The PI will continue studying free boundary problems in phase transitions and fluid flows in the presence of surface tension. In particular, the PI plans to study (1) phase transitions in two-phase viscous fluid flows with variable surface tension and surface viscosity, (2) phase transitions in two-phase viscous fluid flows with kinetic undercooling, and (3) phase transitions driven by chemical potentials. Additional topics to be investigated are (4) evaporation in the presence of surface tension, and (5) stability and instability of solidifying processes in phase transitions. The dynamics of two-phase flows and the motion of their separating interfaces has been a problem of scientific and industrial interest for centuries. In the absence of phase transitions, which means that the interface is advected with the flow, this problem is fairly well understood. However, if phase transitions are present, the resulting models are fundamentally more complex, as they involve the equations of fluid dynamics in conjunction with those of phase transitions. The results anticipated will on the one hand clarify the existence and uniqueness of solutions, and on the other establish qualitative and asymptotic properties of these solutions. The PI proposes a mathematical approach that is general and flexible, and that will open up the treatment of many more problems.
Over the last decades the subject of free boundary problems has attracted increasing attention, both because of its theoretical interest and because of its numerous applications in the natural and engineering sciences. Typically, a free boundary problem consists of one or more partial differential equations that have to be solved in a domain that is a priori unknown and that has to be determined as part of the problem. Free boundary problems are in general harder to solve, both analytically and numerically, than the underlying differential equations would be in a prescribed domain. They have an inherent nonlinear structure, as two separate solutions cannot be superposed. It is not surprising, then, that the variety of applications and the inherent difficulties of solving free boundary problems have initiated a variety of theoretical methods, and in many cases have determined the direction of the development of these methods. Free boundary problems are important in many fields, such as material sciences, fluid mechanics, hydrodynamics, thermo-mechanics, magneto-dynamics, solid-state physics, plasma physics, geology, chemistry, and the biological and medical sciences. Many manufacturing processes in industry lead to free boundary problems, such as electrochemical machining, viscous sintering, the growth of crystals, injection molding processes, etching processes, solidification processes, casting processes, and chemical vapor deposition processes, to mention a few. The appropriate numerical and analytical treatment is a major challenge, both to the engineer and to the mathematician. The proposed work will have an impact for those real-world problems and applications where surface tension cannot be neglected.
