The purpose of this project is to understand from the point of view of geometry and mathematical analysis certain physical models that incorporate either free boundaries or nonlocal (long-range) effects. In terms of the mathematics, most of these problems consist of classical field equations from physics (e.g., fluid mechanics, electromagnetism, elasticity, kinetic theory) and exhibit at least one of two important characteristics. First, they might involve a "free boundary," namely, an unknown submanifold (interface) along which the field in question has a pointwise constraint (for instance, the temperature across along the interface of a metal might depend on the curvature). These submanifolds have as much physical interest as the other quantities, and their dynamics are strongly coupled to that of the fields. The second characteristic these equations might display is nonlocality, which arises when particles or "agents" interact at large (noninfinitesimal) scales, for example, in the Boltzmann equation or the quasigeostrophic equation. This always leads to equations involving integro-differential operators, such as fractional powers of the Laplacian. The specific models studied in this project present challenging analytical problems that are especially attractive in that they highlight the limits of our understanding of nonlinear partial differential equations. In particular, they pinpoint difficulties such as the following: obtaining useful pointwise bounds for solutions (without using comparison principles, or when equations are supercritical); deriving a priori regularity estimates for equations that are both nonlinear and nonlocal; understanding the physical validity of solutions (well-posedness and breakdown); handling nonlinear effects that dominate diffusion or dispersion (again supercriticality); analyzing multiscales and disordered media (homogenization).
Nonlinear partial differential equations are ubiquitous in the natural sciences, as is well known. For this specific project, the richness of nonlocal equations and free boundary problems cover very diverse natural phenomena, for instance nucleation of phases, surface tension effects in fluids, crystal formation in metallurgy, droplet spreading, ocean-atmosphere interaction, and nonlocal electrostatics. All of these phenomena are relevant to science and engineering, for instance in materials science (composite design, dislocations), nanotechnology (microfluids, droplets), bioengineering (martensite or materials with memory), and biochemistry (nonlocal electrostatics, with great potential in medicine). A sound mathematical understanding of the respective equations would be highly beneficial to the development of these technologies.
