The Navier-Stokes equations can be derived directly from the Boltzmann equation
Is it true that the Navier-Stokes equations can be derived directly from the Boltzmann equation?
Yes. Multiple rigorous mathematical studies and kinetic theory expansions demonstrate that the Navier-Stokes equations can be formally derived from the Boltzmann equation, typically in the hydrodynamic or low Knudsen number limit.
Verdict: SUPPORTED · 7 sources for, 0 against · Last checked
The retrieved literature robustly confirms that the Navier-Stokes equations (and related hydrodynamic systems) can be systematically derived from the Boltzmann equation using methods such as the Chapman-Enskog expansion, Sone expansion, or rigorous hydrodynamic limits (as Knudsen numbers approach zero). While some papers discuss the physical limits or alternative expansions (such as critiques on Hilbert's sixth problem or higher-order Burnett equations), the fundamental premise that Navier-Stokes equations are derived from the Boltzmann equation is consistently supported across kinetic theory.
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Briant, M. (2012). From the Boltzmann equation to the incompressible Navier–Stokes equations on the torus: A quantitative error estimate. doi.org. https://doi.org/10.1016/j.jde.2015.07.022
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McLennan, J. A. (1965). Convergence of the Chapman-Enskog Expansion for the Linearized Boltzmann Equation. doi.org. https://doi.org/10.1063/1.1761467
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Dadzie, S. K. (2013). A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion. doi.org. https://doi.org/10.1017/jfm.2012.546
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Deng, Y., Hani, Z., & Ma, X. (2025). Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory..
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Inamuro, T., Suzuki, K., & Yoshino, M. (2025). Derivation of the incompressible Navier–Stokes equations from the lattice Boltzmann equation: difference between the Chapman–Enskog expansion and the Sone expansion. doi.org. https://doi.org/10.1088/1873-7005/adb49f
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Alonso, R., Lods, B., & Tristani, I. (2021). From Boltzmann Equation for Granular Gases to a Modified Navier–Stokes–Fourier System. doi.org. https://doi.org/10.1007/s10955-022-02915-5
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R, B., R, S., & V, G. (2026). Dynamic Properties in a Collisional Model for Confined Granular Fluids: A Review.. doi.org. https://doi.org/10.3390/e28040454
- Is it true that smooth global solutions to the Navier-Stokes equations always exist? CONTESTED
- Is it true that turbulence arises in fluid dynamics as a nonlinear instability solution of the Navier-Stokes equations? PARTIAL LEAN
- Is it true that Boltzmann's tombstone has the equation S = k log w engraved on it? SUPPORTED
What does the evidence say?
The retrieved literature robustly confirms that the Navier-Stokes equations (and related hydrodynamic systems) can be systematically derived from the Boltzmann equation using methods such as the Chapman-Enskog expansion, Sone expansion, or rigorous hydrodynamic limits (as Knudsen numbers approach zero). While some papers discuss the physical limits or alternative expansions (such as critiques on Hilbert's sixth problem or higher-order Burnett equations), the fundamental premise that Navier-Stokes equations are derived from the Boltzmann equation is consistently supported across kinetic theory.
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