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As I was reading on potential flows (specifically a proof for Blasius' theorem), I came across a part where we had to use Bernoulli's equation, and I recalled that Bernoulli's equation was something that holds for solutions to the incompressible Euler equation (and, if we also assume irrational, then we get a stronger version of Bernoulli's equation). Then it occurred to me that we've been assuming all along that a potential flow was automatically a solution to the incompressible Euler equation.

Then, assuming our potential flow is steady, plugging in a potential flow $u$ into the Euler equation (where I "ignore" body forces and ignore the incompressibility condition, which is automatically given) we end up with

$$\frac{1}{\rho}\nabla p=-\frac{1}{2}\nabla |u|^2.$$ My question then boils down to two points:

a) In what sense exactly are we saying that a potential flow solves the Euler equation? Is it because we can put $p=-\frac{\rho}{2}|u|^2$ and make the equality above hold?

b) Related to my first question - what is a concrete example, then, of a steady incompressible flow $u$ that isn't a solution to the incompressible Euler equation? Does it boil down to finding a flow $u$ so that we can't find a scalar $p$ such that $\nabla p=-\rho(u\cdot\nabla)u$?

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By Helmholtz' theorem, any continuously differentiable vector field that vanishes at infinity can be decomposed into irrotational and soleniodal parts of the form

$$\mathbf{u} = \nabla \phi + \nabla \times \mathbf{a}.$$

In potential flow we assume that the velocity field is irrotational, whence

$$\mathbf{u} = \nabla \phi. $$ Applying the continuity condition we have

$$\nabla \cdot \mathbf{u} = \nabla^2 \phi = 0.$$

With suitable boundary conditions there always exists a solution of Laplace's equation for the velocity potential $\phi$ and hence, the velocity field. The Euler equation ensures conservation of momentum and closes the system of equations so we can solve for the pressure field (always).

The Euler equation applies to the general class of inviscid flows (incompressible or compressible) where incompressible potential flow is a special case. With respect to (b), there is no steady, incompressible, inviscid flow (irrotational or rotational) that does not satisfy the Euler equation. Only if the velocity field solves the viscous Navier-Stokes equations will it fail to satisfy the Euler equation.

Also, there is such a thing as unsteady potential flow.