Guo and Jung established that the diffusive (Hilbert) expansion of the Boltzmann equation around a Maxwellian with non-zero passing flow is valid in a three-dimensional exterior domain up to any time within the lifespan of the limiting incompressible Navier–Stokes solution, for initial data of arbitrary prescribed size. The paper explicitly leaves open whether this expansion is valid globally in time for general large data. The question is whether the dependence on the Navier–Stokes lifespan can be removed. A follow-up paper by Jung establishes global validity only for small data and zero far field, so the large-data, non-zero-passing-flow core remains open.

Background

The Boltzmann equation describes a dilute gas through its one-particle distribution F(t,x,v)F(t,x,v), which evolves by free transport in the position-velocity phase space and a quadratic collision operator Q(F,F)Q(F,F) modeling binary collisions. The Knudsen number ε\varepsilon (the ratio of mean free path to macroscopic length) enters the scaled equation ε ∂tF+v⋅∇xF=ε−1Q(F,F)\varepsilon\,\partial_t F+v\cdot\nabla_x F=\varepsilon^{-1}Q(F,F). In the diffusive scaling relevant to incompressible fluid dynamics, and in the spirit of Hilbert's sixth problem on deriving continuum mechanics from kinetic theory, one asks whether, as ε→0\varepsilon\to0 together with the Mach number, solutions of the Boltzmann equation admit the expansion

F=μu∞+εμu∞(f1+εf2+ε1/2R),F=\mu_{u_\infty}+\varepsilon\sqrt{\mu_{u_\infty}}\left(f_1+\varepsilon f_2+\varepsilon^{1/2}R\right),

where μu∞\mu_{u_\infty} is the Maxwellian (Gaussian equilibrium distribution) with bulk velocity u∞u_\infty, the leading profile f1=u⋅vuμu∞f_1=u\cdot v_{\mathfrak u}\sqrt{\mu_{u_\infty}} (with vu=v−u∞v_{\mathfrak u}=v-u_\infty the shifted velocity, defined precisely below) is built from a solution uu of the incompressible Navier–Stokes equations, f2f_2 is the second-order viscous-stress correction (given explicitly in the Problem Statement below), and RR is a remainder to be controlled.

The geometry here is the fundamental flow-past-an-obstacle configuration: Ω\Omega is a bounded C∞C^\infty set in R3\mathbb{R}^3, the gas occupies the exterior domain Ωc\Omega^c, the boundary condition at the obstacle is diffuse reflection (the incoming trace F∣γ−F|_{\gamma_-} equals PγwFP_\gamma^w F, the Maxwellian-weighted average of the outgoing trace over reflected velocities), and F→μu∞F\to\mu_{u_\infty} as ∣x∣→∞|x|\to\infty. In the small-Mach regime of the source, the velocity at infinity is of order ε\varepsilon: u∞=εuu_\infty=\varepsilon\mathfrak u for a fixed non-zero vector u\mathfrak u, so that vu=v−εuv_{\mathfrak u}=v-\varepsilon\mathfrak u and μu∞=M1,εu,1\mu_{u_\infty}=M_{1,\varepsilon\mathfrak u,1}; i.e. the gas streams past the obstacle with a non-zero passing flow. In this regime u+uu+\mathfrak u solves the incompressible Navier–Stokes system ∂tu+(u+u)⋅∇u+∇p=κΔu\partial_tu+(u+\mathfrak u)\cdot\nabla u+\nabla p=\kappa\Delta u in Ωc\Omega^c with no-slip-type data u=−uu=-\mathfrak u on ∂Ω\partial\Omega and u→0u\to0 at infinity.

Prior results frame the question. For small data, Bardos and Ukai (1991) proved the classical incompressible Navier–Stokes limit in the whole space and periodic settings, and Guo (2006) developed the L2L^2–L∞L^\infty framework, validating the expansion beyond the Navier–Stokes approximation in bounded domains. For the steady flow past an obstacle, Esposito, Guo, and Marra (2018) justified the hydrodynamic limit. For large data, Golse and Saint-Raymond (2004) proved a global-in-time incompressible Navier–Stokes limit, but for DiPerna–Lions renormalized solutions (a weak solution notion in which the equation holds for nonlinear functions of the distribution) converging weakly (under structural assumptions on the collision kernel) to Leray solutions (weak solutions obtained from the energy inequality) — a strictly weaker notion of limit that neither validates the expansion nor controls the remainder, and which is not available in the exterior-obstacle, non-zero-passing-flow setting. On the fluid side, global existence and regularity of the three-dimensional incompressible Navier–Stokes equations for large data is the classical open problem (a Clay Millennium Problem); with non-zero far field, small-data global theory is due to Bae and Roh (2010).

