Applied Analysis of Partial Differential Equations and Related Inverse Problems in Mechanics

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.