# Remarks on the energy inequality of a global $L^{\infty}$ solution to the compressible Euler equations for the isentropic nozzle flow

## TL;DR

This paper establishes the energy inequality for global $L^{inity}$ solutions to the isentropic compressible Euler equations in nozzle flow, ensuring these solutions have finite energy and align with physical principles.

## Contribution

The paper introduces a modified Lax Friedrichs scheme and uses compensated compactness to prove energy inequality for $L^{inity}$ solutions, bridging a gap in previous work.

## Key findings

- Energy inequality is derived for $L^{inity}$ solutions.
- Approximate solutions converge to a weak solution with finite energy.
- Solutions possess physically essential properties like finite energy and propagation.

## Abstract

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.