Tan, Dechun; Zhang, Tong; Zheng, Yuxi Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. (English) Zbl 0804.35077 J. Differ. Equations 112, No. 1, 1-32 (1994). Summary: For simple models of hyperbolic systems of conservation laws, we study a new type of nonlinear hyperbolic wave, a delta-shock wave, which is a Dirac delta function supported on a shock. We prove that delta-shock waves are \(w*\)-limits in \(L^ 1\) of solutions to some reasonable viscous perturbations as the viscosity vanishes. Further, we solve the multiplication problem of a delta function with a discontinuous function to show that delta-shock waves satisfy the equations in the sense of distributions. Under suitable generalized Rankine-Hugoniot and entropy conditions, we establish the existence and uniqueness of solutions involving delta-shock waves for the Riemann problems. The existence of solutions to the Cauchy problem is also investigated. Cited in 173 Documents MSC: 35L65 Hyperbolic conservation laws 35L67 Shocks and singularities for hyperbolic equations 35L45 Initial value problems for first-order hyperbolic systems 35D05 Existence of generalized solutions of PDE (MSC2000) 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B40 Asymptotic behavior of solutions to PDEs Keywords:nonlinear hyperbolic wave; delta-shock wave; viscous perturbations PDFBibTeX XMLCite \textit{D. Tan} et al., J. Differ. Equations 112, No. 1, 1--32 (1994; Zbl 0804.35077) Full Text: DOI