drops on a flat or tilted surface in the quasi-static approximation regime. While many models have been proposed in the study of dynamic capillary drops, the analysis of such models is still in its early stages, due the wide range of behavior with respect to parameters such as the roughness of the surface or the volume of the drop. The aim here is to address the well-posedness and long-time behavior of solutions in various regimes, to classify possible singularities, and to investigate the process's transitional behavior upon the change of parameters. The principal investigator also proposes to study the emergence of congested zones in collective motions, for example the evolution of jammed regions in crowd motions with a density constraint, or tumor growth in the motion of cancer cells with anti-crowding pressure. The plan is to characterize the motion law of the congested zones as well as to investigate the stability and long-time behavior of the solutions. Finally, the principal investigator proposes to study interface homogenization problems in random media, where the inhomogeneity is present either in the advection vector field or in the latent heat. The goal is to understand how the inhomogeneities in the system interact with the geometry of the interfaces to affect the macroscopic behavior of the solutions. New approaches will be introduced to the investigation, using tools such as energy estimates and concentration inequalities.