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What type of PDE are Navier-Stokes equations, and Schrödinger equation?

I mean, are they parabolic, hyperbolic, elliptic PDEs?

Christoph B.
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Ana S. H.
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    With the Schrödinger equation it depends on the Hamiltonian you supply. – dmckee --- ex-moderator kitten Apr 24 '13 at 16:38
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    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 I really don't know much about PDEs theory, but I was wondering this because I use those PDEs as a physics student, so I thought it was important to know something like this, I mean a really basic thing. Also, the answer of this question is not clear in usual undergraduate physics books. Of course, if you know some books that treat this, please tell me. – Ana S. H. Apr 25 '13 at 03:08
  • Unfortunately I can't offer any assistance because I myself, (also a physics student) have never come across these properties. Do they have a physical interpretation or just a mathematical one? – user12345 Apr 25 '13 at 09:02
  • Well, I understand that depending of the type of PDE, there are specific boundary conditions to satisfy the uniqueness of the solution. – Ana S. H. Apr 25 '13 at 12:59
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    @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
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    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
  • @user2018790 Nice! – user12345 Apr 25 '13 at 15:43
  • realted: https://physics.stackexchange.com/q/75363/226902 – Quillo Jul 04 '22 at 17:53

2 Answers2

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Classification into parabolic, elliptic and hyperbolic equations applies to the second order linear partial differential equations with constant coefficients. That is, it applies to the equations that are:

  • linear
  • second order
  • have constant coefficients

Thus one particle non-relativistic time-dependent Schrödinger equation with no external potential or magnetic field can be classified as parabolic (although with complex coefficients - its real-coefficients equivalent is the diffusion equation.) In time-independent case in more than one dimension, Schrödinger equation is elliptic.

Navier-Stokes equation is non-linear, and hence does not fit this classification.

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

freude
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