Mean-field vortex blob error budget with signed particles, force quadrature, and RK4

Assume AC and Countable Choice. Let ${\omega }_t$ be a smooth finite signed vorticity density on $[0,T]\times {\mathbb{R}}^2$ with finite first moment, transported by a sign-preserving characteristic flow $X_t$ satisfying ${\omega }_t=(X_t{)}_{\mathrm{\#}}{\omega }_0$ and ${\dot{X}}_t(x)=(K\ast {\omega }_t)(X_t(x))$, where $K$ is the singular kernel in Smooth mollified Biot-Savart kernel. Assume ${\sup }_{t\le T}\mathrm{\parallel }{\omega }_t{\mathrm{\parallel }}_{\mathrm{\infty }}\le W$, $\int {\sup }_{t\le T}\mathrm{\mid }{X}_t(x)\mathrm{\mid }d\mathrm{\mid }{\omega }_0\mathrm{\mid }(x)<\mathrm{\infty }$, and the flow transports each Jordan part to the corresponding Jordan part. Put $M=\mathrm{\mid }{\omega }_0\mathrm{\mid }({\mathbb{R}}^2)$. Let ${\omega }_N^0={\sum }_{i=1}^N{\mathrm{\Gamma }}_i{\delta }_{x_i^0}$ be a finite signed particle measure whose positive and negative Jordan masses respectively equal those of ${\omega }_0$, with finite first moment, and with consolidated atoms whose positive and negative supports are disjoint; thus ${\sum }_i\mathrm{\mid }{\mathrm{\Gamma }}_i\mathrm{\mid }=M$. Let $E_0=C_{\pm }({\omega }_0,{\omega }_N^0)<\mathrm{\infty }$.

Fix $\epsilon >0$. For every time and force location $(t,x)$, suppose the force integrand $F_{t,x}:R\to {\mathbb{R}}^2$ lies in $C^4(R,{\mathbb{R}}^2)$ and its fourth partial derivatives satisfy the uniform bounds $M_{40},M_{04}$. Let the tensor Simpson evaluator use the even panel counts and widths in Tensor composite Simpson error for a two-dimensional force integral, and define its residual by $r_q(t,x)=Q_y{Q}_x{F}_{t,x}-{\int }_R{F}_{t,x}$. Thus ${\eta }_q=\frac{\sqrt{2}\mathrm{\mid }R\mathrm{\mid }}{180}(h_x^4{M}_{40}+h_y^4{M}_{04})\mathrm{.}$ Assume this residual is the same correction function for every particle, ${\sup }_{t,x}\mathrm{\mid }{r}_q(t,x)\mathrm{\mid }\le {\eta }_q$, and the corrected regularized particle flow preserves the two Jordan parts. Write $x_i^{\epsilon ,q}$ for the exact solution from $x_i^0$ of ${\dot{x}}_i^{\epsilon ,q}(t)={\sum }_{j=1}^N{\mathrm{\Gamma }}_j{K}_{\epsilon }(x_i^{\epsilon ,q}(t)-x_j^{\epsilon ,q}(t))+r_q(t,x_i^{\epsilon ,q}(t)),$ and set ${\omega }_N^{\epsilon ,q}(t)={\sum }_i{\mathrm{\Gamma }}_i{\delta }_{x_i^{\epsilon ,q}(t)}$. Assume the resulting nonautonomous field $f(t,X)$ is jointly $C^4$ on an open time-state domain $D$, with derivatives through order four bounded on a common compact convex tube $\mathcal{T}⋐D$ containing all exact paths and RK4 stages. Let $x_i^h(T)$ be the RK4 states from the same initial data for this ODE, and define ${\omega }_{N,h}^{\epsilon ,q}(T)={\sum }_i{\mathrm{\Gamma }}_i{\delta }_{x_i^h(T)}$. Let $h=T\mathrm{/}n\le h_0$ for an integer $n\ge 1$ and define $C_{RK}:=C_4(e^{L_{\ast }T}-1)\mathrm{/}{L}_{\ast }$ if $L_{\ast }>0$, and $C_{RK}:=C_{4}T$ if $L_{\ast }=0$, where $C_4,L_{\ast },h_0$ are the constants in Fourth-order global error for RK4 on a regularized vortex particle tube. Then $d_{BL}({\omega }_T,{\omega }_{N,h}^{\epsilon ,q}(T))\le e^{2L_{\epsilon }MT}(E_0+MT(C_{\rho }\epsilon W+{\eta }_q))+M{C}_{RK}{h}^4,$ where $L_{\epsilon }$ is a one-variable global Lipschitz constant for $K_{\epsilon }$; the induced two-variable constant for $K_{\epsilon }(x-y)$ is also $L_{\epsilon }$. The terms are initial signed particle approximation, blob regularization, force quadrature, and RK4 time error. The constants may depend on $\epsilon$, particle count, and the tube; no uniform singular limit as $\epsilon \to 0$ is claimed.

The RK4 tube condition in the Statement includes all numerical grid states and every stage generated at each grid step from both the exact solution start and the numerical start; both one-step maps are defined.

This theorem is conditional on a smooth finite-time Euler solution, matched signed initial masses, a common sign-preserving quadrature force field, and the compact RK4 tube. It makes no claim for singular initial vorticity, arbitrary-sign particle sampling without a split coupling, or a rate uniform in $\epsilon$.