A rigorous justification of the Euler and Navier-Stokes equations with geometric effects (Q2834575)

The authors consider two models of the flow of a compressible gas through a nozzle of variable cross section with and without the effect of viscosity. Using the concept of dissipative weak solutions to the Navier-Stokes system and the associated relative energy inequality, they show that the first of the models, described above, can be obtained as the inviscid limit of the 3D Navier-Stokes system considered in a cylinder \(\Omega_{\varepsilon}\) with the diameter \(\varepsilon\), when parameters \(\varepsilon\) and \(\lambda\) both tend to zero. And if \(\lambda=1\) and \(\varepsilon\) tends to zero, one can obtain the second model.NEWLINENEWLINEIt is noticed that the asymptotic analysis for the Navier-Stokes limit leads to the study of the Korn-Poincaré inequality on thin domains, the proof of which is presented. The related results and problems are discussed.