For the Boltzmann equation with hard-sphere collisions in a bounded domain whose boundary is a non-isothermal Maxwell boundary (boundary thermostats at prescribed, possibly space-dependent temperatures, with a mixture of diffuse re-emission and specular reflection), Medina (2026) proved existence, uniqueness, and exponential-in-time stability of a non-equilibrium steady state at each fixed small Knudsen number ε\varepsilon, but with smallness thresholds on the boundary temperature fluctuation and on the initial perturbation that shrink to zero as ε→0\varepsilon\to 0 (the latter, the initial-perturbation bound η(ε)2\eta(\varepsilon)^2 with η(ε)≈ε11/2\eta(\varepsilon)\approx\varepsilon^{11/2}, like ε11\varepsilon^{11}). The paper states that the hydrodynamic limit of non-isothermal problems remains an open question and explicitly asks whether the difficulty of simultaneously obtaining the incompressible Navier–Stokes–Fourier (INSF) limit and long-time convergence is a limitation of current techniques or a genuine mechanism preventing both from holding at once. The open problem is to establish (or refute) both limits together with thresholds and decay constants uniform in ε\varepsilon, and thereby decide the status of the ε\varepsilon-dependence of the thresholds.

Background

The Boltzmann equation describes a dilute gas through a phase-space density F(t,x,v)F(t,x,v) whose evolution is driven by free transport v⋅∇xFv\cdot\nabla_x F and a bilinear collision operator Q(F,F)\mathcal{Q}(F,F) modeling binary elastic collisions. In the diffusive (hydrodynamic) scaling one writes ε∂tF=−v⋅∇xF+ε−1Q(F,F)\varepsilon\partial_t F=-v\cdot\nabla_x F+\varepsilon^{-1}\mathcal{Q}(F,F), where the Knudsen number ε\varepsilon is the ratio of mean free path to macroscopic length; formally, as ε→0\varepsilon\to 0, the fluctuations of FF around a Maxwellian (Gaussian equilibrium) are governed by the incompressible Navier–Stokes–Fourier (INSF) system for the fluid velocity and temperature, as established by Saint-Raymond (2009) in the framework of renormalized (weak) solutions introduced by DiPerna and Lions. This work uses the hard-sphere collision kernel B(∣v−v∗∣,σ)=∣(v−v∗)⋅σ∣\mathcal{B}(|v-v_*|,\sigma)=|(v-v_*)\cdot\sigma|, the standard model of elastic billiard-like collisions. The Maxwell boundary condition models gas–wall interaction: at each boundary point, an incoming particle is either diffusely re-emitted with a Gaussian distribution at the local wall temperature (with probability given by the accommodation coefficient ι∈[0,1]\iota\in[0,1], the fraction of particles absorbed and re-emitted diffusely) or specularly reflected (mirror-like reflection with the normal velocity component reversed). When the wall temperature is not constant, the boundary drives the gas out of equilibrium: mass, momentum and energy fluxes through the wall do not balance, and the steady states are non-equilibrium steady states (NESS), which are not Maxwellian. Hypocoercivity is the linear-algebraic framework that recovers time-decay for kinetic equations whose linearized operator is dissipative only in velocity: transport and collisions together generate an effective dissipation; Bernou, Carrapatoso, Mischler, and Tristani (2023) developed this theory for linear kinetic equations in bounded domains with general Maxwell boundary conditions in the isothermal case. For non-isothermal boundaries, the prior program of Esposito, Guo, Kim, and Marra established the Fourier law (2013) and stationary solutions in the hydrodynamic limit (2018), but only for purely diffusive boundary conditions. On the Maxwell-boundary side, Guo and Zhou (2018, revised with Jung as Guo–Jung–Zhou 2025, arXiv:1809.06763) proved the global-in-time strong diffusive limit to the INSF system for all accommodation coefficients, but their methods give no characterization of the long-time behavior of the fluctuation f1f_1 beyond uniform-in-time bounds; adjacent activity continues with Ouyang and Wu (2024) on diffusive limits in bounded domains. Medina (2025) then treated the isothermal Boltzmann equation with Maxwell boundary conditions near the hydrodynamic limit on smooth and cylindrical domains, and Medina (2026) extended the hypocoercivity framework of Bernou et al. to the non-isothermal Maxwell setting: for each fixed ε∈(0,ε⋆)\varepsilon\in(0,\varepsilon_\star) and boundary temperature fluctuation below a threshold ϑ⋆(ε)\vartheta_\star(\varepsilon) with ϑ⋆(ε)→0\vartheta_\star(\varepsilon)\to 0 as ε→0\varepsilon\to 0, there is a unique NESS F\mathfrak{F}, and any solution with initial perturbation of size at most η(ε)2\eta(\varepsilon)^2 converges to it exponentially in a weighted L∞L^\infty norm, where η(ε)≈ε11/2\eta(\varepsilon)\approx\varepsilon^{11/2} also vanishes with ε\varepsilon. Thus, precisely in the hydrodynamic limit regime, the smallness assumptions degenerate, and long-time asymptotic control is lost before the INSF limit can be taken.

