The 1998 paper of Villani introduced H-solutions — weak solutions of the spatially homogeneous Boltzmann equation without angular cutoff and of the Landau equation with Coulomb potential, defined only through conservation of mass, momentum and energy, an entropy inequality, and a nonlocal entropy-production bound on the square root of the product of pair densities — and proved their existence, explicitly leaving open the completion of the theory. This problem asks to resolve that agenda. Goal A: within the bare H-solution class, prove uniqueness, immediate smoothing up to the Schwartz class, and quantitative convergence to equilibrium, for soft and very soft non-cutoff kernels and for the Landau–Coulomb equation (γ=−3\gamma=-3), or refute the statements by explicit construction. Goal B: derive the Landau–Coulomb equation from N-particle Coulomb dynamics in the weak-coupling regime, with a microscopic justification of Debye screening and the Coulomb logarithm, and a determination of the role of the Balescu–Lenard equation. Twenty-eight years of subsequent literature — entropy-dissipation coercivity, smoothing under extra moment and positivity assumptions, well-posedness in critical spaces such as L3/2L^{3/2}, partial regularity of the singular set, monotonicity and production of the Fisher information, and propagation-of-chaos results that either exclude the Coulomb singularity or are proved for stochastic or Kac-type particle systems rather than Newtonian dynamics — has resolved many neighboring questions without closing either goal. The problem therefore remains open.

Background

Rarefied gases and weakly coupled plasmas are described at the kinetic level by a distribution function f(t,v)≥0f(t,v)\ge 0 giving the density of particles with velocity v∈R3v\in\mathbb{R}^3 at time tt. In the spatially homogeneous setting ff does not depend on the position variable, and the Boltzmann equation reduces to ∂tf=Q(f,f)\partial_t f=Q(f,f), a bilinear collision operator built from a cross-section B(∣v−v∗∣,θ)=Φ(∣v−v∗∣)ζ(θ)B(|v-v^*|,\theta)=\Phi(|v-v^*|)\zeta(\theta), where vv and v∗v^* are the pre-collisional velocities of two particles and θ\theta is the deflection angle. Grad's angular cutoff is the assumption that ζ\zeta is integrable near θ=0\theta=0; it is a mathematical convenience that removes the accumulation of grazing (small-angle) collisions. Physical long-range interactions violate it: for inverse-power repulsion ζ(θ)\zeta(\theta) behaves like θ−2−2s\theta^{-2-2s} with 0<s<10<s<1, which is non-integrable at θ=0\theta=0, so the collision operator must be understood in a renormalized sense and gains, as compensation, a regularizing effect on ff.

The potential part classifies kernels as hard (γ>0\gamma>0), Maxwellian (γ=0\gamma=0) or soft (γ<0\gamma<0); the very soft regime is γ+2s<0\gamma+2s<0, where even velocity moments can decay instead of propagating. The Landau (or Fokker–Planck–Landau) equation is the formal limit of the Boltzmann equation as all collisions become grazing; it is the standard collision model for plasmas and is a nonlocal nonlinear diffusion equation with matrix kernel aij(z)=Λ∣z∣γ+2Πij(z)a_{ij}(z)=\Lambda|z|^{\gamma+2}\Pi_{ij}(z), where Π(z)=I−z⊗z/∣z∣2\Pi(z)=I-z\otimes z/|z|^2 is the projection onto the plane orthogonal to zz. For Coulomb interactions γ=−3\gamma=-3, so the kernel behaves like ∣z∣−1Π(z)|z|^{-1}\Pi(z): it is unbounded at z=0z=0, degenerate in the direction of zz, and sits exactly at the endpoint of the available coercivity mechanisms (entropy dissipation, Fisher information). The corresponding Boltzmann kernel with γ=−3\gamma=-3 is not even well defined without truncating the impact parameter.

The Boltzmann H-theorem states that the entropy H(t)=∫flog⁡f dvH(t)=\int f\log f\,dv is non-increasing along solutions, with entropy production D(f)=−dH/dt≥0D(f)=-dH/dt\ge 0 an explicit nonnegative quadratic functional of ff built from the collision operator. The fundamental discovery of Desvillettes (1995) and Villani (1999) is that when the angular cutoff is removed, D(f)D(f) controls a weighted L2L^2 norm of the gradient of f\sqrt{f}: entropy production substitutes for the coercivity lost by removing the cutoff, and it is the analytic engine behind essentially all later regularity results for non-cutoff kinetic equations.

