Complex and Singular Behavior in Continuum Mechanics Models

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The goal of this project is to study models in the mechanics of fluids and elastic materials characterized by a complex and potentially singular behavior in the response of the physical system. Such models are deeply rooted in applications. The project will investigate, in particular, models of faults and rock slippage in the Earth's crust, models for the behavior of slightly viscous fluids around moving bodies in aero- and hydro-dynamics, as well as around the wing of an airplane or around the propeller of a submarine, and models for effective mixing in fluids, which has applications ranging from industrial to environmental processes. A key aspect of the project is the use of rigorous mathematical analysis, combined with simulations whenever feasible, to obtain quantitative results that can be used in a predictive fashion, with potential direct societal impacts. For instance, one part of the project addresses how data from global positioning systems (GPS) can be used, through a mathematical algorithm, to locate buried faults which would be otherwise inaccessible, and to estimate the relative slip of the rock layers along the fault, a predictor of earthquakes. The project provides training opportunities for graduate and undergraduate students, particularly women and members of underrepresented groups. This project aims to study various aspects of models in elasticity and fluid mechanics in the presence of singularities. Analytical techniques will be employed primarily, but the impetus for the proposed problems comes from applications, such as models of dislocations in geophysics, optimal bounds for mixing in convection-dominated problems, and boundary layer analysis for incompressible fluids. The project consists of three separate, but connected, parts: I. Incompressible Fluid Mechanics: I.a. Optimal mixing with diffusion; I.b. Boundary layer analysis in singular domains; II. Elasticity: direct and inverse problems for models of dislocations in geophysics. The unifying aspects of the project are the presence of singularities due to discontinuities and incompatibilities in the parameters for the underlying model equations and to irregular geometries, and the focus on rigorous, quantitative estimates, such as optimal bounds in mixing and quantitative stability estimates in inverse problems. The project contributes to the development of mathematical approaches to challenging open problems by combining known techniques in a novel way, such as in mixing problems, where geometric analysis, partial differential equations, and optimal control are employed, and in advancing our basic understanding of important physical processes, such as anomalous diffusion in turbulent mixing and modeling of interseismic build-up along faults and microseismicity, which may impact other fields, in particular geophysics and engineering.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.