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Equal-probability mean-field and weighted signed vortex particle systems

Definition

For a globally Lipschitz interaction kernel $K$ and $N \geq 1$ , the equal-probability mean-field particle system is ${\dot{x}}_{i} (t) = \frac{1}{N} \sum_{j = 1}^{N} K (x_{i} (t), x_{j} (t)), i = 1, \dots, N .$ Its empirical probability measure is $μ_{N} (t) = N^{- 1} \sum_{i} δ_{x_{i} (t)}$ .

For a mollified two-dimensional Biot-Savart kernel $K_{ε}$ from Smooth mollified Biot-Savart kernel and real weights $Γ_{1}, \dots, Γ_{N}$ , the weighted signed vortex particle system is ${\dot{x}}_{i} (t) = \sum_{j = 1}^{N} Γ_{j} K_{ε} (x_{i} (t) - x_{j} (t)), i = 1, \dots, N,$ with vorticity measure $ω_{N} (t) = \sum_{i} Γ_{i} δ_{x_{i} (t)}$ . If $Γ_{i} = 1 / N$ and $K (x, y) = K_{ε} (x - y)$ , the first system is the probability special case. The weighted signed system is a first-order Biot-Savart model.

Depends on

Used by

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Sources

François Golse, Mean-Field Limits in Statistical Dynamics, Lecture 1 (standard reference, not scraped)
J. Liu and X. Xin, Convergence of Vortex Methods for Weak Solutions to the 2-D Euler Equations with Vortex Sheet Data (standard reference, not scraped)