Department Colloquium | On Hilbert’s sixth problem: from particles to waves

In 1900, the mathematician David Hilbert announced a list of 23 outstanding problems for twentieth century mathematics. In his sixth problem, Hilbert called for the derivation of the equations of fluid mechanics—such as the Euler and Navier-Stokes equations—by way of rigorously justifying Boltzmann’s kinetic theory for particle systems. The scope of this program, also known as Hilbert’s program, was precisely framed in the mid-20th century through the works of Grad and Cercignani, who identified the correct limiting process involved: the Boltzmann-Grad limit. In his celebrated work, Lanford (1975) gave the first rigorous derivation of Boltzmann’s equations, albeit only valid for short times. However, Hilbert’s sixth problem requires a long-time extension of Lanford's result, which remained open for decades. In recent joint work with Yu Deng (U Chicago) and Xiao Ma (U Michigan), we extend Lanford’s theorem to long times—specifically for as long as the solution of Boltzmann’s equation exists. This allows for the full execution of Hilbert’s program, and the derivation of the fluid equations in the Boltzmann-Grad limit. The underlying strategy follows an earlier joint work with Yu Deng that resolved a parallel problem, in which colliding particles are replaced by nonlinear waves; thus establishing the mathematical foundations of wave turbulence theory. In this talk, we will review this progress, and discuss some future directions.