Recent Progress on Potential Singularity of the 3D Navier-Stokes Equation and Related Models by Prof Thomas Hou

Held on 20 December 2024, the seminar was jointly organised by IAS and the School of Physics and Mathematics Sciences (SPMS) in collaboration with the Graduate Students' Club of SPMS. Our distinguished speaker was Prof Thomas Hou, the Charles Lee Powell Professor of Applied and Computational Mathematics at Caltech. His lecture provided fresh insights into solving one of the seven Clay Millennium Prize Problems on the Navier-Stokes equation.

In fluid mechanics, the Navier-Stokes equation describes the flow of viscous, incompressible fluids, making it an essential tool in applications like weather prediction, aircraft engineering, and turbulence studies. Despite being formulated nearly two centuries ago, fully understanding its solutions remains challenging due to the equation's non-linear terms. Proving the existence of singularities from smooth initial conditions is one of the prestigious Millennium Prize Problems. A closely related challenge is proving the global smoothness of solutions to the 3D Euler equation, a special case of the Navier-Stokes equation with zero viscosity.

Prof Thomas Hou has made significant progress in the numerical search for solutions to the 3D Euler and Navier-Stokes equations exhibiting singular behaviour. He identified a new class of initial data that could lead to a potential singularity in the axisymmetric Euler equation. Using an adaptive mesh refinement technique, his team successfully resolved the singular profile. This potential blow-up in the Euler equation solutions suggests the possibility of similar singularities in the Navier-Stokes equation. Computer-assisted proofs have become indispensable tools in their research.

Prof Hou adopted a counter-intuitive approach by setting the viscosity relatively high, leading to a dramatic enhancement of maximum vorticity by a factor of 10⁷, far surpassing previous reports. The solution also exhibited nearly self-similar properties locally, an indicator of potential singular behaviour.

Prof Thomas Hou responding to intriguing questions from the audience.

His lecture provided a thorough overview of the Navier-Stokes problem and the current research landscape. The results and numerical techniques he presented represent a major leap forward in tackling this famous problem and could have far-reaching implications for fluid mechanics, enhancing our understanding and modelling of complex fluid flows.

 

Written by Chirantan Mitra (PhD Student, Division of Physics and Applied Physics, SPMS NTU)

"He used analogies to explain complicated concepts!"- Tan Ken-I (PhD Student, MAE NTU)

" The passion from the Professor as well as the well made slides allowed us to follow the logic." - Bai Yuanqing (PhD Student, MAE NTU)

" Prof Hou was able to explain advanced concepts so that an undergrad with minimal math knowledge can grasp his work" - Anonymous ( Undergraduate Student, MAE NTU)