Analysis of Hyperbolic and Mixed-Type PDEs in Conservation Laws and Applications

This research project aims to develop new mathematical methods and techniques to analyze some nonlinear partial differential equations (PDEs) that govern fluid flows and related phenomena. Fluid flows such as gases are important in nature. Their study is crucial for understanding the dynamics in a wide range of scientific and engineering applications, including gas dynamics, material science, geometry, turbulence, and shell theory. While one-dimensional problems in this field are relatively well understood, the theory for multi-dimensional cases remains mathematically underdeveloped. This project seeks to advance the mathematical understanding of multi-dimensional conservation laws and their applications in both fluid dynamics and geometry, and will integrate research and education, therefore also contributing to the development of the future STEM workforce. The research will focus on four core problems: (1) the existence and stability of the transonic contact discontinuity in the three-dimensional axisymmetric nozzle in gas dynamics: this is a free boundary and mixed-type problem, the free boundary is characteristic, and this study will shed light and provide new methods on the general multi-dimensional theory of conservation laws; (2) the existence of a global solution to the transonic flow past a three-dimensional axisymmetric cone in gas dynamics: new ideas and techniques will be developed to solve this mixed-type PDE problem; and (3) the global smooth solution to the Gauss-Codazzi equations of isometric immersion of surfaces: a global smooth solution to the underlying hyperbolic system of balance laws yields a smooth isometric immersion of surfaces, and a longstanding open problem is to find such a global smooth solution when the curvature of the surface has the optimal decay rate and oscillations. By developing novel analytic methods for these important problems, the project will deepen understanding of multi-dimensional PDEs in fluid dynamics and geometry. It will advance knowledge in fundamental areas of mathematics and mechanics while also providing valuable training opportunities for students in applied mathematics. 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.