The main focus of this project is on problems in nonlinear differential equations in which the boundary is unknown and has to be determined: a so-called 'free boundary'. The PI is interested in the free boundary problems with nonlocal structure, where velocities of the free boundaries depend on global characteristics of the solutions. The goal is to gain a better understanding of the following aspects of such problems: the existence and uniqueness of solutions in global time; asymptotic behavior of the free boundary as time reaches zero or infinity; waiting time phenomena; and the regularity properties of the free boundaries. The PI employs the notion of viscosity solutions to address the behavior of aformentioned free boundary problems. This approach has been succesfully used to study nonlinear PDEs and local laws of motions such as 'motion by mean curvature'. The great advantage of this approach is its seamless handling of topological transitions such as pinch-off. The key-step of the theory of viscosity solutions is to establish a comparison principle. This property leads, in turn, to the existence of solutions and allows for the further study of their properties. Techniques coming from quantitative versions of the comparison principle and harmonic analysis are employed to study further properties of solutions.
The classical Stefan problem of melting ice is an example of the free boundary problems. In the Stefan problem, one is modeling the evolution of the polar ice caps, and the question of interest is the location, as a function of time, of the interface between water and ice. The particular problems to which the methods of the present proposal apply also include: the Hele-Shaw problem which models fluid motion in a narrow cell between two parallel plates; flame fronts; and the interface between oil and water in a flow. This is an area where modeling and computation are far ahead of mathematical analysis. The main obstacle for developing a well-defined notion for general motion of interfaces is that initially smooth boundaries moving under smooth velocities may develop singularities in finite time. The hitting and splitting of interfaces in flame propagation is an example. It is natural to wonder whether the continuation of the solutions is uniquely determined after pinch-off, or whether additional constitutive information might be required at the singular time. The impact of the singularity to the other part of the free boundary is another interesting question. The PI plans to investigate these questions in local and global perspectives.
