Optimal regularity of velocity averages for the stationary transport equation with general velocity measures: sharp conditions on the measure, endpoint behavior, and compactness classification (open questions of Golse-Lions-Perthame-Sentis, 1988)

The open questions left by Golse, Lions, Perthame and Sentis (GLPS) in their 1988 paper 'Regularity of the moments of the solution of a transport equation' concern the velocity averaging lemma: if $u$ solves the stationary transport equation $u + v \cdot \nabla_x u = f$ on $\mathbb{R}^N_x \times \mathbb{R}^N_v$ and the velocity variable is averaged against a measure $\mu$ satisfying a power-law nondegeneracy condition, then the moment $\tilde u(x) = \int u(x,v)\,\phi(v)\,d\mu(v)$ gains fractional regularity in $x$. GLPS proved the gain $s < \gamma \min(1/p, 1/p')$ in the Sobolev space $W^{s,p}$ (with $\gamma$ the nondegeneracy exponent of $\mu$, $p'$ the conjugate exponent) and raised the natural sharpness questions. The time-dependent (evolution) variant is absorbed into the modern theory; the sharp exponents are known for the Lebesgue measure ($\gamma = 1$) in all $L^p$, $1 < p < \infty$; and the $p = 1$ endpoint fails in general, with a substantial but incomplete substitute theory. What remains open is the general-measure core: necessary and sufficient conditions on $\mu$ for a given Sobolev gain, the endpoint characterization at $p = 1$ and $p = \infty$, and the classification of compactness versus failure regimes.

Background

The subject is the regularity in the position variable $x$ of velocity averages (also called moments) of solutions of linear transport equations. A moment of a phase-space function $u(x, v)$ is the weighted velocity integral $\tilde u(x) = \int_{\mathbb{R}^N} u(x, v)\,\phi(v)\,d\mu(v)$, where $\mu$ is a positive bounded measure on velocity space and $\phi$ is a test weight. A velocity averaging lemma asserts that if $u$ solves a transport equation, in the sense of distributions, with source $f$, then $\tilde u$ is smoother than a generic function of $x$ alone: averaging in $v$ restores a fraction of an $x$-derivative. Here smoothness is measured in the Sobolev space $W^{s,p}$ (functions with $s$ fractional derivatives in $L^p$) or the Besov space $B^s_{p,r}$ (a finer scale, obtained by real interpolation, that refines $W^{s,p} = B^s_{p,p}$; roughly $B^s_{p,\infty}$ contains functions whose smoothness holds at a weak, modulus-of-continuity level). The fundamental result is due to Golse, Lions, Perthame and Sentis (1988), with a compactness precursor by Golse-Perthame-Sentis (1985) and an independent discovery by Agoshkov (1984) in transport theory. GLPS consider the stationary equation $u + v \cdot \nabla_x u = f$ on $\mathbb{R}^N_x \times \mathbb{R}^N_v$ with $u, f \in L^p$ of the phase space (with respect to the product measure $dx\,d\mu(v)$) and $\mu$ satisfying the power-law nondegeneracy condition (their (2.1)): $\sup_{e \in S^{N-1}} \mu\{v : |v \cdot e| \le \varepsilon\} \le C\varepsilon^{\gamma}$ for all $\varepsilon > 0$, where $S^{N-1}$ is the unit sphere of $\mathbb{R}^N$, with $0 < \gamma < 2$; this says that no hyperplane through the origin carries too much of the measure, with a power-law rate. Their Theorem 1 gives the Hilbertian gain $L^2 \to H^{\gamma/2}$ (where $H^s = W^{s,2}$) together with a dyadic-interpolation inequality, and their Theorem 2 gives $L^p \to W^{s,p}$ for $s < \gamma \min(1/p, 1/p')$, $1 < p < \infty$, $p' = p/(p-1)$. They complement the positive theory with: an $L^1$ compactness criterion for families of moments under uniform integrability (every moment lying below a fixed convex superlinear function, the de la Vallée-Poussin condition); extensions to bounded convex spatial domains with inflow boundary traces (the part of the boundary where the velocity field points inward); Marcinkiewicz weak-$L^r$ estimates (bounds in the weak space of functions $g$ for which $t^r\,|\{|g|>t\}|$ stays bounded as $t$ varies) in mixed spaces; a Hölder gain $C^{0,\gamma}$ in dimension $N = 1$; and counterexamples, notably an $L^1$ example (Example 1, p. 123) showing that $L^1$ moments of $L^1$ solutions can concentrate on rays and need not gain any Sobolev regularity, plus a kernel family for which equicontinuity fails under $L^\infty$ bounds. Velocity averaging lemmas became a basic tool wherever kinetic equations feed macroscopic limits: the hydrodynamic limits of the Boltzmann equation, the DiPerna-Lions weak solutions of Vlasov-Maxwell, kinetic formulations of scalar conservation laws, and dispersion estimates for Schrödinger-type equations; a 2025 survey by Golse, Mauser and Möller covers the classical, quantum and semi-classical variants. The literature that grew out of GLPS developed along several lines: DiPerna-Lions-Meyer and Bézard extended to $L^p$ and Besov scales; Lions and DeVore-Petrova settled the sharp exponents for the Lebesgue measure; Golse-Saint-Raymond, Arsénio-Masmoudi and Arsénio-Lerner built the $L^1$ and duality theory; Jabin-Vega developed real-space methods based on the X-ray transform (the operator integrating a function of position over straight lines, which is the geometry along which the transport equation propagates information); Arsénio and Saint-Raymond connected averaging to hypoellipticity (the phenomenon that solutions of certain degenerate equations are smooth wherever their data are smooth) and compactness; and Erceg, Karlsen and Mitrović reduced the hypotheses on the measure to minimal nondegeneracy assumptions. Terms used below: a velocity measure is the measure $\mu$ over which the velocity averaging is performed; a compactness regime is a set of hypotheses on a family of solutions under which the family of moments is relatively compact (equicontinuous after normalization) in a function space on $x$ alone; an endpoint is the case $p = 1$ or $p = \infty$ of an $L^p$ scale.

