Description:

Title: On Hilbert’s sixth problem: from particles to waves

Abstract: In his sixth problem, Hilbert called for the derivation of the equations of fluid mechanics—such as the Euler and Navier-Stokes equations—from first principles, by rigorously justifying Boltzmann’s kinetic theory. This entails starting from Newton’s laws for a system of N particles and taking successive limits to first obtain Boltzmann’s kinetic equation, and then deriving the equations of fluid mechanics from it.
The major landmark in the early literature is the work of Oscar Lanford (1975), who provided the first rigorous derivation of the Boltzmann equation, albeit only for short times. Hilbert’s sixth problem, however, requires a long-time version of Lanford’s result, which remained open for decades.

In recent joint work with Yu Deng (University of Chicago) and Xiao Ma (University of Michigan), we extend Lanford’s theorem to long times—specifically, for as long as the solution of the Boltzmann equation exists. This allows us to fully carry out Hilbert’s program and derive the fluid equations in the Boltzmann–Grad limit.

The underlying strategy builds on earlier joint work with Yu Deng that resolved a parallel problem in which colliding particles are replaced by nonlinear waves, thereby establishing the mathematical foundations of wave turbulence theory. In this talk, we will review this progress and discuss some future directions.