A RIGOROUS JUSTIFICATION OF THE EULER AND NAVIER-STOKES EQUATIONS WITH GEOMETRIC EFFECTS

We derive the one-dimensional (1D) isentropic Euler and Navier-Stokes equations describing the motion of a gas through a nozzle of variable cross section as the asymptotic limit of the 3D isentropic Navier-Stokes system in a cylinder, the diameter of which tends to zero. Our method is based on the relative energy inequality satisfied by any weak solution of the 3D Navier-Stokes system and a variant of the Korn-Poincare inequality on thin channels that may be of independent interest.