Villani (1998) used this mechanism to construct H-solutions of the spatially homogeneous non-cutoff Boltzmann equation and of the Landau equation with Coulomb potential: weak solutions carrying only the physically conserved or monotone quantities — conservation of mass, momentum and energy; finite entropy with the entropy inequality — plus the entropy-production bound. For the Landau equation the bound is the differential statement Π(v−v∗)∇2F∈L2((0,T)×R3×R3)\Pi(v-v^*)\nabla_2\sqrt{F}\in L^2((0,T)\times\mathbb{R}^3\times\mathbb{R}^3), where F(v,v∗)=f(v)f(v∗)∣v−v∗∣γ+2F(v,v^*)=f(v)f(v^*)|v-v^*|^{\gamma+2}, ∇2=∇v−∇v∗\nabla_2=\nabla_v-\nabla_{v^*} and Π\Pi projects orthogonally to v−v∗v-v^*; for the Boltzmann equation it is the finite-difference analogue ζ(θ)sin⁡(θ/2) ∣F(x′)−F(x)∣/∣x2′−x2∣∈L2(dt dθ dφ dv dv∗)\sqrt{\zeta(\theta)}\sin(\theta/2)\,|\sqrt{F(x')}-\sqrt{F(x)}|/|x_2'-x_2|\in L^2(dt\,d\theta\,d\varphi\,dv\,dv^*), where x=((v+v∗)/2,v−v∗)x=((v+v^*)/2,v-v^*) and x′=(x1,∣x2∣σ)x'=(x_1,|x_2|\sigma) with σ∈S2\sigma\in\mathbb{S}^2 is the post-collisional image of the relative variable. This is a nonlocal, anisotropic regularity statement about ff∗\sqrt{ff^*}: it controls only derivatives orthogonal to the relative velocity, not radial ones, and it does not directly control ∇f\nabla f. Whether it implies genuine smoothness, uniqueness or quantitative relaxation was left open. Existence was proved for −2≤γ<1-2\le\gamma<1 under an angular integrability hypothesis (and down to γ=−3\gamma=-3 under a stronger one), by passing to the grazing-collision limit of mollified Boltzmann equations; the Landau equation with Coulomb potential is obtained as a renormalized grazing limit in which the impact parameter is cut off at a scale ε\varepsilon and the kernel is rescaled by the Coulomb logarithm log⁡ε−1\log\varepsilon^{-1}.

The derivation side concerns the microscopic origin of these equations. The Boltzmann equation itself can be derived from N-particle dynamics (Lanford's theorem for hard spheres). For plasmas the relevant regime is weak coupling — many particles, each pair interacting weakly through long-range Coulomb forces — and the expected limit is the Landau equation or, at the next order of the correlation expansion, the Balescu–Lenard equation, which accounts for dynamical pair correlations. Physically, the divergent Coulomb cross-section is tamed by Debye screening: collective effects are argued to screen the pair potential beyond the Debye length, which justifies cutting off the impact parameter and absorbing the factor log⁡ε−1\log\varepsilon^{-1}. Both the screening and the logarithm are model assumptions imposed at the kinetic level; no rigorous microscopic justification from particle dynamics is known.

For moderately soft kernels the theory is by now essentially complete (existence, uniqueness, smoothness, quantitative rates), but the very soft and Coulomb regimes and the bare H-solution class remain open.

Problem Statement

The problem is to complete the mathematical foundation of the entropy-production-based weak-solution (H-solution) theory for the spatially homogeneous Boltzmann equation without angular cutoff and the spatially homogeneous Landau equation with Coulomb potential, as posed in Villani (1998). It consists of two connected goals belonging to the same program; both must be settled for the problem to count as fully resolved.

