Consider the general (unsteady, compressible) Navier-Stokes equation (plus continuity and energy equations) for a newtonian fluid. No further peculiarities.
Suppose that a domain has been defined and that a solution exists and has been found meaning that one can give a value for the velocity (vector), pressure (scalar) and temperature (scalar) fields at each point in the domain space $\vec{x}$ at all times $t$.
Now subsequently suppose that a region inside the original domain is considered a black box with a certain boundary.
Question: Is it sufficient to set the original solution values as boundary conditions on the boundary of this black box to maintain the same, original solution in the rest of the domain? If yes, how can this fact be demonstrated with some mathematical rigor?
Please see a sketch of the situation to illustrate the question:
