This project concerns the analysis of certain partial differential equations and associated inverse problems arising in mechanics, in particular continuous mechanics of deformable solids and incompressible fluids. The focus of the principal investigator is on problems where mathematics can impact both a theoretical understanding and the practical implementation in important fields such as turbulence in fluids, seismic imaging, and statistical mechanics. The goal of the project is to make qualitative predictions on the behavior of the physical systems under study, and at the same time to develop concrete, yet accurate, approximate models. The project consists of three main parts: (a) Analysis of incompressible fluid flows: (a1) vanishing viscosity limits in flows with symmetry and the associated boundary layer; (a2) transport in two-dimensional inviscid fluids, in relation to enstrophy dissipation and uniqueness of weak solutions. (b) Analysis of elastic solids: (b1) mixed boundary value/interface problems for elastostatics, and more broadly for elliptic operators, in polyhedral domains, with emphasis on the generalized finite element method; (b2) reflection and transmission of elastic waves using wave packet analysis, and applications to seismic imaging. c) Computation of Green's functions for parabolic equations: (c1) closed-form approximate Green's function of degenerate Fokker-Planck equations, and their performance in model calibration; (c2) extension to semi-linear equations. The topics under investigation relate to phenomena not yet fully understood, inherently multiscale, where direct computer simulation is challenging. A refined mathematical analysis is particularly needed in the presence of complexities, in the form for example of nonlinear equations, singular geometries, illposedness and instability as in the case of inverse problems. The principal investigator employs techniques from harmonic and microlocal analysis, combined with differential geometric ideas, to address these challenges and unify the parts of the project into a cohesive research program.
This project addresses several open issues in the mathematical analysis of elastic solids and incompressible fluids. Progress in these areas has potential impact on various disciplines in science and engineering. Turbulence, in part a) above of the project, is amplified near walls, enhancing mixing and transport in fluids with applications in many areas from climate and pollution models to models of fish migration. Elastic imaging, in part b) above of the project, has been used in seismology to study the earth's interior, with applications to earthquake prediction, and in non- invasive medical imaging, in particular elastography. Interface problems, also in part b) of the project, model physical phenomena in composite materials, such as fiber-reinforced polymers and fiberglass, with widespread applications to industry, from aerospace to health. Finally, Fokker-Planck equations, in part c) above of the project, arise in statistical mechanics of many-particle systems, and more generally in probability, with applications to semiconductors, plasma physics, and pricing of contingent claims. Results from the research carried out by the principal investigator are disseminated through participation at professional meetings and collaboration with other scholars, as well as practitioners, both in the US and abroad, further enhancing broader impact. Two current graduate students, one of which is female, are working on problems addressed in the project. In addition, the principal investigator has supervised two undergraduate students, one of which female, in research experiences related to the project.
The primary aim of this project is to understand the behavior of physical systems in mechanics, by studying mathematically the differential equations that quantities describing the system must satisfy. Continuum mechanics models the behavior of fluids and solids, assuming that material properties vary continuously from point to point. In statistical mechanics, certain quantities describing the statistics of particles in the system satisfy differential equations if the number of particles is very large. The project addresses both so-called "forward" as well as "inverse" problems. In the forward problem, the material properties are assumed known and the equations are solved for their unknowns, which are variables describing the state or dynamic evolution of the physical system, such as velocity and pressure in a fluid or displacement in an elastic solid. In the inverse problem, material properties are inferred from indirect measurement made on the physical system, using the knowledge of the solution to the modeling equations gained from the measurements. Inverse problems in mechanics have a wide range of applications, from non-invasive medical diagnostics to seismology, for instance. The project consists in three distinct, but interconnected, parts: a) the study of incompressible fluid flows, especially the behavior of the fluid at small viscosity near rigid walls; b) the study, theoretically and numerically, of composite materials, for which material properties may vary abruptly across interfaces between the different materials; c) development of efficient method to solve a certain class of differential equations, called parabolic equations, in particular Fokker-Planck equations. Parabolic equations describe, for example, changes in temperature when heating is applied to a material. Fokker-Planck equations model the evolution in time of the distribution of particles in statistical mechanics, and more generally of the distribution of events in a random process. All three parts are motivated by applications, e.g. to aerodynamics and ship design in part a), to modeling of crack formation in part b), to pricing of certain financial derivatives in part c). The following are the main outcomes of the project. For part a), the work of the Principal Investigator (PI) and her collaborators provides a quantitative and qualitative understanding of the behavior of certain fluid flows, including shear flows, in pipes and channels at small viscosity. Flows at small viscosity are almost always turbulent and often exhibit a chaotic behavior. The PI has studied two aspects of turbulent flows: mixing properties, giving examples of optimal mixers under certain physical constraints, and sensitivity to initial conditions, an important aspect in weather and climate forecasting, using "bred vectors", a well-known sensitivity algorithm. Viscosity can be significantly affected by the presence of impurities. The PI has rigorously justified Einstein's formula for the effective viscosity coefficient in a dilute suspension of spheres. For part b) of the project, the PI and her collaborators have contributed to the development of the Generalized Finite Element Method, a variant of the Finite Element Method, a popular numerical method, that is well suited for problems in composite materials. The PI has also studied the inverse problem of determining the location of small inclusions in an elastic material from surface measurements, and has derived a formula for the surface deformation that could be used to detect the presence of the inhomogeneities in the interior, without directly probing it. The PI has also obtained some estimates needed to rigorously study the coupling between an incompressible fluid and an elastic body. For part c) of the project, the PI and her collaborators have developed a new method to algorithmically compute approximate solutions to parabolic equations, in particular Fokker-Planck equations that model pricing of certain financial derivative contracts. Having an accurate, yet efficient, method to price derivatives is important in estimating model parameters from real market data, using statistical inference. The results of the projects have been broadly disseminated by the PI to communities of interests in the form of seminars at conferences and professional meetings, and through mutual exchanges between the PI and other researchers. The project has also provided training and professional development opportunities, thus enhancing its broader impacts. Four PhD students of the PI have worked on problems directly related to the project. Two other PhD students and a postdoctoral researcher collaborated with the PI on some aspects of the project. Six undergraduate students were supervised in research experiences related to the project. Of these students, two graduates and one undergraduate are female, contributing to the diversity of the project personnel.
