I am solving Stokes problem: $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ in a domain that's bounded by two surfaces - a cuboid and a small sphere inside it.
Is this problem well-posed?
If boundary conditions are required, they are: $\vec{u}=u_1 (z-z_0) \vec{e}_x$ (shear flow) on the cuboid and $\vec{u}=\vec{0}$ (stationary) on the sphere.
The geometry looks like this:
This question follows this one. I got the conditions but I have no idea how to apply them in this case. As far as I understand, this domain is not really smooth, but does it instantly make this problem ill-posed?