Analysis on the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes

Abstract

This paper concerns the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes. One of the key points is to quantitatively determine the position of the shock front as the gravity is small. To this end, a free boundary problem for the linearized Euler system with small gravity will be proposed. Then one can obtain the information of the approximating position of the shock front. Once the initial approximation of the shock solution is obtained, a nonlinear iteration scheme can be constructed to show that the position of the shock front is a small perturbation of the initial approximation, which establishes a quantitative expression for the shock solution. Such an expression shows the asymptotic behavior as the gravity goes to zero.