Initial boundary value problems for isentropic gas dynamics

Takeno, Shigeharu

doi:10.1017/s0308210500014955

Cited by 7 publications

(6 citation statements)

References 11 publications

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“…Initial boundary value problems associated with (5)0 were studied by S. Takano [7]. On the other hand the initial value problem without boundary conditions for non-homogeneous systems (h ~ 0) was studied by X. Ding et al [3].…”

Section: (R < R)mentioning

confidence: 99%

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Initial boundary value problem for the spherically symmetric motion of isentropic gas

Makino

Takeno

1994

Japan J. Indust. Appl. Math.

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We study the spherically symmetric motion of an ideal gas surrounding a solid ball. This is governed by the compressible Euler equation of isentropic gas dynamics. The associated initial boundary value problem is solved by using the compensated compactness method for initial data containing the vacuum. The constructed weak solutions are temporally local but the class of initial data includes the stationaxy solutions.Let us consider the system of equations (1)(2) p = Ap ~ on t-> 0 and 1 <= r < +c~. Here M , A and ~ are positive constants such t h a t 1 < "7 = 5/3. This system governs the isentropic and spherically symmetric motion of an atmosphere surrounding a solid star with radius 1 and mass M . The variable p means the density, u the velocity, p the pressure and -M / r 2 stands for the gravitational force. The problem is to find a solution of the system (1) (2) which satisfies the initial condition (3) p I~=0 = p~ u I~=0 = ~0(~) and the b o u n d a r y condition (4) ~ tr=l = 0.We are interested in the case where the support of p~ is compact in [1, +ex)). In the article [4] we established the local existence of "tame" solutions. However this

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Section: (R < R)mentioning

confidence: 99%

“…Introducing the vector variable U ----t(p, rn), where m = pu, we can write ( The Riemann problem of (5)0' is the Cauchy problem for the initial condition u Ir=0 = ~ aL (x < 0) (7) uR (x > 0). This problem is solved uniquely in a standard manner.…”

Section: (R < R)mentioning

confidence: 99%

Initial boundary value problem for the spherically symmetric motion of isentropic gas

Makino

Takeno

1994

Japan J. Indust. Appl. Math.

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“…Once we have these generalized Riemann solutions, we can construct the Godunov scheme and prove the existence of the generalized solution for the mixed problem. The following theorem is given by Takeno [22]. More precisely, if the initial data belong to Ac, then the generalized solutions constructed by the Godunov scheme for the mixed problem (5.1) also belong to…”

Section: Extension To Initial Boundary Problemsmentioning

confidence: 98%

Convergence of Second-Order Schemes for Isentropic Gas Dynamics

Liu¹

1993

Mathematics of Computation

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Abstract. Convergence of a second-order shock-capturing scheme for the system of isentropic gas dynamics with L°° initial data is established by analyzing the entropy dissipation measures. This scheme is modified from the classical MUSCL scheme to treat the vacuum problem in gas fluids and to capture local entropy near shock waves. Convergence of this scheme for the piston problem is also discussed.

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“…Liu ([8]) solved the free piston problem for gas dynamics in Lagrange coordinate provided that the total variation of the initial data is sufficiently small by using the Glimm method. In [13], the author recently solved initial boundary value problems, the so-called moving piston problem, by using a compensated compactness theory provided that 1 < ~/ ~ 5/3. Now, we shall prove the following theorem.…”

Section: [ (P(xo)u(xo))=(po(x)uo(x))mentioning

confidence: 99%

“…The proof of this theorem mainly follows [13]. In Section 2 we introduce notations and basic lemmas for the construction of the approximate solution.…”

Section: Where C 1 and C2 Ate Constants The Piston Path X(t) Is Contmentioning

confidence: 99%

Free piston problem for isentropic gas dynamics

Takeno

1995

Japan J. Indust. Appl. Math.

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Initial boundary value problems for isentropic gas dynamics

Abstract: SynopsisFor piston problems for a system of isentropic gas dynamics, convergence theorems of a difference scheme are obtained by compensated compactness theory and by analysis of the difference scheme.

Cited by 7 publications

References 11 publications

Initial boundary value problem for the spherically symmetric motion of isentropic gas

Initial boundary value problem for the spherically symmetric motion of isentropic gas

Convergence of Second-Order Schemes for Isentropic Gas Dynamics

Free piston problem for isentropic gas dynamics

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