Global BV solutions of compressible Euler equations with spherical symmetry and damping.

*(English)*Zbl 0916.35090The authors examine the compressible Euler equations which describe spherically symmetric isentropic flows in a medium with friction. The equation of state has the form \(p= a^2 \rho^\gamma\), \(a,\gamma= \text{const}\), \(\gamma>0\). The main purpose is to investigate the combined effect of spherical symmetry and frictional damping on the total variation of the solution. To this end, the authors employ a fractional-step version of the Glimm finite difference scheme. For the particular case \(\gamma= 1\) this allows to construct an appropriate measure for the wave strengths in terms of Riemann invariants and to prove the uniform boundedness of the BV norm of approximate difference solutions. As a result, this estimate implies the global existence of BV solutions for the original problem.

Reviewer: O.Titow (Berlin)

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

76M20 | Finite difference methods applied to problems in fluid mechanics |

##### Keywords:

fractional step method; measure for wave strengths; isentropic flows; medium with friction; equation of state; Glimm finite difference scheme; Riemann invariants; existence
PDF
BibTeX
XML
Cite

\textit{L. Hsiao} et al., J. Differ. Equations 146, No. 1, 203--225 (1998; Zbl 0916.35090)

Full Text:
DOI

##### References:

[1] | Chen, G.; Glimm, J., Global solutions to the compressible Euler equations with geometrical structure, Comm. math. phys., 180, 153-193, (1996) · Zbl 0857.76073 |

[2] | Dafermos, C., A system of hyperbolic conservation laws with frictional damping, Z. angew. math. phys., 46, 294-307, (1995) · Zbl 0836.35091 |

[3] | Dafermos, C.; Hsiao, L., Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana univ. math. J., 31, (1982) · Zbl 0497.35058 |

[4] | Hsiao, L.; Liu, T.P., Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. math. phys., 143, 599-605, (1992) · Zbl 0763.35058 |

[5] | Hsiao, L.; Luo, T., Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media, J. differential equations, 125, 329-365, (1996) · Zbl 0859.76067 |

[6] | Liu, T.P., Quasilinear hyperbolic systems, Comm. math. phys., 68, 141-172, (1979) · Zbl 0435.35054 |

[7] | M. Luskin, B. Temple, The existence of a global weak solution to nonlinear waterhammer problem, Comm. Pure Appl. Math. 35, 697, 735 · Zbl 0479.35063 |

[8] | Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, (1984), Springer-Verlag Berlin/New York · Zbl 0537.76001 |

[9] | T. Makino, K. Mizohata, S. Ukai, The global weak solutions of compressible Euler equations with spherical symmetry · Zbl 0761.76085 |

[10] | Nishida, T., Global solutions for an initial boundary value problem of a quasilinear hyperbolic system, Proc. jap. acad., 44, 642-646, (1968) · Zbl 0167.10301 |

[11] | Nishida, T.; Smoller, J., Solutions in large for some nonlinear hyperbolic conservation laws, Pure appl. math., 26, 183-200, (1973) · Zbl 0267.35058 |

[12] | Nishida, T.; Smoller, J., Mixed problems for nonlinear conservation laws, J. differential equations, 23, 244-269, (1977) · Zbl 0303.35052 |

[13] | Yang, T., A functional integral approach to shock wave solutions of Euler equations with spherical symmetry, Comm. math. phys., 171, 607-638, (1995) · Zbl 0840.35062 |

[14] | Yang, T., A functional integral approach to shock wave solutions of Euler equations with spherical symmetry (II), J. differential equations, 130, 162-178, (1996) · Zbl 0866.76040 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.