arXiv subjects

Zhengyang Qiao

Publications and source records attributed to Zhengyang Qiao.

2 recordsLinked to original sources

arXiv2026

Commutator Estimates Uniform in the Screening Parameter and Mean-Field Limits for Yukawa Interactions

We study quantitative mean-field limits for classical particles with Yukawa (screened Coulomb) interactions in every fixed dimension $d\ge2$, uniformly as the screening parameter $κ$ tends to zero. Our main result is a first-order commutator estimate in the natural Yukawa modulated energy, with an additive error of order $N^{-2/d}$ for $d\ge3$ and $(1+\log N)/N$ for $d=2$. It requires only a bounded reference density and a Lipschitz transport field, with no uniform lower bound on interparticle distances and no negative power of $κ$. The modified Helmholtz operator $-Δ+κ^2$ creates the main new difficulty. Truncating the potential to a constant inside each truncation ball produces a surface charge and a positive volume charge whose total mass is strictly less than one. We keep the reference density unchanged and control this loss of mass through an exact Green function representation and renormalized energy identities. A stress-energy identity with interface terms and averaging over the truncation radii then give the uniform commutator estimate. Combined with the modulated energy dissipation identity and a normalized quadratic transport cost, this estimate yields weak--strong stability, propagation of chaos, and time-integrated control of the mean-square difference between empirical and mean-field forces. We also prove a quantitative Yukawa-to-Coulomb limit. For smooth product data, $N\to\infty$ and $κ\downarrow0$ may be taken simultaneously with no relation between their rates; in dimension three, a direct comparison at the particle level also holds for general symmetric initial laws with finite initial error.

math.AP↗

arXiv2026

Quantitative mean-field limits for repulsive Coulomb flows at bounded density and Riesz weak--strong stability

We establish quantitative mean-field convergence and propagation of chaos for repulsive Coulomb gradient flows at the bounded-density regularity of the limiting equation. The argument couples the dissipative modulated-energy identity with the normalized quadratic transport cost of the full $N$-particle law. The remaining negative mean-square force-error term is used through an exact completion of squares after mollification: the non-Lipschitz remainder is absorbed by this negative term, while a sharp first-order commutator estimate is applied to the mollified Lipschitz field. For the Coulomb equation, the sharp $L^\infty$ decay gives the density envelope $m(t)=\|ρ_0\|_{L^\infty}/(1+t\|ρ_0\|_{L^\infty})$. A density-adapted transport weight and mollification scale $m(t)^{-1/d}$ yield an Osgood comparison. Thus, for every $d\ge2$ and $ρ_0\in\mathcal P_2(\mathbb R^d)\cap L^\infty(\mathbb R^d)$, we obtain quantitative comparison with the global bounded-density Coulomb solution on every prescribed finite interval. For tensorized initial data, the normalized squared Wasserstein distance of the full $N$-particle law, the expected modulated energy, and the time-integrated mean-square force error are bounded by $N^{-2γ_{T,d}/d}$ for $d\ge3$ and $((1+\log N)/N)^{γ_{T,2}}$ for $d=2$, where $γ_{T,d}=(1+T\|ρ_0\|_{L^\infty})^{-c_d}$. For $d-2<s<d$, we also prove Riesz weak--strong stability for prescribed reference solutions in $L^\infty(0,T;B^{s-d+2}_{\infty,q})$, with Gronwall, Bihari, and Osgood comparisons according to $q$, together with uniqueness in the stated Besov class. Finally, an outlier construction separates modulated-energy convergence and Kac chaos from normalized Wasserstein convergence of the full $N$-particle law.

math.AP↗