The Riemann problem admitting δ-, [formula omitted]-shocks, and vacuum states (the vanishing viscosity approach)

Abstract

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 ( u v ) x = 0 , w t + 2 ( v 2 + u w ) x = 0 with the initial data ( u ( x , 0 ) , v ( x , 0 ) , w ( x , 0 ) ) = { ( u − , v − , w − ) , x < 0 , ( u + , v + , w + ) , x > 0 . This problem admits δ-, δ ′ - shock wave type solutions, and vacuum states. δ ′ -Shock is a new type of singular solutions to systems of conservation laws first introduced in [E.Yu., Panov, V.M. Shelkovich, δ ′ -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 δ-shock Cauchy problem for the first two equations of the above system. Since δ ′ -shocks can be constructed by the vanishing viscosity method, they are “natural” solutions to systems of conservation laws. We describe the formation of the δ ′ -shocks and the vacuum states from smooth solutions of the parabolic problem. The results of this paper as well as those of the above-mentioned paper show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well.