I have been studying the 2D time-independent advection equation on the unit square $[0,1] \times [0,1]$: $$ \frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) = f(x,y) \, , $$ with some appropriate boundary conditions.

A simple analysis (method of characteristics etc.) shows that the derivatives of the solution are discontinuous.

My intuition tells me that this is because of the non-smooth nature of the square boundary. I'm almost certain there will be at least some papers on related results, but I cannot seem to find them. Can anyone point me in the right direction?