The rigorous hydrodynamic limit from the Boltzmann equation to the compressible Euler equations in a half-space is open for the Maxwell reflection boundary condition with a fixed partial accommodation coefficient α∈(0,1)\alpha\in(0,1). In this regime the leading-order viscous boundary layer satisfies nonlinear compressible Prandtl equations with Dirichlet-type wall data induced by matching with the interior Euler trace; the well-posedness of this system in Sobolev-type spaces is unknown, and the coupled Knudsen-viscous-layer Hilbert expansion has not been closed. The problem asks for a local-in-time derivation of the compressible Euler limit (with the boundary condition inherited from the kinetic wall model) for α∈(0,1)\alpha\in(0,1).

Background

The Boltzmann equation describes a rarefied gas through f(t,x,v)f(t,x,v), the density of molecules at position xx with velocity vv; binary collisions are modeled by a bilinear collision operator Q(f,f)Q(f,f) (for hard-sphere molecules, an integral operator over grazing and separating collision pairs). The Knudsen number ε\varepsilon is the ratio of the molecular mean free path to the macroscopic length scale, and the hydrodynamic limit problem asks whether, as ε→0\varepsilon\to 0, solutions of the scaled Boltzmann equation converge to solutions of fluid equations. In the whole space, Caflisch (1980) justified the compressible Euler limit for short-time classical solutions via the Hilbert expansion: ff is expanded as μ+∑k≥1εkFk\mu+\sum_{k\ge 1}\varepsilon^{k}F_{k}, where μ\mu is a local Maxwellian (a Gaussian equilibrium) whose parameters — density ρ\rho, velocity u\mathfrak{u}, temperature TT — solve the compressible Euler system.

In a confined domain the equation needs a boundary condition modeling gas–wall interaction. The Maxwell reflection boundary condition states that the distribution of molecules leaving the wall is the sum of (1−α)(1-\alpha) times the specularly reflected incoming distribution and α\alpha times a wall Maxwellian at the wall's velocity and temperature; the accommodation coefficient α∈[0,1]\alpha\in[0,1] is the fraction of molecules fully thermalized by the wall (α=0\alpha=0 is pure specular reflection, α=1\alpha=1 complete diffuse re-emission). For a gas in the half-space {x3>0}\{x_{3}>0\}, the interior Euler flow must satisfy an impermeability condition u⋅n=0\mathfrak{u}\cdot n=0 at the wall, and the Hilbert expansion must be corrected near the boundary by two coupled layers (formally derived in the monographs of Sone summarized in the sources): the Knudsen layer, a kinetic layer of thickness ε\varepsilon solving half-space steady kinetic equations, and the viscous or Prandtl layer of thickness ε\sqrt{\varepsilon}, whose profiles solve Prandtl-type equations — the compressible analogue of the classical Prandtl boundary-layer equations, in which tangential momentum and temperature diffuse in the normal variable while being advected by the layer's own convective field. When 0<α<10<\alpha<1, Dirichlet-type wall data (prescribed values) are imposed on the leading viscous-layer profiles by matching with the interior Euler trace, with decay as the stretched normal variable tends to infinity, while the Knudsen-layer solvability conditions supply the higher-order slip-type boundary conditions of the interior fluid, with slip coefficients determined by the half-space kinetic problem.

The structure of the coupling depends sharply on α\alpha. As recounted in the sources, for α=0\alpha=0 the Knudsen-layer problem imposes four solvability conditions while the compressible Euler system accepts only one boundary condition, and this mismatch forces a Prandtl layer with Neumann-type data; in the weak-accommodation regime α=O(ε)\alpha=O(\sqrt{\varepsilon}) the induced viscous-layer data are instead of Robin type (a prescribed linear combination of the value and the normal derivative); for α=O(1)\alpha=O(1) the Knudsen-layer problem has a single solvability condition plus a decay condition at infinity. When α=1\alpha=1 with wall data matched to the interior fluid trace, the leading Knudsen layer vanishes and the strong viscous layer is absent. When 0<α<10<\alpha<1, the leading viscous-layer profiles carry Dirichlet-type wall data induced by matching with the interior Euler trace and solve the nonlinear compressible Prandtl equations, while the Knudsen layer supplies the higher-order slip-type fluid boundary conditions. Local well-posedness of this unsteady nonlinear system — even local in time, in Sobolev-type spaces — is not known; Gevrey regularity (a scale between C∞C^{\infty} and analytic) is the strongest class in which it is currently understood.

Rigorous compressible Euler limits in the half-space are known for α=0\alpha=0 (Guo–Huang–Wang 2021, Hilbert expansion with specular boundary condition) and for α=O(ε)\alpha=O(\sqrt{\varepsilon}) (Jiang–Luo–Tang) — in both regimes all boundary layers are linear because they appear only in higher-order terms — and for α=1\alpha=1 with wall data matched (or a small perturbation thereof) to the interior fluid trace, where the strong layers degenerate (Jiang–Luo–Tang, Trans. AMS 2024). Jiang–Wu (2025) proved the acoustic limit — the linearized counterpart, in which the interior system is the acoustic system (the linearization of compressible Euler describing sound waves) with a slip boundary condition — for the full range 0<α≤10<\alpha\le 1 with hard-sphere collisions; in that linear regime the viscous-layer equations reduce to heat equations, so the nonlinear obstruction does not appear. What remains open is the genuinely nonlinear, genuinely partial-accommodation case 0<α<10<\alpha<1: deriving the compressible Euler limit while controlling the nonlinear compressible Prandtl layer at leading order, as formally predicted by Sone's asymptotic program.

