Delta Shock Waves as a Limit of Shock Waves

We discus the existence of delta shock waves obtained as a limit of two shock waves. For that purpose we perturb a prototype of weakly hyperbolic 2 × 2 system (sometimes called the "generalized pressureless gas dynamics model") by an additional term (called the "generalized vanishing pressure"). The obtained perturbed system is strictly hyperbolic and its Riemann problem is solvable. Since it is genuinely nonlinear, its solution consists of shocks and rarefaction waves combination. As perturbation parameter vanishes, the solution converges in the space of distribution. Specially, a solution consisting of two shocks converge to a delta function. Also, we give a formal definition of approximate solution and prove a kind of entropy argument. The paper finishes by a discussion about delta shock interactions for the original system.