= Solution
Use the conventional <Laplace operator> $\Delta=\partial_{x_1}^2+\partial_{x_2}^2$. The <fundamental solution of the Laplace equation> in two dimensions is $G(x)=(2\pi)^{-1}\log|x|$, with $\Delta G=\delta_0$. To check its sign, integrate the outward normal derivative around a circle: $\int_{\partial B_\varepsilon}\partial_n\log|x|\,ds=2\pi$. Applying <integration by parts> against a smooth compactly supported test function and shrinking the circle gives the distributional identity.
The printed <stream function> has $\phi=-G*\omega$. For each <multi-index> $\beta$ with $|\beta|\leq2$, transfer derivatives to the compactly supported $C^2$ function:
$$
\partial^\beta\phi=-G*(\partial^\beta\omega).
$$
The logarithmic singularity is locally integrable. On a bounded region in $x$, the convolution uses one fixed bounded region in the integration variable; splitting off a small disk around the singularity proves <continuity> of these derivatives. Thus $\phi\in C^2$, and <convolution> of the distributional identity gives
$$
\boxed{\Delta\phi=-\omega.}
$$
\b[The asserted plus sign is false for the printed kernel and the conventional Laplacian.] Any nonzero smooth compactly supported $\omega$ is a counterexample. To obtain $\Delta\phi=\omega$, replace the kernel's minus sign by a plus sign, or explicitly adopt the negative <Laplace operator>. Below, retain the printed kernel and velocity convention. With that convention, $\operatorname{curl}u=\Delta\phi=-\omega$, so $\omega$ is the negative of conventional scalar <vorticity>. This sign does not change the transport or flow arguments.
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