Large solutions for systems of hyperbolic conservation laws and wave equations in one and multiple space dimensions

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This research project is aimed at the investigation of two types of mathematical models, which are widely used to describe the wave-like motion in gas dynamics and physics of liquid crystals. The first model considered in this project is the Euler system of differential equations. As one of the most fundamental systems for fluid dynamics, the Euler system has been extensively used in physics and engineering and is among the oldest partial differential equations, first introduced in the 16th century. However, many fundamental theoretical issues pertaining to the Euler equations have remained unresolved. One of the most important open problems is how the solutions that develop strong shock waves behave over long time spans. This project aims at the fundamental understanding of this problem, following the related Principal Investigator's and collaborators’ recent progress. Another mathematical model considered in this project is a wave model describing liquid crystals. The Principal Investigator (PI) and collaborators recently found a new type of cusp-like singular solutions. This opens a new direction in this field. This research project seeks to deepen understanding of the structure of solutions after the singularity formation. This project will include several research-related undergraduate and graduate student training projects. In this project the PI aims to capture features of solutions that include singularities, such as shock wave and cusp singularities, for some nonlinear wave equations. This will answer a very important and fundamental question for these equations: How do solutions behave beyond the formation of a singularity? Both analytical and numerical techniques will be used to enhance the current understanding. The first goal of the project is to better understand the large-data solutions for the compressible Euler equations, which develop shock waves. The PI will study the existence of solutions with large total variation, the generic properties of solutions, and the optimal lower bounds on the density. The second goal is to study the global existence, uniqueness, and stability for solutions of full Poiseuille flow of nematic liquid crystals via the Ericksen-Leslie model. In a recent discovery, the formation of cusp singularities was found by the PI and collaborators. To overcome the challenge caused by the new singularity, many analytical techniques, including a new transformation of coordinates and an optimal transport metric, will be used.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.