EPSCoR Research Fellows: NSF: Analysis on Models of Fluid Mechanics and Mathematical Physics

This Research Infrastructure Improvement EPSCoR Research Fellows project provides a fellowship to an Assistant Professor and training for a graduate student at the University of Nebraska-Lincoln. This work is conducted in collaboration with Hongjie Dong at Brown University. Through the fellowship, the principal investigator (PI) will explore a fundamental question about the mathematical equations used to model real-world phenomena, such as fluid motion. The project lies in the STEM disciplines of applied mathematics and mathematical physics and investigates the existence and uniqueness of solutions to these equations starting from any given initial data. These questions are crucial because reliable predictions depend on the uniqueness of solutions. The graduate student will gain exposure to cutting-edge research and benefit from networking with students and faculty at Brown. This project will enable the PI and a graduate student to conduct mathematically rigorous analysis on prominent partial differential equations in fluid mechanics and mathematical physics at Brown University. The PI will prove the uniqueness of weak solutions of the Cahn-Hilliard equation with minimal regularity assumptions, providing a rare example where convex integration techniques cannot be used to demonstrate non-uniqueness. Additionally, the PI will develop a global solution theory for the surface quasi-geostrophic equation with stochastic forcing, contributing to the broader understanding of singular stochastic partial differential equations. The PI will also receive mentoring in research collaboration, student and postdoctoral supervision, and academic service such as journal editorship. As part of this collaboration, the PI’s fellowship advisor will visit the University of Nebraska-Lincoln to deliver a colloquium and a career development session, extending the project’s benefits to a broader student audience. This project will foster long-term collaboration and student training between Nebraska and Brown University. This project is supported by the EPSCoR Research Infrastructure Improvement Program: EPSCoR Research Fellows, which supports early- and mid-career investigators in eligible jurisdictions to develop collaborations at the nation’s private, government or academic research institutions. 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.