Nonlinear Partial Differential Equations of Mechanics

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This project concerns research into mathematical models of physical systems, such as the dynamics of avalanches, motion of water waves, the underground flow of fluids, and the motion of small liquid drops on a gel. The models are chosen for their novel mathematical interest, and for the physical relevance of the system. Part of the research, notably granular avalanches and droplets, relates the mathematics to physical experiments. Applications contained in this project include carbon sequestration, a means of storing the greenhouse gas carbon dioxide in liquid form in deep underground salt water aquifers; the motion of cells in biology by a process called durotaxis; the formulation of models of granular materials such as soils, agricultural bulk grains, pharmaceutical materials. The mathematical challenge is to find models that facilitate the stable representation of granular flow, for example in draining a grain bin and hopper avoiding failure of the containing structure. The research in this project on nonlinear partial differential equations concerns properties of the equations modeling a variety of applications in mechanics. One aspect is to explore fundamental properties of dispersive equations, generalizations of the KdV equation, via the Whitham equations. The interest here is in equations for which the Whitham system fails to be hyperbolic, or for which the nonlinearity is degenerate, on hypersurfaces. Numerical simulations exhibit a range of nonlinear wave interactions that will be explored analytically. Novel models of flow in porous media and gravity currents in carbon sequestration exhibit unusual effects that were recently explored in student projects. The next steps aim to establish the rigorous mathematics behind these wave effects. Research on granular materials will build on a recent breakthrough in our understanding of how to formulate continuum models for which the time dependent equations are well-posed. This has been a problem in the field for at least thirty years. The continuing research will specify constitutive laws in the new framework and test them against prototype special flows, in numerical simulations, molecular dynamics representations and physical experiments. Finally, a student project on the motion of droplets in durotaxis has made progress on the corresponding static problem and is ready to move onto dynamics, in which the droplet surfs a ridge created through deformation of the underlying flexible substrate. This project is proceeding in tandem with physical experiments on contact lines and surface tension, at the interfaces between water, a gel, and air.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.