  We study the compressible Euler equations in the isentropic nozzle flow. The
global existence of an $L^{\infty}$ solution has been proved in (Tsuge in
Nonlinear Anal. Real World Appl. 209: 217-238 (2017)) for large data and
general nozzle. However, unfortunately, this solution does not possess
finiteness of energy. Although the modified Godunov scheme is introduced in
this paper, we cannot deduce the energy inequality for the approximate
solutions. Therefore, our aim in the present paper is to derive the energy
inequality for an $L^{\infty}$ solution. To do this, we introduce the modified
Lax Friedrichs scheme, which has a recurrence formula consisting of discretized
approximate solutions. We shall first deduce from the formula the energy
inequality. Next, applying the compensated compactness method, the approximate
solution converges to a weak solution. The energy inequality also holds for the
solution as the limit. As a result, since our solutions are $L^{\infty}$, they
possess finite energy and propagation, which are essential to physics.
