A Riemann problem whose viscosity solutions contain δ-measures

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

“…where α, β, γ, η are arbitrary positive constant numbers and ε is a sufficiently small positive number. It is clear to see that if the limit ε→0 is taken, then the perturbed system (1) becomes formally the following simplified pressureless Euler system [1][2][3][4][5] u t þ u 2 2…”

“…The choice of flux perturbations in (1) makes the original system (2) into a strictly hyperbolic system such that all the calculations dealing with ε are performed analytically. In this paper, it remains to be seen whether or not the limits ε → 0 of solutions to the Riemann problem for the system (1) are identical with those for the system (2), which is a special Cauchy problem with the 2 piecewise initial data of the form ðu; vÞðx; 0Þ ¼ ðu − ; v − Þ;…”

“…It is well known such as in Ambrosio 8 and Danilov and Shelkovich 9 that the Cauchy problem for the system (2) may not possess a weak L ∞ −solution in general, so that the generalized solution usually appears for some nonclassical situations. If we restrict ourselves only to consider the Riemann problem (2) and (3), then we find that the first equation of (2) is just the inviscid Burgers (or Hopf) equation whose Riemann solution is the classical entropy solution, namely, a rarefaction wave when u − < u + or a shock wave when u − >u + . The Dirac delta function should be introduced as a part for the density v in the second equation of (2) when the velocity u is discontinuous and contains the Heaviside function for the case u − > u + .…”

“…The Dirac delta function should be introduced as a part for the density v in the second equation of (2) when the velocity u is discontinuous and contains the Heaviside function for the case u − > u + . More precisely, it is universally acknowledged nowadays that the vacuum state and delta shock wave are involved in the solutions to the Riemann problem for the simplified pressureless Euler system 2,5 and also for the pressureless Euler system. [10][11][12] The first task is to construct the solutions to the Riemann problem for the perturbed system (1) with the initial data (3) in all kinds of situations.…”

Delta and singular delta locus for one‐dimensional systems of conservation laws

“…Also, the following systems have the similar form of solutions as (5). Joseph [11] and Oberguggenberger [3] proved that for…”