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With the Schrödinger equation it depends on the Hamiltonian you supply. – dmckee Apr 24 '13 at 16:38
With the Navier-Stokes equations it depends on the Mach number. Subsonic flows are elliptic, supersonic flows are parabolic. – OSE Apr 24 '13 at 16:40
What's the motivation behind the question? I'm just curious. – user12345 Apr 24 '13 at 17:02
@user12345 For the Navier-Stokes equations, there is a physical significance. (I cannot speak for the Schrodinger equations) For instance in supersonic flow, disturbances do not travel upstream. This is nice if you are trying to use a Pitot tube in the flow because it will not strongly affect what you are trying to measure. I am currently doing subsonic research and it is sometimes extremely difficult to make sure that any measurement probes are not changing the flow field. – OSE Apr 25 '13 at 15:06
One of my colleagues spent the last month or so trying to figure out why the wake behind a cylindrical roughness element on a flat plate was at an angle with the freestream direction. As it turns out, the hot-wire holder that he was using had too much blockage and was influencing the flow upstream of the hot-wire. This is entirely a result of the Navier-Stokes equations being elliptic for subsonic flows. – OSE Apr 25 '13 at 15:09

Time-dependent Schrodinger equation is an elliptic PDE if the Hamiltonian is time-independent.

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