In the now 35 years old survey paper ''Kinetic equations from Hamiltonian dynamics'', Herbert Spohn mentions two important unsolved problems in mathematical physics: On p.571 the hydrodynamic limit, and on p.603 the derivation of the nonlinear Boltzmann equation from quantum mechanics.

Where can I find the current state of affairs? In particular, are there now rigorous derivations of the Navier-Stokes equations and/or the Boltzmann equations from either nonrelativistic many-particle quantum mechanics or quantum field theory? If yes, under which assumptions?