Abstract:
This talk provides an overview of the physical intuition behind the equations, highlights key historical developments, and presents examples from various fluid types. Once a symbol of deterministic science, the Navier--Stokes equations now face phenomena such as turbulence, where predictability breaks down.
Few problems in mathematics have so profoundly deepened the interaction among different areas -- analysis, algebra, geometry, numerics, probability and physics.
From a mathematical perspective, we will explore what is currently known and reflect on the major open questions -- culminating in the celebrated Clay Millennium Problem, which continues to inspire both theoretical and applied research. We will also mention some modest contributions by the author, particularly in non-standard models.