I have been awarded a grant from the Office of the Vice President for Research at the University of South Carolina, on a one-year project: Transition between inviscid and viscous primitive equations.

Project Summary

The primitive equations are the fundamental mathematical models governing large-scale atmospheric and oceanic dynamics. They arise from the Navier-Stokes equations under the hydrostatic approximation and form the core of modern climate and weather models. Despite their central role, the mathematical behavior of the primitive equations depends sensitively on the strength and structure of viscosity, and transitions between well-posed and ill-posed regimes remain poorly understood.

This project investigates the transition between inviscid and viscous primitive equations using fractional horizontal dissipation, which provides a continuous interpolation between these regimes. Recent joint work by the PI established the first rigorous results in two dimensions, identifying a sharp threshold separating local well-posedness from ill-posedness and proving global regularity in certain subcritical regimes. However, the underlying transition mechanisms and their robustness remain largely unexplored.

The proposed research aims to extend this framework in three directions:
(1) resolving the open transition regime between local and global regularity,
(2) understanding the effect of vertical viscosity, and
(3) initiating the study of three-dimensional models, where new mechanisms such as vorticity stretching arise.

The project will develop analytical tools that clarify how viscosity stabilizes geophysical flows, produce publishable results, and lay the groundwork for external funding applications to NSF-DMS and related programs.

USC Office of Research Awards Announcement Page