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

We study the compressible Euler equations in the isentropic nozzle flow. The global existence of an $L^\infty$ solution has been proved in [N. Tsuge, Nonlinear Anal. Real World Appl., 209:217–238, 2017] for large data and general nozzles. 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.

Keywords

the compressible Euler equation, nozzle flow, compensated compactness, finite energy solutions, the modified Lax Friedrichs scheme

2010 Mathematics Subject Classification

Primary 35L03, 35L65, 35Q31, 76N10, 76N15. Secondary 35A01, 35B35, 35B50, 35L60, 76H05, 76M20.

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N. Tsuge’s research is partially supported by Grant-in-Aid for Scientific Research (C) 17K05315, Japan.

Received 31 January 2020

Accepted 20 February 2021

Published 2 August 2021