For the spatially inhomogeneous Boltzmann equation without Grad's angular cutoff, every journal-published large-data existence result in the entropy space carries a non-negative defect measure: the renormalized equation holds only as an inequality, a construction introduced by Alexandre and Villani (CPAM 2002) to complete the program proposed in Lions' 1998 Comptes Rendus note 'Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire'. Luo and Nie (2026, preprint) proved the defect measure vanishes for hard potentials (0≤γ≤10\le\gamma\le1) and showed their method fails for soft potentials (−3<γ<0-3<\gamma<0). The open problem is the surviving core of the 1998 program: whether global existence of renormalized solutions in the standard DiPerna–Lions sense — an equality with no defect measure — holds for large data and soft-potential, grazing-singular kernels.

Background

The Boltzmann equation ∂tf+v⋅∇xf=Q(f,f)\partial_t f+v\cdot\nabla_x f=Q(f,f) governs the evolution of the particle density f(t,x,v)f(t,x,v) of a dilute gas, where xx is position, vv is velocity, and the collision operator QQ acts only on the velocity variable. In the σ\sigma-representation, Q(f,f)=∫R3∫S2B(v−v∗,σ)(f′f∗′−ff∗) dσ dv∗Q(f,f)=\int_{\mathbb{R}^3}\int_{S^2}B(v-v_*,\sigma)(f'f'_*-ff_*)\,d\sigma\,dv_*, where (v′,v∗′)(v',v_*') are the post-collisional velocities determined by momentum and energy conservation and BB is the collision kernel encoding the intermolecular potential. For inverse power law potentials of index ss, one has B(z,σ)=∣z∣γb(cos⁡θ)B(z,\sigma)=|z|^{\gamma}b(\cos\theta) with cos⁡θ=(z⋅σ)/∣z∣\cos\theta=(z\cdot\sigma)/|z|, γ=1−4/(s−1)\gamma=1-4/(s-1), and angular behavior sin⁡θ b(cos⁡θ)∼Kθ−1−ν\sin\theta\,b(\cos\theta)\sim K\theta^{-1-\nu}, ν=2/(s−1)\nu=2/(s-1), as the deflection angle θ→0\theta\to0. Grazing collisions (θ≈0\theta\approx0) therefore dominate: ∫S2B dσ=∞\int_{S^2}B\,d\sigma=\infty, the singularity being non-integrable in the angular variable. Grad's angular cutoff truncates this singularity and makes QQ a bounded operator; removing it (the 'non-cutoff' or 'long-range interaction' regime) is physically correct but mathematically singular, since the gain and loss parts of QQ are individually infinite and must be handled together. The regime γ>0\gamma>0 is called hard potentials, γ=0\gamma=0 Maxwell molecules (where BB does not depend on the relative speed), and −3<γ<0-3<\gamma<0 soft potentials; γ=−3\gamma=-3 corresponds to the Coulomb potential, whose grazing limit is the Landau equation of plasma physics.

For the cutoff equation, DiPerna and Lions (1989) introduced renormalized solutions: for large initial data in the entropy space ∫f0(1+∣x∣2+∣v∣2+∣log⁡f0∣) dx dv<∞\int f_0(1+|x|^2+|v|^2+|\log f_0|)\,dx\,dv<\infty, the equation holds in the sense of distributions after composition with admissible nonlinearities β\beta (bounded C2C^2 functions on [0,∞)[0,\infty) with 0<β′(z)≤C/(1+z)0<\beta'(z)\le C/(1+z) and β′′(z)≤0\beta''(z)\le0 for z≥0z\ge0), which is necessary because Q(f,f)Q(f,f) is not defined for mere L1L^1 data.

Lions' 1998 Comptes Rendus note obtained the first regularity and compactness results for non-cutoff kernels. Its Theorem 1 states that if g=fg=\sqrt{f} lies in L2L^2 with a finite collision-dissipation norm built from ∫B∣g′g∗′−gg∗∣2\int B|g'g'_*-gg_*|^2, then gg belongs locally to the Besov space B∞s,2B^{s,2}_{\infty} (a scale of spaces measuring fractional smoothness) with exponent s=N−12α−1α+N−2s=\frac{N-1}{2}\frac{\alpha-1}{\alpha+N-2} — here NN is the space dimension — for kernels bounded below by c⟨z⟩−m∣cos⁡θ∣−αc\langle z\rangle^{-m}|\cos\theta|^{-\alpha} with m>Nm>N and α>1\alpha>1. The note itself flagged that this exponent is not sharp, and that the marginal case α=1\alpha=1 was covered only by estimates with a logarithmic factor. Its compactness theorem (Theorem 2) states that a sequence fnf_n converging weakly in L1L^1 with uniformly bounded entropy dissipation D(fn)D(f_n) (the non-negative collisional entropy production entering the Boltzmann H-theorem) converges strongly in L1L^1 provided the velocity averages ∫fnφ(v) dv\int\sqrt{f_n}\varphi(v)\,dv converge in L1L^1 locally — a hypothesis (hypothesis (8) in the note) that is natural but was 'expected but certainly not known' for solutions of the non-cutoff equation. Velocity averaging lemmas (standard tools of kinetic theory stating that vv-averages of solutions of transport equations gain regularity) would supply time-space compactness once velocity regularity is available. The note left open the intended application: an existence theory of weak or renormalized solutions for singular (non-cutoff) kernels.

