• Title/Summary/Keyword: method of vanishing viscosity

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DELTA-SHOCK FOR THE NONHOMOGENEOUS PRESSURELESS EULER SYSTEM

  • Shiwei Li;Jianli Zhao
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.3
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    • pp.699-715
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    • 2024
  • We study the Riemann problem for the pressureless Euler system with the source term depending on the time. By means of the variable substitution, two kinds of Riemann solutions including deltashock and vacuum are constructed. The generalized Rankine-Hugoniot relation and entropy condition of the delta-shock are clarified. Because of the source term, the Riemann solutions are non-self-similar. Moreover, we propose a time-dependent viscous system to show all of the existence, uniqueness and stability of solutions involving the delta-shock by the vanishing viscosity method.

THE DELTA STANDING WAVE SOLUTION FOR THE LINEAR SCALAR CONSERVATION LAW WITH DISCONTINUOUS COEFFICIENTS USING A SELF-SIMILAR VISCOUS REGULARIZATION

  • LI, XIUMEI;SHEN, CHUN
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.6
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    • pp.1945-1962
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    • 2015
  • This paper is mainly concerned with the formation of delta standing wave for the scalar conservation law with a linear flux function involving discontinuous coefficients by using the self-similar viscosity vanishing method. More precisely, we use the self-similar viscosity to smooth out the discontinuous coefficient such that the existence of approximate viscous solutions to the delta standing wave for the Riemann problem is established and then the convergence to the delta standing wave solution is also obtained when the viscosity parameter tends to zero. In addition, the Riemann problem is also solved with the standard method and the instability of Riemann solutions with respect to the specific small perturbation of initial data is pointed out in some particular situations.