Nonlinear Partial Differential Equations in Conservation Laws and Applications

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This research project is devoted to developing new mathematical methods and techniques for studying some nonlinear partial differential equations governing the fluid flow and related applications. Their study is crucial for understanding the dynamics of many applications critical for STEM including gas dynamics, engineering, materials science, geometry, fluid turbulence, shell theory, active systems in biology/biophysics, stochastic dynamics, and so on. While the one-dimensional problems are rather well understood, the general theory for the multi-dimensional case is mathematically underdeveloped. The research project will advance knowledge of the fundamental areas of mathematics and mechanics as well as applications; it will also provide opportunities for students, including those from underrepresented groups and women, to receive training in these important areas through participation in the active research in applied mathematics. The goal of the project is to investigate some nonlinear partial differential equations arising from multi-dimensional conservation laws and related applications. The research program focuses on the following topics:(1) the existence and stability of transonic contact discontinuity in gas dynamics: this is a free boundary and mixed-type problem, and this study will provide new methods and shed light on the general multi-dimensional theory of conservation laws;(2) the global smooth solutions to the Gauss-Codazzi equations of isometric immersion of surfaces: a global smooth solution to the Gauss-Codazzi equations yields a smooth isometric immersion of surfaces, and it is challenging to find such a global smooth solution for general surfaces;(3) the weak and strong solutions to the stochastic compressible Navier-Stokes equations with various types of noise: the stochastic problems for the compressible flows are underdeveloped and widely open, and many fundamental problems are difficult; and(4) global solutions to the system of active hydrodynamics in biology: active systems arise in many practical applications in biology/biophysics, and its modeling and analysis are challenging due to its complexity, while fundamental mathematical problems are widely open.The purpose of this research is to develop novel analytic methods and efficient techniques for solving some important problems in multi-dimensional inviscid and viscous compressible flows and applications, and to gain insights into the general multi-dimensional problems of conservation laws and emerging real-world applications.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.