Same-sign smooth vortex density and probability blob error budget

Assume Countable Choice. Let ${\omega }_0\in C_c^{\mathrm{\infty }}({\mathbb{R}}^2)$ be nonnegative with mass one, and suppose the smooth Euler flow satisfies ${\omega }_t=(X_t{)}_{\mathrm{\#}}{\omega }_0$, ${\dot{X}}_t=(K\ast {\omega }_t)(X_t)$, and ${\sup }_{t\le T}\mathrm{\parallel }{\omega }_t{\mathrm{\parallel }}_{\mathrm{\infty }}\le W$, with $\int {\sup }_{t\le T}\mathrm{\mid }{X}_t(x)\mathrm{\mid }d{\omega }_0(x)<\mathrm{\infty }$. Approximate it by $N\ge 1$ distinct initial positions $x_1^0,\dots ,x_N^0$ with equal weights $1\mathrm{/}N$, define ${\mu }_N^0=N^{-1}{\sum }_i{\delta }_{x_i^0}$, and let $E_0=W_1({\omega }_0,{\mu }_N^0)$.

Let $R=[a,b]\times [c,d]$ with $a<b$ and $c<d$, let $n_x,n_y$ be even integers at least two, and put $h_x=(b-a)\mathrm{/}{n}_x$ and $h_y=(d-c)\mathrm{/}{n}_y$. For every time $t\in [0,T]$ and force location $z\in {\mathbb{R}}^2$, let $F_{t,z}\in C^4(R,{\mathbb{R}}^2)$ and assume the uniform bounds ${\sup }_{t\in [0,T],z\in {\mathbb{R}}^2}{\max }_{(x,y)\in R}\mathrm{\mid }{\mathrm{\partial }}_x^4{F}_{t,z}(x,y)\mathrm{\mid }\le M_{40},{\sup }_{t\in [0,T],z\in {\mathbb{R}}^2}{\max }_{(x,y)\in R}\mathrm{\mid }{\mathrm{\partial }}_y^4{F}_{t,z}(x,y)\mathrm{\mid }\le M_{04}\mathrm{.}$ Define $r_q(t,z)=Q_y{Q}_x{F}_{t,z}-{\int }_R{F}_{t,z}$ and ${\eta }_q=\frac{\sqrt{2}\mathrm{\mid }R\mathrm{\mid }}{180}(h_x^4{M}_{40}+h_y^4{M}_{04}),\mathrm{\mid }R\mathrm{\mid }=(b-a)(d-c),$ where $Q_x,Q_y$ are the composite Simpson rules in the two rectangle coordinates. By Tensor composite Simpson error for a two-dimensional force integral, $\mathrm{\mid }{r}_q(t,z)\mathrm{\mid }\le {\eta }_q$ uniformly. Assume this common residual is smooth in $(t,z)$, and let the exact particles solve ${\dot{x}}_i=\frac{1}{N}{\sum }_j{K}_{\epsilon }(x_i-x_j)+r_q(t,x_i)$. Assume the resulting nonautonomous field 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 the exact paths and RK4 stages. For an integer $n\ge 1$ with $h=T\mathrm{/}n\le h_0$, let $x_i^h(T)$ be their RK4 approximations from the same initial nodes, with all paths and stages in that tube. Define ${\omega }_{N,h}^{\epsilon ,q}(T)=N^{-1}{\sum }_i{\delta }_{x_i^h(T)}$. If $C_{RK}$ is the coefficient from Fourth-order global error for RK4 on a regularized vortex particle tube for this $h_0$, then

$W_1({\omega }_T,{\omega }_{N,h}^{\epsilon ,q}(T))\le e^{2L_{\epsilon }T}(E_0+T(C_{\rho }\epsilon W+{\eta }_q))+C_{RK}{h}^4\mathrm{.}$

The RK4 tube hypothesis includes all numerical grid states and every stage generated at each grid step from both the exact start and the numerical start, and both one-step maps are defined, as required by Fourth-order global error for RK4 on a regularized vortex particle tube.