Viscosity Solutions: Beyond the Wellposedness Theory

This project studies some nonlinear partial differential equations (PDE) that appear naturally in chemistry, physics, and engineering and which arise, for example, in the study of crystal growth, combustion, coagulation-fragmentation processes, game theory, and optimal control theory. These equations have connections with a host of other areas of mathematics, including the calculus of variations, differential games, dynamical systems, geometry, homogenization theory, and probability.

The main goal of the project is to discover new underlying principles and general methods to understand the properties of solutions of the PDE under investigation. A key object of the research is a crystal growth model in which the crystal grows in both the horizontal direction, by adatoms, and the vertical direction, by dislocations or nucleation in a supersaturated media.

To make practical use of the model, it is important to understand the qualitative and quantitative aspects of the growth speed and the shape of the crystal. The mentoring of graduate students in research is an important educational component of the project.

The work of the project involves two themes. The first is about critical Coagulation-Fragmentation equations and their connections with Hamilton-Jacobi equations. The Principal Investigator (PI) is interested in regularity and large-time behavior results for Hamilton-Jacobi equations which give implications on the existence of mass-conserving solutions of Coagulation-Fragmentation equations and their behavior.

The second involves level-set mean curvature flow equations with driving and source terms and applications in crystal growths and turbulent combustions. The focus is on the regularity, the large-time average, and the large-time behavior of the solutions. The PI and his collaborators have recently developed new approaches which led to solutions to several open problems in these and related areas.

The new approaches are expected to be developed further in this project, thereby bringing fresh perspectives on and insights into the study of nonlinear PDE and viscosity solutions.

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.