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We are interested in how regular a transport velocity field must be in order to control Riesz-type commutators. Estimates for these commutators play a central role in the analysis of the mean-field limit and fluctuations for systems of particles with pairwise Riesz interactions through modulated energy techniques, which we start by reviewing. In these applications, the transport field is generated by the limiting PDE or by its adjoint linearized flow. Relaxing the regularity demanded of the transport field to control the commutator enlarges the class of limiting densities and fluctuation observables accessible to the method, especially at scaling-critical regularities. Our first new result shows that the usual $L^{\infty }$ assumption on the gradient of the velocity field cannot, in general, be relaxed to a BMO assumption. We construct counterexamples in all dimensions and all Riesz singularities $-2<\mathsf{s}<\mathsf{d}$ , except for the one-dimensional logarithmic endpoint $\mathsf{s}=0$ . At this exceptional endpoint, such a relaxation is possible, a fact related to the classical Coifman–Rochberg–Weiss commutator bound for the Hilbert transform. Our second result identifies a trade-off between the singularity of the interaction potential and the required regularity of the velocity field. Roughly speaking, smoother (less singular) interactions require stronger velocity control if one wants a commutator estimate in the natural energy seminorm determined by the potential. We formulate this principle for a broad class of potentials and show that, in the sub-Coulomb Riesz regime, the velocity regularity appearing in the known commutator inequality is sharp. Despite these negative findings, we show as our third result that adefectivecommutator estimate holds for almost-Lipschitz transport fields. Such a defective estimate, which is a consequence of the celebrated Brezis–Wainger–Hansson inequality, allows us to prove rates of convergence when the mean-field density belongs to the scaling-critical Sobolev space. 

Abstract We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super‐Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean‐field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right‐hand side exhibiting a recursive structure in terms of lower‐order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean‐field limits, quasi‐neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here the expected ‐rate in the modulated energy distance for the mean‐field convergence of first‐order Hamiltonian and gradient flows. 

This paper is intended as a companion to the author’s talk “Commutator estimates and mean-field limits for Coulomb/Riesz gases” at the 2025Journées équations aux dérivées partiellesin Aussois. The goal is to provide a concise, accessible account of sharp commutator estimates recently obtained for modulated energies associated to Coulomb/Riesz interactions and how these estimates lead to optimal results for mean-field and supercritical mean-field limits of Coulomb/Riesz gas dynamics via the modulated-energy method. The exposition centers on the works [129, 131] with Serfaty and [84] with Hess-Childs and Serfaty. 

We study so-called supercritical mean-field limits of trapped particle systems moving according to Newton’s second law with either Coulomb/super-Coulomb or regular interactions, from which we derive a $\mathsf{d}$ -dimensional generalization of thelake equation, which coincides with the incompressible Euler equation in the simplest setting, for monokinetic data. This supercritical mean-field limit may also be interpreted as a combined mean-field and quasineutral limit, and our assumptions on the rates of these respective limits are shown to be optimal. Our work provides a mathematical basis for theuniversalityof the lake equation in this scaling limit—a new observation—in the sense that the dependence on the interaction and confinement is only through the limiting spatial density of the particles. Our proof is based on a modulated-energy method and takes advantage of regularity theory for the obstacle problem for the fractional Laplacian. 

We consider mean-field limits for overdamped Langevin dynamics of $N$ particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to prove the mean-field convergence or propagation of chaos for a certain class of interactions, including Riesz kernels. We show here that generation of chaos, i.e. exponential in time convergence to a tensorized (or iid) state starting from a nontensorized one, can be deduced from the modulated free energy method provided a uniform-in- $N$ “modulated logarithmic Sobolev inequality” holds. Proving such an inequality is a question of independent interest, which is generally difficult. As an illustration, we show that uniform modulated logarithmic Sobolev inequalities can be proven for a class of situations in one dimension. 

We consider overdamped Langevin dynamics for the attractive log gas on the torus ${\mathbb{T}}^{\mathsf{d}}$ , for $\mathsf{d}\ge <\#comment/>1$ . In dimension $\mathsf{d}=2$ , this model coincides with a periodic version of the parabolic-elliptic Patlak-Keller-Segel model of chemotaxis. The attractive log gas (for our choice of units) is well-known to have a critical inverse temperature ${\mathrm{\beta <\#comment/>}}_{\mathrm{c}}=2\mathsf{d}$ corresponding to when the free energy is bounded from below. Moreover, it is well-known that the uniform distribution is always a stationary state regardless of the temperature. We identify another temperature threshold ${\mathrm{\beta <\#comment/>}}_{\mathrm{s}}$ sharply corresponding to the nonlinear stability of the uniform distribution. We show that for $\mathrm{\beta <\#comment/>}>{\mathrm{\beta <\#comment/>}}_{\mathrm{s}}$ , the uniform distribution does not minimize the free energy and moreover is nonlinearly unstable, while for $\mathrm{\beta <\#comment/>}>{\mathrm{\beta <\#comment/>}}_{\mathrm{s}}$ , it is stable. We also show that there exists ${\mathrm{\beta <\#comment/>}}_{\mathrm{u}}$ for which uniqueness of equilibria holds for $\mathrm{\beta <\#comment/>}>{\mathrm{\beta <\#comment/>}}_{\mathrm{u}}$ . Related to the above findings, we establish a uniform-in-time rate for entropic propagation of chaos for a range of $\mathrm{\beta <\#comment/>}>{\mathrm{\beta <\#comment/>}}_{\mathrm{s}}$ . To our knowledge, this is the first such result for singular attractive interactions and affirmatively answers a question of Bresch et al. [Duke Math. J. 172 (2023), pp. 2591–2641]. The proof of the convergence is through the modulated free energy method, in particular relying on amodulated logarithmic Hardy-Littlewood-Sobolev (mLHLS) inequality. Unlike Bresch et al. [Duke Math. J. 172 (2023), pp. 2591–2641], we show that such an inequality holds without truncation of the potential—the avoidance of the truncation being essential to a uniform-in-time result—at sufficiently high temperature and provide a counterexample to the mLHLS inequality when $\mathrm{\beta <\#comment/>}>{\mathrm{\beta <\#comment/>}}_{\mathrm{s}}$ . As a byproduct, we show that it is impossible to have a uniform-in-time rate of propagation of chaos if $\mathrm{\beta <\#comment/>}>{\mathrm{\beta <\#comment/>}}_{\mathrm{s}}$ . 

Abstract Motivated by the possibility of noise to cure equations of finite-time blowup, the recent work [ 90] by the second and third named authors showed that with quantifiable high probability, random diffusion restores global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as the Patlak–Keller–Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from [ 90] in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $$t\rightarrow \infty $$.