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Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. (English) Zbl 0804.35077
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.
##### MSC:
 35L65 Conservation laws 35L67 Shocks and singularities 35L45 First order hyperbolic systems, initial value problems 35D05 Existence of generalized solutions of PDE (MSC2000) 35B30 Dependence of solutions of PDE on initial and boundary data, parameters 35B40 Asymptotic behavior of solutions of PDE