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Computing the potential from the continuity equation

  1. Mar 20, 2009 #1
    Dear All,
    I need help on the following issue. Assuming the flow to be potential, I want to compute the potential given the density at all times, that is :

    From the continuity equation:

    [tex]
    \partial _t \rho + \nabla \cdot \left( {\rho \nabla \phi } \right) =
    0
    [/tex]

    One can write down an elliptic PDE for finding the potential at a
    fixed time t (this is a PDE in space only):

    [tex]
    - \rho \Delta \phi - \nabla \rho \cdot \nabla \phi = \partial
    _t \rho
    [/tex]

    If one wants to impose the homogeneous Neumann boundary conditions,
    what are the results about the existence and the regularity of the
    solution?

    If you have any knowledge about this problem, please help.

    Thanks.

    MF
     
  2. jcsd
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