The rigorous derivation of the compressible Euler equations from the Boltzmann equation on a bounded domain requires constructing a Knudsen boundary layer — a steady kinetic half-space problem in the stretched normal variable. For a two-dimensional disk with specular reflection, the combination of non-zero boundary curvature and non-zero tangential velocity of the Euler flow corrupts this layer problem with geometric correction terms whose coefficients grow in the normal variable and with a drift term that defeats the standard energy method. Huang, Ouyang and Wang (arXiv:2504.14932, 2025) justified the Hilbert expansion to the compressible Euler equations in this setting only for a layer truncated to a finite normal interval and extended by a cutoff, under the technical restriction a<2/3\mathfrak{a}<2/3 on the truncation length d=ε−ad=\varepsilon^{-\mathfrak{a}}, and only on the lifespan of the smooth Euler solution. The open problem is to remove this truncation: construct the steady Knudsen boundary layer with geometric correction and non-zero tangential velocity on the entire physically meaningful normal domain η∈(0,ε−2)\eta\in(0,\varepsilon^{-2}) (in the disk η=(1−r)/ε2\eta=(1-r)/\varepsilon^2, and the geometric coefficient ε2/(1−ε2η)\varepsilon^2/(1-\varepsilon^2\eta) is regular exactly on this interval, its singular endpoint corresponding to the disk center r=0r=0), with existence, uniqueness, and exponential space decay, and thereby justify the Hilbert expansion without the a<2/3\mathfrak{a}<2/3 restriction and cutoff artifact.

Background

The Boltzmann equation governs the evolution of the phase-space density F(t,x,v)F(t,x,v) of a dilute gas; the Knudsen number Kn is the ratio of the mean free path to the macroscopic length scale. In 1912 Hilbert introduced a systematic expansion of FF in powers of Kn; at each order the collision invariants — the functions whose inner product with the collision term QQ vanishes identically, corresponding to conservation of mass, momentum and energy — force the leading coefficients to satisfy macroscopic equations, and for the rescaled equation ∂tF+v⋅∇xF=ε−2Q(F,F)\partial_t F + v\cdot\nabla_x F = \varepsilon^{-2}Q(F,F) (with ε\varepsilon the square root of the Knudsen number, the convention of the source paper) the zeroth-order coefficient is a local Maxwellian — the equilibrium velocity distribution determined pointwise by the local density, bulk velocity and temperature — governed by the compressible Euler equations. Caflisch (following Grad and Ukai–Asano) made this expansion rigorous: as ε→0\varepsilon\to 0, smooth solutions of the Boltzmann equation converge to the smooth Euler solution on its lifespan. Carrying this program out on a bounded domain with a physical boundary is far harder, because the Euler solution cannot satisfy the kinetic boundary condition and a Knudsen boundary layer — a stationary kinetic problem on the half-line of stretched normal distance η\eta (a Milne-type problem) — must be constructed at every order of the expansion. The specular reflection boundary condition, F(t,x,v)=F(t,x,v−2(v⋅n(x))n(x))F(t,x,v)=F(t,x,v-2(v\cdot n(x))n(x)) on the boundary with nn the inward unit normal (gas molecules bounce elastically off the wall), is the natural condition for an isothermal wall and conserves energy. For the flat half-space, Guo–Huang–Wang (Arch. Ration. Mech. Anal. 2021) justified the Hilbert expansion with specular reflection, later extended to soft potentials by Ouyang–Wang (J. Differential Equations 2025); the required flat Knudsen layer solutions with exponential space decay were built by Huang–Jiang–Wang (Acta Math. Appl. Sin. Engl. Ser. 2023), following classical half-space theory. None of these works contains geometric terms, because a plane boundary has zero curvature. On a curved domain the boundary layer equation acquires a geometric correction: in polar coordinates near the boundary of the unit disk, with η\eta the stretched normal variable, the layer satisfies (equation (1.32) of the source) a stationary Boltzmann equation of the schematic form v1 ∂ηf  −  ε21−ε2η(v22 ∂v1f−v1v2 ∂v2f)  +  uτ02T0 ε21−ε2η v1v2 f  +  Lf=S,v_1\,\partial_\eta f \;-\; \frac{\varepsilon^2}{1-\varepsilon^2\eta}\Big(v_2^2\,\partial_{v_1} f - v_1 v_2\,\partial_{v_2} f\Big) \;+\; \frac{u_\tau^0}{2T^0}\,\frac{\varepsilon^2}{1-\varepsilon^2\eta}\, v_1 v_2\, f \;+\; L f = S, where v1v_1 is the normal velocity, LL (denoted L0\mathbf{L}_0 in the source) is the linearized collision operator, SS is a source determined by lower-order coefficients, T0T^0 is the wall temperature, and uτ0u_\tau^0 is the tangential velocity of the Euler flow at the boundary. In the disk the stretched variable is η=(1−r)/ε2\eta=(1-r)/\varepsilon^2, so the geometric coefficient ε2/(1−ε2η)\varepsilon^2/(1-\varepsilon^2\eta) is regular precisely for η<ε−2\eta<\varepsilon^{-2}, its singular endpoint η=ε−2\eta=\varepsilon^{-2} corresponding to the disk center r=0r=0. Wu (J. Differential Equations 2016) solved this geometrically corrected layer problem with inflow boundary data and justified the incompressible Navier–Stokes limit in the disk — but only in the case uτ0=0u_\tau^0=0, an assumption reasonable for diffusive limits yet untenable for the compressible Euler limit, where a generic smooth flow has non-zero tangential boundary velocity. When uτ0≠0u_\tau^0\neq 0, the drift term uτ02T0ε21−ε2ηv1v2 f\frac{u_\tau^0}{2T^0}\frac{\varepsilon^2}{1-\varepsilon^2\eta}v_1v_2\,f produces, in the L2L^2 energy estimate, a contribution whose analogue in shear-flow and Couette-flow problems of the Boltzmann equation is known to permit at best polynomial velocity decay — insufficient to construct layers of arbitrary order; the authors state that because of this term 'it is hard to solve it in η∈(0,+∞)\eta\in(0,+\infty)'. Huang–Ouyang–Wang (2025) circumvented this by truncating the layer to η∈(0,d)\eta\in(0,d) with d=ε−ad=\varepsilon^{-\mathfrak a}, a<2/3\mathfrak a<2/3 (a restriction they flag as 'due to some technical requirement in the proof'), solving the truncated problem with a refined L2L^2–L∞L^\infty method under moment solvability conditions on the boundary data, extending the layer by a cutoff, and closing the remainder estimate — thereby justifying the Euler limit in the disk, but only with this truncation artifact and only on the Euler lifespan t∈[0,τδ]t\in[0,\tau^\delta].