Problem Statement

Consider the scaled Boltzmann equation with hard-sphere (or, more generally, angular-cutoff) collisions in the half-space R+3\mathbb{R}^{3}_{+}, under the Maxwell reflection boundary condition with a fixed accommodation coefficient α∈(0,1)\alpha\in(0,1) independent of the Knudsen number ε\varepsilon.

Derive rigorously, for smooth initial data compatible with the expansion and on a time interval independent of ε\varepsilon, the compressible Euler limit as ε→0\varepsilon\to 0: convergence of the solutions of the scaled Boltzmann equation to the local Maxwellian determined by a local-in-time classical solution of the compressible Euler system (with the impermeability condition u⋅n=0\mathfrak{u}\cdot n=0 at the wall), together with the identification of the boundary condition that the limiting Euler system inherits from the solvability of the α∈(0,1)\alpha\in(0,1) half-space Knudsen-layer problem. The local well-posedness of the nonlinear compressible Prandtl boundary-layer system that arises in this regime, in a function class sufficient to close the construction (open in Sobolev-type spaces), and remainder estimates uniform in ε\varepsilon on that time interval, are part of the problem in this regime.

References

  1. Ning Jiang and Yulong Wu, 'Kinetic-fluid boundary layers and acoustic limit for the Boltzmann equation with general Maxwell reflection boundary condition', Kinetic and Related Models 18 (2025) 633-663, DOI: 10.3934/krm.2024029 (preprint arXiv:2501.08707), https://doi.org/10.3934/krm.2024029
  2. Ning Jiang, Yi-Long Luo, and Shaojun Tang, 'Compressible Euler limit from Boltzmann equation with complete diffusive boundary condition in half-space', Transactions of the American Mathematical Society 377 (2024) 5323-5359, DOI: 10.1090/tran/9197 (preprint arXiv:2104.11964), https://doi.org/10.1090/tran/9197
  3. Ning Jiang, Yi-Long Luo, and Shaojun Tang, 'Compressible Euler limit from Boltzmann equation with Maxwell reflection boundary condition in half-space', arXiv:2101.11199 (2021), https://arxiv.org/abs/2101.11199
  4. Yan Guo, Feimin Huang, and Yong Wang, 'Hilbert Expansion of the Boltzmann Equation with Specular Boundary Condition in Half-Space', Archive for Rational Mechanics and Analysis 241 (2021) 231-309, DOI: 10.1007/s00205-021-01651-6, https://doi.org/10.1007/s00205-021-01651-6
  5. Russel E. Caflisch, 'The fluid dynamic limit of the nonlinear Boltzmann equation', Communications on Pure and Applied Mathematics 33 (1980) 651-666, DOI: 10.1002/cpa.3160330506, https://doi.org/10.1002/cpa.3160330506
  6. Ling-Bing He, Ning Jiang, and Yulong Wu, 'Boltzmann boundary layer equation with Maxwell reflection boundary condition and applications to fluid limits', arXiv:2501.08733 (2025), https://arxiv.org/abs/2501.08733
  7. Ning Jiang, Yi-Long Luo, and Yulong Wu, 'Knudsen boundary layer equations for full ranges of cutoff collision kernels: Maxwell reflection boundary with all accommodation coefficients in [0,1]', arXiv:2407.02852 (2024), https://arxiv.org/abs/2407.02852
  8. Ya-Guang Wang and Yi-Lei Zhao, 'Well-posedness of the compressible boundary layer equations with data in the Gevrey class', arXiv:2604.15878 (2026), https://arxiv.org/abs/2604.15878
  9. Yan Guo and Yong Wang, 'Steady Prandtl Layer for Compressible Fluids', arXiv:2509.13988 (2025), https://arxiv.org/abs/2509.13988
  10. Yan Guo and Yong Wang, 'Validity of Prandtl Expansion for Steady Compressible Navier-Stokes-Fourier Flows', arXiv:2501.13360 (2025), https://arxiv.org/abs/2501.13360
  11. Ning Jiang, Chao Wang, Yulong Wu, and Zhifei Zhang, 'Incompressible Euler limit from the Boltzmann equation with Maxwell reflection boundary condition in the half-space', arXiv:2506.18420 (2025), https://arxiv.org/abs/2506.18420
  12. Yoshio Sone, 'Kinetic Theory and Fluid Dynamics', Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston (2002), ISBN 0-8176-4284-6, DOI: 10.1007/978-1-4612-0061-1, https://doi.org/10.1007/978-1-4612-0061-1
  13. Ning Jiang and Yulong Wu, 'Compressible Navier-Stokes system with slip boundary from Boltzmann equations with reflection boundary: derivations and justifications', arXiv:2501.08715 (2025), https://arxiv.org/abs/2501.08715