In this paper we are concerned with the existence of transonic shocks for twodimensional steady isothermal Euler flows in a horizontal flat nozzle under vertical gravity. In particular, we focus on the contribution of the vertical gravity in determining the position of the shock front. For steady horizontal flows, the existence of normal shocks with the position of the shock front being arbitrary in the nozzle can be easily established. This paper will try to determine the position of the shock front as the state of the flow at the entrance of the nozzle and the pressure at the exit are slightly perturbed. Mathematically, it can be formulated as a free boundary problem for the steady Euler system with vertical gravity, and the position of the shock front is the very free boundary that need to be determined. Since the unperturbed normal shock solutions give no information on the position of the shock front, one of the key difficulties is to find where the shock front may appear. To overcome this difficulty, this paper proposes a free boundary problem of the linearized Euler system with vertical gravity, whose solution could be an initial approximation for the shock solution with the free boundary being the approximation for the shock front. Due to the existence of the vertical gravity, difficulties arise in solving the boundary value problem in the approximate subsonic domain behind the shock front. The linearized Euler system is elliptichyperbolic composite for subsonic flows, and the elliptic part and the hyperbolic part are coupled in the 0-order terms depending on the acceleration of gravity g. Moreover, the coefficients are not constants since the unperturbed shock solution depends on the vertical variable. New ideas and techniques are developed to deal with these difficulties and, under certain sufficient conditions on the perturbation, the existence of the solution to the proposed free boundary problem is established as the acceleration of gravity g > 0 and the perturbation are sufficiently small. Then, with the obtained initial approximation of the shock solution, a nonlinear iteration scheme can be constructed which leads to a transonic shock solution with the position of the shock front being close to the initial approximating position.
