Transonic Flows in Multidimensional Conservation Laws

The project explores new mathematical frameworks for transonic flows in multidimensional conservation laws. The conservation principles are fundamental to fluid mechanics and are widely applied in various engineering contexts. Empirical modeling is often utilized, and these equations may be part of a larger system that includes phenomena such as multiphase flow and flow in porous media. They are also essential for modeling aerodynamics to assess whether an aircraft can fly at relatively high speeds in relation to the speed of sound in the surrounding air, while also considering both economic and environmental factors. This project aims to develop schematics that identify ansatz, advancing our mathematical understanding of multidimensional flows by providing physical insights. The project involves workforce development at both undergraduate and graduate levels by providing cutting-edge research opportunities. This project aims to invigorate mathematical research at the institution in a rural and remote area and energize the department by fostering a culture of investigation in applied mathematics in general. The overarching goal of this project is to develop a unified mathematical framework for solving transonic problems for the compressible Euler system. Specifically, the project aims to develop a new method that utilizes variational inequality approaches to formulate the boundary conditions for transonic problems, particularly when there are transitions involving transonic shocks in some parts and sonic in others. It aims to develop a novel mathematical analysis to understand the solution structures of the governing flow system by finding ansatz and leveraging the co-area formulas. The project bridges to the next project of finding the correct differential forms and admissible boundary conditions for the multidimensional transonic system. Finally, the project explores deep learning methods for conservation laws, assessing their robustness, accuracy, stability, and convergence in relation to capturing transonic shocks and sonic curves. The project includes the development of a workforce in a rural and remote area, applied mathematics curricula, redesigning courses for graduate programs, and open-access software products. 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.