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.
['Peter Bella', 'Eduard Feireisl', 'Marta Lewicka', 'Antonin Novotny']
A rigorous justification of the Euler and Navier-Stokes equations with geometric effects