Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions

Abstract

We investigate the interaction of delta shock waves and contact discontinuities for the Chaplygin Euler equations of compressible fluid flow with split delta functions. The perturbed Riemann problem when initial data are three piecewise constant states is constructively solved, and the global structure and large time‐asymptotic behaviors of solutions are discussed case by case via deriving how the solution continues beyond points of interaction. It is shown that the Riemann solutions are stable for such small perturbations with initial data by letting perturbed parameter ε tend to zero. Moreover, some numerical simulations completely coinciding with the theoretical analysis are also exhibited.

Comment on the paper “Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions, Yu Zhang, Yanyan Zhang, Jinhuan Wang Mathematical Methods in the Applied Sciences , 2018; 41 :7678–7697”

“…Let us now check the reference Caprino et al cited by Zhang et al Equation (1.4) in Caprino et al is …”

“…This equation, which resembles Equation (3), is also dimensionally incorrect, because the units of p are kgm −1 sec −2 , whereas the units of 1 2 ρ 2 are kg 2 m −6 . Thus, there are equations in previous literature papers with incorrect equations of state.…”

“…In summary, some of the references that the authors invoke to justify their equations do not support their arguments (although these references are physically correct), and some of them do support their arguments (but these references are physically incorrect). From a mathematical point of view, the analysis in Zhang et al may be correct. However, from a physical point of view, the studied equations are not valid, and the results do not represent any real physical problem.…”

“…The physical problem in the above paper is described by the following equations (equations 1 and 2 in Zhang et al): where ρ (kgm −3 ) is the fluid density, u (msec −1 ) is the fluid velocity, and p (kgm −1 sec −2 ) is the pressure. The units of the term are m 2 sec −2 , whereas the units of pressure p are kgm −1 sec −2 .…”

“…The oldest reference that Zhang et al mentioned in justifying their equations is the work of Earnshaw in 1858. However, there is nothing in it that supports the authors' equations.…”