Math 2.0

Statistical mechanics, structure of matter Problem

Hilbert's sixth problem and the hydrodynamic limit

Builds on Boltzmann equationEuler equationsNavier–Stokes equationsChapman–Enskog theoryDiPerna–Lions renormalized solutions of the Boltzmann equation

Problem

Hilbert's sixth problem asks for a mathematical, axiomatic treatment of the physical theories in which mathematics plays a central role. Within it, Hilbert asked for a rigorous account of the limits from the atomistic picture of a gas to the motion of continua. In its standard modern form, the problem is the following. Start from Newtonian dynamics of NN hard spheres of diameter dd with random initial data. First derive the Boltzmann equation in the Boltzmann–Grad limit N→∞N \to \infty, d→0d \to 0, Nd2N d^2 fixed. Then derive the compressible Euler equations and the incompressible Navier–Stokes equations from the Boltzmann equation as the Knudsen number tends to 00, the hydrodynamic limit.

Remarks

The formal derivation of fluid equations from the Boltzmann equation is given by the Hilbert expansion and by Chapman–Enskog theory. Lanford proved in 1975 that the Boltzmann equation is the limit of hard-sphere dynamics for a short time, a fraction of the mean free time. Bardos, Golse and Levermore started in the early 1990s a program to derive weak solutions of fluid equations from the DiPerna–Lions renormalized solutions of the Boltzmann equation. Golse and Saint-Raymond completed its main step in 2004, obtaining Leray solutions of the incompressible Navier–Stokes equations in this way. In 2024 and 2025, Yu Deng, Zaher Hani and Xiao Ma announced a derivation of the Boltzmann equation from hard-sphere dynamics for long times. Combined with hydrodynamic limits, their work gives a derivation of fluid equations from hard-sphere dynamics in certain settings.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Boltzmann equation, concept, no evidence levelBoltzmann equationconceptChapman–Enskog theory, concept, no evidence levelChapman–Enskog theoryconceptEuler equations, concept, no evidence levelEuler equationsconceptMaxwell–Boltzmann distribution, concept, no evidence levelMaxwell–Boltzmann dis…conceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptH-theorem, statement, no evidence levelH-theoremstatementDiPerna–Lions renormalized solutions of the Boltzmann equation, statement, no evidence levelDiPerna–Lions renorma…statementHilbert's sixth problem and the hydrodynamic limit, problem, no evidence levelHilbert's sixth probl…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Boltzmann equation concept
  2. Chapman–Enskog theory concept
  3. Euler equations concept
  4. Maxwell–Boltzmann distribution concept
  5. Navier–Stokes equations concept
  6. H-theorem statement
  7. DiPerna–Lions renormalized solutions of the Boltzmann equation statement
  8. Hilbert's sixth problem and the hydrodynamic limit problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_sixth_problem
About this page
TagT1KAGHT39X
URIm2://T1KAGHT39X
Kindproblem
WritingW0 stub
MSC82C40, 76P05, 35Q20
StatusThis copy of the site has no build data. Run m2 status T1KAGHT39X.
AboutBoltzmann equation, Navier–Stokes equations
UsesBoltzmann equation, Euler equations, Navier–Stokes equations, Chapman–Enskog theory, DiPerna–Lions renormalized solutions of the Boltzmann equation
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.