For parabolic or hyperbolic PDEs, does the BC ever contain a time derivative? I am curious if the boundary condition of a parabolic or hyperbolic PDE ever contains the time derivative of a function. I could see this happening if one of the terms in the PDE had both a spatial and time derivative on it. Then when looking at the weak form of the PDE you might get a time derivative on the boundary term. I suppose I could create such a PDE, but is there any literature where this happens?
 A: Theoretically, you can indeed create such a PDE and for sure you can even design the boundary condition with time derivative in a way that the existence of some kind of solution is easy to prove. However, just because such PDEs exist does not mean they are useful. Time dependent PDEs describe evolution of some quantity according to some laws. Analogously, if you allow the boundary conditions to be so complicated, they should describe evolution of that quantity on the boundary under some local laws. In other words, you should be able to provide some justification for your boundary conditions, other than that they annihilate some boundary integral in the weak formulation (because there are always many ways how to write down the weak formulation).
I can think of two examples from fluid mechanics, where the boundary conditions involve the time derivative of the velocity:

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*Dynamic wall slip, observed experimentally in some molten polymers, see Hatzikiriakos and also the recent work Abbatiello et. al. for the corresponding analysis.


*Outflow boundary conditions, see, e.g., Section 2.1. in Li et. al. and references therein. Note, however, that in this case the boundary conditions are only heuristical and to this day there is no agreement on what outflow boundary condition is the ``right'' one.
