From Differential Inclusions to Variational Problems: Theory and Applications

Worcester Polytechnic Institute, Worcester MA

Investigators

Abstract

Singularities are ubiquitous in nature. Examples include turbulence in fluid dynamics, folds in thin films, and defects in liquid crystals. The physical behavior around singularities is generally highly complex, posing great challenges to the efforts of understanding their formation, structure, and influence on neighboring regions. The knowledge of the nature of singularities is fundamental to predict a system’s behavior or to develop effective applications for a material. This project considers systems in continuum mechanics and materials science that can be described directly or indirectly by a class of partial differential equations (PDE). The main goals include developing novel mathematical methods for analyzing this class of PDE and applying such new methods to better understand the nature of their singularities. This project will offer research and training opportunities to graduate and undergraduate students. Many nonlinear PDE modeling physical systems and materials can be formulated as differential inclusions. The nature of singularities in these problems is closely related to the rigidity and flexibility properties of the relevant differential inclusions. This project contains two main themes. The first theme aims to extend the general theory for rigidity and flexibility of differential inclusions, and to develop new analytical tools to study the rigidity and flexibility of scalar and systems of conservation laws viewed as differential inclusions. Specifically, the Eikonal equation and a two-by-two system of conservation laws for isentropic elasticity will be investigated as model problems. The investigator will combine methods from differential inclusions and hyperbolic conservation laws to advance the understanding of the structure of entropy solutions for systems and generalized entropy solutions for scalar equations. The second theme seeks to develop novel perspectives and analytical methods incorporating entropies, to study second order scalar and multi-valued problems in the calculus of variations, which are closely related to the Eikonal equation. Such problems arise in various physical settings, including thin films and layered elastic materials, liquid crystals, and self-organized convection patterns. The investigator plans to meld tools from variational analysis and those developed in the first part of this project to characterize the low-energy states. This analysis will inform the complex singularity structures in different physical settings of broad practical interest. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.