Quantitative Diffusive Limits for Singular Nonlocal Transport

2026-09-10

AI Summary

研究非局部传输方程的扩散极限问题，通过证明解随参数b趋近于0时收敛至热流，并分析N-粒子动力学，给出了解的一致收敛速度。

Abstract

We study the nonlocal continuity equation \[ \partial_tμ_b =\operatorname{div}\!\left( μ_b\nabla\log\bigl((I-b^2Δ)^{-1}μ_b\bigr) \right) \] on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as $b \to 0$, its global solution converges to heat flow $μ(t)$ at the sharp, uniform-in-time rate \[ \sup_{t\ge0}\|μ_b(t)-μ(t)\|_{L^1}\le Cb^2. \] The key estimate is the uniform dissipation of a $b$-weighted higher-order resolvent energy, which yields exponential relaxation despite the absence of a Wasserstein gradient-flow structure. On the circle, we also analyze the corresponding deterministic $N$-particle dynamics. A weak--strong modulated energy argument gives \[ \mathbb E\!\left[ \sup_{t\ge0}W_1(μ_b^N(t),μ_b(t)) \right] \le C(Nb)^{-1/2} \] for iid initialization. Consequently, the choice $b\asymp N^{-1/5}$ approximates heat flow uniformly in time at rate $N^{-2/5}$.

Problem

• nonlocal continuity equation
• Riemannian manifold
• heat flow
• uniform-in-time rate
• deterministic N-particle dynamics

Innovation

• nonlocal continuity equation
• uniform dissipation
• higher-order resolvent energy
• modulated energy argument
• deterministic N-particle dynamics