Goal A — well-posedness and regularity in the bare H-solution class. Consider the spatially homogeneous Landau equation with Coulomb potential,

∂tf=∇v⋅∫R3a(v−v∗)(∇f(v) f(v∗)−f(v) ∇f(v∗))dv∗,aij(z)=Λ∣z∣−1Πij(z),Π(z)=I−z⊗z/∣z∣2,\partial_t f=\nabla_v\cdot\int_{\mathbb{R}^3} a(v-v^*)\left(\nabla f(v)\,f(v^*)-f(v)\,\nabla f(v^*)\right)dv^*,\qquad a_{ij}(z)=\Lambda|z|^{-1}\Pi_{ij}(z),\qquad\Pi(z)=I-z\otimes z/|z|^2,

and the spatially homogeneous Boltzmann equation without angular cutoff with cross-section B=∣v−v∗∣γζ(θ)B=|v-v^*|^{\gamma}\zeta(\theta), ζ(θ)∼θ−2−2s\zeta(\theta)\sim\theta^{-2-2s}, 0<s<10<s<1, in the soft and very soft regimes, in the ranges of γ\gamma where H-solutions were constructed in Villani (1998) or can be constructed by the same mollification and grazing-limit method. Let an H-solution be a solution carrying only: conservation of mass, momentum and energy; finite entropy with the entropy inequality; and the entropy-production bound — for the Landau equation, Π(v−v∗)∇2F∈L2((0,T)×R3×R3)\Pi(v-v^*)\nabla_2\sqrt{F}\in L^2((0,T)\times\mathbb{R}^3\times\mathbb{R}^3) with F=f(v)f(v∗)∣v−v∗∣γ+2F=f(v)f(v^*)|v-v^*|^{\gamma+2} (γ=−3\gamma=-3 for Coulomb), and for the Boltzmann equation the finite-difference analogue ζ(θ)sin⁡(θ/2) ∣F(x′)−F(x)∣/∣x2′−x2∣∈L2(dt dθ dφ dv dv∗)\sqrt{\zeta(\theta)}\sin(\theta/2)\,|\sqrt{F(x')}-\sqrt{F(x)}|/|x_2'-x_2|\in L^2(dt\,d\theta\,d\varphi\,dv\,dv^*), where x=((v+v∗)/2,v−v∗)x=((v+v^*)/2,v-v^*) and x′=(x1,∣x2∣σ)x'=(x_1,|x_2|\sigma) with σ∈S2\sigma\in\mathbb{S}^2. Prove, for all admissible initial data (nonnegative, finite mass, finite energy, finite entropy): (A1) Uniqueness: any two H-solutions with the same initial data coincide. (A2) Regularization: every H-solution belongs, for every t>0t>0, to C∞C^\infty in the velocity variable and indeed to the Schwartz class (the smooth functions all of whose derivatives decay faster than any power), with quantitative bounds on finitely many seminorms over compact subintervals of (0,T)(0,T). (A3) Quantitative convergence to equilibrium: f(t,⋅)f(t,\cdot) converges as t→∞t\to\infty to the Maxwellian (Gaussian) with the same conserved moments, at an explicit exponential or polynomial rate depending only on the initial mass, energy and entropy. Alternatively, refute: construct, within the admissible class, either non-unique H-solutions emanating from the same initial data, or an H-solution that fails to be smooth at some positive time, or an H-solution that converges to equilibrium slower than any explicit rate.

Goal B — microscopic foundation of the Coulomb collision model. Derive the spatially homogeneous Landau–Coulomb equation, or identify the correct limiting equation, from an N-particle system of classical (deterministic Hamiltonian) dynamics with Coulomb pair interactions under the weak-coupling plasma scaling: (B1) specify the particle system, the scaling of the interaction and of time, and a topology in which convergence to the Landau–Coulomb solution is proved; (B2) implement screening at the level of the particle dynamics: either justify microscopically the truncation of the Coulomb potential at the Debye length together with the Coulomb-logarithm renormalization, or prove that the unrenormalized limit exists and identify the correct normalization; (B3) determine the role of the Balescu–Lenard correction: in which scaling, if any, the limiting equation is Balescu–Lenard rather than Landau, and in which scaling the difference is negligible.

Scope and non-goals. The problem is restricted to the spatially homogeneous case and to the entropy-production-based solution framework of the source. Results that hold only near Maxwellian equilibrium, under additional moment, positivity or integrability assumptions beyond the H-solution bounds (for example data in L∞L^\infty, in the critical space L3/2L^{3/2}, or in H3−1/2∩L71H^{-1/2}_3\cap L^1_7), or for non-Coulomb kernels, are partial progress toward Goal A and do not by themselves resolve it; consistency results, truncated hierarchies, chaos ansätze, and propagation-of-chaos theorems for stochastic or Kac-type particle systems rather than Newtonian Coulomb dynamics are partial progress toward Goal B and do not by themselves resolve it.

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