One-dimensional nonlinear chromatography system and delta-shock waves

Guodong Wang

doi:10.1007/s00033-013-0300-x

Abstract

The Riemann problem for the nonlinear chromatography system is considered. Existence and admissibility of δ-shock type solution in both variables are established for this system. By the interactions of δ-shock wave with elementary waves, the generalized Riemann problem for this system is presented, the global solutions are constructed, and the large time-asymptotic behavior of the solutions are analyzed. Moreover, by studying the limits of the solutions as perturbed parameter \({\varepsilon}\) tends to zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data.

Highlights

We are concerned with the following conservation laws:
The motivation comes from the fact that the delta-shock wave was captured numerically and experimentally by Mazzotti et al [22,23,24] in the Riemann solutions for the local equilibrium model of two-component nonlinear chromatography, which consists of the following conservation laws where u and v are the concentrations of the adsorbing species, and u ≥ 0, v ≥ 0, 1−u+v > 0, a2 > a1 > 0
Korchinski [18] introduced the concept of the Dirac function into the classical weak solution when he studied the Riemann problem for the following system ut +

Summary

Introduction

We are concerned with the following conservation laws:. where u, v are the nonnegative functions of the variables (x, t) ∈ R × R+, which express the concentrations of the two adsorbing species and u ≥ 0, v ≥ 0, 1 − u + v > 0. The motivation comes from the fact that the delta-shock wave was captured numerically and experimentally by Mazzotti et al [22,23,24] in the Riemann solutions for the local equilibrium model of two-component nonlinear chromatography, which consists of the following conservation laws. Korchinski [18] introduced the concept of the Dirac function into the classical weak solution when he studied the Riemann problem for the following system ut +. Discovered that the form of Dirac delta functions supported on shocks was used as parts in their Riemann solutions for certain initial data. There is another well-known example, i.e. the transport equations ρt + (ρu)x = 0, (ρu)t + (ρu2)x = 0,. We prove that the solutions of the perturbed initial value problem converge to the corresponding Riemann solutions as ε → 0, which shows the stability of the Riemann solutions for the small perturbation, and analyze the large time-asymptotic behavior of the solutions

Elementary waves

The nonclassical solutions

Interactions of δ-shock wave with elementary waves

P Pm and one can obtain d2 x dt2