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 transon