The subsequent line of work: Alexandre (1999) introduced a definition of renormalized solutions without angular cutoff; Alexandre, Desvillettes, Villani and Wennberg (2000) proved that a bound on the entropy dissipation controls a weighted fractional Sobolev norm of f\sqrt{f}, supplying the velocity-regularity input; and Alexandre and Villani (Communications on Pure and Applied Mathematics 55 (2002) 30–70) combined these ingredients with velocity averaging to construct global renormalized solutions for large data under general assumptions covering singularities in both the angular and the relative-velocity variables, thereby proving Lions' conjecture on the appearance of strong compactness. The price is the defect measure: the construction passes to the limit using Fatou-type lower semicontinuity, so the limit solves ∂tβ(f)+v⋅∇xβ(f)≥β′(f)Q(f,f)\partial_t\beta(f)+v\cdot\nabla_x\beta(f)\ge\beta'(f)Q(f,f) — an inequality whose unknown non-negative gap (the defect measure) is controlled by, but not proven to vanish in, the entropy dissipation. Arsénio and Masmoudi (2012) showed that such renormalized solutions FF nevertheless enjoy explicit velocity smoothness, F1+F∈Wt,x,vs,p\frac{F}{1+F}\in W^{s,p}_{t,x,v} locally. The sharp-regularity side of the 1998 note was meanwhile overtaken by sharper anisotropic entropy-production and coercivity estimates (Gressman–Strain 2011; Alexandre–Morimoto–Ukai–Xu–Yang 2011; Silvestre 2016). Most recently, Luo and Nie (2026, preprint) proved that the defect measure vanishes for hard potentials 0≤γ≤10\le\gamma\le1 — giving standard-sense renormalized solutions — and constructed a counterexample showing their approach fails for soft potentials, noting that under the assumptions actually made in the CPAM 2002 paper the collision integral for soft kernels is uncontrolled.

Problem Statement

For the spatially inhomogeneous Boltzmann equation ∂tf+v⋅∇xf=Q(f,f)\partial_t f+v\cdot\nabla_x f=Q(f,f) in Rx3×Rv3\mathbb{R}^3_x\times\mathbb{R}^3_v without angular cutoff, in the soft-potential long-range regime — concretely, for inverse power law collision kernels B(v−v∗,σ)=∣v−v∗∣γb(cos⁡θ)B(v-v_*,\sigma)=|v-v_*|^{\gamma}b(\cos\theta) with −3<γ<0-3<\gamma<0 and angular behavior sin⁡θ b(cos⁡θ)∼Kθ−1−ν\sin\theta\,b(\cos\theta)\sim K\theta^{-1-\nu} as θ→0\theta\to0 with 0<ν<20<\nu<2 — determine whether the defect measure vanishes for soft potentials: either prove the global existence of renormalized solutions in the standard DiPerna–Lions sense for large initial data in the entropy space ∫f0(1+∣x∣2+∣v∣2+∣log⁡f0∣) dx dv<∞\int f_0(1+|x|^2+|v|^2+|\log f_0|)\,dx\,dv<\infty; that is, a non-negative ff such that for every admissible renormalization β\beta (a bounded C2C^2 function on [0,∞)[0,\infty) with 0<β′(z)≤C/(1+z)0<\beta'(z)\le C/(1+z) and β′′(z)≤0\beta''(z)\le0 for z≥0z\ge0) the identity

∂tβ(f)+v⋅∇xβ(f)=β′(f)Q(f,f)\partial_t\beta(f)+v\cdot\nabla_x\beta(f)=\beta'(f)Q(f,f)

holds as an equality in the sense of distributions, with no defect measure (no non-negative measure may be added to the right-hand side to restore the balance, as happens in the Alexandre–Villani construction where only the inequality ≥\ge is obtained), together with the entropy inequality of the DiPerna–Lions theory (the integrated form of the H-theorem), or exhibit a counterexample/obstruction showing that defect-free global renormalized solutions fail to exist for some admissible large data in this class. Here β′(f)Q(f,f)\beta'(f)Q(f,f) is understood through the renormalized collision operator of Alexandre (1999) / Alexandre–Villani (2002): it is defined as a distribution (in a negative-index Sobolev space, via the singular-integral decomposition of the collision operator) from entropy-level bounds alone, and 'no defect measure' means that the non-negative Radon measure by which the Alexandre–Villani inequality formulation departs from equality vanishes identically, so that the inequality becomes an equality in the sense of distributions. In the terms of the 1998 source note, this is the unresolved remainder of its program: Alexandre and Villani (2002) achieved the existence with a non-negative defect measure, and Luo and Nie (2026) removed the defect measure only for hard potentials 0≤γ≤10\le\gamma\le1 while showing their method fails for soft potentials. Answers restricted to hard potentials, to small perturbative data, to bounded domains with boundary conditions, or to solutions carrying a defect measure do not answer the question.

References

  1. Pierre-Louis Lions, 'Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire', Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 326 (1998), no. 1, 37-41. DOI: 10.1016/S0764-4442(97)82709-7, URL: https://doi.org/10.1016/S0764-4442(97)82709-7 (primary text paywalled)
  2. R. J. DiPerna and P. L. Lions, 'On the Cauchy Problem for Boltzmann Equations: Global Existence and Weak Stability', The Annals of Mathematics 130 (1989), no. 2, 321-366. DOI: 10.2307/1971423, URL: https://doi.org/10.2307/1971423
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