Potential flow

Remarks

A potential flow with $\varphi$ of class $C^2$ is irrotational, since the curl of a gradient is $0$. If it is also incompressible, then $\nabla \cdot u = \Delta \varphi = 0$, so $\varphi$ solves Laplace's equation at each time.

Example: on $\Omega = \R^2$, $\varphi(x, y) = (x^2 - y^2)/2$ gives $u = (x, -y)$, the flow near a stagnation point. It is incompressible, and indeed $\Delta \varphi = 1 - 1 = 0$.

Non-example: the rigid rotation $u = (-y, x)$ has curl $2 \ne 0$, so it is not the gradient of any function.

On a simply connected domain, a $C^1$ irrotational flow is a potential flow. On other domains this can fail. The point vortex $u = (-y, x)/(x^2 + y^2)$ on $\R^2 \setminus \set{0}$ is irrotational. But its circulation around the unit circle is $2\pi \ne 0$, so it has no single-valued potential (see a curl-free field that is not conservative).