Velocity vector $\vec{v}$ given as the gradient of a potential function $\phi$, solution of the Laplace's equation. Is this enough information to define $\phi$ ?

So far we know that the potential function $\phi$ satisfies the Laplace's equation. Is this enough information to obtain $\phi$?

Laplace's equation has an infinite set of solutions, but which of these solutions satisfy our problem?

To answer this question, we need additional information. Where we can get this additional information about $\phi$ or $\vec{v}$ ?