This project focuses on the analysis of partial differential equations arising in continuum mechanics and related fields, such as statistical mechanics and probability. Both theoretical and computational aspects are addressed. Its aim is to help advance our understanding of physical phenomena in many-particle systems and impact real-life applications. Three main areas of investigation are considered. (a) Incompressible fluid mechanics: boundary layers for linearized flows and helically-symmetric flows are studied. (b) Elasticity: the principal Investigator continues to model elasticity in polyhedral domains and to develop suitable numerical methods, in particular the Generalized Finite Element Method (GFEM); she is investigating how to obtain size estimates of inclusions from boundary measurements. (c) Solution methods for evolution equations: the investigator is further developing Green's function methods for parabolic equations, in particular Fokker-Planck equations; she also continues her work on a wave-packet solution method for variable-speed scattering problems, with applications to seismic imaging. Common themes, such as the investigation of the effect of boundaries and interfaces on continuum systems, and the use of specific techniques, such as scaling and localization, make the project a cohesive research program. Employing refined analytical tools, microlocal and harmonic analysis in particular, is warranted by the complexity of the problems studied, which feature nonlinearities in the partial differential equations, ill-posedness and instabilities, and singular geometries.
The aim of this project is to advance our knowledge of complex phenomena occurring in the mechanics of fluids and elastic solids, by utilizing a rigorous analysis of the underlying mathematical models and by devising efficient, yet accurate, computational tools to simulate them. Some of these phenomena, such as turbulence in fluids, are a common occurrence, yet they still lack a thorough understanding. A relevant trait of the project is the interplay between theoretical and computational methods, with each providing its own avenue for investigation that can shed light on different aspects of the same phenomenon. Progress on each part of the project has the potential to impact real-life applications. Vorticity created by viscous flows at container walls enhances mixing and transport in fluids with applications for example to climate and environmental modeling, and industrial processes (part a) of the project). Helically-symmetric flows arise, for instance, in modeling of blood flow (part a)). Problems with interfaces appear naturally in a variety of applications, such as determining the elastic properties of composite materials and modeling of biological processes (part(b) of the project). Imaging by elastic waves is used as a non-invasive medical diagnostic tool and in probing the earth's interior, that is, in seismic imaging, for earthquake prediction (parts (b) and (c) of the project). Parabolic equations of the Fokker-Planck type arise in probability with applications, for example, to plasma physics and economics (part (c) of the project). The project provides training opportunities for both graduate and undergraduate students.
