The Riemann problem admitting δ -, 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 lawsut+(u2)x=0,vt+2(uv)x=0,wt+2(v2+uw)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′δ′-shock wave type solutions, and vacuum states  . δ′δ′-Shock is a new type of singular solutions to systems of conservation laws first introduced in [E.Yu., Panov, V.M. Shelkovich, δ′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>0t>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′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′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.