Quantum fluids and Navier-Stokes system What would become of the Navier-Stokes system for incompressible fluids in quantum mechanics? The classical Navier-Stokes system for incompressible fluids is
$$
\partial_t v+\mathbb P\bigl((v\cdot \nabla) v\bigr)-\nu \Delta v=0,\quad v(t=0)=v_0,
$$
where $v$ is a time-dependent vectoriel field in $\mathbb R^3$, $\mathbb P$ is the Leray projection onto vector fields with null divergence, $\nu$ is a positive parameter (the kinematic viscosity), $v_0$ a given divergence-free vector field. The derivation of the above equations is based upon the application of Newton's law $F=m\gamma$.
What is happening if one applies quantum mechanics?
 A: I don't quite understand why the question and the first answer were down-voted. This strikes me as a good question.
Indeed, historically the Navier-Stokes (and Euler) equations were derived using Newton's equation. Furthermore, this derivation is still found in many text books. This raises the question whether the equations are applicable to quantum fluids, for example liquid helium (even above the lambda transition), the electron fluid in a conductor, the neutron fluid in a neutron star, ultracold atomic gases, spin liquids, etc. Empirically, we know that these systems are described by fluid dynamics.
The main observation is that the fluid dynamic equations are more general than the original, classical, derivation. Fluid dynamics follows from 1) conservation laws, 2) the fact that a system probed at sufficiently long time and distance scales is in approximate local thermal equilibrium, 3) the fact that fluxes of conserved charges can be expressed in terms of gradients of thermodynamic variables, and 4) applicable symmetries (Gallilean invariance, for example). All of these statements apply to quantum systems as well as classical ones.
There is one new ingredient in systems that undergo a phase transition to a superfluid state, for example liquid helium below the lambda transition. In this case a new hydrodynamic variable emerges, the superfluid velocity. This variable satifies a classical equation of motion, but it is also subject to a constraint, the fact that circulation is quantized (this corresponds to superfluid vortices).
A: Fluid mechanics was quantized by Lev Landau in his  "THE THEORY OF SUPERFLUIDITY OF HELIUM II"
Physical  Review  Vol 60, p356 (1941). I am not sure whether there are online versions that are not behind a paywall.
