Given the vorticity form of the Euler equations in $2D$ with stream function $\psi$
\begin{align} \omega_t + \nabla^\perp \psi \cdot \nabla\omega &= 0 \\ \Delta \psi = \omega \end{align}
we know that $v =: \nabla^\perp \psi$ with $\text{curl } v = \omega$ solves the classical Euler equations
\begin{align} v_t + (v \cdot \nabla) v &= -\nabla p \\ \text{div } v &= 0 \\ v(0, \cdot) = v_0 \end{align}
for some pressure term $p$.
Question: If instead we're given a perturbation of the vorticity-stream formulation such that $\Delta \psi = \omega + \varepsilon \phi$ --
Is there a clean way of showing that one can 'invert' the perturbed vorticity formulation to an Euler system with forcing term of the 'same size' as the perturbation?