Guo and Jung (2024, published 2025) proved that in the exterior domain with non-zero passing flow, for initial data F0=μu∞+εμu∞(f1,0+εf2,0)F_0=\mu_{u_\infty}+\varepsilon\sqrt{\mu_{u_\infty}}(f_{1,0}+\varepsilon f_{2,0}) of any prescribed size M>0M>0, the expansion is valid on any time interval [0,T0][0,T_0] for which the Navier–Stokes solution exists and satisfies the a priori bound ∥u(t)∥Lp≲∥u0∥L1+∥u0∥L∞\|u(t)\|_{L^p}\lesssim\|u_0\|_{L^1}+\|u_0\|_{L^\infty}. Their Remark 3 notes that for small data the Navier–Stokes solution is global and hence the limit holds for all time, while for large data global Navier–Stokes existence is a well-known open problem, so the limit is justified only on the fluid lifespan. A companion paper of Jung (2025) established global-in-time validity of the diffusive expansion in an exterior domain — but for small data and zero far field (F→μF\to\mu at infinity), introducing an L2L^2–L3L^3–L6L^6 splitting to overcome the lack of a Poincaré inequality in unbounded domains. What remains unresolved is precisely the large-data, non-zero-passing-flow case; since the leading profile f1f_1 is defined from the Navier–Stokes solution, global validity of the expansion for arbitrary finite-energy data would directly imply global regularity of the three-dimensional incompressible Navier–Stokes equations with non-zero far field in exterior domains.

Problem Statement

In the setting of Guo–Jung's theorem — Ω\Omega a bounded C∞C^\infty open set in R3\mathbb{R}^3, the gas in the exterior domain Ωc\Omega^c under the Boltzmann equation ε ∂tF+v⋅∇xF=ε−1Q(F,F)\varepsilon\,\partial_t F+v\cdot\nabla_x F=\varepsilon^{-1}Q(F,F) with a hard-potential collision kernel under Grad's angular cutoff (0≤θ≤10\le\theta\le1), diffuse reflection F∣γ−=PγwFF|_{\gamma_-}=P_\gamma^w F at the obstacle, and far-field condition F→μu∞F\to\mu_{u_\infty} with u∞=εu≠0u_\infty=\varepsilon\mathfrak u\neq0 (the velocity at infinity is of order ε\varepsilon in the small-Mach regime, μu∞=M1,εu,1\mu_{u_\infty}=M_{1,\varepsilon\mathfrak u,1}), and initial data F0=μu∞+εμu∞(f1,0+εf2,0)≥0F_0=\mu_{u_\infty}+\varepsilon\sqrt{\mu_{u_\infty}}(f_{1,0}+\varepsilon f_{2,0})\ge0 of arbitrary prescribed size M>0M>0, where ∇⋅u0=0\nabla\cdot u_0=0 and MM bounds ∣u∣|\mathfrak u|, ∥u0∥Lp\|u_0\|_{L^p}, ∥∇u0∥Lp\|\nabla u_0\|_{L^p} for p∈[1,∞]p\in[1,\infty], and the remainder-data quantity ∥R0∥Lx,v2+ε−1∥(I−Pu)R0∥Lx,v2+∥∂tR0∥Lx,v2+ε1/2∥ωR0∥Lx,v∞\|R_0\|_{L^2_{x,v}}+\varepsilon^{-1}\|(\mathbf I-\mathbf P_{\mathfrak u})R_0\|_{L^2_{x,v}}+\|\partial_tR_0\|_{L^2_{x,v}}+\varepsilon^{1/2}\|\omega R_0\|_{L^\infty_{x,v}} with ω(v)=eβ∣v∣2\omega(v)=e^{\beta|v|^2}, 0<β≪10<\beta\ll1 (with R0R_0 the initial remainder determined from F0F_0 through the expansion form, and ∂tR0\partial_tR_0 obtained from the remainder equation, as in the source's Remark 4, so that these quantities are fixed by the data rather than free additional data; this mirrors the source's Theorem 2, whose data form (29) f0=f1,0+εf2,0f_0=f_{1,0}+\varepsilon f_{2,0} coexists with the remainder-data bound (35)) — establish (or disprove) the global-in-time validity of the diffusive expansion: for each such MM there exists ε0(M)>0\varepsilon_0(M)>0 such that for all 0<ε<ε0(M)0<\varepsilon<\varepsilon_0(M) the problem has a unique solution on [0,∞)[0,\infty) of the form

