The Riemann problem admitting δ-, δ′-shocks, and vacuum states (the vanishing viscosity approach)

In this paper, using the vanishing viscosity method, we construct a solution of the Riemann problem for the system of conservation laws u(t) + (u(2))(x) = 0, v(t) + 2(uv)(x) = 0, w(t) + 2(v(2) + uw)(x) = 0 with the initial data [GRAPHICS] This problem admits delta-, delta'-shock wave type solutions, and vacutan states. delta'-Shock is a new type of singular solutions to systems of conservation laws first introduced in [E.Yu., Panov, VM. Shelkovich, delta'-Shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations 228 (2006) 49-86]. It is a distributional solution of the Riemann problem such that for t > 0 its second component v may contain Dirac measures, the third component w may contain a linear combination of Dirac measures and their derivatives, while the first component u has bounded variation. Using the above mentioned results, we also solve the delta-shock Cauchy problem for the first two equations of the above system. Since delta'-shocks can be constructed by the vanishing viscosity method, they are "natural" solutions to systems of conservation laws. We describe the formation of the delta'-shocks and the vacuum states from smooth solutions of the parabolic problem.