This question might seem silly but I'll try to make it clear. It's a question (I think) about partial differential equations systems in general, but since currently I'm studying fluid mechanics I'll ask on that context.
The equations for an incompressible flow are:
$$\begin{cases}\nabla\cdot\mathbf{u} &= 0 \\ \dfrac{D\rho}{Dt} &= 0\\ \rho \dfrac{D\mathbf{u}}{Dt} +\nabla p - (\lambda+\mu)\nabla(\nabla\cdot \mathbf{u}) - \mu\nabla^2\mathbf{u} &= 0\end{cases}$$
Where we have to find $u_1,u_2,u_3$ the components of $\mathbf{u}$, $\rho$ and $p$. The point is: these are the equations, regardless of the flow under study. So, how does this connect to the real situation I have?
For example, in Newtonian Mechanics the equation is $\mathbf{F} = m\mathbf{a}$, but here we know the connection with the problem at hand: we plug the forces there and we solve the equation.
Now, in fluid mechanics, if I consider pipe flow, or channel flow, or flow past a sphere, or inside some complicated region $D\subset \mathbb{R}^3$ the equations are just the same. Nothing changes, there's not anything connecting the equations to the problem at hand but we might expect the solutions be quite different on each case.
In that case, since there are $4$ equations and $4$ unknowns, it seems the solutions would always have to be the same. So I ask: what connects these equations to the real situation? Is it just the boundary conditions? I believe my problem is that I don't know yet how existence and uniqueness works for partial differential equations systems.