Another look at regularity in transport-commutator estimates

Elias Hess-Childs; Matthew Rosenzweig; Sylvia Serfaty

doi:10.5802/crmath.837

Article de recherche - Physique mathématique, Équations aux dérivées partielles

Another look at regularity in transport-commutator estimates
[Retour sur la question de la régularité dans les estimées de commutateur-transport]

Elias Hess-Childs ¹ ; Matthew Rosenzweig ¹ ; Sylvia Serfaty ^2,^3,⁴

¹Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA, USA
²Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France
³Institut Universitaire de France
⁴Courant Institute of Mathematical Sciences, New York University, USA

Comptes Rendus. Mathématique, Volume 364 (2026), pp. 437-478

Cet article fait partie du numéro thématique De génération en génération : l'héritage mathématique de Haïm Brezis coordonné par : Henri Berestycki et al..

Abstract (VO)
Résumé

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 a defective commutator 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.

On s’intéresse à la régularité que doit présenter un champ de vitesse de transport pour permettre le contrôle des commutateurs de type Riesz. Les estimées de commutateur jouent un rôle central dans l’analyse de la limite de champ moyen et des fluctuations pour des systèmes de particules en interaction par paires de type Riesz au moyen de techniques d’énergie modulée, que nous commençons par passer en revue. Dans ces applications, le champ de transport est engendré par l’EDP limite ou par son flot linéarisé adjoint. Affaiblir la régularité exigée sur le champ de transport pour contrôler le commutateur élargit la classe des densités limites et des observables de fluctuation accessibles à la méthode, en particulier aux régularités critiques pour le changement d’échelle.

Notre premier résultat nouveau montre que l’hypothèse usuelle de type $L^{\infty }$ sur le gradient du champ de vitesse ne peut, en général, être affaiblie en une hypothèse de type BMO. Nous construisons des contre-exemples en toute dimension et pour toutes les singularités de Riesz $-2 < \mathsf {s} < \mathsf {d}$, à l’exception du cas limite logarithmique unidimensionnel $\mathsf {s} = 0$. En ce point limite exceptionnel, un tel affaiblissement est possible, fait lié à l’estimation classique de Coifman–Rochberg–Weiss pour le commutateur de la transformée de Hilbert. Notre deuxième résultat met en évidence un compromis entre la singularité du potentiel d’interaction et la régularité requise sur le champ de vitesse. En substance, des interactions plus régulières exigent un contrôle plus fort du champ de vitesse si l’on souhaite obtenir une estimée de commutateur dans la semi-norme d’énergie naturelle déterminée par le potentiel. Nous formulons ce principe pour une large classe de potentiels et montrons que, dans le régime de Riesz sous-coulombien, la régularité du champ de vitesse apparaissant dans l’inégalité de commutateur connue est optimale. Malgré ces résultats négatifs, nous montrons, comme troisième résultat, qu’une estimation de commutateur avec défaut est valable pour des champs de transport presque lipschitziens. Une telle estimation avec défaut, qui découle de la célèbre inégalité de Brezis–Wainger–Hansson, nous permet d’établir des taux de convergence lorsque la densité de champ moyen appartient à l’espace de Sobolev critique pour le changement d’échelle.

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Reçu le : 2026-01-05
Révisé le : 2026-03-31
Accepté le : 2026-05-19
Publié le : 2026-07-27

DOI : 10.5802/crmath.837

Classification : 42B20, 82C22
Keywords: Commutator estimates, Riesz interactions, Coulomb interactions, modulated energy, mean-field limits
Mots-clés : Estimées de commutateurs, interactions de Riesz, interaction coulombienne, énergie modulée, limites de champ moyen

Note : Article soumis sur invitation

Affiliations des auteurs :

Elias Hess-Childs ¹ ; Matthew Rosenzweig ¹ ; Sylvia Serfaty ^{2
,

3
,

4}

¹ Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA, USA
² Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France
³ Institut Universitaire de France
⁴ Courant Institute of Mathematical Sciences, New York University, USA

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CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Elias Hess-Childs; Matthew Rosenzweig; Sylvia Serfaty. Another look at regularity in transport-commutator estimates. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 437-478. doi: 10.5802/crmath.837

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     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {364},
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