Guo and Jung (arXiv:2410.22503; Communications in Information and Systems, 2025) proved that the diffusive (small Knudsen–Mach) expansion of the unsteady Boltzmann equation around a far-field Maxwellian with non-zero mean velocity of order ε\varepsilon — a passing flow — in the exterior of a bounded obstacle, with diffuse-reflection boundary condition, is valid up to the critical time, i.e. the lifespan of the associated incompressible Navier–Stokes solution in the exterior domain, and their theorem allows initial perturbations of arbitrary finite size. Whether the expansion can be justified globally in time for large initial data is left open: the obstruction identified by the authors is the global-regularity problem for the three-dimensional incompressible Navier–Stokes equations in an exterior domain with non-zero far-field velocity, and the expansion is currently known to hold only for the lifespan of the fluid solution.

Background

The Boltzmann equation describes a dilute gas through a distribution function F(t,x,v)F(t,x,v) giving the density of particles at time tt, position xx, and velocity vv; binary collisions are encoded by the quadratic collision operator Q(F,F)Q(F,F), taken here for hard interaction potentials with Grad's angular cutoff. Its nondimensional form contains the Knudsen number (mean free path relative to a macroscopic length) and the Mach number (typical flow speed relative to thermal speed); when both are small and of comparable size ε\varepsilon, the rescaled equation ε∂tF+v⋅∇xF=ε−1Q(F,F)\varepsilon\partial_tF+v\cdot\nabla_xF=\varepsilon^{-1}Q(F,F) is expected, as ε→0\varepsilon\to0, to be governed at leading order not by compressible but by incompressible fluid dynamics — the diffusive or incompressible (Navier–)Stokes–Fourier limit. Justifying such hydrodynamic limits rigorously is the kinetic-theoretic part of Hilbert's sixth problem.

Two features make the problem considered here especially hard. First, the gas fills an exterior domain Ωc=R3∖Ω‾\Omega^c=\mathbb{R}^3\setminus\overline{\Omega}, the region outside a bounded obstacle Ω\Omega — the geometrically natural setting for a flow past a body. On the boundary ∂Ω\partial\Omega the gas obeys the diffuse-reflection (accommodation) condition: incoming particles are re-emitted with a Maxwellian distribution (a Gaussian equilibrium in velocity) determined by the wall, an operator usually denoted PγwP_{\gamma_w}. Diffuse reflection makes the trace estimates delicate because of the grazing set, where particle velocities are nearly tangent to the wall. Second, the gas is not at rest at infinity: F→μuF\to\mu_u as ∣x∣→∞|x|\to\infty, where μu(v)=μ(v−εu)\mu_u(v)=\mu(v-\varepsilon u) is a Maxwellian with non-zero mean velocity εu\varepsilon u of order ε\varepsilon, with uu a fixed constant vector — a passing flow in the small-Mach regime — so the linearized collision operator around μu\mu_u loses coercivity in the direction of the flow and the fluid limit inherits a drift.

The diffusive expansion is a Hilbert-type ansatz F=μu+εμu (f1+εf2+ε1/2R),F=\mu_u+\varepsilon\sqrt{\mu_u}\,\bigl(f_1+\varepsilon f_2+\varepsilon^{1/2}R\bigr), where f1=u⋅vu μuf_1=\mathbf{u}\cdot v_u\,\sqrt{\mu_u} with vu=v−εuv_u=v-\varepsilon u is the leading hydrodynamic profile with ∇⋅u=0\nabla\cdot\mathbf{u}=0, f2f_2 collects the strain and second-order collision terms, and the remainder RR must be proven uniformly bounded as ε→0\varepsilon\to0. The coefficient u\mathbf{u} solves the incompressible Navier–Stokes system in the exterior domain with drift uu: ∂tu+(u+u)⋅∇u+∇p=κΔu,∇⋅u=0,\partial_t\mathbf{u}+(u+\mathbf{u})\cdot\nabla\mathbf{u}+\nabla p=\kappa\Delta\mathbf{u},\qquad \nabla\cdot\mathbf{u}=0, with total velocity u+uu+\mathbf{u} vanishing on ∂Ω\partial\Omega (no-slip at a body at rest) and tending to uu at infinity. The paper establishing the state of the art, Guo–Jung (arXiv:2410.22503; published in Communications in Information and Systems in 2025), proves that whenever this Navier–Stokes solution exists on a time interval [0,T0][0,T_0] with LpL^p bounds on u\mathbf{u}, the kinetic problem has a unique solution admitting the expansion with RR under control on [0,T0][0,T_0], for initial data of the well-prepared diffusive form of arbitrary finite size. Their method extends the L2L^2–L∞L^\infty energy framework for Boltzmann boundary-value problems to unbounded domains through a new L3L^3–L6L^6 splitting, since the lack of a Poincaré inequality in an exterior domain precludes the usual coercivity arguments.