F=μu∞+εμu∞(f1+εf2+ε1/2R),F=\mu_{u_\infty}+\varepsilon\sqrt{\mu_{u_\infty}}\left(f_1+\varepsilon f_2+\varepsilon^{1/2}R\right),

where f1=u⋅vuμu∞f_1=u\cdot v_{\mathfrak u}\sqrt{\mu_{u_\infty}} with vu=v−εuv_{\mathfrak u}=v-\varepsilon\mathfrak u, and u+uu+\mathfrak u solves the incompressible Navier–Stokes system ∂tu+(u+u)⋅∇u+∇p=κΔu\partial_tu+(u+\mathfrak u)\cdot\nabla u+\nabla p=\kappa\Delta u in Ωc\Omega^c (u=−uu=-\mathfrak u on ∂Ω\partial\Omega, u→0u\to0 at infinity); f2=12∑i,j=13Aij[∂xiuj+∂xjui]+Lu−1[Γ(f1,f1)]f_2=\tfrac12\sum_{i,j=1}^{3}\mathscr A_{ij}[\partial_{x_i}u_j+\partial_{x_j}u_i]+L_{\mathfrak u}^{-1}[\Gamma(f_1,f_1)] with Aij=−Lu−1(μ (vu,ivu,j−∣vu∣23δij))\mathscr A_{ij}=-L_{\mathfrak u}^{-1}\left(\sqrt{\mu}\,(v_{\mathfrak u,i}v_{\mathfrak u,j}-\tfrac{|v_{\mathfrak u}|^2}{3}\delta_{ij})\right), where μ\mu is the unit rest Maxwellian, LuL_{\mathfrak u} is the linearized collision operator around μu∞\mu_{u_\infty}, and Γ\Gamma is the associated bilinear collision form (the source's eqs. (15)–(17)); and the remainder RR satisfies, uniformly in t≥0t\ge0, the quantitative bounds that Guo–Jung's proof establishes on [0,T0][0,T_0] via their iteration space HH (arXiv:2410.22503, Section 4), with the exp⁡(MT0)\exp(MT_0)-type growth removed: the energy–dissipation bound e(t)+D(t)≤C(M)e(t)+\mathscr D(t)\le C(M), the weighted L∞L^\infty bound ∥ε1/2ωR(t)∥Lx,v∞≤C(M)\|\varepsilon^{1/2}\omega R(t)\|_{L^\infty_{x,v}}\le C(M) with ω(v)=eβ∣v∣2\omega(v)=e^{\beta|v|^2}, 0<β≪10<\beta\ll1, the macroscopic bound ∥PuR∥Lt∞Lx6≤C(M)\|\mathbf P_{\mathfrak u}R\|_{L^\infty_tL^6_x}\le C(M), and the microscopic smoothing bound ∥S1R∥Lt2Lx3\|S_1R\|_{L^2_tL^3_x} finite with T0T_0-independent local-in-time constants. This requires removing the dependence of the validity time on the Navier–Stokes lifespan; any such resolution still produces the global Navier–Stokes profile, since f1f_1 is defined from it.

References

  1. Yan Guo and Junhwa Jung, "Diffusive expansion of the Boltzmann equation for the flow past an obstacle", Communications in Information and Systems 25(2), 301–326 (2025). DOI: 10.4310/cis.250823003724; arXiv:2410.22503, https://arxiv.org/abs/2410.22503
  2. Junhwa Jung, "Global Diffusive Expansion of Boltzmann Equation in Exterior Domain", SIAM Journal on Mathematical Analysis 57, 1781–1831 (2025). DOI: 10.1137/24M1669992; arXiv:2308.03984, https://arxiv.org/abs/2308.03984
  3. François Golse and Laure Saint-Raymond, "The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels", Inventiones mathematicae 155, 81–161 (2004). DOI: 10.1007/s00222-003-0316-5, https://doi.org/10.1007/s00222-003-0316-5
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