A rigorous justification of the Euler and Navier-Stokes equations with geometric effects
Peter Bella, Eduard Feireisl, Marta Lewicka, and Antonín Novotný
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Submission date: 19. Nov. 2015
published in: SIAM journal on mathematical analysis, 48 (2016) 6, p. 3907-3930
DOI number (of the published article): 10.1137/15M1048963
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We derive the 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 Korn-Poincaré’s inequality on thin channels that may be of independent interest.