Spherically symmetric flow of the compressible Euler equations -For the case including the origin-

Tsuge, Naoki

doi:10.1215/kjm/1250283586

Cited by 12 publications

(9 citation statements)

References 7 publications

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“…By (4.15) and the compactness framework established in [11,13,16], we can prove that there exists a subsequence of (ρ ε , m ε ) (still denoted by (ρ ε , m ε )) such that (ρ ε , m ε ) → (ρ, m) in L p loc (Π T ), p ≥ 1. ρ are integrable near the origin with respect to x. As in [5,6,26,27], we can prove that (ρ, m) is an entropy solution to the problem (1.3) and the test function Φ(x, t) can contain the origin. Therefore, the proof of Theorem 1.2 is completed.…”

Section: H −1mentioning

confidence: 60%

“…As in [5,6,26,27], we can prove that (ρ, m) is an entropy solution to the initial-boundary value problem (1.4) and m| x=1 = 0 in the sense of the divergence-measure fields introduced in [3,4]. Therefore Theorem 1.1 is completed.…”

Section: H −1mentioning

confidence: 66%

“…When α becomes smaller, the increasing rate of −x −α to 0 decreases, which means that the range of negative velocity is larger. In Tsuge [26], the initial data satisfies m 0…”

Section: Introductionmentioning

confidence: 99%

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Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry

Huang

Li²,

Yuan

2019

Nonlinearity

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We are concerned with spherically symmetric solutions to the Euler equations for the multi-dimensional compressible fluids, which have many applications in diverse real physical situations. The system can be reduced to one dimensional isentropic gas dynamics with geometric source terms. Due to the presence of the singularity at the origin, there are few papers devoted to this problem. The present paper proves two existence theorems of global entropy solutions. The first one focuses on the case excluding the origin in which the negative velocity is allowed, and the second one is corresponding to the case including the origin with non-negative velocity. The L ∞ compensated compactness framework and vanishing viscosity method are applied to prove the convergence of approximate solutions. In the second case, we show that if the blast wave initially moves outwards and the initial densities and velocities decay to zero with certain rates near origin, then the densities and velocities tend to zero with the same rates near the origin for any positive time. In particular, the entropy solutions in two existence theorems are uniformly bounded with respect to time.

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Section: H −1mentioning

confidence: 60%

Section: H −1mentioning

confidence: 66%

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Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry

Huang

Li²,

Yuan

2019

Nonlinearity

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“…Chen-Wang [6] applied the shock capturing schemes to the compressible Euler-Possion equations with geometrical structure in semiconductor devices. More interesting and relevent results can be found in [3,7,8,20,23,30,33,34] and references therein. Almost all the results in previous work developed numerical scheme to construct approximate solutions and used lengthy estimates.…”

Section: Introductionmentioning

confidence: 86%

Global solutions to rotating motion of isentropic flows with cylindrical symmetry

Yuan¹

2020

Preprint

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We are concerned with global weak solutions to the isentropic compressible Euler equations with cylindrically symmetric rotating structure, in which the origin is included. Due to the presence of the singularity at the origin, only the case excluding the origin | x| ≥ 1 has been considered by Chen-Glimm [4]. The convergence and consistency of the approximate solutions are proved by using L ∞ compensated compactness framework and vanishing viscosity method. Moreover, if the blast wave initially moves outwards, and if initial density and velocity decay to zero at certain algebraic rate near the origin, then the density and velocity decay at the same rate for any positive time. 2 and adiabatic exponent γ > 1.

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“…Moreover, Chen and Glimm [6] claimed the global existence of solutions for (1.3) with a general nozzle and arbitrary L ∞ data. However, the author of the present paper could not understand some parts of this proof (see Tsuge [20,Section 8]). On the other hand, Glimm et al [11,Section 6] obtained the results of numerical tests for a Laval nozzle by using a random choice method.…”

Section: Remark 11 1 It Holds That A(x) A(0) That Is A(x)mentioning

confidence: 93%

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

Tsuge

2012

Arch Rational Mech Anal

Self Cite

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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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Spherically symmetric flow of the compressible Euler equations -For the case including the origin-

Cited by 12 publications

References 7 publications

Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry

Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry

Global solutions to rotating motion of isentropic flows with cylindrical symmetry

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

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