A distributional product approach to the delta shock wave solution for the one-dimensional zero-pressure gas dynamics system. (October 2018)

• Record Type: Journal Article
• Authors: Shen, Chun; Sun, Meina
• Is Part Of: International journal of non-linear mechanics. Volume 105(2018)
• Page Start: 105
• Page End: 112
• Publication Date: 2018-10
• Subjects: Product of distributions -- Zero-pressure gas dynamics -- Delta shock wave -- Riemann problem -- Coulomb-like friction term -- Hyperbolic conservation law; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2018.06.008
• Languages: English
• ISSNs: 0020-7462

Abstract

Abstract: The Riemann problem for the one-dimensional zero-pressure gas dynamics system is considered in the frame of α − solutions based on a solution concept defined in the setting of a product of distributions. The reformulated form of the zero-pressure gas dynamics system is provided and consequently the unique α − solution is obtained within a convenient class of distributions including the Dirac delta measure. It is shown that our constructed α − solution is reasonable compared with the known results using other methods. Furthermore, the result is generalized for the one-dimensional zero-pressure gas dynamics system with the Coulomb-like friction term, which enables us to see that the α − solution is not self-similar any more. It is shown that the time evolution of the delta shock wave discontinuity is represented by a parabolic curve under the influence of the Coulomb-like friction term. Highlights: The Riemann problem for the one-dimensional zero-pressure gas dynamics system is considered in the frame of α − solutions based on a solution concept defined in the setting of a product of distributions. The reformulated form of the zero-pressure gas dynamics system is provided. Then, the unique α − solution is obtained within a convenient class of distributions including the Dirac delta measure. The result is generalized for the one-dimensional zero-pressure gas dynamics system with the Coulomb-like friction term, in which the time evolution of the delta shock wave discontinuity is represented by a parabolic curve under the influence of the Coulomb-like friction term.