On the unique solvability of the div–curl problem in unbounded domains and energy estimates of solutions

We study the problem of reconstructing a solenoidal vector field from a vortex function with a no-slip condition on the boundary of an external two-dimensional domain. A solvability criterion is obtained as a condition for the orthogonality of the vortex function to harmonic functions. We also obtain some estimates of the solution in the spaces $L_2$ and $H_1$.