Characteristic solution for a finite signed regularized vorticity

Assume AC. Let ${\omega }_0$ be a finite signed Borel measure on ${\mathbb{R}}^2$, set $M=\mathrm{\mid }{\omega }_0\mathrm{\mid }({\mathbb{R}}^2)$, and let ${\omega }_0={\omega }_0^{+}-{\omega }_0^{-}$ be its Jordan decomposition. The Jordan total-variation formula gives $M={\omega }_0^{+}({\mathbb{R}}^2)+{\omega }_0^{-}({\mathbb{R}}^2)$. If $M=0$, its characteristic solution is the constant zero measure curve. If $M>0$, set $E={\mathbb{R}}^2\times \{+1,-1\}$ with sigma-algebra $\mathcal{B}({\mathbb{R}}^2)\otimes 2^{\{+1,-1\}}$, and define the probability measure ${\lambda }_0$ by ${\lambda }_0(A\times \{s\})={\omega }_0^s(A)\mathrm{/}M$ for Borel $A$ and $s\in \{+1,-1\}$; its total mass is one by the displayed formula. A signed characteristic solution for the regularized kernel $K_{\epsilon }$ on $[0,T]$ is a jointly measurable path map $Z:[0,T]\times E\to {\mathbb{R}}^2$, modulo ${\lambda }_0$-almost everywhere equality, with absolutely continuous paths for almost every label, $Z_0(x,s)=x$, and

$Z_t(x,s)=x+M{\int }_0^t{\int }_{E}r{K}_{\epsilon }(Z_u(x,s)-Z_u(y,r))d{\lambda }_0(y,r)du,$

such that its time-supremum function is measurable and ${\int }_E{\sup }_{0\le t\le T}\mathrm{\mid }{Z}_t(x,s)\mathrm{\mid }d{\lambda }_0(x,s)<\mathrm{\infty }$. It represents the signed measure curve

${\omega }_t=M{\int }_{E}s{\delta }_{Z_t(x,s)}d{\lambda }_0(x,s)\mathrm{.}$

Here ${\omega }_0^s$ denotes ${\omega }_0^{+}$ when $s=+1$ and ${\omega }_0^{-}$ when $s=-1$. The total variation mass is $M$ at time zero, and the sign coordinate records each Jordan part separately.