A Micro-Local and Fourier-Analytical Approach to Some Non-Linear Problems in Fluid Mechanics and Elasticity

Project Abstract: 0405803 A Mazzucato, Pennsylvania State University

A Micro-Local and Fourier-Analytical Approach to Some Non-Linear Problems in Fluid Mechanics and Elasticity

The investigator A. L. Mazzucato will address several questions in the

mathematical investigation of fluid flows and elasticity using methods

from Fourier and micro-local analysis. Micro-local analysis seeks to

identify points and directions along which a solution to partial

differential equations looses regularity, by localizing it in both space and frequency. Modern techniques in Fourier analysis consist in

decomposing a signal by testing it against a given set of waves or

wave-forms at different length scales, so that relevant information can be extracted accurately and efficiently. Turbulent flows, for example,

exhibits a complex behavior at both large and small scales. The coupling between different scales is often due to the non-linearity of the underlying equations. The investigator will concentrate on the following problems. She will study dissipation of enstrophy, the squared vorticity, for the two-dimensional Euler equations, which model inviscid fluid flow, by considering transport by irregular vector fields. Understanding how

enstrophy is dissipated is important for two-dimensional turbulence.

She will analyze certain weak solutions to the Navier-Stokes

equations, which describe the motion of viscous fluids, with globally

infinite energy by using generalized energy inequalities. Allowing for weaker control at infinity could in turn lead to refined estimates on the local behavior of solutions. She will investigate existence of mild solutions to the Navier-Stokes equations by semi-group methods in polyhedral domains, which are domains of particular interest in numerical simulations. Finally, the investigator will continue studying the inverse problem of unique identification of elastic properties by dynamic

surface measurements, exploiting the covariance of the elasticity

equations under coordinate changes. The determination of elastic

parameters has significant applications in medical imaging.

The present project stresses the inter-disciplinary nature of the analysis of partial differential equations, which mathematically model physical phenomena. Theoretical tools developed to discern subtle properties of these equations have been successfully employed in real-life problems. Micro-local analysis studies how singularities are propagated by differential equations. Changes in material properties cause singularities to form in waves and can hence be determined when direct measurement is not possible, as in seismology, oil exploration, and medical diagnostics. Fourier analysis examines the content of a signal at a given frequency or length scale. Understanding crucial aspects of turbulent flows, for

example concentration and dissipation of energy and vorticity, at

different scales has an impact in disciplines ranging from aerodynamics, to meteorology, to human physiology. The need for numerical simulation and design has underlined the role of complex geometries, where a refined mathematical analysis is often necessary for a qualitative understanding.

With this project the investigator also aims at strengthening her

collaborative effort with other female researchers both in the United

States and abroad.