= Solution
The displayed evolution law uses the usual assumptions of <incompressible flow>, constant <density> and conservative body force. With $\boldsymbol\omega=\nabla\times\mathbf u$, the <Euler equation> is
$$
\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u=-\nabla(p/\rho+\Phi).
$$
Use $(\mathbf u\cdot\nabla)\mathbf u=\nabla(|\mathbf u|^2/2)-\mathbf u\times\boldsymbol\omega$ and take a <curl>. <Gradients> have zero <curl>, and
$$
\partial_t\boldsymbol\omega=\nabla\times(\mathbf u\times\boldsymbol\omega)
=(\boldsymbol\omega\cdot\nabla)\mathbf u-(\mathbf u\cdot\nabla)\boldsymbol\omega
+\mathbf u\nabla\cdot\boldsymbol\omega-\boldsymbol\omega\nabla\cdot\mathbf u.
$$
Here $\nabla\cdot\boldsymbol\omega=0$ because it is a <curl>, and $\nabla\cdot\mathbf u=0$ by incompressibility. The <vorticity equation> is therefore
$$
\boxed{\frac{D\boldsymbol\omega}{Dt}=(\boldsymbol\omega\cdot\nabla)\mathbf u.}
$$
The <material derivative> combines local change with advection by fluid particles. The right-hand side is <vortex stretching> and tilting: spatial variation of velocity along a vortex line changes its strength and direction. Along each particle path this is a homogeneous linear equation for <vorticity>. For a smooth velocity field its solution with zero initial <vorticity> remains zero by uniqueness. Thus an initially <irrotational flow> remains irrotational while these assumptions and smoothness hold.
For a plane flow $\mathbf u=(u(x,y,t),v(x,y,t),0)$, the <vorticity> is $\boldsymbol\omega=(0,0,\zeta)$ and $(\boldsymbol\omega\cdot\nabla)\mathbf u=\zeta\partial_z\mathbf u=0$. Hence $D\zeta/Dt=0$. Every particle retains its initial value, so an initially uniform value gives
$$
\boxed{\boldsymbol\omega(x,y,t)=\boldsymbol\omega_0\quad\text{for all later times}.}
$$
Inviscid motion alone is insufficient for the printed simplified law: compressible variable-density motion generally adds $-\boldsymbol\omega\nabla\cdot\mathbf u+\nabla\rho\times\nabla p/\rho^2$. Those terms vanish under the assumptions used above.
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