Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory (2503.01800v1)

Abstract: In this paper, we rigorously derive the fundamental PDEs of fluid mechanics, such as the compressible Euler and incompressible Navier-Stokes-Fourier equations, starting from the hard sphere particle systems undergoing elastic collisions. This resolves Hilbert's sixth problem, as it pertains to the program of deriving the fluid equations from Newton's laws by way of Boltzmann's kinetic theory. The proof relies on the derivation of Boltzmann's equation on 2D and 3D tori, which is an extension of our previous work (arXiv:2408.07818).

Hilbert's Sixth Problem and the Derivation of Fluid Equations from Kinetic Theory

The paper "Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory" by Yu Deng, Zaher Hani, and Xiao Ma provides a rigorous mathematical resolution to a long-standing question posed by David Hilbert in 1900, known as Hilbert's sixth problem. This problem involves deriving the laws of continuum mechanics from Newton's laws via statistical mechanics, specifically through Boltzmann's kinetic theory of gases. The authors focus on rigorously justifying the limiting processes that connect the microscopic dynamics of particle systems to macroscopic fluid equations.

Contribution and Theoretical Framework

The paper is set against the backdrop of kinetic theory and fluid dynamics. It addresses the two critical limiting processes: the kinetic limit and the hydrodynamic limit. The kinetic limit transitions the description from Newtonian particle systems, modeled by the Boltzmann equation, to mesoscopic descriptions. In contrast, the hydrodynamic limit derives macroscopic fluid equations, like the Navier-Stokes and Euler equations, from solutions of the Boltzmann equation.

The authors' primary contribution is the rigorous derivation of the compressible Euler and incompressible Navier-Stokes-Fourier equations from the underlying hard-sphere dynamics governed by Newton's laws. This derivation resolves Hilbert's original problem, providing a clear pathway from atomistic views to continuum physics.

Methodology and Proof

The proof strategy involves establishing a rigorous kinetic limit for the Boltzmann equation over long timescales to connect it effectively to fluid dynamics equations. One of the central complications in this endeavor is handling the interactions and collisions of particles in a domain like a torus ($\mathbb{T}^d$), which introduces topological complexities absent in unbounded domains. The authors introduce sophisticated analytical tools and methods for dealing with these, such as layered cluster expansions, delicate integral estimates, and new combinatorial algorithms for managing the double collision possibilities inherent to periodic domains.

In particular, the proof extends previous work on Euclidean spaces to periodic domains, where unique challenges such as potential double collisions must be managed. The paper provides a detailed account of the collisions and recollisions that particles undergo in such periodic settings.

Numerical and Bold Claims

The paper's results, validated with precise estimates, encompass long-time validity for the Boltzmann equation in any dimension up to three (with expectations for higher dimensions). Strong claims are embedded within the theoretical results, like the $O(1)$ time derivation and existence results for solution structures, even when allowing for large timescales pertinent to realistic physical and engineering scenarios. Results are quantitative and involve bounds that relate time intervals' durations to system size in controlled manners.

Implications and Future Directions

Practically, this research has substantial implications for modeling gas dynamics in confined geometries. Theoretically, it strengthens the understanding of kinetic theory and its utility in deriving continuum equations, which is central to statistical mechanics. The work presents a connective rigor that could be extended to more complex particle interactions or potentially different physical systems described collectively.

Future developments may involve relaxing assumptions (e.g., on initial conditions or boundary constraints) or addressing unresolved aspects in higher dimensions. This paper sets a robust foundation for such inquiries, potentially influencing studies across physics, mathematics, and engineering domains related to fluid dynamics and statistical mechanics.

Overall, "Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory" enriches the landscape of mathematical physics by rigorously bridging foundational mechanics with emergent hydrodynamic phenomena.