Approximation of generalized Riemann solutions to compressible Euler-Poisson equations of isothermal flows in spherically symmetric space-times

Article Sidebar

PDF
Published: Mar 30, 2017
DOI: https://doi.org/10.5556/j.tkjm.48.2017.2274

Main Article Content

John Meng-Kai Hong
Reyna Marsya Quita

Abstract

In this paper, we consider the compressible Euler-Poisson system in spherically symmetric space-times. This system, which describes the conservation of mass and momentum of physical quantity with attracting gravitational potential, can be written as a $3\times 3$ mixed-system of partial differential systems or a $2\times 2$ hyperbolic system of balance laws with $global$ source. We show that, by the equation for the conservation of mass, Euler-Poisson equations can be transformed into a standard $3\times 3$ hyperbolic system of balance laws with $local$ source. The generalized approximate solutions to the Riemann problem of Euler-Poisson equations, which is the building block of generalized Glimm scheme for solving initial-boundary value problems, are provided as the superposition of Lax's type weak solutions of the associated homogeneous conservation laws and the perturbation terms solved by the linearized hyperbolic system with coefficients depending on such Lax solutions.

Article Details

How to Cite
Hong, J. M.-K., & Quita, R. M. (2017). Approximation of generalized Riemann solutions to compressible Euler-Poisson equations of isothermal flows in spherically symmetric space-times. Tamkang Journal of Mathematics, 48(1), 73–94. https://doi.org/10.5556/j.tkjm.48.2017.2274
Issue
Vol. 48 No. 1 (2017)
Section
Papers
Author Biographies

John Meng-Kai Hong

Department ofMathematics, National Central University, Taoyuan 32001, Taiwan.

Reyna Marsya Quita

Department ofMathematics, National Central University, Taoyuan 32001, Taiwan.

References

S. W. Chou, J. M. Hong and Y. C. Su, An extension of Glimm's method to the gas dynamical model of transonic flows, Nonlinearity, 26(2013), 1581--1597.

S. W. Chou, J. M. Hong and Y. C. Su, Global entropy solutions of the general nonlinear hyperbolic balance laws with time-evolution flux and source, Mathods Appl. Anal., 19(2012), 43--76.

S. W. Chou, J. M. Hong and Y. C. Su, The initial-boundary value problem of hyperbolic integro-differential systems of nonlinear balance laws, Nonlinear Anal. 75

(2012), 5933--5960.

C. M. Dafermos and L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana Univ. Math. J. 31(1982), 471--491.

J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Commun. Pure Appl. Math.18(1965), 697--715.

J. B. Goodman, Initial boundary value problems for hyperbolic systems of conservation laws, Thesis (Ph.D.)--Stanford University, 1983.

J. Groah, J. Smoller and B. Temple, Shock Wave Interactions in General Relativity, Monographs in Mathematics, Springer, Berlin, New York, 2007.

J. M. Hong, An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by "weaker than weak'' solutions of the Riemann problem, J. Diff. Equ. 222(2006), 515--549.

J. M. Hong and P. G. LeFloch, A version of Glimm method based on generalized Riemann problems, J. Portugal Math., 64(2007), 199--236.

J. M. Hong and Y.-C. Su, Generalized Glimm scheme to the initial-boundary value problem of hyperbolic systems of balance laws, Nonlinear Analysis: Theory, Methods and Applications, 72(2010), 635--650.

E. Isaacson and B. Temple, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Anal., 52(1992), 1260--1278.

P. D. Lax, Hyperbolic system of conservation laws II,Commun. Pure Appl. Math., 10(1957), 537--566.

P. G. LeFloch and P. A. Raviart, Asymptotic expansion for the solution of the generalized Riemann problem, Part $1$, Ann. Inst. H. Poincare, Nonlinear Analysis, 5(1988), 179--209.

T.-P. Liu, Quasilinear hyperbolic systems, Commun. Math. Phys.,68(1979), 141--172.

M. Luskin and B. Temple, The existence of global weak solution to the nonlinear water-hammer problem, Commun. Pure Appl. Math., 35(1982), 697--735.

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer-Verlag, Berlin, New York, 1994.

B. Temple, Global solution of the Cauchy problem for a class of 2$times$2 nonstrictly hyperbolic conservation laws, Adv. Appl. Math., 3(1982), 335--375.