This project includes three directions of research. The first concerns the study of some partial differential equations that arise in the modeling of the motion of liquid droplets on a solid support (e.g., a water drop sliding down an inclined plane). This part of the research focuses on two particular equations: the thin film equation and the quasi-static approximation. The main feature of both of these models is the presence of a moving contact line (the boundary of the contact region between the drop and the solid support) whose motion is not known a priori. These models are thus examples of "free boundary problems," whose mathematical analysis is very challenging. The research focuses on the questions of existence of solutions, their regularity, and their long-time behavior (especially their convergence to traveling-wave-type solutions). The second direction of research concerns certain nonlocal, third-order parabolic equations that arise, in particular, in the modeling of hydraulic fractures. These equations are reminiscent of the thin film equation, but involve nonlocal singular integral operators (such as the half-Laplacian). The project aims at developing a full existence and regularity theory for such equations. Though parts of the theory developed over the years for the thin film equation seem to adapt readily to this equation, there are important differences due to the nonlocal character of the operator. As a consequence, the existence of solutions is not presently known in many physically important cases. The last direction of research concerns the study of anomalous diffusion phenomena. This is part of a broad program initiated by the principal investigator to study anomalous diffusion regimes arising as limits of kinetic-type models. He intends to push this program to study anomalous heat conduction in chains of anharmonic oscillators. In such chains, heat is transported by vibrations that can be modeled as a gas of phonons, whose evolution is modeled by the Boltzmann phonon equation. By studying asymptotic regimes for this equation, the principal investigator seeks to derive a nonlinear anomalous Fourier law for heat conduction.
Accurately modeling the motion of liquid droplets is an important problem in fluid mechanics with many applications in engineering. The physical phenomena are extremely complex (the motion of the fluid inside the droplet and its behavior at the edge of the droplet both involve very complicated equations), and many simplified models have been proposed. This project focuses on the mathematical analysis of some of those models with the aim of better understanding their fundamental properties. Ultimately, the goal is to compare these properties with experiments to validate (or invalidate) the various models. Another aspect of the project involves equations that arise in the modeling of hydraulic fracture. (Hydraulic fracturing, or "fracking," consists in propagating rock fractures by the injection of fluids with very high pressure. It is involved, for instance, in the extraction of shale gas.) The project addresses some fundamental questions concerning these equations, such as the existence and regularity of solutions. This is important, since without a proper mathematical theory it is very difficult to develop accurate and trustworthy numerical methods. The project will thus lead to a better understanding of the properties of these widely used models and provide a framework for developing accurate computer-based numerical simulations. Finally, this research program includes the training and mentoring of students. Indeed, this proposal offers many opportunities for both graduate and undergraduate students to work on accessible research projects with physically relevant applications.