Problem Statement

Construct the global steady Knudsen boundary layer with geometric correction and non-zero tangential velocity for the Boltzmann equation on the unit disk with specular reflection, and thereby complete the non-truncated Hilbert-expansion justification of the compressible Euler limit. Concretely: for the Knudsen-layer equation of Huang–Ouyang–Wang (their equation (1.32), a stationary Boltzmann equation in the stretched normal variable η\eta with curvature terms of coefficient ε2/(1−ε2η)\varepsilon^2/(1-\varepsilon^2\eta) and a drift term proportional to the boundary tangential velocity uτ0≠0u_\tau^0\neq 0, with linearized collision operator LL (denoted L0\mathbf{L}_0 in the source) and source SS), prove existence, uniqueness, and exponential decay in η\eta of solutions on the entire physically meaningful normal domain η∈(0,ε−2)\eta\in(0,\varepsilon^{-2}) — the geometric coefficient ε2/(1−ε2η)\varepsilon^2/(1-\varepsilon^2\eta) is regular precisely on this interval, and its singular endpoint η=ε−2\eta=\varepsilon^{-2} corresponds to the disk center r=0r=0; the equation (1.32) is required to hold on the sub-domain where the layer is non-negligible, its exponential decay in η\eta rendering the blow-up of the geometric coefficient at the singular endpoint irrelevant and confining the layer effectively to a neighborhood of the boundary, away from this endpoint — with prescribed boundary data at η=0\eta=0 satisfying the moment solvability conditions (the three flux-type moment conditions on the boundary datum fbf_b, equation (1.37) of the source, under which the stationary half-space problem is solvable) and with behavior controlled near the grazing set {v1=0}\{v_1=0\} (the velocities tangent to the wall, where the transport coefficient v1v_1 degenerates). The target is the un-truncated layer theorem (existence, uniqueness and exponential decay on the entire normal domain η∈(0,ε−2)\eta\in(0,\varepsilon^{-2})) together with its use at every order of the Hilbert expansion to remove the truncated domain η∈(0,d)\eta\in(0,d), d=ε−ad=\varepsilon^{-\mathfrak a} with a<2/3\mathfrak a<2/3, and the cutoff extension of the layers, so that the convergence of Boltzmann solutions with specular boundary condition to the compressible Euler solution in the disk is justified without these truncation artifacts.

References

  1. Feimin Huang, Jing Ouyang, Yong Wang: "Hilbert expansion of the Boltzmann equation on a 2-dimensional disk with specular boundary condition", arXiv:2504.14932 (2025). DOI: 10.48550/arXiv.2504.14932, https://arxiv.org/abs/2504.14932
  2. Yan Guo, Feimin Huang, 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
  3. Jing Ouyang, Yong Wang: "Hilbert expansion of Boltzmann equation with soft potentials and specular boundary condition in half-space", Journal of Differential Equations 414 (2025) 143–200. DOI: 10.1016/j.jde.2024.09.001, https://doi.org/10.1016/j.jde.2024.09.001 (also arXiv:2310.02337, https://arxiv.org/abs/2310.02337)
  4. Fei-min Huang, Zai-hong Jiang, Yong Wang: "Boundary Layer Solution of the Boltzmann Equation for Specular Boundary Condition", Acta Mathematicae Applicatae Sinica, English Series 39 (2023) 65–94. DOI: 10.1007/s10255-023-1031-1, https://doi.org/10.1007/s10255-023-1031-1 (published online 2022-12-28)
  5. Lei Wu: "Hydrodynamic limit with geometric correction of stationary Boltzmann equation", Journal of Differential Equations 260 (2016) 7152–7249. DOI: 10.1016/j.jde.2016.01.024, https://doi.org/10.1016/j.jde.2016.01.024 (also arXiv:1409.3945, https://arxiv.org/abs/1409.3945)
  6. Lei Wu: "Boundary layer of the Boltzmann equation in 2-dimensional convex domains", Analysis & PDE 14 (2021) 1363–1428. DOI: 10.2140/apde.2021.14.1363, https://doi.org/10.2140/apde.2021.14.1363 (preprint arXiv:1805.07771, https://arxiv.org/abs/1805.07771)
  7. Ling-Bing He, Ning Jiang, 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
  8. Ning Jiang, 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