Gravitational effect on transonic shocks in finite divergent nozzle

In this paper, we adopted the steady compressible Euler system with a gravitational term as the governing equations, and focused on the effect of gravity on the transonic shock position in a three-dimensional spherically symmetric divergent nozzle. For a fixed supersonic inflow at the nozzle inlet, we proved that if the outlet pressure fell within an appropriate interval and the gravitational parameter $K$ was sufficiently small, the shock position was uniquely determined by $K$ and the outlet pressure, and the shock position was a strictly decreasing and continuously differentiable function of $K$.