  In this paper we are concerned with the existence of transonic shocks for 2-D
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. To this end, 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 elliptic-hyperbolic 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.
