Smooth mollified Biot-Savart kernel

In this definition only, use the local one-based coordinate names $z_1:=z(0)$ and $z_2:=z(1)$ for $z\in {\mathbb{R}}^2$, whose coordinate domain is $\{0,1\}$ as in The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Let $z^{\perp }=(-z_2,z_1)$ and $G(z)=\frac{1}{2\pi }\log \mathrm{\mid }z\mathrm{\mid },K(z)={\mathrm{\nabla }}^{\perp }G(z)=\frac{z^{\perp }}{2\pi \mathrm{\mid }z{\mathrm{\mid }}^2}(z\mathrm{\ne }0)\mathrm{.}$ The locally integrable value assigned to $K(0)$ is immaterial. Fix a radial, nonnegative smooth bump $\rho$ supported in the unit ball with $\int \rho =1$; its scaled family is the mollifier family of The mollifier family generated by a unit-mass smooth bump. Put ${\rho }_{\epsilon }(z)={\epsilon }^{-2}\rho (z\mathrm{/}\epsilon )$, $G_{\epsilon }=G\ast {\rho }_{\epsilon }$, and $K_{\epsilon }=K\ast {\rho }_{\epsilon }\mathrm{.}$ The kernel is a regularized interaction; its smoothness and global Lipschitz property are proved in a separate lemma. It does not define the singular physical interaction at a collision.