Stability, Uniqueness, and Existence for Solutions of Hyperbolic Conservation Laws and Nonlinear Wave Equations

The compressible Euler system, first introduced in the sixteenth century, is amongst the oldest partial differential equation models, and it is extensively used in physics and engineering, for instance, to model gas dynamics. However, fundamental theoretical issues, relevant to its applications, remain unresolved. This project will bring insights in two of these still open issues: the existence and stability of solutions that develop strong shock waves, that is small regions where large changes in physical properties occur, over long-time spans. The investigator will also consider a wave model to study the time-evolution of the structure of defects that form in liquid crystals, which are materials used for example in display devices, where understanding and controlling the structure of defects has technological ramifications. This project will also offer research-related training opportunities for undergraduate and graduate students. In this project, the investigator will address a long-standing and fundamental question for compressible Euler equations and for a wave model for nematic liquid crystals: How do solutions behave beyond the formation of a singularity, such as shock waves? Both analytical and numerical techniques will be used to enhance the current understanding. The first goal of the project is to study the stability of solutions of the compressible Euler equations, which develop shock waves, and the existence of solutions with large total variation. A second goal is to study the singularity formation, global existence, and stability for solutions of Poiseuille flow of a nematic liquid crystal in a tube, via the wave-type Ericksen-Leslie model. To overcome the challenge caused by the singularity formation, the analytical techniques used will include a new transformation of coordinates and an optimal transport metric. 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.