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arXiv · 2607.27583

Incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation

Young-Pil Choi·Byung-Hoon Hwang·Ju-Hwan Hyun
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Abstract

We derive the incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation with Bose-Einstein or Fermi-Dirac statistics. The model has a self-consistent collision structure, with the local density acting as the collision frequency and the bulk velocity and temperature determined by nonlinear quantum-weighted moments of the distribution. We work near a global quantum equilibrium under the diffusive scaling and keep the quantum parameter fixed. Uniform estimates with respect to the Knudsen number yield strong microscopic relaxation and identify the limiting infinitesimal quantum equilibrium. Using the local conservation laws, we prove the incompressibility condition, the Boussinesq relation, and strong compactness of the divergence-free velocity component and a quantum-adapted thermal mode, while the acoustic modes vanish locally by a dispersive estimate. The limiting viscous stress tensor and heat flux are identified by solving auxiliary equations for the linearized quantum Fokker-Planck operator and by expanding the local quantum equilibrium manifold. The resulting incompressible Navier-Stokes-Fourier system retains the effect of quantum statistics through its normalization constants and transport coefficients.

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Young-Pil Choi, Byung-Hoon Hwang, Ju-Hwan Hyun. 2026-07-30. Incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation. https://arxiv.org/abs/2607.27583

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