Mathematicians have recently solved a problem that has eluded scientists for over 125 years, bridging three key theories that describe the motion of fluids. This milestone addresses a long-standing challenge in unifying different levels of fluid dynamics.
The Problem Posed by David Hilbert
According to Live Science, the significance of this discovery lies in its connection to the famous list of unsolved mathematical problems proposed by legendary mathematician David Hilbert in 1900.
At the International Congress of Mathematicians, held at the Sorbonne in Paris, Hilbert presented 23 challenging problems, one of which was focused on axiomatizing physics. His sixth problem called for determining the minimum mathematical assumptions that underpin all physical theories.
Hilbert’s problem was ambitious, in fact, that mathematicians have been working on its resolution for over a century.
The task of grounding physics in mathematics is not easy, and the path to achieving this goal has been paved with incremental steps and advancements.
Uniting Three Theories of Fluid Dynamics
In March 2025, mathematicians Yu Deng from the University of Chicago and Zaher Hani and Xiao Ma from the University of Michigan published a paper suggesting that they had cracked a crucial part of this problem.
They claim to have found a way to unify three fundamental theories that describe fluid motion at different scales.
These theories, while immensely useful in engineering applications such as aircraft design and weather prediction, have historically rested on assumptions that hadn’t been rigorously proven until now.
The breakthrough doesn’t alter the theories themselves but provides a solid mathematical foundation for them.
From Microscopic to Macroscopic: The Three Theories
The breakthrough addresses the need to unite three perspectives of fluid motion, each describing the same phenomenon but from different scales.
At the microscopic level, fluids are made up of individual particles, and Newton’s laws of motion are effective in modeling their behavior.
However, this microscopic view becomes less practical when considering the collective behavior of vast numbers of particles.
In 1872, Ludwig Boltzmann tackled this challenge by developing the Boltzmann equation, which uses a statistical approach to model the typical behavior of particles in a fluid.
At the macroscopic level, fluids are treated as a continuous substance rather than a collection of individual particles. Here, the Euler and Navier-Stokes equations come into play, accurately describing fluid movement and how physical properties interrelate without considering particles at all.
The Mathematical Link
Physicists have long struggled to unify theories explaining fluid dynamics at different scales. Deng, Hani, and Ma’s breakthrough addresses this challenge by linking the statistical behavior of individual particles to the collective behavior of fluids.
Their proof involves three major steps. This work unifies Newton’s laws, Boltzmann’s equation, and the Euler and Navier-Stokes equations, marking a significant step toward solving a key problem in mathematical physics.
If confirmed, it would provide a rigorous foundation for future advancements in physics, as envisioned by Hilbert over a century ago.