The neighboring literature delimits what is known. For small initial data, the associated Navier–Stokes problem in the exterior domain with non-zero far-field velocity is globally well posed (classical small-data stability theory, e.g. Bae and Roh, 2010), so the Guo–Jung theorem already yields global-in-time validity of the expansion in the small-data regime. A companion paper of Jung (arXiv:2308.03984; SIAM Journal on Mathematical Analysis, 2025) proves a genuinely global diffusive expansion to the Navier–Stokes–Fourier system in an exterior domain, but only for a far-field Maxwellian at rest and only for small perturbations. In bounded domains, Ouyang and Wu (arXiv:2304.01129) proved incompressible Navier–Stokes–Fourier limits with in-flow boundary conditions. For arbitrary large data with no boundary, Golse and Saint-Raymond (2004) obtained the incompressible Navier–Stokes limit from DiPerna–Lions renormalized solutions — global in time by construction — but only in the whole space or periodic setting, with distributional convergence to Leray solutions and no remainder control. The steady (time-independent) flow past an obstacle was treated by Esposito, Guo, and Marra (2018). Most spectacularly, Deng, Hani, and Ma (arXiv:2503.01800, 2025) resolved Hilbert's sixth problem on two- and three-dimensional tori by deriving the Euler and Navier–Stokes–Fourier equations from hard-sphere dynamics — but their setting has no boundary, no exterior geometry, and no diffuse-reflection condition. The exterior-domain, non-zero-passing-flow, large-data, global-in-time question thus marks the current frontier.

Problem Statement

Let Ω⊂R3\Omega\subset\mathbb{R}^3 be a bounded domain with smooth boundary (an obstacle), Ωc=R3∖Ω‾\Omega^c=\mathbb{R}^3\setminus\overline{\Omega} the exterior domain, and QQ the Boltzmann collision operator for hard potentials with Grad's angular cutoff. Consider the unsteady Boltzmann equation in the diffusive scaling with small Knudsen and Mach numbers, ε∂tF+v⋅∇xF=1εQ(F,F)in R+×Ωc×R3,\varepsilon\partial_tF+v\cdot\nabla_xF=\frac{1}{\varepsilon}Q(F,F)\quad\text{in }\mathbb{R}_+\times\Omega^c\times\mathbb{R}^3, together with the diffuse-reflection boundary condition F∣γ−=Pγw(F)F|_{\gamma_-}=P_{\gamma_w}(F) on the incoming set of the boundary, the far-field condition F(t,x,v)→μu(v)F(t,x,v)\to\mu_u(v) as ∣x∣→∞|x|\to\infty where μu(v)=μ(v−εu)\mu_u(v)=\mu(v-\varepsilon u) is a Maxwellian with non-zero mean velocity εu\varepsilon u of order ε\varepsilon (the passing flow, with uu a fixed constant vector), and a non-negative initial datum of the well-prepared diffusive form F0=μu+εμu (f1,0+εf2,0+ε1/2R0),F_0=\mu_u+\varepsilon\sqrt{\mu_u}\,\bigl(f_{1,0}+\varepsilon f_{2,0}+\varepsilon^{1/2}R_0\bigr), where f1,0=u0⋅vu μuf_{1,0}=u_0\cdot v_u\,\sqrt{\mu_u} with vu=v−εuv_u=v-\varepsilon u, ∇⋅u0=0\nabla\cdot u_0=0, and f2,0f_{2,0} is the associated second-order profile, and where the size of the datum, measured in the norms of Guo–Jung's Theorem 2 (∥R0∥Lx,v2\|R_0\|_{L^2_{x,v}}, ε−1∥(I−Pu)R0∥Lx,v2\varepsilon^{-1}\|(I-P_u)R_0\|_{L^2_{x,v}}, ∥∂tR0∥Lx,v2\|\partial_tR_0\|_{L^2_{x,v}}, ε1/2∥ωR0∥Lx,v∞\varepsilon^{1/2}\|\omega R_0\|_{L^\infty_{x,v}}, and LpL^p norms of f1,0,f2,0f_{1,0},f_{2,0} for p∈[1,∞]p\in[1,\infty]), is an arbitrary finite constant MM — not small.