Problem Statement

For the Boltzmann equation ε∂tF=−v⋅∇xF+ε−1Q(F,F)\varepsilon\partial_t F=-v\cdot\nabla_x F+\varepsilon^{-1}\mathcal{Q}(F,F) with the hard-sphere kernel in a bounded domain with non-isothermal Maxwell boundary conditions (boundary temperature with fluctuation ϑ0\vartheta_0 around a reference temperature), in each of the source's two geometric settings — a smooth bounded C2C^2 domain with accommodation coefficient bounded below by ι0>0\iota_0>0, or a right circular cylinder with diffusive bases and specular lateral wall — determine whether the hydrodynamic limit and the long-time asymptotic behavior can be achieved simultaneously (a resolution in only one of the two settings is a partial result; the full question is resolved only when both settings are settled): prove that, for well-prepared initial data (data whose fluctuation matches the leading profiles of the expansion, so that no transient initial layer forms; this well-prepared setting is the one in which the INSF limit of the Hilbert-expansion fluctuation is formulated), the fluctuation predicted by the Hilbert expansion — the formal power series of FF in powers of ε\varepsilon about a spatially varying Maxwellian — converges as ε→0\varepsilon\to 0 to a solution of the incompressible Navier–Stokes–Fourier system while solutions converge exponentially in time to the unique non-equilibrium steady state with smallness thresholds and decay constants uniform in ε\varepsilon (in particular not vanishing as ε→0\varepsilon\to 0), or establish that a genuine mechanism prevents both from holding simultaneously. In particular, determine whether the ε\varepsilon-dependence of the thresholds in Medina (2026) — the fluctuation threshold ϑ⋆(ε)→0\vartheta_\star(\varepsilon)\to 0 and the perturbation-size threshold η(ε)2\eta(\varepsilon)^2 with η(ε)≈ε11/2\eta(\varepsilon)\approx\varepsilon^{11/2} — is a technical artifact of the current hypocoercive L2L^2–L∞L^\infty framework or is forced by the physics of the non-isothermal Maxwell boundary. The following secondary directions, explicitly raised by the source, are optional extensions that do not by themselves resolve the task: whether the smallness assumption on the boundary temperature fluctuation can be relaxed, whether the estimates are sharp, and how far the framework extends beyond hard spheres to weaker collision kernels.

References

  1. R. Medina, "The Boltzmann equation with non-isothermal Maxwell boundary conditions", 2026, arXiv:2604.13572, DOI 10.48550/arxiv.2604.13572, https://arxiv.org/abs/2604.13572
  2. Richard Medina Rodriguez, "The Boltzmann equation on smooth and cylindrical domains with Maxwell boundary conditions", 2025, arXiv:2510.13260, https://arxiv.org/abs/2510.13260
  3. Yan Guo, Junhwa Jung, Fujun Zhou, "Strong diffusive limit of the Boltzmann equation with Maxwell boundary condition", 2025 (v1 2018 as Guo–Zhou, "Boltzmann Diffusive Limit with Maxwell Boundary Condition"), arXiv:1809.06763, https://arxiv.org/abs/1809.06763
  4. Armand Bernou, Kleber Carrapatoso, Stéphane Mischler, Isabelle Tristani, "Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition", Ann. Inst. H. Poincaré C Anal. Non Linéaire 40(2):287–338, 2023, DOI 10.4171/aihpc/44, https://doi.org/10.4171/aihpc/44
  5. R. Esposito, Y. Guo, C. Kim, R. Marra, "Non-isothermal boundary in the Boltzmann theory and Fourier law", Comm. Math. Phys. 323(1):177–239, 2013, DOI 10.1007/s00220-013-1766-2, https://doi.org/10.1007/s00220-013-1766-2
  6. Raffaele Esposito, Yan Guo, Chanwoo Kim, Rossana Marra, "Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit", Ann. PDE 4(1), Paper No. 1, 2018, DOI 10.1007/s40818-017-0037-5, https://doi.org/10.1007/s40818-017-0037-5
  7. Laure Saint-Raymond, "Hydrodynamic Limits of the Boltzmann Equation", Lecture Notes in Mathematics 1971, Springer-Verlag, Berlin, 2009, DOI 10.1007/978-3-540-92847-8, https://doi.org/10.1007/978-3-540-92847-8
  8. Zhimeng Ouyang, Lei Wu, "Diffusive Limit of the Boltzmann Equation in Bounded Domains", Comm. Math. Phys. 405, 2024, DOI 10.1007/s00220-024-05152-x, https://doi.org/10.1007/s00220-024-05152-x