Aspects of Fluid Mechanics and Elasticity from the Point of View of Microlocal and Fourier Analysis

Mathematical aspects of fluid mechanics and elasticity will be investigated using Fourier and microlocal analysis techniques. Despite recent developments, fundamental questions remain open in understanding fluid flow and elastic behavior in solids, in particular with respect to turbulence, elastic wave propagation, and singularity formation. A main goal is to obtain qualitative, but physically relevant, information from properties of solutions to the underlying differential equations. The complex phenomena observed in physical systems correspond to ill-posedness of the equations, in the form of instability, irregularity, and non-uniqueness of the corresponding solutions. Microlocal and Fourier analysis have proven effective tools for this investigation, as they encode the smoothness, size, and oscillations in a signal accurately and efficiently. Microlocal analysis provides crucial directional information in the presence of complex geometries, such as corners and cracks. Three main problems will be addressed. The first is dissipation of enstrophy, the mean square of vorticity, for incompressible 2D and quasi-geostrophic flows, and local decay of the energy spectrum for incompressible 3D flows using the Wigner transform. The second is

anisotropic static elasticity on curved polyhedral domains with cracks. The third is identification of density and anisotropic elastic constants in the interior of a body from dynamic surface displacement-traction measurements.

The proposed research consists of problems where the exchange between mathematics and other sciences has been fruitful. Fluid turbulence is a fundamental occurrence, which still lacks a complete understanding. It affects the way fluids transport and mix other substances with implications in global climate models, fish migration, and industrial design, for example. The mechanism by which vortices form and transfer energy at different length scales is central to turbulence and is one of the problems under study. Modeling of slow crack formation is important for structural stability in engineering. Mathematical analysis proposed in the second problem under study validates the results of computer simulations, which can be used to predict failure in elastic materials under mechanical stress. Identification of elastic response in materials from remote measurements gives rise to non-invasive, diagnostic imaging of the human body, and imaging of the earth's crust in seismology and oil exploration. The investigation proposed in the third problem aims at determining a priori when sufficient information in the data exists for image reconstruction.

The overall goal of the proposal is to exploit mathematical results to advance understanding of physical phenomena with impact on real-life applications.