Regularity problem of Axisymmetric Naiver-Stokes equations

The motions of fluids like water, air and blood are captured by a set of equations called the Navier–Stokes equations. Starting with a smooth initial state, the equations promise to tell you the future. But there’s a catch: in three dimensions, nobody knows whether that promise holds. The regularity problem asks a deceptively simple question—if you start with a perfectly smooth flow, will the equations ever spontaneously produce a singularity, an infinite spike in speed or a loss of smoothness, in finite time?

My research focuses on the regularity problem of the axisymmetric case, which can be thought of as 2.5-dimensional. An example would be a tornado. In theory, when the nonlinearity in the Navier-Stokes equation dominates viscosity, air along the z-axis will approach infinite speed. 

I study under what conditions no finite-time blow-up will occur.

Mathematical modelling

Mathematical modelling applies ODEs and PDEs to model the spread of a pandemic. Ordinary differential equations (ODEs) are the backbone of the classic SIR model—they divide people into compartments and track the compartments across time, assuming everyone mixes evenly. In real life, people also move, commute, and cluster in cities. That’s where partial differential equations (PDEs) step in. They add the dimensions of space and even age, turning the model into a dynamic map where the disease ripples across neighbourhoods, diffuses along travel routes, and hits some places harder than others.