Guo and Jung proved that the diffusive expansion is valid up to the critical time: if the solution u\mathbf{u} of the associated incompressible Navier–Stokes system in Ωc\Omega^c, ∂tu+(u+u)⋅∇u+∇p=κΔu,∇⋅u=0,u=−u on ∂Ω,u→0 as ∣x∣→∞,\partial_t\mathbf{u}+(u+\mathbf{u})\cdot\nabla\mathbf{u}+\nabla p=\kappa\Delta\mathbf{u},\quad\nabla\cdot\mathbf{u}=0,\quad \mathbf{u}=-u\ \text{on }\partial\Omega,\quad \mathbf{u}\to0\ \text{as }|x|\to\infty, exists on [0,T0][0,T_0] with ∥u(t)∥Lp≲∥u0∥L1+∥u0∥L∞\|\mathbf{u}(t)\|_{L^p}\lesssim\|u_0\|_{L^1}+\|u_0\|_{L^\infty} for p∈[1,∞]p\in[1,\infty], then for ε>0\varepsilon>0 small enough (depending on T0T_0 and MM) the kinetic problem has a unique solution admitting the expansion F=μu+εμu(f1+εf2+ε1/2R)F=\mu_u+\varepsilon\sqrt{\mu_u}(f_1+\varepsilon f_2+\varepsilon^{1/2}R) with the remainder RR controlled on [0,T0][0,T_0]. For small initial data this holds for all times, since the exterior Navier–Stokes problem is then globally well posed.

The open question left by that work is whether the diffusive expansion is globally valid in time for arbitrary large initial data in this exterior-domain, nonzero-passing-flow regime: determine whether the expansion is valid for every T0>0T_0>0 (equivalently, on all of R+\mathbb{R}_+) for data of arbitrary finite size MM — a resolution being either a proof of global-in-time validity for all data of finite size MM, or the exhibition of data of finite size MM for which the expansion fails. The obstruction identified in the source (Guo–Jung, Remark 3) is on the fluid side: for large initial data the global existence of the above drift Navier–Stokes system is a well-known open problem, whereas any global-in-time Navier–Stokes solution with the required LpL^p bounds would, through the Guo–Jung theorem, yield the desired validity with T0T_0 arbitrary; at present the expansion is known to hold only for the lifespan of the fluid solution.

References

  1. Yan Guo, Junhwa Jung, Diffusive Expansion of the Boltzmann equation for the flow past an obstacle, arXiv:2410.22503 (2024), DOI 10.48550/arXiv.2410.22503, https://arxiv.org/abs/2410.22503; published as Communications in Information and Systems 25(2), 301–326 (2025), DOI 10.4310/cis.250823003724, https://doi.org/10.4310/cis.250823003724
  2. Junhwa Jung, Global Diffusive Expansion of Boltzmann Equation in Exterior Domain, arXiv:2308.03984 (2023, v3 2025), DOI 10.48550/arXiv.2308.03984, https://arxiv.org/abs/2308.03984; published in SIAM Journal on Mathematical Analysis 57(2), 1781–1831 (2025), DOI 10.1137/24m1669992, https://doi.org/10.1137/24m1669992
  3. Zhimeng Ouyang, Lei Wu, Diffusive Limit of the Boltzmann Equation in Bounded Domains, arXiv:2304.01129 (2023), DOI 10.48550/arXiv.2304.01129, https://arxiv.org/abs/2304.01129
  4. François Golse, Laure Saint-Raymond, The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels, Inventiones mathematicae 155(1), 81–161 (2004), DOI 10.1007/s00222-003-0316-5, https://doi.org/10.1007/s00222-003-0316-5
  5. Raffaele Esposito, Yan Guo, Rossana Marra, Hydrodynamic Limit of a Kinetic Gas Flow Past an Obstacle, Communications in Mathematical Physics 364(2), 765–823 (2018), DOI 10.1007/s00220-018-3173-1, https://doi.org/10.1007/s00220-018-3173-1
  6. Hyeong-Ohk Bae, Jaiok Roh, Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains, Journal of Mathematical Fluid Mechanics 14(1), 117–139 (published online 2010; print issue 2012), DOI 10.1007/s00021-010-0040-z, https://doi.org/10.1007/s00021-010-0040-z
  7. Yu Deng, Zaher Hani, Xiao Ma, Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory, arXiv:2503.01800 (2025), DOI 10.48550/arXiv.2503.01800, https://arxiv.org/abs/2503.01800