Htykuut

I want to express a divergence-free vector $vec{w}(x,y)$ with non-zero curl in the domain $x,y geq 0$ as
$$ vec{w} = b nabla c ; .$$
The vector is normal to the boundary $partial Omega$. Furthermore, the value of $c$ on the boundary is zero ($c=0$ on $x,y=0$). Based on a helpful answer I received to this question in another post, I have made some progress. However, I am now unsure about the uniqueness of the problem.

To illustrate the problem, I will assume the model
$$ vec{w} = frac{1}{sqrt{x^2+y^2}} left( begin{array}{c} y \ -x end{array} right) ; .$$
This vector has the properties that it is divergence-free, has non-zero curl, and is normal to the boundary as mentioned above. From the theory of first-order linear PDE's it follows that $c$ has the general solution
$$ c = phi left( frac{x}{y} right) $$
where $phi$ is an arbitrary function of the variable $eta equiv x/y$. From this follows the solution for $b$ as
$$ b = frac{y^2}{sqrt{x^2+y^2}} frac{1}{phi^{prime}} $$
where the prime indicates a derivative. Taking the gradient results in
$$ nabla b = left( begin{array}{c}
-frac{eta}{(1+eta^2)^{3/2}} frac{1}{phi^{prime}} - frac{phi^{primeprime}}{{phi^{prime}}^2} frac{1}{sqrt{1+eta^2}} \
frac{2}{sqrt{1+eta^2}} frac{1}{phi^{prime}} - frac{1}{(1+eta^2)^{3/2}} frac{1}{phi^{prime}} + frac{phi^{primeprime}}{{phi^{prime}}^2} frac{eta}{sqrt{1+eta^2}}
end{array} right) ; . $$
These solutions satisfy the rotational and the divergence-free condition and the equation $vec{w} = b nabla c$. In order to impose the boundary conditions for $c$, one needs $phi(0) = 0$ and $phi(eta) = 0$ as $eta rightarrow infty$, as well as that $phi^{primeprime} / phi^{prime}$ remains finite at $eta=0$ and $eta=infty$.

Question:

As can be seen from the existence of an arbitrary function, the conditions so far do not suffice to determine $b$ and $c$. What further conditions can I reasonably impose to uniquely determine $b$ and $c$?
The term $b dot c$ can be regarded as a physical source term where $dot{()}$ indicates a derivative. Ideally, this source term would indicate no spurious sources. I have considered setting $nabla^2 b = 0$ which gives a 3rd order ODE but I see no formal justification for doing so.