$ \delta$- and $ \delta'$-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes

Approximate solutions to the initial value problem for some compressible flows

“…The need for the search of mathematical solutions to the initial value problem for compressible fluids in several dimensions when shocks show up has been recently pointed out by Lax and Serre [25,31]; our results provide sequences of approximate solutions with full mathematical proofs that tend to satisfy the equations. We have been motivated by the fact our approximate solutions are weak asymptotic solutions as considered by Albeverio et al [1][2][3], Danilov et al [15][16][17][18], Shelkovich et al [30,32,33], as an extension of the Maslov-Whitham asymptotic analysis. Exact or approximate solutions with full proofs in the case of spherical symmetry had already been obtained in Joseph et al [4,22] and Kunzinger et al [23,24] for some of these systems or related systems.…”

“…These sequences of approximate solutions have been introduced under the name of weak asymptotic solutions by Danilov et al [15], as an extension of Maslov asymptotic analysis, and they have proved to be an efficient mathematical tool to study creation and superposition of singular solutions to various nonlinear PDEs, in particular δ-waves: [1][2][3][15][16][17][18]30,32,33]. A weak asymptotic solution for the system…”

“…Colombeau ZAMP tool in itself by various authors following Maslov asymptotic analysis [1][2][3][15][16][17][18]30,32,33], and also, they could be a provisional substitute of solutions, since one has to cope with an absence of known usual weak solutions in 2-D and 3-D in the presence of shocks. The proof that these sequences tend to satisfy the equations permits to explain results observed from computing in [9,25,26].…”

Formation of Delta Standing Wave for a Scalar Conservation Law with a Linear Flux Function Involving Discontinuous Coefficients

“…Thus the delta standing wave is introduced in the Riemann solution when k l > 0 > k r , which is a Dirac delta function supported on the standing wave discontinuity at x = 0. It is worthwhile to notice that the concept of delta standing wave is analogous to the concept of delta contact discontinuity [24,25,32] or the delta shock wave [22,28,35] which is a weighted Dirac delta function supported on the contact discontinuity or on the shock wave. In order to define the delta standing wave type solutions, let us introduce the following definition as in [21,29].…”

One-dimensional nonlinear chromatography system and delta-shock waves

“…The transport equations (1.6) have been analyzed extensively, see [5,8,14,15,[19][20][21]35,41] and so on. Recently, the weak asymptotics method was widely used to study the δ-shock wave type solution by Danilov et al [12,13,29,32,40] in the case of systems which are linear with respect to one of unknown functions. In the same papers, it is introduced a concept allowing functions of the form…”