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

Question

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

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

Answer 1

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.

Answer 2

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