具有幂核的非局部吸引-排斥输运方程
The nonlocal attraction-repulsion transport equation with power kernels
- Technical University of Munich（慕尼黑工业大学）
- Munich Data Science Institute（慕尼黑数据科学研究所）
- Munich Center for Machine Learning（慕尼黑机器学习中心）
- Hunan University（湖南大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究\(\mathbb{R}^d\)上非局部连续性方程，通过平方半径正则化建立全局拉格朗日适定性，给出初始数据紧支且吸引占优时的相关结论，刻画零通量稳态并给出显式例子，证明有界能量和矩界的全局解收敛到零通量稳态。
AI中文摘要：
我们研究\(\mathbb{R}^d\)上的一个非局部连续性方程，其中概率密度由对规定背景测度\(\omega\)的吸引与粒子间的自排斥竞争驱动，分别由幂律核\(\psi_a(x)=|x|^{1 + a}\)和\(\psi_r(x)=|x|^{1 + r}\)控制，指数\(a,r\in[0,1)\)。我们通过平方半径正则化建立全局拉格朗日适定性，得到一致\(L^\infty\)和矩界、\(W^{n,\infty}\)正则性以及拉格朗日类中的唯一性。当初始数据紧支且吸引占优时，我们证明支撑始终一致有界；一个反例表明\(a = r>1\)时不成立。对于吸引占优的非二次范围\(0\leq r\leq a<1\)，我们通过涉及分数拉普拉斯算子的自由边界问题刻画零通量稳态，将平稳性条件简化为分数外部狄利克雷问题。这种刻画使我们能够在\(d\in\{1,2,3\}\)维中展示平稳测度的显式例子。数值粒子模拟证实与理论平稳分布一致。最后，我们证明每个具有有界能量和一致矩界的全局解收敛到零通量稳态。
英文摘要：
We study a nonlocal continuity equation on $\mathbb{R}^d$ in which a probability density is driven by the competition between attraction toward a prescribed background measure $ω$ and self-repulsion among particles, governed respectively by the power-law kernels $ψ_a(x) = |x|^{1+a}$ and $ψ_r(x) = |x|^{1+r}$ with exponents $a, r \in [0,1)$. We establish global Lagrangian well-posedness via a squared-radius regularization, obtaining {finite-time $L^\infty$ and moment bounds}, $W^{n,\infty}$ regularity, and uniqueness in the Lagrangian class. When the initial data is compactly supported and attraction dominates ($a > r$, or $a = r$ with $ω(\mathbb{R}^d) > 1$), we prove that the support remains uniformly bounded at all time; a counterexample shows this fails for $a = r > 1$. For the attractive-dominant nonquadratic range $0 \leq r \leq a < 1$, we characterize zero-flux stationary states via a free-boundary problem involving a fractional Laplacian operator, reducing the stationarity condition to a fractional exterior Dirichlet problem. This characterization allows us to exhibit explicit examples of stationary measures in dimensions $d \in \{1,2,3\}$. Numerical particle simulations {are consistent} with the theoretical stationary profiles. Finally, for $0 \le r \le a < 1$, we prove that every global solution whose energy is bounded from below and whose moments are bounded uniformly in time converges, along a sequence of times, to a zero-flux stationary state. Full convergence holds when, in addition, the support remains uniformly bounded (and, for $r = 0$, the density remains uniformly bounded) and the omega-limit set contains a single stationary state.
