Taking the curl of the vorticity produces \[ \nabla \times \boldsymbol{\omega} = \nabla \times (\nabla \times \mathbf{u}) = \nabla (\nabla \cdot \mathbf{u}) - \nabla^2 \mathbf{u} = - \nabla^2 \mathbf{u} \] where the second equality follows from an identity of vector calculus \(\nabla \times (\nabla \times \mathbf{u}) = \nabla (\nabla \cdot \mathbf{u}) - \nabla^2 \mathbf{u}\)

The two ends of this extended equality form a Poisson equation, and its solution is the vorticity-induced portion of the fluid velocity \[ \mathbf{u}(\mathbf{x}, t) = -\frac{1}{4\pi} \int_{V'} \frac{1}{|\mathbf{x} - \mathbf{x}'|} \left( \nabla' \times \boldsymbol{\omega}(\mathbf{x}', t) \right) d^3 x' \] where \(V'\) encloses the vorticity of interest and \(\nabla'\) operates on the \(\mathbf{x}'\) coordinates

Rewriting the integrand $$ \frac{1}{|\mathbf{x} - \mathbf{x}'|} \left( \nabla' \times \boldsymbol{\omega}(\mathbf{x}', t) \right) = \nabla' \times \left( \frac{\boldsymbol{\omega}(\mathbf{x}', t)}{|\mathbf{x} - \mathbf{x}'|} \right) - \nabla'\left( \frac{1}{|\mathbf{x} - \mathbf{x}'|} \right) \times \boldsymbol{\omega}(\mathbf{x}', t) $$ $$ = \nabla' \times \left( \frac{\boldsymbol{\omega}(\mathbf{x}', t)}{|\mathbf{x} - \mathbf{x}'|} \right) + \left( \frac{\mathbf{x} - \mathbf{x}'}{|\mathbf{x} - \mathbf{x}'|^3} \right) \times \boldsymbol{\omega}(\mathbf{x}', t) $$ to obtain $$ \mathbf{u}(\mathbf{x}, t) = -\frac{1}{4\pi} \int_{V'} \nabla' \times \left( \frac{\boldsymbol{\omega}(\mathbf{x}', t)}{|\mathbf{x} - \mathbf{x}'|} \right) d^3 x' + \frac{1}{4\pi} \int_{V'} \frac{\boldsymbol{\omega}(\mathbf{x}', t) \times (\mathbf{x} - \mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|^3} d^3 x' $$