TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generated

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Exact weak transport identity for weighted signed vortex particles

Statement

Let $x_{i} : [0, T] \to R^{2}$ be $C^{1}$ trajectories with fixed real weights $Γ_{i}$ , and let $g : [0, T] \times R^{2} \to R^{2}$ be a common continuous force correction. Suppose ${\dot{x}}_{i} (t) = \sum_{j = 1}^{N} Γ_{j} K_{ε} (x_{i} (t) - x_{j} (t)) + g (t, x_{i} (t)),$ where $K_{ε}$ is the mollified kernel in Equal-probability mean-field and weighted signed vortex particle systems. For $ω_{N} (t) = \sum_{i} Γ_{i} δ_{x_{i} (t)}$ and every $φ \in C_{c}^{1} (R^{2})$ , $\int φ d ω_{N} (t) - \int φ d ω_{N} (0) = \int_{0}^{t} \int \nabla φ (x) \cdot (\int K_{ε} (x - y) d ω_{N} (s) (y) + g (s, x)) d ω_{N} (s) (x) d s .$ In particular, the uncorrected regularized particle system satisfies the same weak identity with $g = 0$ .

Facts & Assumptions

Given: The weighted particle paths, fixed weights, common continuous force correction, and $φ \in C_{c}^{1} (R^{2})$ .

[F1]

The weighted system and its signed particle measure use the finite sums ${\dot{x}}_{i} = \sum_{j} Γ_{j} K_{ε} (x_{i} - x_{j})$ and $ω_{N} = \sum_{i} Γ_{i} δ_{x_{i}}$ (Equal-probability mean-field and weighted signed vortex particle systems).

[F2]

Differentiable compositions obey the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

1.1givenF1F2algebra

By [F2] and the ODE in the Statement, $\frac{d}{d t} \sum_{i} Γ_{i} φ (x_{i} (t)) = \sum_{i, j} Γ_{i} Γ_{j} \nabla φ (x_{i} (t)) \cdot K_{ε} (x_{i} (t) - x_{j} (t)) + \sum_{i} Γ_{i} \nabla φ (x_{i} (t)) \cdot g (t, x_{i} (t)) .$ All sums are finite, so integrating this equality in time is valid.

2.1F1step 1.1algebra∎

Expanding the two signed-measure integrals on the right of the Statement gives exactly the two finite sums in step 1.1. The left side is the weighted test sum at times $t$ and $0$ , so integration of step 1.1 proves the identity.

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Sources

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)
François Golse, Mean-Field Limits in Statistical Dynamics, Lecture 1 (standard reference, not scraped)