Problem Statement

Let $\mu$ be a positive bounded measure on $\mathbb{R}^N_v$ satisfying the GLPS power-law nondegeneracy condition $\sup_{e \in S^{N-1}} \mu\{v : |v \cdot e| \le \varepsilon\} \le C\varepsilon^{\gamma}$ for all $\varepsilon > 0$ and some $0 < \gamma < 2$, and let $\phi$ be a smooth compactly supported velocity weight. For solutions $u \in L^p(\mathbb{R}^N_x \times \mathbb{R}^N_v;\,dx\,d\mu(v))$ (in the sense of distributions) of the stationary transport equation $u + v \cdot \nabla_x u = f$ with source $f \in L^p(\mathbb{R}^N_x \times \mathbb{R}^N_v;\,dx\,d\mu(v))$, the moment $\tilde u(x) = \int u(x, v)\,\phi(v)\,d\mu(v)$ belongs to the Sobolev space $W^{s,p}(\mathbb{R}^N_x)$ for every $s < \gamma\min(1/p, 1/p')$, $1 < p < \infty$ (GLPS Theorem 2). The surviving open core of the questions raised in the source paper is: (i) Sharpness for general measures: determine necessary and sufficient conditions on the measure $\mu$ for the moment to lie in $W^{s,p}$. This part requires both of: (a) the sharp gain that holds uniformly over all measures satisfying the power-law nondegeneracy condition with exponent $\gamma$ — in particular, whether the bound $s < \gamma\min(1/p, 1/p')$ is the best possible such uniform gain, that is, the worst case over the class — and (b) whether the fine geometry of a fixed measure $\mu$ can force a gain strictly larger than that uniform bound, together with what geometric property of $\mu$ governs the maximal exponent (the nondegeneracy condition is only an upper bound on the concentration of $\mu$, so a single measure may satisfy it with several exponents $\gamma$, as the Lebesgue case and the mixed-integrability theory illustrate). (ii) Endpoints: characterize the behavior at $p = 1$, where the naive conclusion fails by the known ray-concentration counterexample, by identifying the exact substitute space (weak-type, Besov $B^{s}_{1,\infty}$, or otherwise) for moments of $L^1$ solutions; and characterize the $p = \infty$ endpoint, for which no result is known. (iii) Compactness classification: for families of solutions bounded in $L^p(\mathbb{R}^N_x \times \mathbb{R}^N_v;\,dx\,d\mu(v))$ together with their sources, classify the regimes in which the corresponding family of moments is relatively compact (equicontinuous) in the natural function space on $x$ alone — relative compactness in $L^p(\mathbb{R}^N_x)$, weak relative compactness at the endpoint $p = 1$, and strong compactness in the Sobolev space $W^{s,p}(\mathbb{R}^N_x)$ furnished by the averaging estimate with $s < \gamma\min(1/p, 1/p')$ — versus the regimes in which concentration or oscillation destroys compactness, completing the counterexample program of the source paper. Each of (i)-(iii) is an independent open problem: a complete answer to any one of them resolves that corresponding part of the open core, and the open core as a whole is addressed only when all three parts are answered. The time-dependent variant, the Lebesgue-measure sharpness for $1 < p < \infty$, and the failure of the naive $p = 1$ conclusion are settled in the literature and are not part of the open core; no other restriction is imposed on $N$, $\mu$, $\phi$, or the integrability class.

References

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20. D. Arsénio, N. Masmoudi, 'Maximal gain of regularity in velocity averaging lemmas', Anal. PDE 12(2) (2019) 333-388. https://doi.org/10.2140/apde.2019.12.333
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24. T. Erceg, K. H. Karlsen, D. Mitrović, 'Regularity of velocity averages in kinetic equations with heterogeneity', arXiv:2507.04102 (2025). https://arxiv.org/abs/2507.04102 (regularity of velocity averages for equations with $x$-dependent drift under a quantitative non-degeneracy condition)
25. F. Golse, N. J. Mauser, J. Möller, 'Velocity Averaging Lemmas: Classical, Quantum and Semi-Classical' (survey), arXiv:2512.01529 (2025). https://arxiv.org/abs/2512.01529 (reaffirms the $L^1$ failure citing GLPS Example 1, p. 123)