Concentration Phenomena of Riemann Solutions to a Logarithmic Perturbed Model

Introducing a logarithmic pressure, we analyze the phenomenon of concentration and the formation of delta-shocks for the generalized Chaplygin gas dynamics. We first solve the Riemann problem for the logarithmic perturbed model and construct the solutions with four kinds of structures R1+R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{1}+R_{2}$\end{document}, R1+S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{1}+S_{2}$\end{document}, S1+R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{1}+R_{2}$\end{document} and S1+S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{1}+S_{2}$\end{document}. Then it is shown that when the logarithmic pressure vanishes, the limits of the Riemann solutions for the logarithmic perturbed model are just these of the generalized Chaplygin gas dynamics. In particular, when the initial data satisfy some certain conditions, the S1+S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{1}+S_{2}$\end{document} solution of the logarithmic perturbed model tends to the delta-shock solution of the generalized Chaplygin gas dynamics. Finally, some numerical results exhibit the process of formation of delta-shocks.