For the spatially homogeneous Boltzmann equation with a shear deformation force and hard-potential collision kernels (0<γ≤10<\gamma\le1, Grad angular cutoff), Duan and Liu (2022) proved global existence for small shear strength and a uniform-in-time Hilbert-type asymptotic expansion of the solution up to order α2\alpha^2 under a homoenergetic self-similar scaling, with remainder decaying like (1+γϱ0α2t)−2(1+\gamma\varrho_0\alpha^2 t)^{-2}. The paper explicitly leaves open the completion of this program: constructing the expansion to arbitrary finite order nn with remainder decay (1+γϱ0α2t)−n(1+\gamma\varrho_0\alpha^2 t)^{-n}, and obtaining estimates uniform in γ\gamma as γ→0+\gamma\to0^+ so that, for orders n>1/γn>1/\gamma, the limit lim⁡γ→0+(1+γϱ0α2t)−1/γ=e−ϱ00α2t\lim_{\gamma\to0^+}(1+\gamma\varrho_0\alpha^2 t)^{-1/\gamma}=e^{-\varrho_{00}\alpha^2 t}, with ϱ00=lim⁡γ→0+ϱ0\varrho_{00}=\lim_{\gamma\to0^+}\varrho_0, recovers the exponential-in-time Maxwell-molecule relaxation established previously for γ=0\gamma=0. No published or preprinted work has constructed such higher-order expansions for hard potentials or proved the uniform Maxwell-molecule limit; the problem remains open.

Background

The Boltzmann equation ∂tF+v⋅∇xF=Q(F,F)\partial_t F+v\cdot\nabla_x F=Q(F,F) governs the velocity distribution F(t,x,v)F(t,x,v) of a rarefied gas, where the bilinear collision operator QQ encodes binary elastic collisions. The uniform shear flow (USF) is the state in which the density is constant and the macroscopic velocity field is linear in position, u(x)=αAxu(x)=\alpha A x, with a traceless deformation matrix AA and shear strength α>0\alpha>0. In a Lagrangian frame following this flow the distribution becomes spatially homogeneous, F(t,x,v)=F(t,v−αAx)F(t,x,v)=F(t,v-\alpha A x), and satisfies the kinetic equation ∂tF−α∇v⋅(AvF)=Q(F,F)\partial_t F-\alpha\nabla_v\cdot(AvF)=Q(F,F), i.e. the homogeneous Boltzmann equation with a linear deformation (shear) force. Such solutions are examples of homoenergetic solutions of the Boltzmann equation, introduced by Galkin and Truesdell; the shearing motion produces viscous heating, so the temperature grows without bound and the state is driven far from equilibrium — the feature that makes USF harder to analyze than perturbations of a global Maxwellian (a Maxwellian is the Gaussian equilibrium distribution μ\mu determined by the conserved mass, momentum, and energy). The collision kernel is taken of hard-potential form B(ω,v−v∗)=∣v−v∗∣γB0(cos⁡θ)B(\omega,v-v_*)=|v-v_*|^{\gamma}B_0(\cos\theta) with 0<γ≤10<\gamma\le1 under Grad's angular cutoff assumption (the angular part B0B_0 is bounded and cutoff away from grazing collisions); the exponent γ\gamma measures how the collision rate grows with relative speed, with γ=1\gamma=1 corresponding to hard spheres. A central difficulty for γ>0\gamma>0 is that the collision frequency of the associated linearized operator LL (obtained by linearizing QQ around μ\mu) grows with velocity, while the shear force continuously pumps energy into high velocities. Duan and Liu (2022) overcame this by a homoenergetic self-similar scaling of time and velocity, θ3/2(t)F(t,θ1/2(t)v)\theta^{3/2}(t)F(t,\theta^{1/2}(t)v), where θ(t)\theta(t) is the time-dependent temperature, which captures the heating law θ(t)∼(1+γϱ0α2t)2/γ\theta(t)\sim(1+\gamma\varrho_0\alpha^2 t)^{2/\gamma} with ϱ0>0\varrho_0>0 an explicitly determined constant depending only on the deformation matrix and LL. Within this scaling they justified a Hilbert-type expansion (an asymptotic expansion in powers of the small shear strength, in the spirit of Hilbert's hydrodynamic expansion) of the form G=μ+μ{αG1+α2G2+αmGR}G=\mu+\sqrt{\mu}\{\alpha G_1+\alpha^2 G_2+\alpha^m G_R\} with m>2m>2, where G1,G2G_1,G_2 are explicitly determinable functions orthogonal to ker⁡L\ker L (the null space of LL, spanned by the collision invariants, i.e. the locally conserved mass, momentum, and energy densities) that decay in time like (1+γϱ0α2t)−1(1+\gamma\varrho_0\alpha^2 t)^{-1} and (1+γϱ0α2t)−2(1+\gamma\varrho_0\alpha^2 t)^{-2}, and the remainder GRG_R is controlled uniformly in time. Their motivation draws on the Maxwell-molecule case (γ=0\gamma=0, where the collision rate is independent of relative speed): in earlier work (Arch. Ration. Mech. Anal. 2021) the same authors constructed self-similar USF profiles for Maxwell molecules and proved exponential-in-time stability of size proportional to α2\alpha^2. They also built on the homoenergetic-solution program of James, Nota, and Velázquez, who formally and rigorously analyzed long-time asymptotics of such flows for different regimes of dominance between the deformation and collision terms.

