Vanishing Shear Viscosity Limit for the Navier–Stokes Equations with Cylindrical Symmetry: Boundary Layer and Optimal Convergence Rate

Abstract

We consider the initial boundary-value problem for the nonisentropic compressible Navier–Stokes equations with cylindrical symmetry and degenerate heat conductivity coefficient. The vanishing shear viscosity limit with strong boundary layer effect for arbitrarily large data is investigated. Compared with the existing result in [X. Qin et al. Arch. Rational Mech. Anal., 216 (2015) 1049–1086], where the optimal convergence rate in terms of shear viscosity is obtained for the angular and axial velocity components, we derive the optimal convergence rate. The main challenges include that the presence of the strong boundary layer effect will prevent the convergence over the whole spatial domain, and that the quantities of a nonisentropic flow interact strongly with each other not only through the pressure term but also through the strongly nonlinear term in the temperature equation which would possibly slow down the convergence rate. Our main strategy is to find some new functions via asymptotic matching methods such that both the strong boundary layer effect and those quantities decaying at a lower speed can be canceled.