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Quoc-Hung Nguyen

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Preprint Jul 2026

Singular mean-field limits via a multiscale mollification metric

We consider a general class of first order ODE systems for the evolution of $N$ interacting particles (in Euclidean space $\mathbb{R}^d$) in a mean-field regime. The class of interactions treated includes singular interactions of inverse power type up to power $d+1$, attractive or repulsive, and not necessarily deriving from a potential -- unlike, for instance, the modulated energy method. We introduce a new method to prove quantitative convergence of the discrete system to solutions of the mean-field equation. It relies on studying the evolution of a metric encoding a multiscale control of the difference between the empirical measure and its limit, via mollification by heat kernels. We prove that the desired convergence holds (i) up to the maximal time of existence of the smooth solution to the limiting equation if the singularity is sub-coulombic in any dimension, or coulombic in dimensions 1 and 2 (where, to do so, we introduce a notion of weak solution to the ODE system), or (ii) for short time in the case of Coulomb singularity in dimension 3 and above and (iii) up to a short $N$-dependent timescale for super-coulombic interactions in all dimensions. The latter two results are demonstrated to be optimal as we prove that collisions occur within the same timescale for a class of attractive interactions.

Quoc-Hung Nguyen, Sylvia Serfaty · 2 citations

Preprint Aug 2026

Higher Regularity of Homogeneous Gradient Compositions for $p$-Laplace-Type Equations

In this paper, we study higher regularity of homogeneous functions of the gradient of solutions to the inhomogeneous $p$-Laplace equation $\operatorname{div}(|Du|^{p-2}Du)=f$. Although a solution need not be of class $C^2$ across its critical set, its gradient is locally H\"older continuous. Suppose that $Du\in C^{0,\alpha}_{\rm loc}$ with $\alpha\le 1/(p-1)$, and let $\Phi$ be smooth away from the origin and positively homogeneous of degree $m$. We prove that $\Phi(Du)\in C^k_{\rm loc}$ whenever $m>k/\alpha$. Moreover, all its derivatives of order at most $k$ vanish on the critical set. The proof uses the intrinsic scale $r\simeq |Du|^{1/\alpha}$, Schauder estimates for a normalized uniformly elliptic equation, and an extension lemma across the critical set. We also obtain corresponding results for autonomous anisotropic equations and for elliptic and parabolic $p$-Laplace systems, under the appropriate H\"older assumption on the gradient. Finally, the same argument gives $C^k$ regularity criteria for high powers of nonnegative solutions to the porous medium equation.

Quoc-Hung Nguyen, Le Xuan Truong · 0 citations