Problem Statement

For the spatially homogeneous Boltzmann equation with shear deformation force ∂tF−α∇v⋅(AvF)=Q(F,F)\partial_t F-\alpha\nabla_v\cdot(AvF)=Q(F,F) with trA=0\mathrm{tr}A=0, hard-potential collision kernel B(ω,v−v∗)=∣v−v∗∣γB0(cos⁡θ)B(\omega,v-v_*)=|v-v_*|^{\gamma}B_0(\cos\theta), 0<γ≤10<\gamma\le1, under Grad's angular cutoff, small shear strength α>0\alpha>0, and finite-mass-energy initial data perturbed of order αm\alpha^m, m>nm>n: construct, for every finite integer order nn (the source theory covers n=2n=2), the Hilbert-type expansion under the homoenergetic self-similar scaling, G=μ+μ{αG1+α2G2+⋯+αnGn+αmGR},G=\mu+\sqrt{\mu}\{\alpha G_1+\alpha^2 G_2+\cdots+\alpha^n G_n+\alpha^m G_R\}, where the correctors GiG_i are explicitly determined functions orthogonal to ker⁡L\ker L decaying in time like (1+γϱ0α2t)−i(1+\gamma\varrho_0\alpha^2 t)^{-i}, and prove uniform-in-time estimates on the remainder of order αm(1+γϱ0α2t)−n\alpha^m(1+\gamma\varrho_0\alpha^2 t)^{-n}, with constants uniform over 0<γ≤10<\gamma\le1 in such a way that, for orders n>1/γn>1/\gamma, the decay of GRG_R as γ→0+\gamma\to0^+ is recovered through the limit lim⁡γ→0+(1+γϱ0α2t)−1/γ=e−ϱ00α2t\lim_{\gamma\to0^+}(1+\gamma\varrho_0\alpha^2 t)^{-1/\gamma}=e^{-\varrho_{00}\alpha^2 t} with ϱ00=lim⁡γ→0+ϱ0\varrho_{00}=\lim_{\gamma\to0^+}\varrho_0, thereby recovering, and making continuous with, the γ=0\gamma=0 Maxwell-molecule theory of Duan–Liu (Arch. Ration. Mech. Anal. 2021). The problem asks for the rigorous derivation and uniform control of the expansion for all 0<γ≤10<\gamma\le1 and arbitrary finite order, jointly with the quantified uniformity in γ\gamma as γ→0+\gamma\to0^+.

References

  1. Renjun Duan and Shuangqian Liu, Uniform shear flow via the Boltzmann equation with hard potentials, arXiv:2210.12340, DOI:10.48550/arXiv.2210.12340, https://arxiv.org/abs/2210.12340; published as Discrete Contin. Dyn. Syst. 45 (2025), no. 10, 4071–4118, DOI:10.3934/dcds.2025049, https://doi.org/10.3934/dcds.2025049
  2. Renjun Duan and Shuangqian Liu, The Boltzmann equation for uniform shear flow, arXiv:2008.02551, https://arxiv.org/abs/2008.02551; published as Arch. Ration. Mech. Anal. 242 (2021), no. 3, 1947–2002, DOI:10.1007/s00205-021-01717-5, https://doi.org/10.1007/s00205-021-01717-5
  3. Renjun Duan, Shuangqian Liu, and Tong Yang, The Boltzmann equation for plane Couette flow, arXiv:2107.02458, https://arxiv.org/abs/2107.02458; published as J. Eur. Math. Soc. (JEMS) 27 (2025), no. 3, 1107–1183, DOI:10.4171/jems/1390, https://doi.org/10.4171/jems/1390
  4. Renjun Duan, Shuangqian Liu, and Shunlin Shen, Global stability of the inhomogeneous sheared Boltzmann equation in torus, arXiv:2509.13921 (2025), https://arxiv.org/abs/2509.13921
  5. Shuangqian Liu, Yating Wang, Anita Yang, and Xueying Zhang, The stability of the Boltzmann equation with deformations, Calc. Var. Partial Differential Equations 64 (2025), no. 8, DOI:10.1007/s00526-025-03136-w, https://doi.org/10.1007/s00526-025-03136-w
  6. José A. Carrillo, Kam Fai Chan, Renjun Duan, and Zongguang Li, Inelastic Boltzmann equation under shear heating, arXiv:2505.13960 (2025), https://arxiv.org/abs/2505.13960; published as Kinet. Relat. Models 21 (2026), 1–40, DOI:10.3934/krm.2026002, https://doi.org/10.3934/krm.2026002
  7. Richard D. James, Alessia Nota, and Juan J. L. Velázquez, Self-similar profiles for homoenergetic solutions of the Boltzmann equation: particle velocity distribution and entropy, arXiv:1710.03653, https://arxiv.org/abs/1710.03653; published as Arch. Ration. Mech. Anal. 231 (2019), no. 2, 787–843, DOI:10.1007/s00205-018-1289-2, https://doi.org/10.1007/s00205-018-1289-2
  8. Richard D. James, Alessia Nota, and Juan J. L. Velázquez, Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Collision-dominated case, arXiv:1808.06941, https://arxiv.org/abs/1808.06941; published as J. Nonlinear Sci. 29 (2019), no. 5, 1943–1973, DOI:10.1007/s00332-019-09535-6, https://doi.org/10.1007/s00332-019-09535-6