Spherically symmetric solutions to the compressible Euler equation with an asymptotic γ-law

Hsu, Cheng Hsiung; Makino, Tetu

doi:10.1007/bf03167459

Cited by 11 publications

(1 citation statement)

References 13 publications

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“…However, we cannot justify the above argument by the existing difference scheme such as the Godunov or Lax-Friedrichs scheme (cf. [6], [7], [9] and [15]). Therefore, we introduce the modified Godunov scheme in Section 3.…”

Section: We Assume the Following There Exists A Nonnegative Functionmentioning

confidence: 99%

Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the modified Godunov scheme and the generalized invariant regions

Tsuge

2017

Preprint

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We study the motion of isentropic gas in nozzles. This is a major subject in fluid dynamics. In fact, the nozzle is utilized to increase the thrust of rocket engines. Moreover, the nozzle flow is closely related to astrophysics. These phenomena are governed by the compressible Euler equations, which are one of crucial equations in inhomogeneous conservation laws.In this paper, we consider its unsteady flow and devote to proving the global existence and stability of solutions to the Cauchy problem for the general nozzle. The theorem has been proved in (Tsuge in Arch. Ration. Mech. Anal. 209:365-400 (2013)). However, this result is limited to small data. Our aim in the present paper is to remove this restriction, that is, we consider large data. Although the subject is important in Mathematics, Physics and engineering, it remained open for a long time. The problem seems to rely on a bounded estimate of approximate solutions, because we have only method to investigate the behavior with respect to the time variable. To solve this, we first introduce a generalized invariant region. Compared with the existing ones, its upper and lower bounds are extended constants to functions of the space variable. However, we cannot apply the new invariant region to the traditional difference method. Therefore, we invent the modified Godunov scheme. The approximate solutions consist of some functions corresponding to the upper and lower bounds of the invariant regions. These methods enable us to investigate the behavior of approximate solutions with respect to the space variable. The ideas are also applicable to other nonlinear problems involving similar difficulties.

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Section: We Assume the Following There Exists A Nonnegative Functionmentioning

confidence: 99%

Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the modified Godunov scheme and the generalized invariant regions

Tsuge

2017

Preprint

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Existence of Global Solutions for Unsteady Isentropic Gas Flow in a Laval Nozzle

Tsuge

2012

Arch Rational Mech Anal

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In this paper, we study the motion of isentropic gas in the Laval nozzle. The Laval nozzle is the most important type of nozzle utilized in some turbines. In particular, we consider unsteady flows, including transonic gas flows, and prove the existence of global solutions for the Cauchy problem. In spite of its importance, this problem has received little attention until now. The most difficult point is to obtain bounded estimates for approximate solutions. To overcome this, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation and yield a sharper bounded estimate. As a result, we find an invariant region for our solutions. Finally, in order to prove their convergence, we use the compensated compactness framework.

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Existence of periodic solutions to an isothermal relativistic Euler system

Wang

2021

Acta Math Sci

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Spherically symmetric solutions to the compressible Euler equation with an asymptotic γ-law

Abstract: This work investigates the spherically symmetric solutions of compressible Euler equation with an asymptotic -y-law. We generalize the method of [6,7,8] to show the existence of weak solution of the equation with initial data containing the vacuum state.

Cited by 11 publications

References 13 publications

Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the modified Godunov scheme and the generalized invariant regions

Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow for large data: Application of the modified Godunov scheme and the generalized invariant regions

Existence of Global Solutions for Unsteady Isentropic Gas Flow in a Laval Nozzle

Existence of periodic solutions to an isothermal relativistic Euler system

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