Recent progress in fluid dynamics is nipping at the heels of one of math’s most daunting problems. This was the challenge laid down by David Hilbert more than a hundred years ago. Unparalleled since 1900, when Hilbert first proposed his own ten unsolved problems. These challenges laid down the path for the mathematical landscape of the 20th century to follow. His sixth problem—to axiomatize physics—seems naïve. You’d hardly be alone in thinking so—it’s long been considered one of the field’s loftiest challenges by many.
What Hilbert meant by his call for “axiomatizing” physics was to find the absolutely minimal mathematical assumptions that form the basis for all theories in the discipline. He had an implicit influence from Ludwig Boltzmann’s work on gases, especially in the original description of the problem. This linkage underscores the significance of Boltzmann’s contributions, particularly his 1872 formulation of what is now known as the Boltzmann equation.
Within that grand challenge, Hilbert laid out concrete subgoals that researchers have further translated into tangible steps toward solving that challenge. At the heart of this problem lies the unification of three distinct physical theories that explain fluid motion: the microscopic, mesoscopic, and macroscopic analyses. Each one is telling us something slightly different about the same underlying reality, which is the fluid flow.
At the microscopic level, we view fluids as large aggregates of particles. We find that Newton’s laws of motion govern precisely the trajectories of these particles. At the mesoscopic region the Boltzmann equation becomes the fundamental description. It does a great job modeling the emergent group behavior of trillions of particles. Lastly, at the largest scale, the Euler or Navier-Stokes equations govern the flow of fluids and interactions of their constituent physical properties.
It is this rational ascent from level to level that is central to Hilbert’s ideal. The Boltzmann equation is the basis for the Euler and Navier-Stokes equations. This connection is not arbitrary but is instead rooted in Newton’s laws of motion. This majestic layered structure is a daunting obstacle course to any mathematician or physicist at any level.
Mathematicians Yu Deng, Zaher Hani, and Xiao Ma have scored a major victory in defense of Hilberts sixth problem. On just one of its major aims, they say they’ve broken the code. Their joint work has produced a stunning new proof that unifies the three major theories of fluid dynamics. This rigorous journey derives the macroscopic theory from the mesoscopic one. This synthesis builds an easily understandable picture of fluid behavior. It unites largely unknown microscopic laws to experimentally verified macroscopic response principles in a single derivation.
The implications of this breakthrough are profound. Deng, Hani, and Ma have mathematically united the understanding of fluid dynamics by unifying different concepts. Their contributions have both deepened our understanding of mathematics and enriched the fabric of physics. Their work demonstrates the interconnectedness between various scientific principles. It shows how advances in pure mathematics can revolutionize our understanding of the natural world.
